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d5670ef45a7bf277 | Decoherence and the Transition from Quantum to Classical
Wojciech Zurek Physics Today Oct 91 (abridged)
Quantum mechanics works exceedingly well in all practical applications. No example of conflict between its predictions and experiment is known. Without quantum physics we could not explain the behavior of solids, the structure and function of DNA, the color of the stars, the action of lasem or the properties of superfluids. Yet well over half a century after its inception, the debate about the relation of quantum mechanics to the familiar physical world continues. How can a theory that can account with precision for everything we can measure still be deemed lacking.?
What is wrong with quantum theory?
The only "failure" of quantum theory is its inability to provide a natural framework that can accommodate our prejudices about the workings of the universe. States of quantum systems evolve according to the deterministic, linear Schrödinger equation,
That is, just as in classical mechanics, given the initial state of the system and its Hamiltonian H, one can compute the state at an arbitrary time. This deterministic evolution of has been verified in carefully controlled experiments. Moreover, there is no indication of a border between quantum and classical behavior at which equation 1 fails. There is, however, a very poorly controlled experiment with results so tangible and immediate that it has an enormous power to convince: Our perceptions are often difficult to reconcile with the predictions of equation 1.
Why? Given almost any initial condition the universe described by evolves into a state that simultaneously contains many alternatives never seen to coexist in our world. Moreover, while the ultimate evidence for the choice of one such option resides in our elusive "consciousness," there is every indication that the choice occurs long before consciousness ever gets involved. Thus at the root of our unease with quantum mechanics is the clash between the principle of superposition-the consequence of the linearity of equation 1-and the everyday classical reality in which this principle appears to be violated. The problem of measurement has a long and fascinating history. The first widely accepted explanation of how a single outcome emerges from the many possibilities was the Copenhagen interpretation, proposed by Niels Bohr, who insisted that a classical apparatus is necessary to carry out measurements. Thus quantum theory was not to be universal. The key feature of the Copenhagen interpretation is the dividing line between quantum and classical. Bohr emphasized that the border must be mobile, so that even the "ultimate apparatus"-the human nervous system-can be measured and analyzed as a quantum object, provided that a suitable classical device is available to carry out the task. In the absence of a crisp criterion to distinguish between quantum and classical, an identifir-ation of the "classical" with the "macroscopic" has often been tentatively accepted. The inadequacy of this approach has become apparent as a result of relatively recent developments: A cryogenic version of the Weber bar-a gravity wave detector-must be treated as a quantum harmonic oscillator even though it can weigh a ton.' Nonclassical squeezed states can describe oscillations of suitably prepared electromagnetic fields with macroscopic numbers of photons. Superconducting Josephson junctions have quantum states associated with currents involving macroscopic numbers of electrons, and yet they can tunnel between the minima of the effective potential.'
If macroscopic systems cannot always be safely placed on the classical side of the boundary, might there be no boundary at all? The many-worlds interpretation (or, more accurately, the many-universes interpretation) claims to do away with the boundary.' The many-worlds interpretation was developed in the 1950s by Hugh Everett III with the encouragement of John Archibald Wheeler. In this interpretation all of the universe is described by quantum theory. Superpositions evolve forever according to the Schrödinger equation. Each time a suitable interaction takes place between any two quantum systems, the wavefunction of the universe splits, so that it develops ever more "branches." Everett's work was initially almost unnoticed. It was taken out of mothballs over a decade later by Bryce DeWitt. The many-worlds interpretation is a natural choice for quantum cosmology, which describes the whole universe by means of a state vector. There is nothing more macroscopic than the universe. It can have no a priori classical subsystems. There can be no observer "on the outside." In this context, classicality has to be an emergent property of ne selected observables or systems.
At a first glance, the two interpretations - many-worlds and Copenhagen-have little in common. The Copenhagen interpretation demands an a priori "classical domain" with a border that enforces a classical "embargo" by letting through just one potential outcome. The many-worlds interpretation aims to abolish the need for the border altogether: Every potential outcome is accommodated by the ever proliferating branches of the wavefunction of the universe. The similarity of the difficulties faced by these two viewpoints nevertheless becomes apparent when we ask the obvious question "Why do I, the observer, perceive only one of the outcomes?" Quantum theory, with its freedom to rotate bases in Hilbert space, does not even clearly define which states of the universe correspond to branches. Yet our perception of a reality with alternatives and not a coherent superposition of alternatives demands an explanation of when, where and how it is decided what the observer actually perceives. Considered in this context, the many-worlds interpretation in its original version does not abolish the border but pushes it all the way to the boundary between the physical universe and consciousness. Needless to say, this is a very uncomfortable place to do physics.
In spite of the profound difficulties and the lack of a breakthrough for some time, recent years have seen a growing consensus that progress is being made in dealing with the measurement problem. The key (and uncontroversial) fact has been known almost since the inception of quantum theory, but its significance for the transition from quantum to classical is being recognized only now: Macroscopic quantum systems are never isolated from their environments. Therefore, as H. Dieter Zeh emphasized,' they should not be expected to follow Schrödinger's equation, which is applicable only to a closed system. As a result systems usually regarded as classical suffer (or benefit) from the natural loss of quantum coherence, which "leaks out" into the environment.' The resulting "decoherence" cannot be ignored when one addresses the problem of the reduction of wavepackets: It imposes in effect, the required embargo on the potential outcomes, only allowing the observer to maintain records of alternatives and to be aware of only one branch.
Correlations and Measurements
(This section gives an abbreviated paraphrase of Zurek's argument)
Supposing we consider a measurement of electron spin made in a Stern-Gerlach magnet - an asymmetrical magnetic field created by a pointed and rounded pair of magnets that differentially deflects electrons or atoms of opposite spins.
This correlated state involves two branches of the detector, one in which it (actively in our case) measures spin up and the other (passively) spin down. This is precisely the splitting of the wave function into two branches advanced by Everett to articulate the many-worlds description of quantum mechanics.
However in the real world, we know the alternatives are distinct outcomes rather than a mere superposition of states. Even if we do not know what the outcomes are we can safely assume that one of them will have actually occurred and when we do experience the result it is one or another outcome, not simply a superposition of outcomes. Everett got around this by pointing out that the observer (consider d) cannot detect the splitting since in either branch a consistent result is registered. Von Neumann was well aware of these difficulties and postulated that in addition to the unitary evolution of the wave function there is a non-unitary 'reduction of the state vector or wve function which converts the superposition into a mixture by cancelling the correlating off-diagonal terms of the pure density matrix.
The key advantage of this interpretation is that it enables us to interpret the coefficients as classical probabilities. However, as we have seen with the EPR pair-splitting experiments, the quantum system has not made any decisions about its nature until measurement has taken place. This explains the off-diagonal terms which are essential to maintain the fully undetermined state of the quantum system which has not yet even decided whether the electrons are spin up or spin down or perhaps instead are right or left or some other basis combination altogether.
One way to explain how this additional information is disposed of is to include the interaction of the system with the environment in other ways. Consider a system S detector D and environment E. If the environment canalso interact and become correlated with the apparatus, we have the following transition:
This final state extends the correlation beyond the system-detector pair. When the states of the environment corresponding to the spin up and spin down states of the detector are orthogonal, we can take the trace over the uncontrolled degrees of freedom to get the same results as the reduced matrix. Essentially whenever the observable is a constant of motion of the detector-environment Hamiltonian, the observable will be reduced from a superposition to a mixture. In practice, the interaction of the particle carrying the quantum spin states with a photon and the large number of degrees of freedom of the open environment can make this loss of coherence irreversible.
For large classical objects decoherence would be virtually instantaneous because of the high probability of interaction of such a system with some environmental quantum. Zurek then constructs a quantitative model to illustrate the bulk effects of decoherence over time.
The gradual cancellation of the off-diagonal elements with decoherence
Decoherence, histories, and the universe
The universe is, of course, a closed system. As far as quantum phase information is concerned, it is practically the only system that is effectively closed. Of course, an observer inhabiting a quantum universe can monitor only very few observables, and decoherence can arise when the unobserved degrees of freedom are "traced over."" A more fundamental issue, however, is that of the emergence of the effective classicality as a feature of the universe that is more or less independent of special observers, or of coarse grainings such as a fixed separation of the universe into observed and unobserved systems. The quantum mechanics of the universe must allow for possible large quantum fluctuations of space-time geometry at certain epochs and scales. In particular, it may include important effects of quantum gravity early in the expansion of the universe. Nontrivial issues such as the emergence of the usual notion of time in quantum mechanics must then be addressed. Here we shall neglect such considerations and simply treat the universe as a closed system with a simple initial condition. Significant progress in the study of decoherence in this context has been reported by Murray Gell-Mann and James B. Hartle," who are pursuing a program suitable for quantum cosmology that may be called the manyhistories interpretation. The many-histories interpretation builds on the foundation of Everett's many-worlds interpretation, but with the addition of three crucial ingredients: the notion of sets of alternative coarsegrained histories of a quantum system, the decoherence of the histories in a set, and their approximate determinism near the effectively classical limit. A set of coarse-grained alternatives for a quantum system at a given time can be represented by a set of mutually exclusive projection operators, each corresponding to a different range of values for some properties of the system at that time. (A completely fine-grained set of alternatives would be a complete set of commuting operators.) An exhaustive set of mutually exclusive coarse-grained alternative histories can be obtained, each one represented by a time-ordered sequence of such projection operators.
The definition of consistent histories for a closed quantum system was first proposed by Robert Griffiths. He demonstrated that when the sequences of projection operators satisfy a certain condition (the vanishing of the real part of every interference term between sequences), the histories characterized by these sequences can be assigned classical probabilities-in other words, the probabilities of alternative histories can be added. Griffiths's idea was further extended by Roland Omnes who developed the "logical interpretation" of quantum mechanics by demonstrating how the rules of ordinary logic can be recovered when making statements about properties that satisfy the Griffiths criterion.
Recently Gell-Mann and Hartle pointed out that in practice somewhat stronger conditions than Griffiths's tend to hold whenever histories decohere. The strongest condition is connected with the idea of records and the crucial fact that noncommuting projection operators in a historical sequence can be registered through commuting operators designating records. They defined a decoherence functional in terms of which the Griffiths criterion and the stronger versions of decoherence are easily stated.
Given the initial state of the universe (perhaps a mixed state) and the time evolution dictated by the quantum field theory of all the elementary particles and their interactions, one can in principle predict probabilities for any set of alternative decohering ed histories of the universe. Gell-Mann and Hartle raise the question of which sets exhibit the classicality of familiar experience. Decoherence is a precondition for such classicality; the remaining criterion, approximate determinism, is not yet defined with precision and generality. Within the many-histories program, one is studying the stringent requirements put on the coarseness of histories by their classicality. Besides the familiar and comparativelv trinial indeterminacy imposed bv the uncertainty principle, there is the further coarse graining required for decoherence of histories. Still further coarseness-for example, that encountered in following hydrodynamic variables averaged over macroscopic scales-can supply the high inertia that resists the noise associated with the mechanics of decoherence and so permits decohering histories to exhibit approximate predictability. Thus the effectively classical domain through which quantum mechanics can be perceived necessarily involves a much greater indeterminacy than is generally attributed to the quantum character of natural phenomena.
Quantum theory of classical reality
We have seen how classical reality emerges from the substrate of quantum physics: Open quantum systems are forced into states described by localized wavepackets. These essentially classical states obey classical equations of motion, although with damping and fluctuations of possibly quantum origin. What else is there to explain? The origin of the question about the interpretation of quantum physics can be traced to the clash between predictions of the Schr6dinger equation and our perceptions. It is therefore useful to conclude this paper bv revisiting the source of the problem-our awareness of definite outcomes. If the mental processes that produce this awareness were essentially unphysical, there would be no hope of addressing the ultimate question-why do we perceive just one of the quantum altematives?-within the context of phvsics. Indeed, one might be tempted to follow Eugene Wigner in giving consciousness the last word in collapsing the state vector .
I shall assume the opposite. That is, I shall examine the idea that the higher mental processes adl correspond to well-defined but, at present, poorly understood information processing functions that are carried out by physical systems, our brains. Described in this manner, awareness becomes susceptible to physical analysis. In particular, the process of decoherence is bound to affect the states of the brain: Relevant observables of individual neurons, inc!uding chemical concentrations and electrical potentials, are macroscopic. They obey classical, dissipative equations of motion. Thus any quantum superposition of the states of neurons will be destroyed far too quickly for us to become conscious of quantum goings-on: Decoherence applies to our own "state of mind." One might still ask why the Preferred basis of neurons becomes correlated with the classical observables in the familiar universe. The selection of available interaction Hamiltonians is limited and must constrain the choices of the detectable observables. There is, however, another process that must have played a decisive role: Our senses did not evolve for the purpose of verifying quantum mechanics. Rather, they developed through a process in -hich survival of the fittest played a central role. And when nothing can be gained from prediction, there is no evolutionary reason for perception. Moreover, only classi cal States are robust in spite of decoherence and therefore have predictable consequences. Hence one might argue that we had to evolve to perceive classical reality. There is little doubt that the process of decoherence sketched in this paper is an important fragment central to the understanding of the big picture-the transition from quantum to classical: Decoherence destroys superposi tiOnsThe environment induces, in effect, a superselec tion rule that prevents certain superpositions from being observed. Only states that survive this process can become classical.
There is even less doubt that the rough outline of this big picture will be further extended. Much work needs to be done both on technical issues (such as studying more re alistic models that could lead to experiments) and on issues that require new conceptual input (such as defining what constitutes a "system" or answering the question of how an observer fits into the big picture). Decoherence is of use within the framework of either of the two major interpretations: It can supply a definition of the branches in Everett's many-worlds interpretation, but it can also delineate the border that is so central to Bohr's point of view. And if there is one lesson to be learned from what we already know about such matters, it is undoubtedly the key role played by information and its transfer in the quantum universe. The natural sciences were built on a tacit assumption: Information about a physical system can be acquired without influencing the system's state. Until recently, information was regarded as unphysical, a mere record of the tangible, material universe, existing beyond and essentially decoupled from the domain governed by the laws of physics. This view is no longer tenable. Quantum theory has helped to put an end to such Laplacean dreams of a mechanical universe. The dividing line between what is and what is known to be has been blurred forever. Conscious observers have lost their monopoly on acquiring and storing information. The environment can also monitor a system, and the only difference from a man-made apparatus is that the records maintained by the environment are nearly impossible to decipher. Nevertheless, such monitoring causes decoherence, which allows the familiar approximation known as classical objective reality-a perception of a selected subset of all conceivable quantum states evolving in a largely predictable manner-to emerge from the quantum substrate.
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2cb28e4df088a9cd | Friday, April 29, 2011
Octonions and quantum physics
Peter Woit reports in "This Week's Hype" string model hype about classical number fields, in particular the possible role of octonions. It would be nice to write a comment and tell how elegantly classical number fields appear in TGD framework and make dual descriptions in terms of 8-D Minkowski space (sub-space of complexified octonions) and M4× CP2 unique. Unfortunately, Peter Woit wathces over his territory so jealously against invasion of anything which stinks like a good idea that it does not bother to take the risk of getting biten. Therefore John Baez - who as a Name is allowed to make intelligent comments- must continue to live in the illusion that no-one does anything to understand the role of octonions in physics.
The non-associativity of octonions is the basic problem if one attempts to build octonionic quantum mechanics. Nothing like this is tried in TGD. Instead, classical number fields appear at the level of classical physics (see this). Space-time surfaces as classical correlates of quantum physics are conjectured to decompose to associative (/quaternionic/Minkowskia)n and co-associative (/co-quaternionic/Euclidian) regions so that the weakness of octonionic quantum mechanics would turn into a strength making classical physics completely unique purely number theoretical. More precisely, the induced spinor structure for 8-D imbedding space has a special representation in terms of octonionic gamma matrices and the induced gamma matrices (not strictly speaking matrices anymore) are conjectured to span a quaternionic or co-quaternionic subspace of octonions over complex numbers at each point of the preferred extremal of Kähler action.
Addition: Also Motl has comments about octonions. The usual flood of insults and extremely arrogant super-stringy attitude towards anyone who does not regard superstrings as the laws of Moses for physics and dares to ask whether some aspects of super-strings might be part of a more successful physical theory. John Baez was the target of the aggression at this time. Maybe it is high time to Lubos to realize that the glamour of Harward does not last forever: we also remember that the exit of Lubos from Harward was not graceful. Some real output would be desperately needed if Lubos wants to keep his position as a blog authority and we have been waiting for years. My comment about the role of classical number fields in physics of course goes un-noticed: Lubos reads nothing which he has decided to represent crackpot theory. In any case, Lubos does a valuable work: he teaches us to tolerate people behaving like complete idiots. Learning this is after all the only manner to build a better world;-).
Thursday, April 28, 2011
About GRT limit of TGD
TGD should have General Relativity type theory as an appropriate limit. Therefore it is interesting to see what one obtains when one applies TGD picture by replacing space-times as 4-surfaces with abstract geometries as in Einstein's theory and assumes holography in the sense that space-times satisfy besides Einstein-Maxwell equations also conditions guaranteeing Bohr orbit like property. The resulting picture could be also regarded as quantized GRT type limit of quantum TGD obtained by dropping the condition that space-times are surfaces. This limit could also provide totally new insights to the quantization of GRT.
Several pleasant surprises were in store.
1. Essentially the same formalism could apply to GRT limit of TGD as TGD itself meaning that Einstein-Maxwell system can be described as almost topological QFT with holography implying that action reduces to 3-D Chern-Simons action with a metric dependent constraint term expressing the weak form of electric-magnetic duality and quantizing electric charge.
2. The existence of this limit gives valuable information also about TGD itself. In particular, the interpretation of the weak form of electric-magnetic duality is sharpened. The space-time regions with Minkowskian signature would be those in which only electromagnetic and gravitational interactions make themselves visible and regions with Euclidian signature would be the interiors of generalized Feynman graphs in which electroweak and color interactions become manifest. In particular, Weinberg angle should vanish in the Minkowskian phase so that electromagnetic field reduces to induced Kähler field identifiable as Maxwell field of Einstein-Maxwell system. This conforms with the finding that Kähler coupling strength equals to fine structure constant within the very tight constraints available.
3. The limit also suggests how one could understand the extremely small value of cosmological constant characterizing the cosmology according to GRT in terms of CP2 geometry providing idealization for the space-time region with Euclidian signature of metric representing generalized Feynman graphs also in GRT framework.
4. Non-Euclidian regions could correspond also to blackhole like regions in TGD framework, where only part of the interior of black hole is imbeddable. Black holes would naturally correspond to gigantic values of gravitational Planck constant implying that the Compton length of black-hole is of order of Schwartschild radius. Black hole would be elementary parton with very large fermion and antifermion numbers and large Planck constant and consist of dark matter in TGD sense. This picture is mathematically consistent since at event horizon the determinant of four-metric vanishes so that it is light-like just as it is at wormhole throats. Consistency with experimental factors is also achieved: about the interiors of blackholes we know nothing so that nothing prevents from assuming that it has Euclidian signature of metric: especially so if this explains the mysterious cosmological constant and standard model quantum numbers.
GRT is a more general theory than TGD in the sense that much more general space-times are allowed than in TGD - this leads also to difficulties - and one could also argue that the mathematical existence of WCW Kähler geometry actually forces the restriction of these geometries to those imbeddable in M4× CP2 so that the quantization of GRT type theory would lead to TGD.
1. The conceptual framework of TGD
There are several reasons to expect that something analogous to thermodynamics results from quantum TGD. The following summarizes the basic picture, which will be applied to a proposal about how to quantize (or rather de-quantize!) Einstein-Maxwell system with quantum states identified as the modes of classical WCW spinor field with spinors identifiable in terms of Clifford algebra of WCW generated by second quantized induced spinor fields of H.
1. In TGD framework quantum theory can be regarded as a "complex square root" of thermodynamics in the sense that zero energy states can be described in terms of what I call M-matrices which are products of hermitian square roots of density matrices and unitary S-matrix so that the moduli squared gives rise to a density matrix. The mutually orthogonal Hermitian square roots of density matrices span a Lie algebra of a subgroup of the unitary group and the M-matrices define a Kac-Moody type algebra with generators proportional to powers of S assuming that they commute with S. Therefore this algebra acts as symmetries of the theory.
What is nice that this algebra consists of generators multi-local with respect to partonic 2-surfaces and represents therefore a generalization of Yangian algebra. The algebra of M-matrices makes sense if causal diamonds (double light-cones) have sizes coming as integer multiples of CP2 size. U-matrix has as its rows the M-matrices. One can look how much of this structure could make sense in GRT framework.
2. In TGD framework one is forced to geometrize WCW consisting of 3-surfaces to which one can assign a unique space-time surfaces as analogs of Bohr orbits and identified as preferred extremals of Kähler action (Maxwell action essentially). The 3-surfaces could be identified as the intersections space-time surface with the future and past light-like boundaries causal diamond (CDs analogous to Penrose diagrams). The preferred extremals associated with the preferred 3-surfaces allow to realize General Coordinate Invariance (GCI) and its natural to assign quantum states with these.
GCI in strong sense implies even stronger form of holography. Space-time regions with Euclidian signature of metric are unavoidable in TGD framework and have interpretation as particle like structure and are identified as lines of generalized Feynman diagrams. The light-like 3-surfaces at which the signature of the induced metric changes define equally good candidates for 3-surfaces with which to assign quantum numbers. If one accepts both identifications then the intersections of the ends of space-time surfaces with these light-like surfaces should code for physics. In other words, partonic 2-surfaces plus their 4-D tangent space-data would be enough and holography would be more or less what the holography of ordinary visual perception is!
In the sequel the 3-surfaces at the ends of space-time and and light-like 3-surfaces with degenerate 4-metric will be referred to as preferred 3-surfaces.
3. WCW spinor fields are proportional to a real exponent of Kähler function of WCW defined as Kähler action for a preferred extremal so that one has indeed square root of thermodynamics also in this sense with Kähler essential one half of Hamiltonian and Kähler coupling strength playing the role of dimensionless temperature in "vibrational" degrees of freedom. One should be able to identify the counterpart of Kähler function also in General Relativity and if one has Einstein-Maxwell system one could hope that the Kähler function is just the Maxwell action for a preferred extremal and therefore formally identical with the Kähler function in TGD framework.
Fermionic degrees of freedom correspond to spinor degrees of freedom and are representable in terms of oscillator operators for second quantized induced spinor fields. This means geometrization of fermionic statistics. There is no quantization at WCW level and everything is classical so that one has "quantum without quantum" as far as quantum states are considered.
4. The dynamics of the theory must be consistent with holography. This means that the Kähler action for preferred extremal must reduce to an integral over 3-surface. Kähler action density decomposes to a sum of two terms. The first term is jαAα and second term a boundary term reducing to integral over light-like 3-surfaces and ends of the space-time surface. The first term must vanish and this is achieved if the Kähler current jα is proportional to Abelian instanton current
jα ∝ *jαα β γ δAβJγ δ
since the contraction involves Aα twice. This is at least part of the definition of preferred extremal property but not quite enough. Note that in Einstein-Maxwell system without matter jα vanishes identically so that the action reduces automatically to a surface term.
5. The action would reduce reduce to terms which should make sense at light-like 3-surfaces. This means that only Abelian Chern-Simons term is allowed. This is guaranteed if the weak form of electric-magnetic duality stating
at preferred at light-like throats with degenerate four-metric and at the ends of space-time surface. These conditions reduce the action to Chern-Simons action with a constraint term realizing what I call weak form of electric-magnetic duality. One obtains almost topological QFT since the constraint term depends on metric. This is of course what one wants.
Here the constant k is integer multiple of basic value which is proportional to gK2 from the quantization of Kähler electric charge which corresponds to U(1) part of electromagnetic charge. Fractional charges for quarks require k=ngK2/3. Physical particles correspond to several Kähler magnetically charged wormhole throats with vanishing net magnetic charge but with non-vanishing Kähler electric proportional to the sum ∑i εi kiQm,i, with εi=+/- 1 determined by the direction of the normal component of the magnetic flux for i:th throat.
The first guess is that the length of magnetic flux tube associated with the particle is of order Compton length or perhaps corresponds to weak length scale as was the original proposal. The screening of weak isospin can be understood as magnetic confinement such that neutrino pair at the second end of magnetic flux tube screens the weak charged leaving only electromagnetic charge. Also color confinement could be understood in terms of flux tubes of length of order hadronic size scales. Compton length hypothesis is enough to understand color confinement and weak screening.
Note that 1/gK2 factor in Kähler action is compensated by the proportionality of Chern-Simons action to gK2. This need not mean the absence of non-perturbative effects coming as powers of 1/gK2 since the constraint expressing electric magnetic duality depends on gK2 and might introduce non-analytic dependence on gK2.
6. In TGD the space-like regions replace black holes and a concrete model for them is as deformations of CP2 type vacuum extremals which are just warped imbeddings of CP2 to M4× CP2 with random light-like random curve as M4 projection: the light-like randomness gives Virasoro conditions. This reflects as a special case the conformal symmetries of light-like 3-surfaces and those assignable to the light-like ends of the CDs.
One could hope that this picture more or less applies for the GRT limit of quantum TGD.
2. What one wants?
What one wants is at least following.
1. Euclidian regions of the space-time should reduce to metrically deformed pieces of CP2. Since CP2 spinor structure does not exist without the coupling of the spinors to Kähler gauge potential of CP2 one must have Maxwell field. CP2 is gravitational instanton and constant curvature space so that cosmological constant is non-vanishing unless one adds a constant term to the Maxwell action, which is non-vanishing only in Euclidian regions. It is matter of taste, whether one regards V0 as term in Maxwell action or as cosmological constant term in gravitational part of the action. CP2 radius is determined by the value of this term so that it would define a fundamental constant.
This raises an interesting question. Could one say that one has a small value of cosmological constant defined as the average value of cosmological constant assignable to the Euclidian regions of space-time? The average value would be proportional to the fraction of 3-space populated by Euclidian regions (particles and possibly also macroscopic Euclidian regions). The value of cosmological constant would be positive as is the observed value. In TGD framework the proposed explanation for the apparent cosmological constant is different but one must remain open minded. In fact, I have proposed the description in terms of cosmological constant also as a proper description in the approximation to TGD provided by GRT like theory. The answer to the question is far from obvious since the cosmological constant is associated with Euclidian rather than Minkowskian regions: all depends on the boundary conditions at the wormhole throats where the signature of the metric changes.
2. One can also consider the addition of Higgs term to the action in the hope that this could allow to get rid of constant term which is non-vanishing only in Euclidian regions. It turns turns out that only free action for Higgs field is possible from the condition that the sum of Higgs action and curvature scalar reduces to a surface term and that one must also now add to the action the constant term in Euclidian regions. Conformal invariance requires that Higgs is massless.
The conceptual problem is that the surface term from Higgs does not correspond to topological action since it is expressible as as flux of Φ∇ Φ. Hence the simplest possibility is that Kähler action contains a constant term in Euclidian regions just as in TGD, where curvature scalar is however absent. Einstein-Maxwell field equations however apply that it vanishes and is effectively absent also in GRT quantized like TGD.
3. Reissner-Nordström solutions are obtained as regions exterior to CP2 type regions. In black hole horizon the metric becomes light-like and the solution can be glued to a deformed CP2 type region with metric becoming degenerate at the 3-surface involved. This surface corresponds to wormhole throat in TGD framework. Blackhole is replaced with CP2 type region. In TGD black hole solutions indeed fail to be imbeddable at certain radius so that deformed CP2 type vacuum extremal is much more natural object than black hole. In the recent framework the finite size of CP2 means that macroscopic size for the Euclidian regions requires large deformation of CP2 type solution.
Remark: In TGD framework large value of hbar and space-time as 4-surface property changes the situation. The generalization of Nottale's formula for gravitational Planck constant in the case of self gravitating system gives hbargr= GM2/v0, where v0/c<1 has interpretation as velocity type parameter perhaps identifiable as a rotation velocity of matter in black hole horizon. This gives for the Compton length associated with mass M the value LC= hbargr/M= GM/v0. For v0=c/2 one obtains Scwartschild radius as Compton length. The interpretation would be that one has CP2 type vacuum extremal in the interior up to some macroscopic value of Minkowski distance. One can whether even the large voids containing galaxies at their boundaries could correspond to Euclidian blackhole like regions of space-time surface at the level of dark matter.
4. The geometry of CP2 allows to understand standard model symmetries when one considers space-times as surfaces. This is not necessarily the case for GRT limit.
1. In the recent case one has different situation color quantum numbers make sense only inside the Euclidian regions and momentum quantum numbers in Minkowskian regions. This is in conflict with the assumption that quarks can carry both momentum and color. On the other, color confinement could be used to argue that this is not a problem.
2. One could assume that spinors are actually 8-component M4× CP2 spinors but this would be somewhat ad hoc assumption in general relativistic context. Also the existence of this kind of spinor structure is not obvious for general solutions of Einstein-Maxwell equations unless one just assumes it.
3. It is far from clear whether the symplectic transformations of CP2 could be interpreted as isometries of WCW in general relativity like theory. These symmetries certainly act in non-trivial manner on Euclidian regions but it is highly questionable whether this could give rise to a genuine symmetry. Same applies to Kac-Moody symmetries assigned to isometries of M4× CP2 in TGD framework. These symmetries are absolutely essential for the existence of WCW Kähler geometry in infinite-D context as already the uniqueness of the loop space Kähler geometries demonstrates (maximal group of isometries is required by the existence of Riemann connection).
Note that a generalization of Equivalence Principle follows in TGD framework from the assumption that coset representations of super-conformal symplectic algebra and super Kac-Moody algebra define conformally invariant physical states. The equality of gravitational and inertial masses follows from the condition that the actions of the super-generators of two algebras are identical. This also justifies the use p-adic thermodynamics for the scaling generator of either super-conformal algebra without a loss of conformal invariance.
5. One could argue that GRT limit does not make sense since in Minkowskian regions the theory knows nothing about the color and electroweak quantum numbers: there is only metric and Maxwell field. On the other hand, in TGD one has color confinement and weak screening by magnetic confinement. If the functional integral over Euclidian regions representing generalized Feynman diagrams is enough to construct scattering amplitudes, pure Einstein-Maxwell system in Minkowskian regions might be enough. All experimental data is expressible in terms of classical em and gravitational fields. If Weinberg angle vanishes in Minkowskian regions, electromagnetic field reduces to Kähler form and the interpretation of the Maxwell field as em field should make sense. The very tight empirical constraints on the value of Kähler coupling strength αK indeed allow its identification as fine structure constant at electron length scale.
6. One can worry about the almost total disappearance of the metric from the theory. This is not a problem in TGD framework since all elementary particles correspond to many-fermion states. For instance, gauge bosons are identified as pairs of fermion and antifermion associated with opposite throats of a wormhole connecting two space-time sheets with Minkowskian signature of the induced metric. Similar picture should make sense also now.
7. TGD possesses also approximate super-symmetries and one can argue that also these symmetries should be possessed by the GRT limit. All modes of induced spinor field generate a badly broken SUSY with rather large value of N (number of spinor modes) and right-handed neutrino and its antiparticle give rise to N=2 SUSY with R-parity breaking induced by the mixing of left- and right handed neutrinos induced by the modified Dirac equation. This picture is consistent with the existing data from LHC and there are characteristic signatures -such as the decay of super partner to partner and neutrino- allowing to test it. These super-symmetries might make sense if one replaces ordinary space-time spinors with 8-D spinors.
Note that the possible inconsistency of Minkowskian and Euclidian 4-D spinor structures might force the use of 8-D Minkowskian spinor structure.
3. Preferred extremal property for Einstein-Maxwell system
Consider now the preferred extremal property defined to be such that the action reduces to Chern-Simons action at space-like 3-surfaces at the ends of space-time surface and at light-like wormhole throats.
1. In Maxwell-Einstein system the field equations imply
jα=0 .
so that the Maxwell action for extremals reduces automatically to a surface term assignable to the preferred 3-surfaces. Note that Higgs field could in principle serve as a source of Kähler field but its presence does not look like a good idea since it is not present in the field equations of TGD and because the resulting boundary term is not topological.
2. The condition
J=k× *J
at preferred 3-surfaces guarantees that the surface term from Kähler action reduces to Abelian Chern-Simons term and one has hopes about almost topological QFT.
Since CP2 type regions carry magnetic monopole charge and since the weak form of electric-magnetic duality implies that electric charge is proportional to the magnetic charge, one has electric charge without electric charge as Wheeler would express it. The identification of elementary building blocks as magnetic monopoles leads in TGD context to the picture about particle as Kähler magnetic flux tubes having opposite magnetic charges at their ends. It is not quite clear what the length of the tubes is. One possibility is Compton length and second possibility is weak length scale and the color confinement length scale. Note that in TGD the physical charges reside at the wormhole throats and correspond to massless fermions.
3. CP2 is constant curvature space and satisfies Einstein equations with cosmological constant. The simplest manner to realize this is to add to the action constant volume term which is non-vanishing only in Euclidian regions. This term could be also interpreted as part of Maxwell action so that it is somewhat a matter of taste whether one speaks about cosmological constant or not. In any case, this would mean that the action contains a constant potential term
V= V0× (1+sign(g))/2 ,
where sign(g)=-1 holds true in Minkowskian regions and sign(g)=1 holds true in Euclidian regions.
Note that for a piece of CP2 V0 term can be expressed is proportional to Maxwell action and by self-duality this is proportional to instanton action reducible to a Chern-Simons term so that V0 is indeed harmless from the point of view of holography.
4. For Einstein-Maxwell system with similar constant potential in Euclidian regions curvature scalar vanishes automatically as a trace of energy momentum tensor so that no interior or surface term results and the only surface term corresponds to a pure Chern-Simons term for Maxwell field. This is exactly the situation also in quantum TGD. The constraint term guaranteeing the weak form of electric-magnetic duality implies that the metric couples to the dynamics and the theory does not reduce to a purely topological QFT.
5. In TGD framework a non-trivial theory is obtained only if one assumes that Kähler function corresponds apart from sign to either the Kähler action in the Euclidian regions or its negative in Minkowskian regions. This is required also by number theoretic vision. This implies a beautiful duality between field descriptions and particle descriptions.
This also guarantees that the Kähler function reducing to Chern-Simons term is negative definite: this is essential for the existence of the functional integral and unitarity of the theory. This is due to the fact that Kähler action density as a sum of magnetic and electric energy densities is positive definite in Euclidian regions. This duality would be very much analogous to that implied by the possibility to perform Wick rotation in QFTs. Therefore it seems natural to postulate similar duality also in the proposed variant of quantized General Relativity.
6. The Kähler function of the WCW would be given by Chern-Simons term with a constraint expressing the weak form of electric-magnetic duality both in TGD and General Relativity. One should be able regard also in GRT framework WCW as a union of symmetric spaces with Kähler structure possessing therefore a maximal group of isometries. This is an absolutely essential prerequisite for the existence of WCW Kähler geometry. The symmetric spaces in the union are labelled by zero modes which do not contribute to the line element and would represent classical degrees of freedom essential for quantum measurement theory. In TGD the induced CP2 Kähler form would represents such degrees of freedom and the quantum fluctuating degrees of freedom would correspond to symplectic group of δ M4+/-× CP2.
The difference between TGD and GRT would be that light-like 3-surfaces for all possible space-times containing Euclidian and Minkowskian regions would be considered for GRT type theory. In TGD these space-times are representable as surfaces of M4× CP2. In TGD framework the imbeddability assumption is crucial for the mathematical existence of the theory since it eliminates space-times with non-physical characteristics. The problem posed by arbitrarily large values of cosmological constants is one of the basic problems solved by this assumption. Also mass density is sub-critical for cosmologies with infinite duration and critical cosmologies are unique apart from their duration and quantum critical cosmologies replace inflationary cosmologies.
7. Note that one could consider assigning the gravitational analog of Chern-Simons term with the preferred 3-surfaces: this kind of term is discussed by Witten in this classic work about Jones polynomial. This term is a non-abelian version of Chern-Simons term and one must replace curvature tensor with its contraction with sigma matrices so that 4-D spinor structure is necessarily involved. The objection is that this term contains second derivatives. In TGD spinor structure is induced from that of M4× CP2 and this kind of term need not make sense as such since gamma matrices are expressed in terms of imbedding space gamma matrices: among other things this resolves the problems caused by the non-existence of spinor structure for generic 4-geometries. The coupling to the metric however results from the constraint term expressing weak form of electric-magnetic duality.
The difference between TGD and GRT would be basically due to the factor of scattering amplitudes coming from the duality expressing electric-magnetic duality and due to the fact that induced metric in terms of H-coordinates and Maxwell potential is expressible in terms of CP2 coordinates. The latter implies topological field quantization and many-sheeted space-time crucial for the interpretation of quantum TGD.
4. Could ZEO and the notion of CD make sense in GRT framework?
The notion of CD is crucial in ZEO and one can ask whether the notion generalizes to GRT context. In the previous arguments related to EG the notion of ZEO plays a fundamental role since it allows to replace S-matrix with M-matrix defining "complex square root" of density matrix.
1. In TGD framework CDs are Cartesian products of Minkowskian causal diamonds of M4 with CP2. The existence of double light-cones in curved space-time would be required and its is not clear whether this makes sense generally. TGD suggest that the scales of these diamonds defined in terms of the proper time distance between the tips are integer multiples of CP2 scale defined in terms of the fundamental constant V0 (the more restrictive assumption allowing only 2n multiples would explain p-adic length scale hypothesis but would not allow the generalization of Kac-Moody algebra spanned by M-matrices). The difference between boundaries of GRT CDs and wormhole throats would be that four-metric would not be degenerate at CDs.
2. The conformal symmetries of light-cone boundary and light-like wormhole throats generalize also now since they are due to the metric 2-dimensionality of light-like 3-surfaces. It is however far from clear whether one can have anything something analogous to conformal variants of symplectic algebra of δ M4+/-× CP2 and isometry algebra of M4× CP2.
Could one perhaps identify four-momenta as parameters associated with the representations of the conformal algebras involved? This hope might be unrealistic in TGD framework: the basic idea behind TGD indeed is that Poincare invariance lost in GRT is retained if space-times are surfaces in H=M4× CP2. The reason is that that super-Kac-Moody symmetries correspond to localized isometries of H whereas the super-conformal algebra associated with the symplectic group is assignable to the light-like boundaries δ M4+/-× CP2 of CD of H rather than space-time surface.
3. One could of course argue that some physical conditions on GRT -most naturally just the highly non-trivial mathematical existence of WCW Kähler geometry and spinor structure- could force the representability of physically acceptable 4-geometries as surfaces M4× CP2. If so, then also CDs would the same CDs as in TGD and quantization of GRT would lead to TGD and all the huge symmetries would emerge from quantum GRT alone.
The first objection is that the induced spinor structure in TGD is not consistent with that natural in GRT. Second objection is that in TGD framework Einstein-Maxwell equations are not true in general and Einstein's equations can be assumed only in long length scales for the vacuum extremals of Kähler action. The Einstein tensor would characterize the energy momentum tensor assignable to the topologically condensed matter around these vacuum extremals and neither geometrically nor topologically visible in the resolution defined by very long length scale. If Maxwell field corresponds to em field in Minkowskian regions, the vacuum extremal property would make sense in scales where matter is electromagnetic neutral and em radiation is absent.
5. What can one conclude?
The previous considerations suggest that a surprisingly large piece of TGD can be applied also in GRT framework and raise the possibility about quantization of Einstein-Maxwell system in terms of Kähler geometry of WCW consisting of 3-geometries instead of 3-surfaces. One can even consider a new manner to understand TGD as resulting from the quantization of GRT in terms of WCW Kähler geometry in the space of 3-metrics realizing holography and making classical theory an exact part of quantum theory. Since the space-times allowed by TGD define a subset of those allowed by GRT one can ask whether the quantization of GRT leads to TGD or at least sub-theory of TGD. The arguments represented above however suggest that this is not the case.
The generalization of S-matrix to a complex of U-matrix, S-matrix and algebra of M-matrices forced by ZEO gives a natural justification for the modification of EG allowing gravitons and giving up the rather nebulous idea about emergent space-time. Whether ZEO crucial for EG makes sense in GRT picture is not clear. A promising signal is that the generalization of EG to all interactions in TGD framework leads to a concrete interpretation of gravitational entropy and temperature, to a more precise view about how the arrow of geometric time emerges, to a more concrete realization of the old idea that matter antimatter asymmetry could be due to different arrows of geometric time for matter and antimatter, and to the idea that the small value of cosmological constant could correspond to the small fraction of non-Euclidian regions of space-time with cosmological constant characterized by CP2 size scale.
The above considerations were inspired by the attempt to understand what is good and what is bad in the entropic gravity scenario of Verlinde in TGD framework with the basic idea being that quantum TGD as a square root of thermodynamics must predict something analogous to thermalization of the lines of generalize Feynman graphs. The above interpretation for the lines of Feynman graphs as analogs of blackholes indeed allows to understand blackhole temperature and entropy as a manifestation of this underlying thermodynamics. The generalization of blackhole thermodynamics implies that both virtual gravitons and gauge bosons are thermalized. For details see the article TGD inspired vision about entropic gravity.
Tuesday, April 26, 2011
D0 reports a new 3 sigma bump with mass around 325 GeV
It seems that experimentalists have gone totally crazy. Maybe new physics is indeed emerging from LHC and they want to publish every data bit in the hope of getting paid visit to Stockholm. CDF and ATLAS have told about bumps and now Lubos tells about a new 3 sigma bump reported by D0 collaboration at mass 325 GeV producing muon in its decay producing W boson plus jets. The proposed identification of bump is in terms of decay of t' quark producing W boson.
Lubos mentions also second mysterious bump at 324.8 GeV or 325.0 GeV reported by CDF collaboration and discussed by Tommaso Dorigo towards the end of the last year. The decays of these particles produce 4 muons through the decays of two Z bosons to two muons. What is peculiar is that two mass values differing by .2 GeV are reported. The proposed explanation is in terms of Higgs decaying to two Z bosons. TGD based view about new physics suggests strongly that the three of four particles forming a multiplet is in question.
One can consider several explanations in TGD framework without forgetting that these bumps very probably disappear. Consider first the D0 anomaly alone.
1. TGD predicts also higher generations but there is a nice argument based on conformal invariance and saying that higher particle families are heavy. What "heavy" means is not clear. It could of mean heavier that intermediate gauge boson mass scale. This explanation does not look convincing to me.
2. Another interpretation would be in terms of scaled up variant of top quark. The mass of top is around 170 GeV and p-adic length scale hypothesis would predict that the mass should equal to a multiple of half octave of top quark mass. Single octave would give mass of 340 GeV. The deviation from predicted mass would be 5 per cent. This quark could correspond to t quark of scaled up hadron physics predicted by TGD and discussed in previous postings (see this, this, abd this).
The prediction of the scaled up hadron physics allows to ask whether a common explanation for all these particles as decay products of kaons of M89 hadron physics could exist. Could charged kaon produceneutral pion and single W boson and therefore muon just as the 300 GeV charged pion would produce W boson plus neutral pion decaying to two jets. This explanation excludes the the interpretation of ATLAS bump as neutral pion and CDF bump as charged kaon but CDF and D0 bumps could live peacefully together.
If there indeed are two slightly different masses one can can ask whether the two different masses could be due to CP breaking. The mass difference between short-lived and long-lived ordinary kaon is however extremely small- 3.5×10-12 MeV- and scaling by a factor 512 would give quite too small mass difference. That CP (or even CPT) breaking should be so large for the scaled up version of hadron physics looks odd. As a matter fact, the splitting is of the same order as electromagnetic splitting between mesons with different charges obtained by scaling with factor 512 from the mass splitting of order 1 MeV for ordinary mesons.
Addition: The newest rumor is that ATLAS rumor about too photo-philic Higgs with exactly the same mass of 115 GeV as the hegemony wanted it to have was not more than a rumor. Sorry Lubos;-).
For me the newest rumor is a relief since it makes it more easier to find room for the remaining rumors in zoomed up hadron physics. The CDF rumor about 145 GeV bump would be interpreted in terms of charged pion. The latest D0 rumor weighting 325 GeV and producing W bosons, and the earlier CDF rumor having two slightly different masses around 325 GeV and producing two Z bosons would in turn be interpreted in terms of scaled up charged and neutral kaons.
However, if a strict scaling would hold true for the meson masses, one could conclude that either 145 GeV or 325 rumour is only a humor since the mass scale ratio 325/145 ≈ 2.24 is smaller than the mass scale ratio for ordinary kaon and pion about 490/140≈ 3.5. This would leave only one or two rumors to be killed. Probably they suffer a natural death within week or two in any case. I have not taken main stream theorists seriously for decades but believed that experimentalists are somehow more rooted in reality. Has the hype disease infected also experimentalists? This would be sad. Addition: The latest rumor about Atlas by Peter Woit tells that New Science has received inner information that ATLAS bump has not been found in other experiments. Tommaso in turn claims that this cannot be true! From which some reader concludes between lines that ATLAS has observed photo-philic Higgs after all!! When physics blogs came, I thought that they would provide forums for a genuine discussion about new ideas and could also serve some kind of educational function: for instance, about statistical methods of particle physics. I was wrong: they are forums for a chat about what names have said, for boosting the ego of the blogger, for the endlessly boring n sigma talk, and speculations around rumors and counter rumors. Does the situation in the web of so called respected blogs reflect the situation also in experimental particle physics? I sincerely hope that this is not the case.
Objection against zero energy ontology and quantum classical correspondence
The motivation for requiring geometry and topology of space-time as correlates for quantum states is the belief that quantum measurement theory requires the representability of the outcome of quantum measurement in terms of classical physics -and if one believes in geometrization- one ends up with generalization of Einstein's vision.
There is however a counter argument against this view and second one against zero energy ontology in which one assigns eigenstates of four-momentum with causal diamonds (CDs).
1. One can argue that momentum eigenstates for which particle regarded as a topological inhomogenuity of space-time surface, which is non-localized cannot allow a space-time correlate.
2. Even worse, CDs have finite size so that strict four-momentum eigenstates strictly are not possible.
On the other hand, the paradoxical fact is that we are able to perceive momentum eigenstates and they look localized to us. This cannot be understood in the framework of standard Poincare symmetry.
The resolution of the objections and of the apparent paradox could rely on conformal symmetry assignable to light-like 3-surfaces implying a generalization of Poincare symmetry and other symmetries with their Kac-Moody variants for which symmetry transformations become local.
1. Poincare group is replaced by its Kac-Moody variant so that all non-constant translations act as gauge symmetries. Translations which are constant in the interior of CD and trivial at the boundaries of CDs are physically equivalent with constant translations. Hence the latter objection can be circumvented.
2. The same argument allows also a localization of momentum eigenstates at the boundaries of CD. In the interior the state is non-local. Classically the momentum eigenstate assigned with the partonic 2-surface is characterized by its 4-D tangent space data coding for momentum classically. The modified Dirac equation and Kähhler action indeed contain and additional term representing coupling to four-momenta of particles. Formally this corresponds only to a gauge transform linear in momentum but Kahler gauge potential has U(1) gauge symmetry only as a spin glass like degenary, not as a gauge symmetry so that space-time surface depends on momenta.
3. Conscious observer corresponds in TGD inspired theory of consciousness to CD and the sensory data of the observer come from partonic 2-surfaces at the boundaries of CD and its sub-CDs. This implies classicality of sensory experience and momentum eigenstates look classical for conscious perceiver.
The usual argument resolving the paradox is based on the notion of wave packet and also this notion could be involved. The notion of finite measurement resolution is key notion of TGD and it is quite possible that one can require the localization of momentum eigenstates at the boundaries of CDs only modulo finite measurement resolution for the position of the partonic 2-surfaces.
For background see the chapter Construction of Quantum Theory: M-matrix of "Towards M-matrix".
How arrow of geometric time is selected at quantum level?
I have discussed in the chapter About the Nature of Time of "Matter, Mind, Quantum" how the arrow of geometric time as a correlate for the experienced arrow of geometric time might be selected in TGD Universe. The discussion does not touch the question what arrow of time means at the level of quantum states. Therefore the notion of negative energy signal propagating backwards in geometric time crucial for TGD inspired quantum biology remains somewhat fuzzy.
The recent progress in the understanding of the basic properties of zero energy states makes it possible to understand what arrow of geometric time and the notion of negative energy state and signals propagating to the direction of geomeric past mean at the level of zero energy states. This understanding has surprisingly non-trivial philosophical implications. In the following I shall briefly the quantum view about arrow of time.
Arrow of time as an inherent property of zero energy states?
The basic idea can be expressed in very conscise form. In positive energy ontology arrow of time characterizes dynamics. In zero energy ontology arrow of time characterizes quantum states.
1. The breaking of time reversal invariance (see this) means that zero energy states can be localized with respect to particle number and other quantum numbers only for future or past light-like boundary of CD but not both. M-matrix generalizing S-matrix provides the time-like entanglement coefficients expressing the state at the second boundary as quantum superposition of states with well-defined particle numbers and other quantum numbers. But only at the second end of CD since one cannot choose freely the states at both boundaries: if this were the case the counterpart of Schrödinger equation would be completely non-deterministic. This is what the breaking of time reversal symmetry means. It occurs spontaneously and assigns to the arrow of subjective time geometric arrow of time.
This picture gives a precise meaning to the arrow of geometric time and therefore also for the otherwise fuzzy notion of negative energy signals propagating backwards in space-time playing key role in TGD based models of memory, metabolism, and intentional action (see this).
2. Quantum jump begins with the unitary U-process between zero energy states generating a superposition of zero energy states. After that follows state function reduction cascade proceeding from the level of CD to the level of sub-CDs forming a fractal hierarchy. The reductions cannot take independently at both light-like boundaries of CD as is also clear from the fact that scattering state leads from a prepared state to a quantum superposition of prepared states.
The first guess is that the cascade takes place for the second boundary of CD only so that the arrow of geometric time would be same in all scales. This need not be the case always: the geometric arrow of time seems to change in some situations: phase conjugate laser light and spontaneous self-assembly of bio-molecules are good examples about this (see this and this). In fact, one of the defining properties of living matter could be just the possibility that the arrow of geometric time is not same in all scales (size scales of CDs) so that memory, metabolism, and intentional action become possible. In any case, the second end remains a superposition of quantum states.
The lack of quantum measurements at the second end of space-times could explain why the conscious percepts are sharply localized in time at the second end of CD. This could also allow to understand memories as reductions occurring at the second, non-standard, end of sub-CDs in the geometric past.
3. The correspondence between the reduced state and the quantum superposition of states at the opposite boundary of CD allows an interpretation in terms of logical implication arrow with all statements present in the superposition implying the statement represented by the reduced state. Only implication arrow rather than equivalence is possible unless the M-matrix is diagonal meaning that there are no interactions. If it is possible to diagonalize M-matrix then in diagonal basis one has equivalences. It must be however emphasized that the physically preferred state basis fixed as in terms of eigenstates of density matrix does not allow diagonal M-matrix. Number theoretic conditions required that the density matrix corresponds to fixed algebraic extension of rationals can also make possible the diagonalization without leaving the extension and this condition might be highly relevant in the TGD inspired view about cognition relying on p-adic number fields and their algebraic extensions (see this).
4. In classical logic implication corresponds to the inclusion of subset by subset. In quantum case it corresponds to the inclusion for sub-space of state space. The inclusions of hyper-finite factors (WCW spinors define HFF of type II1) realize the notion of finite measurement resolution, which would suggest that inclusion arrow has also interpretation in terms of finite measurement resolution.
All quantum states equivalent with a given state in the resolution used imply it. Finite measurement resolution would mean that there would infinite number of instances always in the quantum superposition representing the rule A → B. Ironically, both finite measurement resolution and dissipation implying the arrow of geometric time and usually regarded as something negative from the point of view of information processing would be absolutely essential element of logical thinking in this framework.
5. Conscious theorem proving would has as correlate to building of sequences zero energy states representing A → B, B→ C, C → D with basic building bricks representing simple basic rules. These sequences would represent more complex truths.
Does state function-state preparation sequence correspond to alternating arrow of geometric time?
The state function reduction at light-like boundary of CD implies delocalization at the opposite boundary. This inspires so fascinating questions.
1. Could the state function reduction process take place alternately at the two boundaries of CD so that a kind of flip-flop in which the arrow of geometric time changes back and forth would result, and have interpretation as an alternating sequence of state function reductions and state preparations in the framework of positive energy ontology?
2. State function reductions are needed for sensory percepts. Could the sleep-wake-up period correspond to this kind of process so that during what we call sleep the past boundary of our personal CD would be in wake-up state? Could dreams and memories represent sharing of mental images of this kind of consciousness? Could it be that in the time scale of entire life cycle death is accompanied by birth at the second boundary of personal CD? Could this quantum physics representation for endless sequence of deaths and rebirths? Could the fact that old people often spend they last years in childhood have interpretation in this framework?
3. State preparation-reduction cycle might characterize only living matter whereas for inanimate matter second choice for the arrow of time would be dominant between two U-processes. TGD based reformulation of entropic gravity idea of Verlinde in terms of ZEO does not assume the absence of gravitons and the emergence of space-time (see this). The formulation leads to the proposal that thermodynamical stability selects the arrow of the geometric time and that it could be different for matter and antimatter implying that matter and antimatter reside at different space-time sheets. This would explain the apparent absence of antimatter and also support the view that the arrow alternates only in living matter.
The arrow of geometric time and the arrow of logical implication
If physics is mathematics in the sense that there is nothing behind quantum states regarded as purely mathematical objects, Boolean logic must have a direct manifestation in the structure of physical states. Physical states should represent quantal Boolean statements which get their meaning via quantum jumps. In TGD framework WCW ("world of classical worlds") spinor fields represent quantum states of the Universe and WCW spinors correspond to fermionic Fock states for second quantized induced spinor fields at space-time surface. Fock state basis has interpretation in terms of Boolean algebra. In positive energy ontology the problem is that fermion number as a super-selection rule would allow very limited number of Boolean statements to be represented. In ZEO the situation changes.
The fermionic parts of positive and negative energy parts can be seen as quantum superpositions of Boolean statements with fermion number in given mode (equal to 0 or 1) representing yes/no or true/false. Also various spin like quantum numbers associated with oscillator operators have same interpretation. Zero energy state could be seen as quantum superposition of pairs of elements of Boolean algebras associated with positive and negative energy parts of the zero energy state.
The first - and incorrect - interpretation is that zero energy state represents a quantum superposition of equivalent statements a↔ b and thus abstraction A<---> B involving several instances of A and B. The breaking of time reversal invariance allowing localization to definite fermionic quantum numbers at single end of CD only however implies that quantum states can only represent abstraction of logical implication to A→ B rather than equivalence. p-Adic physics for various primes p (see this) would represent correlates for cognition and intentionality.
For background see the chapter About the Nature of Time of "Matter, Mind, Quantum".
Monday, April 25, 2011
Water memory made visible
Water memory is one of those phenomena crucially important for understanding living matter whose existence is stubbornly forbidden by skeptics who say that water is just H2O and nothing else since this is what they learned in the elementary school.
The latest demonstrations of water memory is by the research group of HIV Nobelist Montagnier giving also strong support for a completely new realization of genetic realized somehow by water (see this) but finnish skeptics concluded that Montagnier and his group are either swindlers or that the group knows nothing about basics of experimental biology. Only a complete idiot can have the self-confidence and ignorance possessed by the most pathological finnish skeptics.
In our neighboring country Sweden skeptics have totally different attitude towards truth. For instance, two swedish physics professors admitted that the recent demonstrations of cold fusion by Italian researchers (see this) suggest strongly that new kind of nuclear reactions taking place at low temperatures are involved. The other professor leads swedish skeptics society (see this).
TGD based view about dark matter leads to a model of dark nucleon with size of order DNA triplet in which nucleon states consisting of three quarks are in one-one correspondence with DNA,RNA,tRNA, and aminoacids and vertebrate genetic code has a simple and beatiful realization (see this). This supports the view that genetic code is realized at the nuclear physics level for dark matter in water, and that the chemical realization emerged much later. This would have profound implications for the understanding of evolution and also for what happens in the cellular water of living organisms also now.
Fischer Gabor sent me Youtube video making water memory directly visible. For instance, water droplets remember the person who prepared them or the flower dropped to the water by the structure of the droplets made visible by the method used by the researchers. Essentially holographic memory is in question. Maybe this video might open some eyes to see the fascinating reality in all its beauty. Enjoy!
Thursday, April 21, 2011
TGD based view about entropic gravity
I discussed entropic gravity of Verlinde for some time ago in rather critical spirit but made also clear that quantum TGD in the framework of zero energy ontology could be called square root of thermodynamics so that thermodynamics- or its square root- should emerge at the level of the lines of generalized Feynman diagrams. The intolerable-to-me features of entropic gravity idea are the claimed absence of gravitons and the nonsense talk about the emergence of dimensions assuming at the same time basic formulas of general relativity.
I returned to the topic later again with a boost given by one of the few people in the finnish academic establishment who have regarded me as a life form with some indications about genuine intelligence. What demonstrates the power of a good idea is that just posing some naturally occurring questions led rapidly to a TGD inspired phenomenology of EG allowing to see what is good and what is bad in EG hypothesis and also to see possible far reaching connections with apparently completely unrelated basic problems of recent day physics.
Consider first the phenomenology of EG in TGD framework.
1. Gravitating bodies can be seen as sources of virtual and real gravitons propagating along flux tubes. The gravitons at flux tubes are thermalized and thus characterized by temperature and entorpy when the wavelength is much shorter than the distance between the source and receiver. One can say that massive object serves as a heat source. One could also say that the pair of bodies connected by flux tubes serves as a heat source for the flux tubes with temperature determined by reduced mass so that their is a complete symmetry between the two bodies.
2. The expression for the gravitonic entropy of the flux tube is naturally proportional to the length of flux tube at a given "holographic screen" - and for the gravitonic temperature-naturally proportional to the inverse of distance squared in absence of other heat sources from standard Laplace equation- are consistent with their forms at the non-relativistic limit discussed by Sabine Hossenfelder in very transparent manner. In general case, the stringy slicing for the preferred extremals of Kähler action provide the preferred coordinates in which gravitational potential and the counterpart of the radial coordinate can be identified.
3. EG generalizes to all interactions but negative temperatures mean a severe problem. This in turn suggests a direct connection with matter-antimatter asymmetry. Could thermally stable matter and antimatter correspond in zero energy ontology to different arrows of geometric time and appear therefore in different space-time regions? I have made this question also earlier but with a motivation coming directly from the formalism of quantum TGD.
This approach leads to the question whether the mathematical formalism of quantum TGD could make sense also in General Relativity when appropriately modified. In particular, do the notions of zero energy ontology and causal diamond and the identification of generalized Feynman diagrams as space-time regions of Euclidian signature of the metric make sense? Does the Kähler geometry for world of classical worlds realizing holography in strong sense lead to a formulation of GRT as almost topological QFT characterized by Chern-Simons action with a constraint depending on metric?
1. Einstein-Maxwell theory generalizes Kähler action and the conditions guaranteing reduction of action to 3-D "boundary term" are realized automatically by Einstein-Maxwell equations and the weak form of electric-magnetic duality leads to Chern-Simons action.
2. One distinction beween GRT and TGD is the possibility of space-time regions of Euclidian signature of the induced metric in TGD representing the lines of generalized Feynman diagrams. The deformations of CP2 type vacuum extremals with Euclidian signature of the induced metric represent these lines replace black holes in TGD Universe. Black hole horizons are big particles and are suggested to possess gigantic effective value of Planck constant for which Schwartshild radius is essentially the Compton length for gravitational Planck constant so that black hole becomes indeed a particle in quantum sense. Blackholes represent dark matter in TGD sense.
3. CP2 type vacuum extremals are solutions of Einstein's equations with a unique value of cosmological constant fixing CP2 radius and this constant can be non-vanishing only in regions of Euclidian signature. The average value of the cosmological constant would be proportional to the ratio of the three-volume of Euclidian regions to the whole volue of 3-space and therefore very small. Could this be equivalent with the smallness of the actual cosmological constant? To answer the question one should understand the interaction between Euclidian and Minkowskian regions. I have proposed alternative manners to understand apparent cosmological constant in TGD Universe. Negative pressure could be understood in terms of the magnetic energy of magnetic flux tubes. On the other hand, quantum critical cosmology replacing inflation in TGD framework characterized by single parameter - its duration- corresponds to "negative pressure". These explanations need not be mutually exclusive.
At the formal level the formalism for WCW Kähler geometry generalizes as such to almost topological quantum field theory but the conditions of mathematical existence are extremely powerful and the conjecture is that this requires sub-manifold property.
1. The number of physically allowed space-times is much larger in GRT than in TGD framework and this leads to space-time with over-critical and arbitrarily large mass density and other problems plaguing GRT. M-theory exponentiates the problem and leads to landscape misery. The natural conjecture is that one cannot do without assuming that physically acceptable metrics are representable as surfaces in M4× CP2.
2. CP2 type regions give rise to electroweak quantum numbers and Minkowskian regions to four-momentum and spin. This almost gives standard model quantum numbers just from Einstei-Maxwell system! It is however far from clear whether one obtains both of them at the wormhole throats between the Minkowskian and Euclidian regions (perhaps from the representations of super-conformal algebras associated with light-like 3-surfaces by their geometric 2-dimensionality). Since both are needed it seems that one must replace geometry with sub-manifold geometry. Also electroweak spin is obtained naturally only if spinors are induced spinors of the 8-D imbedding space rather than 4-D spinors for which also the existence of spinor structure poses problems in the general case.
For more details see the article TGD inspired vision about entropic gravitation.
Friday, April 08, 2011
New 150 GeV boson stimulates emotions and bad rhetorics
This bump at 150 GeV manages to generate strong emotional responses. Very understandable. All predictions of the hegemony which has dominated particle physics the last thirty years seem to fail and anomalies suggesting unexpected new physics are emerging. It is intolerable that the theory of this TGD guy who has tried to talk sense for thirty years and been ruthlessly silenced and ridiculed can explain the 150 GeV anomaly elegantly using 15 years old predictions of his theory.
The latest example about highly emotional response is from Lubos. I glue below my two comments to the posting of Lubos, the response of Lubos and my response to it: I do not know whether Lubos allows it to appear in the blog. Note that Lubos carefully avoids of saying anything about the contents of my comments since he simply cannot make any reasonable counter argument. Lubos also argues against completely nonsensical statements that he has put to my mouth: another telltale signature of the rhetoric of a poor loser. Draw your own conclusions.
My first comment
TGD suggests two explanations for the possible new particle. Exotic octet of weak bosons is the first guess: it fails because there is no preference to decays to quarks.
Second explanation is in terms of a decay of charged pion of scaled up variant of ordinary hadron physics: scaling factor is 512 for the mass scale from the ratio for the Mersenne primes M_107 and M_89 labeling corresponding p-adic mass scales. According to the recent view not identical with the original one, pions would be produced abundantly and ρ meson or the first p-adic octave of charged pion would decay to W and neutral pion in turn producing quark jets. One signature of the new hadron physics would be monochromatic photon pairs with photon energy in the range 60-80 GeV. The naive scaling argument from ordinary pion mass would give mass of 71.4 GeV. p-Adic scaling with 2 is possible and produces mass 143.4 GeV to be compared with 145 GeV mentioned by Lubos.
Maybe the most dramatic prediction of TGD will be verified within next years! For details see my blog posting .
My second comment
Internal consistency arguments force to conclude that new physics is in TeV scale and the people in CDF are high rank professionals. Therefore I would be cautious in making skeptic or even cynical comments about their skills and even motivations unless I were a similar top professional myself.
Those who predict take this kind of potential discoveries quite seriously for understandable reasons. Both the forward-backward asymmetry in ttbar production and the new particle candidate can be understood in terms of scaled up variant of hadron physics predicted by TGD as I explain in detail at my blog.
Personally I cannot take seriously any model postulating ad hoc particle with hoc couplings to explain single anomaly. Principle is needed. I though for decades ago that after the advent of superstrings theorists would start to predict entire new branches of physics instead of single particle with couplings tinkered to explain single experimental anomaly.
In any case, LHC will certainly tell within few years what is the truth. We can only wait.
Don't be silly, Matti. Most of similar 3-sigma bumps supporting "previously unexpected physics" that have ever been promoted by similar teams turned have been showed to be flukes or mistakes. I have surely done similar things at the top global level so if you ask me whether I consider myself competent to judge the likelihood that this is just hogwash resulting from some rather silly errors, the answer is a resounding Yes. Your encouragement to irrationally worship people who are at least as fallible as I am and who have done lots of very problematic and complex manipulations doesn't belong to science. Science just doesn't operate and cannot operate in this way, by intimidating researchers by the "expertise" of other experts. Science can only get settled if the arguments are being verified and multiplied, not by mindless agreement with some people who are promoted to infallible holy fathers (and, in this case, also mothers).
Your TGD crackpot junk will be left without comments.
Also, it's nonsense that the LHC will need "years" to decide about similar effects. First of all, the D0 Collaboration - the second team at the Fermilab - will publicize its own verdict within weeks. And the LHC could already have the answer in their collected data, too. If it doesn't, it will have the answer this year. The more likely answer is that the effect is bunk. But if it is not bunk, it is not because of infallibility of the CDF folks who have contributed to this paper.
My response
Dear Lubos,
I am just saying that those people who have theories able to predict something (not very many of them!) are quite interested in these bumps, at I have a high respect to the work of the people doing the hard work with experiments and analyzing their results, and that I do not see why this respect could be somehow crackpottish. Certainly this respect does not mean a blind belief to the correctness of their analysis. We are all human beings and most of us are doing their best.
The person who takes the scientific discussion as a battle rather than exchange and comparison of ideas must fight against the temptation to use as the last weapon the crackpot claim. I can understand that for a fanatic string model aficionado the failure of the cherished theory is extremely traumatic experience. But still: I am disappointed that you could not resist this temptation. You are one of the *very* few blog physicists whom I can take seriously and I would respect you much more if you would make at least a single argument about the explanation provided by TGD. Why it is wrong? Why it is nonsensical? No emotional bursts: just answers to these questions in the spirit of normal scientific argumentation. Just arguments about content instead of crackpot rhetorics.
Dear Matti, apologies but your comment that followed my comment above was so atrocious that I had to use it to ban you.
My comment
Dear Lubos,
it is amusing that you are telling someone that his posting is too atrocius;-)! It was not. I just told that your put my mouth something I never said as anyone can directly check. I also asked you tell tell why my proposal is wrong instead of labeling me as a crackpot. This is just ordinary scientific discussion.
I added to my blog the comments including the comment that you deleted so that anyone can see what is involved: see .
I added also a simple estimate of the decay width of the pion of M89 hadron physics (dominating contribution comes from the box diagram with 3 gluons and one quark decaying to W at edges). The order of magnitude for the decay rate is around 20 GeV as required if one assumes flavor octet explaining also top quark asymmetry.
If you respect the rules of normal scientific debate you should tell what is wrong with the proposed mechanism for associated production of W boson and quark pair from pions produced abundantly. You could also tell what is wrong with the estimate for the decay width: what makes standard calculation crackpottish? The estimate can be found at .
You can of course delete also this posting but I will add it to my blog so that everyone can see what is involved.
With Best Regards.
Matti Pitkanen
I could not get this comment through. The blog program told that it has more than 3000 characters. It had about 1000. Perhaps this is the manner to realize the ban. I am really surprised. The briliant Lubos Motl who has been talking about intellectual honesty is afraid of a real scientific debate and uses this kind of tricks to avoid it?! Why so? If the opponent is just a miserable crackpot it should be extremely easy to demonstrate that his arguments are wrong!
Wednesday, April 06, 2011
New particle at mass about 150 GeV?
Tommaso tells about the newest result from CDF. The eprint of CDF collaboration (the first name in the long list of names is T. Aaltonen who comes from Finland) reports evidence for a new resonance like state, presumably a boson with mass around 150 GeV. The interpretation as Higgs is definitely excluded. Nature seems to be mercilessly humiliating the arrogant theoreticians;-). Tommaso promised to represent further comments already today and we are eagerly waiting! Also this posting is expected to develop in steps.
This posting has been updated a couple of times and reflects the evolution of my confused picture about what is involved. As I said: Nature seems to mercilessly humiliate arrogant theorists, me included. I shall confess below all my silly mistakes: enjoy!
First impressions
For the inhabitant of the TGD Universe the most obvious identification of the new particle would be as an exotic weak boson. The TGD based explanation of family replication phenomenon predicts that gauge bosons come in singlets and octets of a dynamical SU(3) symmetry associated with three fermion generations (fermion families correspond to topologies of partonic wormhole throats characterized by the number of handles attached to sphere). Exotic Z or W boson could be in question.
If the symmetry breaking between octet and singlet is due to different value of p-adic prime alone then the mass would come as an multiple of half-octave of the mass of Z or W. For W boson one would obtain 160 GeV consistent with 150 GeV. Z would give 180 GeV mass which is perhaps too high. The Weinberg angle could be however different for the singlet and octet so that the naive p-adic scaling need not hold true exactly.
Note that the strange forward backward asymmetry in the production of top quark pairs might be understood in terms of exotic gluon octet whose existence means neutral flavor changing currents.
One day later
Bloggers have reacted intensively to the possibility of a new particle. Tommaso has now a nice detailed analysis about the intricacies of the analysis of the data leading to the identification of the bump. Also Lubos and Resonaances have commented the new particle. Its existence have been actually known for months in physics circles. The flow of eprints to arXiv explaining the new particle has begun.
People are already now talking about an entirely new interaction. I have done this for more than decade! Actually I have talked about entire hierarchy of scaled up variants of hadron physics (Aaaarrrrgggghhh!; do not get scared: it was an expression of extreme irritation by some colleague who believes that physics proceeds by infinitesimal steps) associated with Mersenne primes and strongly suggested by p-adic length scale hypothesis!
Why an exotic weak boson a la TGD cannot be in question
From the additional data bits leaking via the blogs I can conclude that the new particle cannot be exotic weak boson but more plausibly the basic signature for what I call M89 hadron physics and for which the proton mass is by a factor 512 higher than for the ordinary hadron physics. Pions are abundantly produced in any hadron physics and the signature of any hadron physics are the weak and electromagnetic decays of pions.
The extremely important data bit that I did not have yesterday is that the decays to two jets favor quark pairs over lepton pairs. A model assuming exotic Z -called Z'- produced together with W and decaying preferentially to quark pairs has been proposed as an explanation. Neither ordinary nor the exotic weak gauge bosons of TGD Universe have this kind of preference to decay to quark pairs so that my first guess was wrong.
The resonance appears to be produced in association with W boson. Now comes the confession! This led on my side to an extremely stupid misunderstanding lasting for weeks. I thought that it is the 150 GeV bump which decays to W boson and dijet and forgot to check this when more data came. Stupid me! Ironically, it turned out that later evidence for the production of Wjj state in a decay of resonance with mass slightly below 150 GeV emerged so that the stupid error might have contained a seed of truth.
Remark: It has turned out that bump does not disappear and the most recent analysis assigns 4.1 sigma signicance to it. The mass of the bump would be at 147+/- 5 GeV. Also some evidence that the entire Wjj system results in the decay of a resonance with mass slightly below 300 GeV has emerged.
Is a scaled up copy of hadron physics in question?
The natural explanation for preference of quark pairs would be that strong interactions are somehow involved. This suggests a state analogous to a charged pion decaying to W boson and two gluons annihilating to the quark pair (box diagram). This kind of proposal is indeed made in Technicolor at the Tevatron and has as its analog second fundamental prediction of TGD that p-adically scaled up variants of hadron physics should exist and one of them is waiting to be discovered in TeV region. This prediction emerged already for about 15 years ago as I carried out p-adic mass calculations and discovered that Mersenne primes define fundamental mass scales (see this).
Sidestep: Also colored excitations of leptons and therefore leptohadron physics are predicted (see this). What is amusing that CDF discovered towards the end of 2008 what became known as CDF anomaly giving support for tau-pions. The evidence for electro-pions and mu-pions had emerged already earlier (for details see the link above). All these facts have been buried underground because they simply do not fit to the standard model wisdom. TGD based view about dark matter is indeed needed to circumvent the fact that the lifetimes of weak bosons do not allow new light particles. There is a long series of postings in my blog about CDF anomaly: see for instance this. At that time I did of course my best to inform colleagues about the predicted scaled up version of hadron physics. The only visible outcome of my efforts was that I lost my right to use the computer of Helsinki University since finnish colleagues got really angry! In any case, it would be nice if CDF would have discovered two new hadron physics without even knowing it!
Back to the topic: TGD indeed predicts p-adically scaled up copy of hadron physics in TeV region and the lightest hadron of this physics is a pion like state produced abundantly in the hadronic reactions. Ordinary hadron physics corresponds to Mersenne prime M107=2107-1 whereas the scaled up copy would correspond to M89. The mass scale would be 512 times the mass scale 1 GeV of ordinary hadron physics so that the mass of M89 proton should be about 512 GeV. The mass of the M89 pion would be by a naive scaling 71.7 GeV and about two times smaller than the observed mass in the range 120-160 GeV and with the most probable value around 145 GeV as Lubos reports. 2*71.7 GeV = 143.4 GeV would be the guess of the believer in the p-adic scaling hypothesis and the assumption that pion mass is solely due to quarks. It is important to notice that this scaling works precisely only if CKM mixing matrix is same for the scaled up quarks and if charged pion consisting of u-d quark pair is in question. The well-known current algebra hypothesis that pion is massless in the first approximation would mean that pion mass is solely due to the quark masses whereas proton mass is dominated by other contributions if one assumes that also valence quarks are current quarks with rather small masses. The alternative which also works is that valence quarks are constituent quarks with much higher mass scale.
The killer prediction for the scaled up hadron physics hypothesis are gamma pairs with gamma energy in the range 60-80 GeV. The naivest assumption would give gamma energy of 71.7 GeV. My guess based on deep ignorance about the experimental side is that this signature should be easily testable: one should scan the energy range 60-80 GeV for mono-chromatic gamma pairs.
The simplest identification of the 150 GeV resonance
The picture about CDF resonance has become clearer during the last weeks (see the postings Theorists vs. the CDF bump and More details about the CDF bump. One of the results is that leptophobic Z' can explain only 60 per cent of the production rate.
Situation is coming also clearer for me. A really cold shower came as I found an incredibly silly misunderstanding in my earlier model which assumed that Wjj results from 150 GeV resonance that I identified as charged pion of M89 hadron physics. It is of course jj which results from 150 GeV bump. This is unforgivable sloppiness. Ironically, there is now however evidence that my erratic assumption was correct in the sense that the entire Wjj might results from a resonance with mass slightly below 300 GeV. This suggests that its mass is in good accuracy two times the mass of 150 GeV bump for which best estimate is 147+/-5 GeV.
This brings in mind the explanation for the two and half year old CDF anomaly in which tau-pions with masses coming as octaves of basic tau-pion played a key role (masses were in good approximation 2k× m(&piτ), m(&piτ)≈ 2m&tau:, k=1,2. The same mechanism would explain the discrepancy between the DAMA and Xenon100 experiments. Could this mechanism be at work also now so that 300 GeV bump would correspond to the first octave M89 pion which would have mass 150 GeV. This would mean that the one first octave of charged M89 pion decaying to W and neutral M89 pion with mass slightly below 150 GeV in turn decaying to two jets. Parity conservation would force the decay via emission of W boson. Parity conservation would prevent the decays to two pions. The nasty question is why the octaves of pion are realized as resonances in ordinary hadron physics. One could indeed imagine the mother particle to be ρ meson of M89 hadron physics: in this case derivative coupling would make the decay rate small near the threshold. One can also ask whether the lightest state of M89 pion could be actually around 73 GeV as the naivest possible scaling of pion mass predicts. If so then the situation would be very similar to that in the case of tau-pion.
Connection with the top pair backward-forward asymmetry?
The predicted exotic octet of gluons proposed as an explanation of the anomalous backward-forward asymmetry in top pair production could actually correspond to the gluons of the scaled up variant of hadron physics. M107 hadron physics would correspond to ordinary gluons only and M89 only to the exotic octet of gluons only so that a strict scaled up copy would not be in question. Could it be that given Mersenne prime tolerates only single hadron physics or leptohadron physics?
In any case, this would give a connection with the TGD based explanation of the backward-forward asymmetry in the production of top pairs. In the collision incoming quark of proton and antiquark of antiproton would topologically condense at M89 hadronic space-time sheet and scatter by the exchange of exotic octet of gluons: the exchange between quark and antiquark would not destroy the information about directions of incoming and outgoing beams as s-channgel annihilation would do and one would obtain the large asymmetry.
Yesterday I generated irritation in learned colleagues by writing: "It would be nice if LHC would add to the Particle Data Tables both gluonic and electroweak octets and TGD to the text books;-)". Remaining in super-optimistic mood I would like to induce even more irritation by writing: "It would be nice if LHC would add to the Particle Data Tables not only exotic gluonic and electroweak octets but entire new hadron physics - and as a side product TGD to the text books;-;.). Good physics is fun! Enjoy!
For more about new physics predicted by TGD see the chapter p-Adic mass calculations: New Physics of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy". For reader's convenience I have added a short pdf article Is the new boson reported by CDF pion of M89 hadron physics? at my homepage. |
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Rogue wave observation in a Water Wave Tank
Chabchoub, A; Hoffmann, N P; Akhmediev, Nail
The conventional definition of rogue waves in the ocean is that their heights, from crest to trough, are more than about twice the significant wave height, which is the average wave height of the largest one-third of nearby waves. When modeling deep water waves using the nonlinear Schrödinger equation, the most likely candidate satisfying this criterion is the so-called Peregrine solution. It is localized in both space and time, thus describing a unique wave event. Until now, experiments...[Show more]
CollectionsANU Research Publications
Date published: 2011
Type: Journal article
Source: Physical Review Letters
DOI: 10.1103/PhysRevLett.106.204502
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01_Chabchoub_Rogue_wave_observation_in_a_2011.pdf3.64 MBAdobe PDF Request a copy
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2e478ae58d8cbb8c | Wednesday, April 29, 2015
What could be the origin of p-adic length scale hypothesis?
The argument would explain the existence of preferred p-adic primes. It does not yet explain p-adic length scale hypothesis stating that p-adic primes near powers of 2 are favored. A possible generalization of this hypothesis is that primes near powers of prime are favored. There indeed exists evidence for the realization of 3-adic time scale hierarchies in living matter (see this) and in music both 2-adicity and 3-adicity could be present, this is discussed in TGD inspired theory of music harmony and genetic code (see this).
The weak form of NMP might come in rescue here.
1. Entanglement negentropy for a negentropic entanglement characterized by n-dimensional projection operator is the log(Np(n) for some p whose power divides n. The maximum negentropy is obtained if the power of p is the largest power of prime divisor of p, and this can be taken as definition of number theoretic entanglement negentropy. If the largest divisor is pk, one has N= k× log(p). The entanglement negentropy per entangled state is N/n=klog(p)/n and is maximal for n=pk. Hence powers of prime are favoured which means that p-adic length scale hierarchies with scales coming as powers of p are negentropically favored and should be generated by NMP. Note that n=pk would define a hierarchy of heff/h=pk. During the first years of heff hypothesis I believe that the preferred values obey heff=rk, r integer not far from r= 211. It seems that this belief was not totally wrong.
2. If one accepts this argument, the remaining challenge is to explain why primes near powers of two (or more generally p) are favoured. n=2k gives large entanglement negentropy for the final state. Why primes p=n2= 2k-r would be favored? The reason could be following. n=2k corresponds to p=2, which corresponds to the lowest level in p-adic evolution since it is the simplest p-adic topology and farthest from the real topology and therefore gives the poorest cognitive representation of real preferred extremal as p-adic preferred extermal (Note that p=1 makes formally sense but for it the topology is discrete).
3. Weak form of NMP suggests a more convincing explanation. The density matrix of the state to be reduced is a direct sum over contributions proportional to projection operators. Suppose that the projection operator with largest dimension has dimension n. Strong form of NMP would say that final state is characterized by n-dimensional projection operator. Weak form of NMP allows free will so that all dimensions n-k, k=0,1,...n-1 for final state projection operator are possible. 1-dimensional case corresponds to vanishing entanglement negentropy and ordinary state function reduction isolating the measured system from external world.
4. The negentropy of the final state per state depends on the value of k. It is maximal if n-k is power of prime. For n=2k=Mk+1, where Mk is Mersenne prime n-1 gives the maximum negentropy and also maximal p-adic prime available so that this reduction is favoured by NMP. Mersenne primes would be indeed special. Also the primes n=2k-r near 2k produce large entanglement negentropy and would be favored by NMP.
5. This argument suggests a generalization of p-adic length scale hypothesis so that p=2 can be replaced by any prime.
This argument together with the hypothesis that preferred prime is ramified would correlate the character of the irreducible extension and character of super-conformal symmetry breaking. The integer n characterizing super-symplectic conformal sub-algebra acting as gauge algebra would depends on the irreducible algebraic extension of rational involved so that the hierarchy of quantum criticalities would have number theoretical characterization. Ramified primes could appear as divisors of n and n would be essentially a characteristic of ramification known as discriminant. An interesting question is whether only the ramified primes allow the continuation of string world sheet and partonic 2-surface to a 4-D space-time surface. If this is the case, the assumptions behind p-adic mass calculations would have full first principle justification.
For details see the article The Origin of Preferred p-Adic Primes?.
Tuesday, April 28, 2015
How preferred p-adic primes could be determined?
p-Adic mass calculations allow to conclude that elementary particles correspond to one or possible several preferred primes assigning p-adic effective topology to the real space-time sheets in discretization in some length scale range. TGD inspired theory of consciousness leads to the identification of p-adic physics as physics of cognition. The recent progress leads to the proposal that quantum TGD is adelic: all p-adic number fields are involved and each gives one particular view about physics.
Adelic approach plus the view about evolution as emergence of increasingly complex extensions of rationals leads to a possible answer to th question of the title. The algebraic extensions of rationals are characterized by preferred rational primes, namely those which are ramified when expressed in terms of the primes of the extensions. These primes would be natural candidates for preferred p-adic primes.
1. Earlier attempts
How the preferred primes emerges in this framework? I have made several attempts to answer this question.
1. Classical non-determinism at space-time level for real space-time sheets could in some length scale range involving rational discretization for space-time surface itself or for parameters characterizing it as a preferred extremal correspond to the non-determinism of p-adic differential equations due to the presence of pseudo constants which have vanishing p-adic derivative. Pseudo- constants are functions depend on finite number of pinary digits of its arguments.
2. The quantum criticality of TGD is suggested to be realized in in terms of infinite hierarchies of super-symplectic symmetry breakings in the sense that only a sub-algebra with conformal weights which are n-multiples of those for the entire algebra act as conformal gauge symmetries. This might be true for all conformal algebras involved. One has fractal hierarchy since the sub-algebras in question are isomorphic: only the scale of conformal gauge symmetry increases in the phase transition increasing n. The hierarchies correspond to sequences of integers n(i) such tht n(i) divides n(i+1). These hierarchies would very naturally correspond to hierarchies of inclusions of hyper-finite factors and m(i)= n(i+1)/n(i) could correspond to the integer n characterizing the index of inclusion, which has value n≥ 3. Possible problem is that m(i)=2 would not correspond to Jones inclusion. Why the scaling by power of two would be different? The natural question is whether the primes dividing n(i) or m(i) could define the preferred primes.
3. Negentropic entanglement corresponds to entanglement for which density matrix is projector. For n-dimensional projector any prime p dividing n gives rise to negentropic entanglement in the sense that the number theoretic entanglement entropy defined by Shannon formula by replacing pi in log(pi)= log(1/n) by its p-adic norm Np(1/n) is negative if p divides n and maximal for the prime for which the dividing power of prime is largest power-of-prime factor of n. The identification of p-adic primes as factors of n is highly attractive idea. The obvious question is whether n corresponds to the integer characterizing a level in the hierarchy of conformal symmetry breakings.
4. The adelic picture about TGD led to the question whether the notion of unitary could be generalized. S-matrix would be unitary in adelic sense in the sense that Pm=(SS)mm=1 would generalize to adelic context so that one would have product of real norm and p-adic norms of Pm. In the intersection of the realities and p-adicities Pm for reals would be rational and if real and p-adic Pm correspond to the same rational, the condition would be satisfied. The condition that Pm≤ 1 seems however natural and forces separate unitary in each sector so that this options seems too tricky.
These are the basic ideas that I have discussed hitherto.
2. Could preferred primes characterize algebraic extensions of rationals?
The intuitive feeling is that the notion of preferred prime is something extremely deep and the deepest thing I know is number theory. Does one end up with preferred primes in number theory? This question brought to my mind the notion of ramification of primes (see this) (more precisely, of prime ideals of number field in its extension), which happens only for special primes in a given extension of number field, say rationals. Could this be the mechanism assigning preferred prime(s) to a given elementary system, such as elementary particle? I have not considered their role earlier also their hierarchy is highly relevant in the number theoretical vision about TGD.
1. Stating it very roughly (I hope that mathematicians tolerate this language): As one goes from number field K, say rationals Q, to its algebraic extension L, the original prime ideals in the so called integral closure (see this) over integers of K decompose to products of prime ideals of L (prime is a more rigorous manner to express primeness).
Integral closure for integers of number field K is defined as the set of elements of K, which are roots of some monic polynomial with coefficients, which are integers of K and having the form xn+an-1xn-1+...+a0 . The integral closures of both K and L are considered. For instance, integral closure of algebraic extension of K over K is the extension itself. The integral closure of complex numbers over ordinary integers is the set of algebraic numbers.
2. There are two further basic notions related to ramification and characterizing it. Relative discriminant is the ideal divided by all ramified ideals in K and relative different is the ideal of L divided by all ramified Pi:s. Note that te general ideal is analog of integer and these ideas represent the analogous of product of preferred primes P of K and primes Pi of L dividing them.
3. A physical analogy is provided by decomposition of hadrons to valence quarks. Elementary particles becomes composite of more elementary particles in the extension. The decomposition to these more elementary primes is of form P= ∏ Pie(i), where ei is the ramification index - the physical analog would be the number of elementary particles of type i in the state (see this). Could the ramified rational primes could define the physically preferred primes for a given elementary system?
In TGD framework the extensions of rationals (see this) and p-adic number fields (see this) are unavoidable and interpreted as an evolutionary hierarchy physically and cosmological evolution would have gradually proceeded to more and more complex extensions. One can say that string world sheets and partonic 2-surfaces with parameters of defining functions in increasingly complex extensions of prime emerge during evolution. Therefore ramifications and the preferred primes defined by them are unavoidable. For p-adic number fields the number of extensions is much smaller for instance for p>2 there are only 3 quadratic extensions.
1. In p-adic context a proper definition of counterparts of angle variables as phases allowing definition of the analogs of trigonometric functions requires the introduction of algebraic extension giving rise to some roots of unity. Their number depends on the angular resolution. These roots allow to define the counterparts of ordinary trigonometric functions - the naive generalization based on Taylors series is not periodic - and also allows to defined the counterpart of definite integral in these degrees of freedom as discrete Fourier analysis. For the simplest algebraic extensions defined by xn-1 for which Galois group is abelian are are unramified so that something else is needed. One has decomposition P= ∏ Pie(i), e(i)=1, analogous to n-fermion state so that simplest cyclic extension does not give rise to a ramification and there are no preferred primes.
2. What kind of polynomials could define preferred algebraic extensions of rationals? Irreducible polynomials are certainly an attractive candidate since any polynomial reduces to a product of them. One can say that they define the elementary particles of number theory. Irreducible polynomials have integer coefficients having the property that they do not decompose to products of polynomials with rational coefficients. It would be wrong to say that only these algebraic extensions can appear but there is a temptation to say that one can reduce the study of extensions to their study. One can even consider the possibility that string world sheets associated with products of irreducible polynomials are unstable against decay to those characterize irreducible polynomials.
3. What can one say about irreducible polynomials? Eisenstein criterion states following. If Q(x)= ∑k=0,..,n akxk is n:th order polynomial with integer coefficients and with the property that there exists at least one prime dividing all coefficients ai except an and that p2 does not divide a0, then Q is irreducible. Thus one can assign one or more preferred primes to the algebraic extension defined by an irreducible polynomial Q - in fact any polynomial
allowing ramification. There are also other kinds of irreducible polynomials since Eisenstein's condition is only sufficient but not necessary.
4. Furthermore, in the algebraic extension defined by Q, the primes P having the above mentioned characteristic property decompose to an n :th power of single prime Pi: P= Pin. The primes are maximally/completely ramified. The physical analog P=P0n is Bose-Einstein condensate of n bosons. There is a strong temptation to identify the preferred primes of irreducible polynomials as preferred p-adic primes.
A good illustration is provided by equations x2+1=0 allowing roots x+/-=+/- i and equation x2+2px+p=0 allowing roots x+/-= -p+/-p1/2p-11/2. In the first case the ideals associated with +/- i are different. In the second case these ideals are one and the same since x+= =- x- +p: hence one indeed has ramification. Note that the first example represents also an example of irreducible polynomial, which does not satisfy Eisenstein criterion. In more general case the n conditions on defined by symmetric functions of roots imply that the ideals are one and same when Eisenstein conditions are satisfied.
5. What does this mean in p-adic context? The identity of the ideals can be stated by saying P= P0n for the ideals defined by the primes satisfying the Eisenstein condition. Very loosely one can say that the algebraic extension defined by the root involves n:th root of p-adic prime p. This does not work! Extension would have a number whose n:th power is zero modulo p. On the other hand, the p-adic numbers of the extension modulo p should be finite field but this would not be field anymore since there would exist a number whose n:th power vanishes. The algebraic extension simply does not exist for preferred primes. The physical meaning of this will be considered later.
6. What is so nice that one could readily construct polynomials giving rise to given preferred primes. The complex roots of these polymials could correspond to the points of partonic 2-surfaces carrying fermions and defining the ends of boundaries of string world sheet. It must be however emphasized that the form of the polynomial depends on the choices of the complex coordinate. For instance, the shift x→ x+1 transforms (xn-1)/(x-1) to a polynomial satisfying the Eisenstein criterion. One should be able to fix allowed coordinate changes in such a manner that the extension remains irreducible for all allowed coordinate changes.
Already the integral shift of the complex coordinate affects the situation. It would seem that only the action of the allowed coordinate changes must reduce to the action of Galois group permuting the roots of polynomials. A natural assumption is that the complex coordinate corresponds to a complex coordinate transforming linearly under subgroup of isometries of the imbedding space.
In the general situation one has P= ∏ Pie(i), e(i)≥ 1 so that aso now there are prefered primes so that the appearance of preferred primes is completely general phenomenon.
3. A connection with Langlands program?
In Langlands program (see this) the great vision is that the n-dimensional representations of Galois groups G characterizing algebraic extensions of rationals or more general number fields define n-dimensional adelic representations of adelic Lie groups, in particular the adelic linear group Gl(n,A). This would mean that it is possible to reduce these representations to a number theory for adeles. This would be highly relevant in the vision about TGD as a generalized number theory. I have speculated with this possibility earlier (see this) but the mathematics is so horribly abstract that it takes decade before one can have even hope of building a rough vision.
One can wonder whether the irreducible polynomials could define the preferred extensions K of rationals such that the maximal abelian extensions of the fields K would in turn define the adeles utilized in Langlands program. At least one might hope that everything reduces to the maximally ramified extensions.
At the level of TGD string world sheets with parameters in an extension defined by an irreducible polynomial would define an adele containing various p-adic number fields defined by the primes of the extension. This would define a hierarchy in which the prime ideals of previous level would decompose to those of the higher level. Each irreducible extension of rationals would correspond to some physically preferred p-adic primes.
It should be possible to tell what the preferred character means in terms of the adelic representations. What happens for these representations of Galois group in this case? This is known.
1. For Galois extensions ramification indices are constant: e(i)=e and Galois group acts transitively on ideals Pi dividing P. One obtains an n-dimensional representation of Galois group. Same applies to the subgroup of Galois group G/I where I is subgroup of G leaving Pi invariant. This group is called inertia group. For the maximally ramified case G maps the ideal P0 in P=P0n to itself so that G=I and the action of Galois group is trivial taking P0 to itself, and one obtains singlet representations.
2. The trivial action of Galois group looks like a technical problem for Langlands program and also for TGD unless the singletness of Pi under G has some physical interpretation. One possibility is that Galois group acts as like a gauge group and here the hierarchy of sub-algebras of super-symplectic algebra labelled by integers n is highly suggestive. This raises obvious questions. Could the integer n characterizing the sub-algebra of super-symplectic algebra acting as conformal gauge transformations, define the integer defined by the product of ramified primes? P0n brings in mind the n conformal equivalence classes which remain invariant under the conformal transformations acting as gauge transformiations. . Recalling that relative discriminant is an of K ideal divisible by ramified prime ideals of K, this means that n would correspond to the relative discriminant for K=Q.
Are the preferred primes those which are "physical" in the sense that one can assign to the states satisfying conformal gauge conditions?
4. A connection with infinite primes?
Infinite primes are one of the mathematical outcomes of TGD. There are two kinds of infinite primes. There are the analogs of free many particle states consisting of fermions and bosons labelled by primes of the previous level in the hierarchy. They correspond to states of a supersymmetric arithmetic quantum field theory or actually a hierarchy of them obtained by a repeated second quantization of this theory. A connection between infinite primes representing bound states and and irreducible polynomials is highly suggestive.
1. The infinite prime representing free many-particle state decomposes to a sum of infinite part and finite part having no common finite prime divisors so that prime is obtained. The infinite part is obtained from "fermionic vacuum" X= ∏kpk by dividing away some fermionic primes pi and adding their product so that one has X→ X/m+ m, where m is square free integer. Also m=1 is allowed and is analogous to fermionic vacuum interpreted as Dirac sea without holes. X is infinite prime and pure many-fermion state physically. One can add bosons by multiplying X with any integers having no common denominators with m and its prime decomposition defines the bosonic contents of the state. One can also multiply m by any integers whose prime factors are prime factors of m.
2. There are also infinite primes, which are analogs of bound states and at the lowest level of the hierarchy they correspond to irreducible polynomials P(x) with integer coefficients. At the second levels the bound states would naturally correspond to irreducible polynomials Pn(x) with coefficients Qk(y), which are infinite integers at the previous level of the hierarchy.
3. What is remarkable that bound state infinite primes at given level of hierarchy would define maximally ramified algebraic extensions at previous level. One indeed has infinite hierarchy of infinite primes since the infinite primes at given level are infinite primes in the sense that they are not divisible by the primes of the previous level. The formal construction works as such. Infinite primes correspond to polynomials of single variable at the first level, polynomials of two variables at second level, and so on. Could the Langlands program could be generalized from the extensions of rationals to polynomials of complex argument and that one would obtain infinite hierarchy?
4. Infinite integers in turn could correspond to products of irreducible polynomials defining more general extensions. This raises the conjecture that infinite primes for an extension K of rationals could code for the algebraic extensions of K quite generally. If infinite primes correspond to real quantum states they would thus correspond the extensions of rationals to which the parameters appearing in the functions defining partonic 2-surfaces and string world sheets.
This would support the view that partonic 2-surfaces associated with algebraic extensions defined by infinite integers and thus not irreducible are unstable against decay to partonic 2-surfaces which corresponds to extensions assignable to infinite primes. Infinite composite integer defining intermediate unstable state would decay to its composites. Basic particle physics phenomenology would have number theoretic analog and even more.
5. According to Wikipedia, Eisenstein's criterion (see this) allows generalization and what comes in mind is that it applies in exactly the same form also at the higher levels of the hierarchy. Primes would be only replaced with prime polynomials and the there would be at least one prime polynomial Q(y) dividing the coefficients of Pn(x) except the highest one such that its square would not divide P0. Infinite primes would give rise to an infinite hierarchy of functions of many complex variables. At first level zeros of function would give discrete points at partonic 2-surface. At second level one would obtain 2-D surface: partonic 2-surfaces or string world sheet. At the next level one would obtain 4-D surfaces. What about higher levels? Does one obtain higher dimensional objects or something else. The union of n 2-surfaces can be interpreted also as 2n-dimensional surface and one could think that the hierarchy describes a hierarchy of unions of correlated partonic 2-surfaces. The correlation would be due to the preferred extremal property of Kähler action.
One can ask whether this hierarchy could allow to generalize number theoretical Langlands to the case of function fields using the notion of prime function assignable to infinite prime. What this hierarchy of polynomials of arbitrary many complex arguments means physically is unclear. Do these polynomials describe many-particle states consisting of partonic 2-surface such that there is a correlation between them as sub-manifolds of the same space-time sheet representing a preferred extremals of Kähler action?
This would suggest strongly the generalization of the notion of p-adicity so that it applies to infinite primes.
1. This looks sensible and maybe even practical! Infinite primes can be mapped to prime polynomials so that the generalized p-adic numbers would be power series in prime polynomial - Taylor expansion in the coordinate variable defined by the infinite prime. Note that infinite primes (irreducible polynomials) would give rise to a hierarchy of preferred coordinate variables. In terms of infinite primes this expansion would require that coefficients are smaller than the infinite prime P used. Are the coefficients lower level primes? Or also infinite integers at the same level smaller than the infinite prime in question? This criterion makes sense since one can calculate the ratios of infinite primes as real numbers.
2. I would guess that the definition of infinite-P p-adicity is not a problem since mathematicians have generalized the number theoretical notions to such a level of abstraction much above of a layman like me. The basic question is how to define p-adic norm for the infinite primes (infinite only in real sense, p-adically they have unit norm for all lower level primes) so that it is finite.
3. There exists an extremely general definition of generalized p-adic number fields (see this). One considers Dedekind domain D, which is a generalization of integers for ordinary number field having the property that ideals factorize uniquely to prime ideals. Now D would contain infinite integers. One introduces the field E of fractions consisting of infinite rationals.
Consider element e of E and a general fractional ideal eD as counterpart of ordinary rational and decompose it to a ratio of products of powers of ideals defined by prime ideals, now those defined by infinite primes. The general expression for the p-adic norm of x is x-ord(P), where n defines the total number of ideals P appearing in the factorization of a fractional ideal in E: this number can be also negative for rationals. When the residue field is finite (finite field G(p,1) for p-adic numbers), one can take c to the number of its elements (c=p for p-adic numbers.
Now it seems that this number is not finite since the number of ordinary primes smaller than P is infinite! But this is not a problem since the topology for completion does not depend on the value of c. The simple infinite primes at the first level (free many-particle states) can be mapped to ordinary rationals and q-adic norm suggests itself: could it be that infinite-P p-adicity corresponds to q-adicity discussed by Khrennikov about p-adic analysics. Note however that q-adic numbers are not a field.
Finally, a loosely related question. Could the transition from infinite primes of K to those of L takes place just by replacing the finite primes appearing in infinite prime with the decompositions? The resulting entity is infinite prime if the finite and infinite part contain no common prime divisors in L. This is not the case generally if one can have primes P1 and P2 of K having common divisors as primes of L: in this case one can include P1 to the infinite part of infinite prime and P2 to finite part.
Monday, April 27, 2015
Could adelic approach allow to understand the origin of preferred p-adic primes?
The comment of Crow to the posting Intentions, cognitions, and time stimulated rather interesting ideas about the adelization of quantum TGD.
First two questions.
1. What is Adelic quantum TGD? The basic vision is that scattering amplitudes are obtained by algebraic continuation
to various number fields from the intersection of realities and p-adicities (briefly intersection in what follows) represented at the space-time level by string world sheets and partonic 2-surfaces for which defining parameters (WCW coordinates) are in rational or in in some algebraic extension of p-adic numbers. This principle is a combination of strong form of holography and algebraic continuation as a manner to achieve number theoretic universality.
2. Why Adelic quantum TGD? Adelic approach is free of the earlier assumptions, which require mathematics which need not exist: transformation of p-adic space-time surfaces to real ones as a realization of intentional actions was the questionable assumption, which is un-necessary if cognition and matter are two different aspects of existence as already the success of p-adic mass calculations strongly suggests. It always takes years to develop ability to see things from bigger perspective and distill discoveries from clever inventions. Now adelicity is totally obvious. Being a conservative radical - not radical radical or radical conservative - is the correct strategy which I have been gradually learning. This particular lesson was excellent!
For some years ago Crow sent to me the book of Lapidus about adelic strings. Witten wrote for long time ago an article in which the demonstrated that that the product of real stringy vacuum amplitude and its p-adic variants equals to 1. This is a generalisation of the adelic identity for a rational number stating that the product of the norm of rational number with its p-adic norms equals to one.
The real amplitude in the intersection of realities and p-adicities for all values of parameter is rational number or in an appropriate algebraic extension of rationals. If given p-adic amplitude is just the p-adic norm of real amplitude, one would have the adelic identity. This would however require that p-adic variant of the amplitude is real number-valued: I want p-adic valued amplitudes. A further restriction is that Witten's adelic identity holds for vacuum amplitude. I live in Zero Energy Ontology (ZEO) and want it for entire S-matrix, M-matrix, and/or U-matrix and for all states of the basis in some sense.
Consider first the vacuum amplitude. A weaker form of the identity would be that the p-adic norm of a given p-adic valued amplitude is same as that p-adic norm for the rational-valued real amplitude (this generalizes to algebraic extensions, I dare to guess) in the intersection. This would make sense and give a non-trivial constraint: algebraic continuation would guarantee this constraint. In particular, the p-adic norm of the real amplitude would be inverse of the product of p-adic norms of p-adic amplitudes. Most of these amplitudes should have p-adic norm equal to one in other words, real amplitude is product of finite number of powers of prime. This because the p-adic norms must approach rapidly to unity as p-adic prime increases and for large p-adic primes this means that the norm is exactly unity. Hence the p-adic norm of p-adic amplitude equals to 1 for most primes.
In ZEO one must consider S-, M-, or U-matrix elements. U and S are unitary. M is product of hermitian square root of density matrix times unitary S-matrix. Consider next S-matrix.
1. For S-matrix elements one should have pm=(SS)mm=1. This states the unitarity of S-matrix. Probability is conserved. Could it make sense to generalize this condition and demand that it holds true only adelically that only for the product of real and p-adic norms of pm equals to one: NR(pm)(R)∏p Np(pm(p))=1. This could be actually true identically in the intersection if algebraic continuation principle holds true. Despite the triviality of the adelicity condition, one need not have anymore unitarity separately for reals and p-adic number fields. Notice that the numbers pm would be arbitrary
rationals in the most general cased.
2. Could one even replace Np with canonical identification or some form of it with cutoffs reflecting the length scale cutoffs? Canonical identification behaves for powers of p like p-adic norm and means only
more precise map of p-adics to reals.
3. For a given diagonal element of unit matrix characterizing particular state m one would have a product of real norm and p-adic norms. The number of the norms, which differ from unity would be finite. This condition would give finite number of exceptional p-adic primes, that is assign to a given quantum state m a finite number of preferred p-adic primes! I have been searching for a long time the underlying deep reason for this assignment forced by the p-adic mass calculations and here it might be.
4. Unitarity might thus fail in real sector and in a finite number of p-adic sectors (otherwise the product of p-adic norms would be infinite or zero). In some sense the failures would compensate each other in the adelic picture. The failure of course brings in mind p-adic thermodynamics, which indeed means that adelic SS, or should it be called MM, is not unitary but defines the density matrix defining the p-adic thermal state! Recall that M-matrix is defined as hermitian square root of density matrix and unitary S-matrix.
5. The weakness of these arguments is that states are assumed to be labelled by discrete indices. Finite measurement resolution implies discretization and could justify this.
The p-adic norms of pm or the images of pm under canonical identification in a given number field would define analogs of probabilities. Could one indeed have ∑m pm=1 so that SS would define a density matrix?
1. For the ordinary S-matrix this cannot be the case since the sum of the probabilities pm equals to the dimension N of the state space: ∑ pm=N. In this case one could accept pm>1 both in real and p-adic sectors. For this option adelic unitarity would make sense and would be highly non-trivial condition allowing perhaps to understand how preferred p-adic primes emerge at the fundamental level.
2. If S-matrix is multiplied by a hermitian square root of density matrix to get M-matrix, the situation changes and one indeed obtains ∑ pm=1. MM†=1 does not make sense anymore and must be replaced with MM†=ρ, in special case a projector to a N-dimensional subspace proportional to 1/N. In this case the numbers p(m) would have p-adic norm larger than one for the divisors of N and would define preferred p-adic primes. For these primes the sum Np(p(m)) would not be equal to 1 but to NNp(1/N.
3. Situation is different for hyper-finite factors of type II1 for which the trace of unit matrix equals to one by definition and MM=1 and ∑ pm=1 with sum defined appropriately could make sense. If MM† could be also a projector to an infinite-D subspace. Could the M-matrix using the ordinary definition of dimension of Hilbert space be equivalent with S-matrix for the state space using the definition of dimension assignable to HFFs? Could these notions be dual of each other? Could the adelic S-matrix define the counterpart of M-matrix for HFFs?
This looks like a nice idea but usually good looking ideas do not live long in the crossfire of counter arguments. The following is my own. The reader is encouraged to invent his or her own objections.
1. The most obvious objection against the very attractive direct algebraic continuation} from real to p-adic sector is that if the real norm or real amplitude is small then the p-adic norm of its p-adic counterpart is large so that p-adic variants of pm(p) can become larger than 1 so that probability interpretation fails. As noticed there is no actually no need to pose probability interpretation. The only way to overcome the "problem" is to assume that unitarity holds separately in each sector so that one would have p(m)=1 in all number fields but this would lead to the loss of preferred primes.
2. Should p-adic variants of the real amplitude be defined by canonical identification or its variant with cutoffs? This is mildly suggested by p-adic thermodynamics. In this case it might be possible to satisfy the condition pm(R)∏p Np(pm(p))=1. One can however argue that the adelic condition is an ad hoc condition in this
To sum up, if the above idea survives all the objections, it could give rise to a considerable progress. A first principle understanding of how preferred p-adic primes are assigned to quantum states and thus a first principle justification for p-adic thermodynamics. For the ordinary definition of S-matrix this picture makes sense and also for M-matrix. One would still need the justification of canonical identification map playing a key role in p-adic thermodynamics allowing to map p-adic mass squared to its real counterpart.
Sunday, April 26, 2015
Hierarchies of conformal symmetry breakings, quantum criticalities, Planck constants, and hyper-finite factors
TGD is characterized by various hierarchies. There are fractal hierarchies of quantum criticalities, Planck constants and hyper-finite factors and these hierarchies relate to hierarchies of space-time sheets, and selves. These hierarchies are closely related and this article describes these connections. In this article the recent view about connections between various hierarchies associated with quantum TGD are described.
For details see the article Hierarchies of conformal symmetry breakings, quantum criticalities, Planck constants, and hyper-finite factors.
Updated View about Kähler geometry of WCW
Quantum TGD reduces to a construction of Kähler geometry for what I call the "World of Classical Worlds. It has been clear from the beginning that the gigantic super-conformal symmetries generalizing ordinary super-conformal symmetries are crucial for the existence of WCW Kähler metric. The detailed identification of Kähler function and WCW Kähler metric has however turned out to be a difficult problem. It is now clear that WCW geometry can be understood in terms of the analog of AdS/CFT duality between fermionic and space-time degrees of freedom (or between Minkowskian and Euclidian space-time regions) allowing to express Kähler metric either in terms of Kähler function or in terms of anti-commutators of WCW gamma matrices identifiable as super-conformal Noether super-charges for the symplectic algebra assignable to δ M4+/-× CP2. The string model type description of gravitation emerges and also the TGD based view about dark matter becomes more precise. String tension is however dynamical rather than pregiven and the hierarchy of Planck constants is necessary in order to understand the formation of gravitationally bound states. Also the proposal that sparticles correspond to dark matter becomes much stronger: sparticles actually are dark variants of particles.
A crucial element of the construction is the assumption that super-symplectic and other super-conformal symmetries having the same structure as 2-D super-conformal groups can be seen a broken gauge symmetries such that sub-algebra with conformal weights coming as n-ples of those for full algebra act as gauge symmetries. In particular, the Noether charges of this algebra vanish for preferred extremals- this would realize the strong form of holography implied by strong form of General Coordinate Invariance. This gives rise to an infinite number of hierarchies of conformal gauge symmetry breakings with levels labelled by integers n(i) such that n(i) divides n(i+1) interpreted as hierarchies of dark matter with levels labelled by the value of Planck constant heff=n× h. These hierarchies define also hierarchies of quantum criticalities and are proposed to give rise to inclusion hierarchies of hyperfinite factors of II1 having interpretation in terms of finite cognitive resolution. These hierarchies would be fundamental for the understanding of living matter.
For details see the article Updated view about Kähler geometry of WCW.
Intentions, Cognition, and Time
Intentions involve time in an essential manner and this led to the idea that p-adic-to-real quantum jumps could correspond to a realization of intentions as actions. It however seems that this hypothesis posing strong additional mathematical challenges is not needed if one accepts adelic approach in which real space-time time and its p-adic variants are all present and quantum physics is adelic. I have already earlier developed the first formulation of p-adic space-time surfaces as cognitive charges of real space-time surfaces and also the ideas related to the adelic vision.
The recent view involving strong form of holography would provide dramatically simplified view about how these representations are formed as continuations of representations of strings world sheets and partonic 2-surfaces in the intersection of real and p-adic variants of WCW ("World of Classical Worlds") in the sense that the parameters characterizing these representations are in the algebraic numbers in the algebraic extension of p-adic numbers involved.
For details see the article Intentions, Cognition, and Time
Saturday, April 25, 2015
Good and Evil, Life and Death
In principle the proposed conceptual framework allows already now a consideration of the basic questions relating to concepts like Good and Evil and Life and Death. Of course, too many uncertainties are involved to allow any definite conclusions, and one could also regard the speculations as outputs of the babbling period necessarily accompanying the development of the linguistic and conceptual apparatus making ultimately possible to discuss these questions more seriously.
Even the most hard boiled materialistic sceptic mentions ethics and moral when suffering personal injustice. Is there actual justification for moral laws? Are they only social conventions or is there some hard core involved? Is there some basic ethical principle telling what deeds are good and what deeds are bad?
Second group of questions relates to life and biological death. How should on define life? What happens in the biological death? Is self preserved in the biological death in some form? Is there something deserving to be called soul? Are reincarnations possible? Are we perhaps responsible for our deeds even after our biological death? Could the law of Karma be consistent with physics? Is liberation from the cycle of Karma possible?
In the sequel these questions are discussed from the point of view of TGD inspired theory of consciousness. It must be emphasized that the discussion represents various points of view rather than being a final summary. Also mutually conflicting points of view are considered. The cosmology of consciousness, the concept of self having space-time sheet and causal diamond as its correlates, the vision about the fundamental role of negentropic entanglement, and the hierarchy of Planck constants identified as hierarchy of dark matters and of quantum critical systems, provide the building blocks needed to make guesses about what biological death could mean from subjective point of view.
For details see the article Good and Evil, Life and Death.
Friday, April 24, 2015
Variation of Newton's constant and of length of day
J. D. Anderson et al have published an article discussing the observations suggesting a periodic variation of the measured value of Newton constant and variation of length of day (LOD) (see also this). This article represents TGD based explanation of the observations in terms of a variation of Earth radius. The variation would be due to the pulsations of Earth coupling via gravitational interaction to a dark matter shell with mass about 1.3× 10-4ME introduced to explain Flyby anomaly: the model would predict Δ G/G= 2Δ R/R and Δ LOD/LOD= 2Δ RE/RE with the variations pf G and length of day in opposite phases. The expermental finding Δ RE/RE= MD/ME is natural in this framework but should be deduced from first principles.
The gravitational coupling would be in radial scaling degree of freedom and rigid body rotational degrees of freedom. In rotational degrees of freedom the model is in the lowest order approximation mathematically equivalent with Kepler model. The model for the formation of planets around Sun suggests that the dark matter shell has radius equal to that of Moon's orbit. This leads to a prediction for the oscillation period of Earth radius: the prediction is consistent with the observed 5.9 years period. The dark matter shell would correspond to n=1 Bohr orbit in the earlier model for quantum gravitational bound states based on large value of Planck constant. Also n>1 orbits are suggestive and their existence would provide additional support for TGD view about quantum gravitation.
For details see the chapter Cosmology and Astrophysics in Many-Sheeted Space-Time or the article Variation of Newton's constant and of length of day.
Tuesday, April 21, 2015
Connection between Boolean cognition and emotions
Weak form of NMP allows the state function reduction to occur in 2n-1 manners corresponding to subspaces of the sub-space defined by n-dimensional projector if the density matrix is n-dimensional projector (the outcome corresponding to 0-dimensional subspace and is excluded). If the probability for the outcome of state function reduction is same for all values of the dimension 1≤m ≤n, the probability distribution for outcome is given by binomial distribution B(n,p) for p=1/2 (head and tail are equally probable) and given by p(m)= b(n,m)× 2-n= (n!/m!(n-m)!)×2-n . This gives for the average dimesion E(m)= n/2 so that the negentropy would increase on the average. The world would become gradually better.
One cannot avoid the idea that these different degrees of negentropic entanglement could actually give a realization of Boolean algebra in terms of conscious experiences.
1. Could one speak about a hierarchies of codes of cognition based on the assignment of different degrees of "feeling good" to the Boolean statements? If one assumes that the n:th bit is always 1, all independent statements except one correspond at least two non-vanishing bits and corresponds to negentropic entanglement. Only of statement (only last bit equal to 1) would correspond 1 bit and to state function reduction reducing the entanglement completely (brings in mind the fruit in the tree of Good and Bad Knowlege!).
2. A given hierarchy of breakings of super-symplectic symmetry corresponds to a hierarchy of integers ni+1= ∏k≤ i mk. The codons of the first code would consist of sequences of m1 bits. The codons of the second code consists of m2 codons of the first code and so on. One would have a hierarchy in which codons of previous level become the letters of the code words at the next level of the hierarchy.
In fact, I ended up with almost Boolean algebra for decades ago when considering the hierarchy of genetic codes suggested by the hierarchy of Mersenne primes M(n+1)= MM(n), Mn= 2n-1.
1. The hierarchy starting from M2=3 contains the Mersenne primes 3,7,127,2127-1 and Hilbert conjectured that all these integers are primes. These numbers are almost dimensions of Boolean algebras with n=2,3,7,127 bits. The maximal Boolean sub-algebras have m=n-1=1,2,6,126 bits.
2. The observation that m=6 gives 64 elements led to the proposal that it corresponds to a Boolean algebraic assignable to genetic code and that the sub-algebra represents maximal number of independent statements defining analogs of axioms. The remaining elements would correspond to negations of these statements. I also proposed that the Boolean algebra with m=126=6× 21 bits (21 pieces consisting of 6 bits) corresponds to what I called memetic code obviously realizable as sequences of 21 DNA codons with stop codons included. Emotions and information are closely related and peptides are regarded as both information molecules and molecules of emotion.
3. This hierarchy of codes would have the additional property that the Boolean algebra at n+1:th level can be regarded as the set of statements about statements of the previous level. One would have a hierarchy representing thoughts about thoughts about.... It should be emphasized that there is no need to assume that the Hilbert's conjecture is true.
One can obtain this kind of hierarchies as hierarchies with dimensions m, 2m, 22m,... that is n(i+1)= 2n(i). The conditions that n(i) divides n(i+1) is non-trivial only for at the lowest step and implies that m is power of 2 so that the hierarchies starting from m=2k. This is natural since Boolean algebras are involved. If n corresponds to the size scale of CD, it would come as a power of 2.
p-Adic length scale hypothesis has also led to this conjecture. A related conjecture is that the sizes of CDs correspond to secondary p-adic length scales, which indeed come as powers of two by p-adic length scale hypothesis. In case of electron this predicts that the minimal size of CD associated with electron corresponds to time scale T=.1 seconds, the fundamental time scale in living matter (10 Hz is the fundamental bio-rhythm). It seems that the basic hypothesis of TGD inspired partly by the study of elementary particle mass spectrum and basic bio-scales (there are 4 p-adic length scales defined by Gaussian Mersenne primes in the range between cell membrane thickness 10 nm and and size 2.5 μm of cell nucleus!) follow from the proposed connection between emotions and Boolean cognition.
4. NMP would be in the role of God. Strong NMP as God would force always the optimal choice maximizing negentropy gain and increasing negentropy resources of the Universe. Weak NMP as God allows free choice so that
entropy gain is not be maximal and sinners populate the world. Why the omnipotent God would allow this? The reason is now obvious. Weak form of NMP makes possible the realization of Boolean algebras in terms of degrees of "feels good"! Without the God allowing the possibility to do sin there would be no emotional intelligence!
Hilbert's conjecture relates in interesting manner to space-time dimension. Suppose that Hilbert's conjecture fails and only the four lowest Mersenne integers in the hierarchy are Mersenne primes that is 3,7,127, 2127-1. In TGD one has hierarchy of dimensions associated with space-time surface coming as 0,1,2,4 plus imbedding space dimension 8. The abstraction hierarchy associated with space-time dimensions would correspond discretization of partonic 2-surfaces as point set, discretization of 3-surfaces as a set of strings connecting partonic 2-surfaces characterized by discrete parameters, discretization of space-time surfaces as a collection of string world sheet with discretized parameters, and maybe - discretization of imbedding space by a collection of space-time surfaces. Discretization means that the parameters in question are algebraic numbers in an extension of rationals associated with p-adic numbers.
In TGD framework it is clear why imbedding space cannot be higher-dimensional and why the hierarchy does not continue. Could there be a deeper connection between these two hierarchies. For instance, could it be that higher dimensional manifolds of dimension 2×n can be represented physically only as unions of say n 2-D partonic 2-surfaces (just like 3×N dimensional space can be represented as configuration space of N point-like particles)? Also infinite primes define a hierarchy of abstractions. Could it be that one has also now similar restriction so that the hierarchy would have only finite number of levels, say four. Note that the notion of n-group and n-algebra involves an analogous abstraction hierarchy.
Monday, April 20, 2015
Can one identify quantum physical correlates of ethics and moral?
TGD inspired theory of consciousness involves a bundle of new concepts making it possible to seriously discuss quantum physical correlates of ethics and moral assuming that we live in TGD Universe. In the following I summarize the recent understanding. I do not guarantee that I will agree with myself tomorrow since I am just going through this stuff in the updating of TGD inspired theory of consciousness and quantum biology.
Quantum ethics very briefly
Could physics generalized to a theory of consciousness allow to undersand the physical correlates of ethics and moral. The proposal is that this is the case. The basic ethical principle would be that good deeds help evolution to occur. Evolution should correspond to the increase of negentropic entanglement resources, defining negentropy sources, which I have called Akashic records.
This idea can be criticized.
1. If strong form of NMP prevails, one can worry that TGD Universe does not allow Evil at all, perhaps not even genuine free will! No-one wants Evil but Evil seems to be present in this world.
2. Could one weaken NMP so that it does not force but only allows to make a reduction to a final state characterized by density matrix which is projection operator? Self could choose whether to perform a projection to some sub-space of this subspace, say 1-D ray as in ordinary state function reduction. NMP would be like Christian God allowing the sinner to choose between Good and Evil. The final entanglement negentropy would be measure for the goodness of the deed. This is so if entanglement negentropy is a correlate for love. Deeds which are done with love would be good. Reduction of entanglement would in turn mean loneliness and separation.
3. Or could could think that the definition of good deed is as a selection between deeds, which correspond to the same maximal increase of negentropy so that NMP cannot tell what happens. For instance the density matrix operator is direct sum of projection operators of same dimension but varying coefficients and there is a selection between these. It is difficult to imagine what the criterion for a good deed could be in this case. And how self can know what is the good deed and what is the bad deed.
Good deeds would support evolution. There are many manners to interpret evolution in TGD Universe.
1. p-Adic evolution would mean a gradual increase of the p-adic primes characterizing individual partonic 2-surfaces and therefore their size. The identification of p-adic space-time sheets as representations for cognitions gives additional concreteness to this vision. The earlier proposal that p-adic--real-phase transitions correspond to realization of intentions and formations of cognitions seems however to be wrong. Instead, adelic view that both real and p-adic sectors are present simultaneously and that fermions at string world sheets correspond to the intersection of realities and p-adicities seems more realistic.
The inclusion of phases q=exp(i2π/n) in the algebraic extension of p-adics allows to define the notion of angle in p-adic context but only with a finite resolution since only finite number of angles are represented as phases for a given value of n. The increase of the integers n could be interpreted as the emergence of higher algebraic extensions of p-adic numbers in the intersection of the real and p-adic worlds. These observations suggest that all three views about evolution are closely related.
2. The hierarchy of Planck constants suggests evolution as the gradual increase of the Planck constant characterizing p-adic space-time sheet (or partonic 2-surface for the minimal option). The original vision about this evolution was as a migration to the pages of the book like structure defined by the generalized imbedding space and has therefore quite concrete geometric meaning. It implies longer time scales of long term memory and planned action and macroscopic quantum coherence in longer scales.
The new view is in terms of first quantum jumps to the opposite boundary of CD leading to the death of self and its re-incarnation at the opposite boundary.
3. The vision about life as something in the intersection of real and p-adic words allows to see evolution information theoretically as the increase of number entanglement negentropy implying entanglement in increasing length scales. This option is equivalent with the second view and consistent with the first one if the effective p-adic topology characterizes the real partonic 2-surfaces in the intersection of p-adic and real worlds.
The third kind of evolution would mean also the evolution of spiritual consciousness if the proposed interpretation is correct. In each quantum jump U-process generates a superposition of states in which any sub-system can have both real and algebraic entanglement with the external world. If state function reduction process involves also the choice of the type of entanglement it could be interpreted as a choice between good and evil. The hedonistic complete freedom resulting as the entanglement entropy is reduced to zero on one hand, and the negentropic entanglement implying correlations with the external world and meaning giving up the maximal freedom on the other hand. The selfish option means separation and loneliness. The second option means expansion of consciousness - a fusion to the ocean of consciousness as described by spiritual practices.
In this framework one could understand the physics correlates of ethics and moral. The ethics is simple: evolution of consciousness to higher levels is a good thing. Anything which tends to reduce consciousness represents violence and is a bad thing. Moral rules are related to the relationship between individual and society and presumably develop via self-organization process and are by no means unique. Moral rules however tend to optimize evolution. As blind normative rules they can however become a source of violence identified as any action which reduces the level of consciousness.
There is an entire hierarchy of selves and every self has the selfish desire to survive and moral rules develop as a kind of compromise and evolve all the time. ZEO leads to the notion that I have christened cosmology of consciousness. It forces to extend the concept of society to four-dimensional society. The decisions of "me now" affect both my past and future and time like quantum entanglement makes possible conscious communication in time direction by sharing conscious experiences. One can therefore speak of genuinely four-dimensional society. Besides my next-door neighbors I had better to take into account also my nearest neighbors in past and future (the nearest ones being perhaps copies of me!). If I make wrong decisions those copies of me in future and past will suffer the most. Perhaps my personal hell and paradise are here and are created mostly by me.
What could the quantum correlates of moral be?
We make moral choices all the time. Some deeds are good, some deeds are bad. In the world of materialist there are no moral choices, the deeds are not good or bad, there are just physical events. I am not a materialist so that I cannot avoid questions such as how do the moral rules emerge and how some deeds become good and some deeds bad. Negentropic entanglement is the obvious first guess if one wants to understand emergence of moral.
1. One can start from ordinary quantum entanglement. It corresponds to a superposition of pairs of states. Second state corresponds to the internal state of the self and second state to a state of external world or biological body of self. In negentropic quantum entanglement each is replaced with a pair of sub-spaces of state spaces of self and external world. The dimension of the sub-space depends on the which pair is in question. In state function reduction one of these pairs is selected and deed is done. How to make some of these deeds good and some bad?
2. Obviously the value of heff/h=n gives the criterion in the case that weak form of NMP holds true. Recall that weak form of NMP allows only the possibility to generate negentropic entanglement but does not force it. NMP is like God allowing the possibility to do good but not forcing good deeds.
Self can choose any sub-space of the subspace defined by n-dimensional projector and 1-D subspace corresponds to the standard quantum measurement. For n=1 the state function reduction leads to vanishing negentropy, and separation of self and the target of the action. Negentropy does not increase in this action and self is isolated from the target: kind of price for sin.
For the maximal dimension of this sub-space the negentropy gain is maximal. This deed is good and by the proposed criterion the negentropic entanglement corresponds to love or more generally, positively colored conscious experience. Interestingly, there are 2n possible choices which is the dimension of Boolean algebra consisting of n independent bits. This could relate directly to fermionic oscillator operators defining basis of Boolean algebra. The deed in this sense would be a choice of how loving the attention towards system of external world is.
3. Could the moral rules of society be represented as this kind of entanglement patterns between its members? Here one of course has entire fractal hierarchy of societies corresponding different length scales. Attention and magnetic flux tubes serving as its correlates is the basic element also in TGD inspired quantum biology already at the level of bio-molecules and even elementary particles. The value of heff/h=n associated with the magnetic flux tube connecting members of the pair, would serve as a measure for the ethical value of maximally good deed. Dark phases of matter would correspond to good: usually darkness is associated with bad!
4. These moral rules seem to be universal. There are however also moral rules or should one talk about rules of survival, which are based on negative emotions such as fear. Moral rules as rules of desired behavior are often tailored for the purposes of power holder. How this kind of moral rules could develop? Maybe they cannot be realized in terms of negentropic entanglement. Maybe the superposition of the allowed alternatives for the deed contains only the alternatives allowed by the power holder and the superposition in question corresponds to ordinary entanglement for which the signature is simple: the probabilities of various options are different. This forces the self to choose just one option from the options that power holder accepts. These rules do not allow the generation of loving relationship.
Moral rules seem to be generated by society, up-bringing, culture, civilization. How the moral rules develop? One can try to formulate and answer in terms of quantum physical correlates.
1. Basically the rules should be generated in the state function reductions which correspond to volitional action which corresponds to the first state function reduction to the earlier active boundary of CD. Old self dies and new self is born at the opposite boundary of CD and the arrow of time associated with CD changes.
2. The repeated sequences of state function reductions can generate negentropic entanglement during the quantum evolutions between them. This time evolution would be the analog for the time evolution defined by Hamiltonian - that is energy - associated with ordinary time translation whereas the first state function reduction at the opposite boundary inducing scaling of heff and CD would be accompanied by time evolution defined by conformal scaling generator L0.
Note that the state at passive boundary does not change during the sequence of repeated state function reductions. These repeated reductions however change the parts of zero energy states associated with the new active boundary and generate also negentropic entanglement. As the self dies the moral choices can made if the weak form of NMP is true.
3. Who makes the moral choices? It looks of course very weird that self would apply free will only at the moment of its death or birth! The situation is saved by the fact that self has also sub-selves, which correspond to sub-CDs and represent mental images of self. We know that mental images die as also we do some day and are born again (as also we do some day) and these mental images can generate negentropic resources within CD of self.
One can argue that these mental images do not decide about whether to do maximally ethical choice at the moment of death. The decision must be made by a self at higher level. It is me who decides about the fate of my mental images - to some degree also after their death! I can choose the how negentropic the quantum entanglement characterizing the relationship of my mental image and the world outside it. I realize, that the misused idea of positive thinking seems to unavoidably creep in! I have however no intention to make money with it!
There are still many questions that are waiting for more detailed answer. These questions are also a good manner to detect logical inconsistencies.
1. What is the size of CD characterizing self? For electron it would be at least of the order of Earth size. During the lifetime of CD the size of CD increases and the order of magnitude is measured in light-life time for us. This would allow to understand our usual deeds affecting the environment in terms of our subselves and their entanglement with the external world which is actually our internal world, at least if magnetic bodies are considered.
2. Can one assume that the dynamics inside CD is independent from what happens outside CD. Can one say that the boundaries of CD define the ends of space-time or does space-time continue outside them. Do the boundaries of CD define boundaries for 4-D spotlight of attention or for one particular reality? Does the answer to this question have any relevance if everything physically testable is formulated in term physics of string world sheets associated with space-time surfaces inside CD?
Note that the (average) size of CDs (, which could be in superposition but need not if every repeated state function reduction is followed by a localization in the moduli space of CDs) increases during the life cycle of self. This makes possible generation of negentropic entanglement between more and more distant systems. I have written about the possibility that ZEO could make possible interaction with distant civilizations (see this. The possibility of having communications in both time directions would allow to circumvent the barrier due to the finite light-velocity, and gravitational quantum coherence in cosmic scales would make possible negentropic entanglement.
3. How selves interact? CDs as spot-lights of attention should overlap in order that the interaction is possible. Formation of flux tubes makes possible quantum entanglement. The string world sheets carrying fermions also essential correlates of entanglement and the possibly entanglement is between fermions associated with partonic 2-surfaces. The string world sheets define the intersection of real and p-adic worlds, where cognition and life resides.
Intentions, cognitions, time, and p-adic physics
1. What intentions are?
2. p-Adic physics as physics of only cognition?
Most of the following considerations apply in both cases.
3. Some questions to ponder
b) Could cognitive resolution fix the measurement resolution?
c) What selects the preferred p-adic prime?
The connection with hyper-finite factors suggests itself.
5. Number theoretic universality for cognitive representations
Monday, April 13, 2015
Manifest unitarity and information loss in gravitational collapse
There was a guest posting in the blog of Lubos by Prof. Dejan Stojkovic from Buffalo University. The title of the post was Manifest unitarity and information loss in gravitational collapse. It explained the contents of the article Radiation from a collapsing object is manifestly unitary by Stojkovic and Saini.
The posting
The posting describes calculations carried out for a collapsing spherical mass shell, whose radius approaches its own Scwartschild radius. The metric outside the shell with radius larger than rS is assumed to be Schwartschild metric. In the interior of the shell the metric would be Minkowski metric. The system considered is second quantized massless scalar field. One can calculate the Hamiltonian of the radiation field in terms of eigenmodes of the kinetic and potential parts and by canonical quantization the Schrödinger equation for the eigenmodes reduces to that for a harmonic oscillator with time dependent frequency. Solutions can be developed in terms of solutions of time-independent harmonic oscillator. The average value of the photon number turns out to approach to that associated with a thermal distribution irrespective of initial values at the limit when the of the shell approaches its blackhole radius. The temperature is Hawking temperature. This is of course highly interesting result and should reflect the fact that Minkowski vacuum looks from the point of view of an accelerated system to be in thermal equilibrium. Manifest unitary is just what one expects.
The authors assign a density matrix to the state in the harmonic oscillator basis. Since the state is pure, the density matrix is just a projector to the quantum state since the components of the density matrix are products of the coefficients characterizing the state in the oscillator basis (there are a couple of typos in the formulas, reader certainly notices them). In Hawking's original argument the non-diagonal cross terms are neglected and one obtains a non-pure density matrix. The approach of authors is of course correct since they consider only the situation before the formation of horizon. Hawking consider the situation after the formation of horizon and assumes some un-specified process taking the non-diagonal components of the density matrix to zero. This decoherence hypothesis is one of the strange figments of insane theoretical imagination which plagues recent day theoretical physics.
Authors mention as a criterion for purity of the state the condition that the square of the density matrix has trace equal to one. This states that the density matrix is N-dimensional projector. The criterion alone does not however guarantee the purity of the state for N> 1. This is clear from the fact that the entropy is in this case non-vanishing and equal to log(N). I notice this because negentropic entanglement in TGD framework corresponds to the situation in entanglement matrix is proportional to unit matrix (that is projector). For this kind of states number theoretic counterpart of Shannon entropy makes sense and gives negative entropy meaning that entanglement carries information. Note that unitary 2-body entanglement gives rise to negentropic entanglement.
Authors inform that Hawkins used Bogoliubov transformations between initial Minkowski vacuum and final Schwartschild vacum at the end of collapse which looks like thermal distribution with Hawking temperature in terms from Minkowski space point of view. I think that here comes an essential physical point. The question is about the relationship between two observers - one might call them the observer falling into blackhole and the observer far away approximating space-time with Minkowski space. If the latter observer traces out the degrees of freedom associated with the region below horizon, the outcome is genuine density matrix and information loss. This point is not discussed in the article and authors inform that their next project is to look at the situation after the spherical shell has reached Schwartschild radius and horizon is born. One might say that all that is done concerns the system before the formation of blackhole (if it is formed at all!).
Several poorly defined notions arise when one tries to interpret the results of the calculation.
1. What do we mean with observer? What do we mean with information? For instance, authors define information as difference between maximum entropy and real entropy. Is this definition just an ad hoc manner to get sum well-defined number christened as an information? Can we really reduce the notion of information to thermodynamics? Shouldn't we be very careful in distinguishing between thermodynamical entropy and entanglement entropy? A sub-system possessing entanglement entropy with its complement can be purified by seeing it as a part of the entire system. This entropy relates to pair of systems. Thermal entropy can be naturally assigned to an average representative of ensemble and is single particle observable.
2. Second list of questions relates to quantum gravitation. Is blackhole really a relevant notion or just a singular outcome of a theory exceeding its limits? Does something deserving to be called blackhole collapse really occur? Is quantum theory in its recent form enough to describe what happens in this process or its analog? Do we really understand the quantal description of gravitational binding?
What TGD can say about blackholes?
The usual objection of string theory hegemony is that there are no competing scenarios so that superstring is the only "known" interesting approach to quantum gravitation (knowing in academic sense is not at all the same thing as knowing in the naive layman sense and involves a lot of sociological factors transforming actual knowing to sociological unknowing: in some situations these sociological factors can make a scientist practically blind, deaf, and as it looks - brainless!) . I dare however claim that TGD represents an approach, which leads to a new vision challenging a long list of cherished notions assigned with blackholes.
To my view blackhole science crystallizes huge amount of conceptual sloppiness. People can calculate but are not so good in concetualizing. Therefore one must start the conceptual cleaning from fundamental notions such as information, notions of time (experienced and geometric), observer, etc... In attempt to develop TGD from a bundle of ideas to a real theory I have been forced to carry out this kind of distillation and the following tries to summarize the outcome.
1. TGD provides a fundamental description for the notions of observer and information. Observer is replaced with "self" identified in ZEO by a sequences of quantum jumps occurring at same boundary of CD and leaving it and the part of the zero energy state at it fixed whereas the second boundary of CD is delocalized and superposition for which the average distance between the tips of CDs involve increases: this gives to the experience flow of time and its correlation with the flow of geometric time. The average size of CDs simply increases and this means that the experiences geometric time increases. Self "dies" as the first state function reduction to the opposite boundary takes place and new self assignable it is born.
2. Negentropy Maximizaton Principle favors the generation of entanglement negentropy. For states with projection operator as density matrix the number theoretic negentropy is possible for primes dividing the dimension of the projection and is maximum for the largest power of prime factor of N. Second law is replaced with its opposite but for negentropy which is two-particle observable rather than single particle observable as thermodynamical entropy. Second law follows at ensemble level from the non-determinism of the state function reduction alone.
The notions related to blackhole are also in need of profound reconsideration.
1. Blackhole disappears in TGD framework as a fundamental object and is replaced by a space-time region having Euclidian signature of the induced metric identifiable as wormhole contact, and defining a line of generalized Feynman diagram (here "Feynmann" could be replaced with " twistor" or "Yangian" something even more appropriate). Blackhole horizon is replaced the 3-D light-like region defining the orbit of wormhole throat having degenerate metric in 4-D sense with signature (0,-1,-1,-1). The orbits of wormhole throats are carries of various quantum numbers and the sizes of M4 projections are of order CP2 size in elementary particle scales. This is why I refer to these regions also as light-like parton orbits. The wormhole contacts involved connect to space-time sheets with Minkowskian signature and stability requires that the wormhole contacts carry monopole magnetic flux. This demands at least two wormhole contacts to get closed flux lines. Elementary particles are this kind of pairs but also multiples are possible and valence quarks in baryons could be one example.
2. The connection with GRT picture could emerge as follows. The radial component of Schwartschild-Nordström metric associated with electric charge can be deformed slightly at horizon to transform horizon to light-like surface. In the deep interior CP2 would provide gravitational instanton solution to Maxwell-Einstein system with cosmological constant and having thus Euclidian metric. This is the nearest to TGD description that one can get within GRT framework obtained from TGD at asymptotic regions by replacing many-sheeted space-time with slightly deformed region of Minkowski space and summing the gravitational fields of sheets to get the the gravitational field of M4 region.
All physical systems have space-time sheets with Euclidian signature analogous to blackhole. The analog of blackhole horizon provides a very general definition of "elementary particle".
3. Strong form of general coordinate invariance is central piece of TGD and implies strong form of holography stating that partonic 2-surfaces and their 4-D tangent space data should be enough to code for quantum physics. The magnetic flux tubes and fermionic strings assignable to them are however essential. The localization of induced spinor fields to string world sheets follows from the well-definedness of em charge and also from number theoretical arguments as well as generalization of twistorialization from D=4 to D=8.
One also ends up with the analog of AdS/CFT duality applying to the generalization of conformal invariance in TGD framework. This duality states that one can describe the physics in terms of Kähler action and related bosonic data or in terms of Kähler-Dirac action and related data. In particular, Kähler action is expressible as string world sheet area in effective metric defined by Kähler-Dirac gamma matrices. Furthermore, gravitational binding is describable by strings connecting partonic 2-surfaces. The hierarchy of Planck constants is absolutely essential for the description of gravitationally bound states in thems of gravitational quantum coherence in macroscopic scales. The proportionality of the string area in effective metric to 1/heff2, heff=n× h=hgr=GMm/v0 is absolutely essential for achieving this.
If the stringy action were the ordinary area of string world sheet as in string models, only gravitational bound states with size of order Planck length would be possible. Hence TGD forces to say that superstring models are at completely wrong track concerning the quantum description of gravitation. Even the standard quantum theory lacks something fundamental required by this goal. This something fundamental relates directly to the mathematics of extended super-conformal invariance: these algebras allow infinite number of fractal inclusion hierarchies in which algebras are isomorphic with each other. This allows to realize infinite hierarchies of quantum criticalities. As heff increases, some degrees are reduced from critical gauge degrees of freedom to genuine dynamical degrees of freedom but the system is still critical, albeit in longer scale.
4. A naive model for the TGD analog of blackhole is as a macroscopic wormhole contact surrounded by particle wormhole contacts with throats connected to the large wormhole throats by flux tubes and strings to the large wormhole contact. The macroscopic wormhole contact would carry magnetic charge equal to the sum of those associated with elemenentary particle wormhole throats.
5. What about black hole collapse and blackhole evaporation if blackholes are replaced with wormhole contacts with Euclidian signature of metric? Do they have any counterparts in TGD? Maybe! Any phase transition increasing heff=hgr would occur spontaneously as transitions to lower criticality and could be interpreted as analog of blackhole evaporation. The gravitationally bound object would just increase in size. I have proposed that this phase transition has happened for Earth (Cambrian explosion) and increases its radius by factor 2. This would explain the strange finding that the continents seem to fit nicely together if the radius of Earth is one half of the recent value. These phase transitions would be the quantum counterpart of smooth classical cosmic expansion.
The phase transition reducing heff would not occur spontaneusly and in living systems metabolic energy would be needed to drive them. Indeed, from the condition that heff=hgr= GMm/v0 increases as M and v0 change also gravitational Compton length Lgr=hgr/m= GM/v0 defining the size scale of the gravitational object increases so that the spontaneous increase of hgr means increase of size.
Does TGD predict any process resembling blackhole collapse? In Zero Energy Ontology (ZEO) state function reductions occurring at the same boundary of causal diamond (CD) define the notion of self possessing arrow of time. The first quantum state function reduction at opposite boundary is eventually forced by Negentropy Maximization Principle (NMP) and induces a reversal of geometric time. The expansion of object with a reversed arrow of geometric time with respect to observer looks like collapse. This is indeed what the geometry of causal diamond suggests.
6. The role of strings (and magnetic flux tubes with which they are associated) in the description of gravitational binding (and possibly also other kinds of binding) is crucial in TGD framework. They are present in arbitrary long length scales since the value of gravitational Planck constant heff = hgr = GMm/v0, v0 (v0/c<1) has dimensions of velocity can have huge values as compared with those of ordinary Planck constant. This implies macroscopic quantum gravitational coherence and the fountain effect of superfluidity could be seen as an example of this.
The presence of flux tubes and strings serves as a correlate for quantum entanglement present in all scales is highly suggestive. This entanglement could be negentropic and by NMP and could be transferred but not destroyed. The information would be coded to the relationship between two gravitationally bound systems and instead of entropy one would have enormous negentropy resources. Whether this information can be made conscious is a fascinating problem. Could one generalize the interaction free quantum measurement so that it would give information about this entanglement? Or could the transfer of this information make it conscious?
Also super string camp has become aware about possibility of geometric and topological correlates of entanglement. The GRT based proposal relies on wormhole connections. Much older TGD based proposal applied systematically in quantum biology and TGD inspired theory of consciousness identifies magnetic flux tubes and associated fermionic string world sheets as correlates of negentropic entanglement. |
d3e2916d990807e7 | Branches as hidden nodes in a neural net
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Models of decoherence and branching
[This is akin to a living review, which will hopefully improve from time to time. Last edited 2017-11-26.]
This post will collect some models of decoherence and branching. We don’t have a rigorous definition of branches yet but I crudely define models of branching to be models of decoherenceI take decoherence to mean a model with dynamics taking the form U \approx \sum_i \vert S_i\rangle\langle S_i |\otimes U^{\mathcal{E}}_i for some tensor decomposition \mathcal{H} = \mathcal{S} \otimes \mathcal{E}, where \{\vert S_i\rangle\} is an (approximately) stable orthonormal basis independent of initial state, and where \mathrm{Tr}[ U^{\mathcal{E}}_i \rho^{\mathcal{E} \dagger}_0 U^{\mathcal{E}}_j ] \approx 0 for times t \gtrsim t_D and i \neq j, where \rho^{\mathcal{E}}_0 is the initial state of \mathcal{E} and t_D is some characteristic time scale. which additionally feature some combination of amplification, irreversibility, redundant records, and/or outcomes with an intuitive macroscopic interpretation. I have the following desiderata for models, which tend to be in tension with computational tractability:
• physically realistic
• symmetric (e.g., translationally)
• no ad-hoc system-environment distinction
• Ehrenfest evolution along classical phase-space trajectories (at least on Lyapunov timescales)
Regarding that last one: we would like to recover “classical behavior” in the sense of classical Hamiltonian flow, which (presumably) means continuous degrees of freedom.In principle you could have discrete degrees of freedom that limit, as \hbar\to 0, to some sort of discrete classical systems, but most people find this unsatisfying. Branching only becomes unambiguous in some large-N limit, so it seems satisfying models are necessarily messy and difficult to numerically simulate.… [continue reading]
Comments on Weingarten’s preferred branch
A senior colleague asked me for thoughts on this paper describing a single-preferred-branch flavor of quantum mechanics, and I thought I’d copy them here. Tl;dr: I did not find an important new idea in it, but this paper nicely illustrates the appeal of Finkelstein’s partial-trace decoherence and the ambiguity inherent in connecting a many-worlds wavefunction to our direct observations.
We propose a method for finding an initial state vector which by ordinary Hamiltonian time evolution follows a single branch of many-worlds quantum mechanics. The resulting deterministic system appears to exhibit random behavior as a result of the successive emergence over time of information present in the initial state but not previously observed.
We start by assuming that a precise wavefunction branch structure has been specified. The idea, basically, is to randomly draw a branch at late times according to the Born probability, then to evolve it backwards in time to the beginning of the universe and take that as your initial condition. The main motivating observation is that, if we assume that all branch splittings are defined by a projective decomposition of some subsystem (‘the system’) which is recorded faithfully elsewhere (‘the environment’), then the lone preferred branch — time-evolving by itself — is an eigenstate of each of the projectors defining the splits. In a sense, Weingarten lays claim to ordered consistency [arxiv:gr-qc/9607073] by assuming partial-trace decoherenceNote on terminology: What Finkelstein called “partial-trace decoherence” is really a specialized form of consistency (i.e., a mathematical criterion for sets of consistent histories) that captures some, but not all, of the properties of the physical and dynamical process of decoherence.[continue reading]
Symmetries and solutions
Here is an underemphasized way to frame the relationship between trajectories and symmetries (in the sense of Noether’s theorem)You can find this presentation in “A short review on Noether’s theorems, gauge symmetries and boundary terms” by Máximo Bañados and Ignacio A. Reyes (H/t Godfrey Miller).. Consider the space of all possible trajectories q(t) for a system, a real-valued Lagrangian functional L[q(t)] on that space, the “directions” \delta q(t) at each point, and the corresponding functional gradient \delta L[q(t)]/\delta q(t) in each direction. Classical solutions are exactly those trajectories q(t) such that the Lagrangian L[q(t)] is stationary for perturbations in any direction \delta q(t), and continuous symmetries are exactly those directions \delta q(t) such that the Lagrangian L[q(t)] is stationary for any trajectory q(t). That is,
(1) \begin{align*} q(t) \mathrm{\,is\, a\,}\mathbf{solution}\quad \qquad &\Leftrightarrow \qquad \frac{\delta L[q(t)]}{\delta q(t)} = 0 \,\,\,\, \forall \delta q(t)\\ \delta q(t) \mathrm{\,is\, a\,}\mathbf{symmetry} \qquad &\Leftrightarrow \qquad \frac{\delta L[q(t)]}{\delta q(t)} = 0 \,\,\,\, \forall q(t). \end{align*}
There are many subtleties obscured in this cartoon presentation, like the fact that a symmetry \delta q(t), being a tangent direction on the manifold of trajectories, can vary with the tangent point q(t) it is attached to (as for rotational symmetries). If you’ve never spent a long afternoon with a good book on the calculus of variations, I recommend it.
(↵ returns to text)
[continue reading]
How to think about Quantum Mechanics—Part 7: Quantum chaos and linear evolution
[Other parts in this series: 1,2,3,4,5,6,7.]
You’re taking a vacation to Granada to enjoy a Spanish ski resort in the Sierra Nevada mountains. But as your plane is coming in for a landing, you look out the window and realize the airport is on a small tropical island. Confused, you ask the flight attendant what’s wrong. “Oh”, she says, looking at your ticket, “you’re trying to get to Granada, but you’re on the plane to Grenada in the Caribbean Sea.” A wave of distress comes over your face, but she reassures you: “Don’t worry, Granada isn’t that far from here. The Hamming distance is only 1!”.
After you’ve recovered from that side-splitting humor, let’s dissect the frog. What’s the basis of the joke? The flight attendant is conflating two different metrics: the geographic distance and the Hamming distance. The distances are completely distinct, as two named locations can be very nearby in one and very far apart in the other.
Now let’s hear another joke from renowned physicist Chris Jarzynski:
The linear Schrödinger equation, however, does not give rise to the sort of nonlinear, chaotic dynamics responsible for ergodicity and mixing in classical many-body systems. This suggests that new concepts are needed to understand thermalization in isolated quantum systems. – C. Jarzynski, “Diverse phenomena, common themes” [PDF]
Ha! Get it? This joke is so good it’s been told by S. Wimberger“Since quantum mechanics is the more fundamental theory we can ask ourselves if there is chaotic motion in quantum systems as well.[continue reading]
Reeh–Schlieder property in a separable Hilbert space
As has been discussed here before, the Reeh–Schlieder theorem is an initially confusing property of the vacuum in quantum field theory. It is difficult to find an illuminating discussion of it in the literature, whether in the context of algebraic QFT (from which it originated) or the more modern QFT grounded in RG and effective theories. I expect this to change once more field theorists get trained in quantum information.
The Reeh–Schlieder theorem states that the vacuum \vert 0 \rangle is cyclic with respect to the algebra \mathcal{A}(\mathcal{O}) of observables localized in some subset \mathcal{O} of Minkowski space. (For a single field \phi(x), the algebra \mathcal{A}(\mathcal{O}) is defined to be generated by all finite smearings \phi_f = \int\! dx\, f(x)\phi(x) for f(x) with support in \mathcal{O}.) Here, “cyclic” means that the subspace \mathcal{H}^{\mathcal{O}} \equiv \mathcal{A}(\mathcal{O})\vert 0 \rangle is dense in \mathcal{H}, i.e., any state \vert \chi \rangle \in \mathcal{H} can be arbitrarily well approximated by a state of the form A \vert 0 \rangle with A \in \mathcal{A}(\mathcal{O}). This is initially surprising because \vert \chi \rangle could be a state with particle excitations localized (essentially) to a region far from \mathcal{O} and that looks (essentially) like the vacuum everywhere else. The resolution derives from the fact the vacuum is highly entangled, such that the every region is entangled with every other region by an exponentially small amount.
One mistake that’s easy to make is to be fooled into thinking that this property can only be found in systems, like a field theory, with an infinite number of degrees of freedom. So let me exhibitMost likely a state with this property already exists in the quantum info literature, but I’ve got a habit of re-inventing the wheel. For my last paper, I spent the better part of a month rediscovering the Shor code… a quantum state with the Reeh–Schlieder property that lives in the tensor product of a finite number of separable Hilbert spaces:
\[\mathcal{H} = \bigotimes_{n=1}^N \mathcal{H}_n, \qquad \mathcal{H}_n = \mathrm{span}\left\{ \vert s \rangle_n \right\}_{s=1}^\infty\]
As emphasized above, a separable Hilbert space is one that has a countable orthonormal basis, and is therefore isomorphic to L^2(\mathbb{R}), the space of square-normalizable functions.… [continue reading]
Legendre transform
The way that most physicists teach and talk about partial differential equations is horrible, and has surprisingly big costs for the typical understanding of the foundations of the field even among professionals. The chief victims are students of thermodynamics and analytical mechanics, and I’ve mentioned before that the preface of Sussman and Wisdom’s Structure and Interpretation of Classical Mechanics is a good starting point for thinking about these issues. As a pointed example, in this blog post I’ll look at how badly the Legendre transform is taught in standard textbooks,I was pleased to note as this essay went to press that my choice of Landau, Goldstein, and Arnold were confirmed as the “standard” suggestions by the top Google results. and compare it to how it could be taught. In a subsequent post, I’ll used this as a springboard for complaining about the way we record and transmit physics knowledge.
Before we begin: turn away from the screen and see if you can remember what the Legendre transform accomplishes mathematically in classical mechanics.If not, can you remember the definition? I couldn’t, a month ago. I don’t just mean that the Legendre transform converts the Lagrangian into the Hamiltonian and vice versa, but rather: what key mathematical/geometric property does the Legendre transform have, compared to the cornucopia of other function transforms, that allows it to connect these two conceptually distinct formulations of mechanics?
(Analogously, the question “What is useful about the Fourier transform for understanding translationally invariant systems?” can be answered by something like “Translationally invariant operations in the spatial domain correspond to multiplication in the Fourier domain” or “The Fourier transform is a change of basis, within the vector space of functions, using translationally invariant basis elements, i.e., the Fourier modes”.)
The status quo
Let’s turn to the canonical text by Goldstein for an example of how the Legendre transform is usually introduced.… [continue reading]
Toward relativistic branches of the wavefunction
I prepared the following extended abstract for the Spacetime and Information Workshop as part of my continuing mission to corrupt physicists while they are still young and impressionable. I reproduce it here for your reading pleasure.
Finding a precise definition of branches in the wavefunction of closed many-body systems is crucial to conceptual clarity in the foundations of quantum mechanics. Toward this goal, we propose amplification, which can be quantified, as the key feature characterizing anthropocentric measurement; this immediately and naturally extends to non-anthropocentric amplification, such as the ubiquitous case of classically chaotic degrees of freedom decohering. Amplification can be formalized as the production of redundant records distributed over spatial disjoint regions, a certain form of multi-partite entanglement in the pure quantum state of a large closed system. If this definition can be made rigorous and shown to be unique, it is then possible to ask many compelling questions about how branches form and evolve.
A recent result shows that branch decompositions are highly constrained just by this requirement that they exhibit redundant local records. The set of all redundantly recorded observables induces a preferred decomposition into simultaneous eigenstates unless their records are highly extended and delicately overlapping, as exemplified by the Shor error-correcting code. A maximum length scale for records is enough to guarantee uniqueness. However, this result is grounded in a preferred tensor decomposition into independent microscopic subsystems associated with spatial locality. This structure breaks down in a relativistic setting on scales smaller than the Compton wavelength of the relevant field. Indeed, a key insight from algebraic quantum field theory is that finite-energy states are never exact eigenstates of local operators, and hence never have exact records that are spatially disjoint, although they can approximate this arbitrarily well on large scales.… [continue reading]
Branches and matrix-product states
I’m happy to use this bully pulpit to advertise that the following paper has been deemed “probably not terrible”, i.e., published.
When the wave function of a large quantum system unitarily evolves away from a low-entropy initial state, there is strong circumstantial evidence it develops “branches”: a decomposition into orthogonal components that is indistinguishable from the corresponding incoherent mixture with feasible observations. Is this decomposition unique? Must the number of branches increase with time? These questions are hard to answer because there is no formal definition of branches, and most intuition is based on toy models with arbitrarily preferred degrees of freedom. Here, assuming only the tensor structure associated with spatial locality, I show that branch decompositions are highly constrained just by the requirement that they exhibit redundant local records. The set of all redundantly recorded observables induces a preferred decomposition into simultaneous eigenstates unless their records are highly extended and delicately overlapping, as exemplified by the Shor error-correcting code. A maximum length scale for records is enough to guarantee uniqueness. Speculatively, objective branch decompositions may speed up numerical simulations of nonstationary many-body states, illuminate the thermalization of closed systems, and demote measurement from fundamental primitive in the quantum formalism.
Here’s the figureThe editor tried to convince me that this figure appeared on the cover for purely aesthetic reasons and this does not mean my letter is the best thing in the issue…but I know better! and caption:
Spatially disjoint regions with the same coloring (e.g., the solid blue regions \mathcal{F}, \mathcal{F}', \ldots) denote different records for the same observable (e.g., \Omega_a = \{\Omega_a^{\mathcal{F}},\Omega_a^{\mathcal{F}'},\ldots\}).
[continue reading]
Comments on Cotler, Penington, & Ranard
One way to think about the relevance of decoherence theory to measurement in quantum mechanics is that it reduces the preferred basis problem to the preferred subsystem problem; merely specifying the system of interest (by delineating it from its environment or measuring apparatus) is enough, in important special cases, to derive the measurement basis. But this immediately prompts the question: what are the preferred systems? I spent some time in grad school with my advisor trying to see if I could identify a preferred system just by looking at a large many-body Hamiltonian, but never got anything worth writing up.
I’m pleased to report that Cotler, Penington, and Ranard have tackled a closely related problem, and made a lot more progress:
Locality from the Spectrum
Jordan S. Cotler, Geoffrey R. Penington, Daniel H. Ranard
Essential to the description of a quantum system are its local degrees of freedom, which enable the interpretation of subsystems and dynamics in the Hilbert space. While a choice of local tensor factorization of the Hilbert space is often implicit in the writing of a Hamiltonian or Lagrangian, the identification of local tensor factors is not intrinsic to the Hilbert space itself. Instead, the only basis-invariant data of a Hamiltonian is its spectrum, which does not manifestly determine the local structure. This ambiguity is highlighted by the existence of dualities, in which the same energy spectrum may describe two systems with very different local degrees of freedom. We argue that in fact, the energy spectrum alone almost always encodes a unique description of local degrees of freedom when such a description exists, allowing one to explicitly identify local subsystems and how they interact.
[continue reading]
Singular value decomposition in bra-ket notation
In linear algebra, and therefore quantum information, the singular value decomposition (SVD) is elementary, ubiquitous, and beautiful. However, I only recently realized that its expression in bra-ket notation is very elegant. The SVD is equivalent to the statement that any operator \hat{M} can be expressed as
(1) \begin{align*} \hat{M} = \sum_i \vert A_i \rangle \lambda_i \langle B_i \vert \end{align*}
where \vert A_i \rangle and \vert B_i \rangle are orthonormal sets of vectors, possibly in Hilbert spaces with different dimensionality, and the \lambda_i \ge 0 are the singular values.
That’s it.… [continue reading]
Comments on Bousso’s communication bound
Bousso has a recent paper bounding the maximum information that can be sent by a signal from first principles in QFT:
I derive a universal upper bound on the capacity of any communication channel between two distant systems. The Holevo quantity, and hence the mutual information, is at most of order E\Delta t/\hbar, where E the average energy of the signal, and \Delta t is the amount of time for which detectors operate. The bound does not depend on the size or mass of the emitting and receiving systems, nor on the nature of the signal. No restrictions on preparing and processing the signal are imposed. As an example, I consider the encoding of information in the transverse or angular position of a signal emitted and received by systems of arbitrarily large cross-section. In the limit of a large message space, quantum effects become important even if individual signals are classical, and the bound is upheld.
Here’s his first figure:
This all stems from vacuum entanglement, an oft-neglected aspect of QFT that Bousso doesn’t emphasize in the paper as the key ingredient.I thank Scott Aaronson for first pointing this out. The gradient term in the Hamiltonian for QFTs means that the value of the field at two nearby locations is always entangled. In particular, the value of \phi(x) and \phi(x+\Delta x) are sometimes considered independent degrees of freedom but, for a state with bounded energy, they can’t actually take arbitrarily different values as \Delta x becomes small, or else the gradient contribution to the Hamiltonian violates the energy bound. Technically this entanglement exists over arbitrary distances, but it is exponentially suppressed on scales larger than the Compton wavelength of the field.… [continue reading]
How to think about Quantum Mechanics—Part 1: Measurements are about bases
[This post was originally “Part 0”, but it’s been moved. Other parts in this series: 1,2,3,4,5,6,7.]
In an ideal world, the formalism that you use to describe a physical system is in a one-to-one correspondence with the physically distinct configurations of the system. But sometimes it can be useful to introduce additional descriptions, in which case it is very important to understand the unphysical over-counting (e.g., gauge freedom). A scalar potential V(x) is a very convenient way of representing the vector force field, F(x) = \partial V(x), but any constant shift in the potential, V(x) \to V(x) + V_0, yields forces and dynamics that are indistinguishable, and hence the value of the potential on an absolute scale is unphysical.
One often hears that a quantum experiment measures an observable, but this is wrong, or very misleading, because it vastly over-counts the physically distinct sorts of measurements that are possible. It is much more precise to say that a given apparatus, with a given setting, simultaneously measures all observables with the same eigenvectors. More compactly, an apparatus measures an orthogonal basis – not an observable.We can also allow for the measured observable to be degenerate, in which case the apparatus simultaneously measures all observables with the same degenerate eigenspaces. To be abstract, you could say it measures a commuting subalgebra, with the nondegenerate case corresponding to the subalgebra having maximum dimensionality (i.e., the same number of dimensions as the Hilbert space). Commuting subalgebras with maximum dimension are in one-to-one correspondence with orthonormal bases, modulo multiplying the vectors by pure phases. You can probably start to see this by just noting that there’s no actual, physical difference between measuring X and X^3; the apparatus that would perform the two measurements are identical.… [continue reading]
Bleg: Classical theory of measurement and amplification
I’m in search of an authoritative reference giving a foundational/information-theoretic approach to classical measurement. What abstract physical properties are necessary and sufficient?
Motivation: The Copenhagen interpretation treats the measurement process as a fundamental primitive, and this persists in most uses of quantum mechanics outside of foundations. Of course, the modern view is that the measurement process is just another physical evolution, where the state of a macroscopic apparatus is conditioned on the state of a microscopic quantum system in some basis determined by their mutual interaction Hamiltonian. The apparent nonunitary aspects of the evolution inferred by the observer arises because the measured system is coupled to the observer himself; the global evolution of the system-apparatus-observer system is formally modeled as unitary (although the philosophical meaningfulness/ontology/reality of the components of the wavefunction corresponding to different measurement outcomes is disputed).
Eventually, we’d like to be able to identify all laboratory measurements as just an anthropocentric subset of wavefunction branching events. I am very interested in finding a mathematically precise criteria for branching.Note that the branches themselves may be only precisely defined in some large-N or thermodynamic limit. Ideally, I would like to find a property that everyone agrees must apply, at the least, to laboratory measurement processes, and (with as little change as possible) use this to find all branches — not just ones that result from laboratory measurements.Right now I find the structure of spatially-redundant information in the many-body wavefunction to be a very promising approach.
It seems sensible to begin with what is necessary for a classical measurement since these ought to be analyzable without all the philosophical baggage that plagues discussion of quantum measurement.… [continue reading]
Comments on an essay by Wigner
[PSA: Happy 4th of July. Juno arrives at Jupiter tonight!]
This is short and worth reading:
The sharp distinction between Initial Conditions and Laws of Nature was initiated by Isaac Newton and I consider this to be one of his most important, if not the most important, accomplishment. Before Newton there was no sharp separation between the two concepts. Kepler, to whom we owe the three precise laws of planetary motion, tried to explain also the size of the planetary orbits, and their periods. After Newton's time the sharp separation of initial conditions and laws of nature was taken for granted and rarely even mentioned. Of course, the first ones are quite arbitrary and their properties are hardly parts of physics while the recognition of the latter ones are the prime purpose of our science. Whether the sharp separation of the two will stay with us permanently is, of course, as uncertain as is all future development but this question will be further discussed later. Perhaps it should be mentioned here that the permanency of the validity of our deterministic laws of nature became questionable as a result of the realization, due initially to D. Zeh, that the states of macroscopic bodies are always under the influence of their environment; in our world they can not be kept separated from it.
This essay has no formal abstract; the above is the second paragraph, which I find to be profound. Here is the PDF. The essay shares the same name and much of the material with Wigner’s 1963 Nobel lecture [PDF].The Nobel lecture has a nice bit contrasting invariance principles with covariance principles, and dynamical invariance principles with geometrical invariance principles.[continue reading] |
685b31b6b5c9cd7d | Can quantum computing be simulated by an optical network? | PhysicsOverflow
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Can quantum computing be simulated by an optical network?
+ 5 like - 0 dislike
On page 43 of http://www.mat.univie.ac.at/~neum/ms/optslides.pdf, we read
8. Simulating quantum mechanics
The simulation of quantum computing by classical fields is essentially achieved by using an optical network in which each quantum level is modelled by a corresponding mode of the electromagnetic field.
The linearity of the Maxwell equations then directly translates into the superposition principle for pure quantum states.
Thus it is possible to simulate arbitrary quantum systems which have a finite number of levels by the Maxwell equations, and hence by a classical model.
Therefore we shall look a little more closely into the reasons for this ability to simulate quantum systems.
...some explanations about second order coherence theory of the Maxwell equations...
As a consequence, it is possible (at least in principle) to simulate with classical electromagnetic waves and suitable classical linear optical networks any quantum system that can be embedded into the single photon quantum system.
Since all Hilbert spaces arising in applications of quantum physics are separable, they have a countable basis, and can be embedded into the single photon quantum system, at least in principle.
Thus it appears that, all quantum systems can be simulated by classical electromagnetic waves!
Of course, a practical realization may be difficult.
How is this optical network supposed to work? Does it really work? I don't see how the fact that all separable Hilbert spaces are isomorphic to each other would allow me to conclude that the time evolution of a single photon by the Schrödinger equation will be able to simulate the time evolution of an arbirary quantum system by the Schrödinger equation. But "optical network" sounds very concrete to me, so maybe some simple examples of how to simulate the time evolution of a two photon quantum systems by them could be help me?
asked Dec 16, 2014 in Theoretical Physics by Thomas Klimpel (70 points) [ no revision ]
I suppose @ArnoldNeumaier (who wrote the lecture notes) could answer this question?
In http://www.mat.univie.ac.at/~neum/ms/hidden.pdf @ArnoldNeumaier writes: "With more beam splitters, through which several narrowly spaced beams are passed, one can produce a cascade of more complex tensor product states. Indeed, Reck et al. [23] showed that (i) any quantum system with only finitely many degrees of freedom can be simulated by a collection of spatially entangled beams; (ii) in the simulated system, there is for any Hermitian operator H an experiment measuring H; (iii) for every unitary operator S, there is an optical arrangement in the simulated system realizing this transformation, assuming lossless beam splitters."
Neither that article, nor its reference [23] can be found in the reference section of the presentation which is quoted in the question. So the answer to the question seems to be "yes!", but one would have to read (and understand) "some" references proving this to be sure.
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647fcaf0f84bbaff | Collect. Czech. Chem. Commun. 2004, 69, 141-176
Convergence Behavior of Symmetry-Adapted Perturbation Expansions for Excited States. A Model Study of Interactions Involving a Triplet Helium Atom
Michał Przybytek, Konrad Patkowski and Bogumił Jeziorski*
The convergence behavior of symmetry-adapted perturbation theory (SAPT) expansions is investigated for an interacting system involving an excited, open-shell monomer. By performing large-order numerical calculations for the interaction of the lowest, 1s2s, triplet state of helium with the ground state of the hydrogen atom we show that the conventional polarization and symmetrized Rayleigh-Schrödinger expansions diverge in this case. This divergence is attributed to the continuum of intruder states appearing when the hydrogen electron is falling on the helium 1s orbital and the 2s electron is ejected from the interacting system. One of the dimer states resulting from the interaction becomes then a resonance, which presents a hard case to treat by a perturbation theory. We show that the SAPT expansions employing the strong symmetry-forcing procedure, such as the Eisenschitz-London- Hirschfelder-van der Avoird or the Amos-Musher theories, can cope with this situation and lead to convergent series when the permutational symmetry of the bound, quartet state is forced. However, these theories suffer from a wrong asymptotic behavior of the second- and higher-order energies when the interatomic distance R grows to infinity, which makes them unsuitable for practical applications. We show that by a suitable regularization of the Coulomb potential and by treating differently the regular, long-range and the singular, short-range parts of the interatomic electron-nucleus attraction terms in the Hamiltonian one obtains a perturbation expansion which has the correct asymptotic behavior in each order and which converges fast for a wide range of interatomic distances.
Keywords: Weak interactions; Hamiltonian; Schrödinger equation; FCI calculations; Amos-Musher theory; Quantum chemistry.
References: 67 live references. |
dd047ccc29ae58b4 | Take the 2-minute tour ×
Electrons in an atom have quantized energy quantity. Can uncertainty principle be applied in this case, then?
How does this work?
As energy is fixed, this seems to disobey $\Delta E \Delta t \geq \hbar/2$...
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You can only measure the energy of an electron in an atom perfectly if you measure it for an infinitely long time. If you do your measurement for a finite time there will be a finite uncertainty due to the uncertainty principle.
This is not some esoteric piece of mathematics, it's a real and measurable effect. For example if you measure the emission spectrum from an atom you are measuring the difference in energy between some excited state and a lower energy state. Because the lifetime of the excited state is short it's energy is uncertain, and consequently the energy of the emitted photon is uncertain. The result is that the lines in the emission spectrum are not infinitely sharp. They have a finite width due to the uncertainty principle.
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Indeed, line width is how we measure the lifetime of very fast transitions. – dmckee Aug 8 '12 at 11:31
Then how do we get quantized values for energy level? – Mark Lucas Aug 8 '12 at 11:48
By solving the time independent Schrödinger equation. The energy levels we calculate are the limits of infinite time. – John Rennie Aug 8 '12 at 12:23
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5684dc72a16e0a3e | zbMATH — the first resource for mathematics
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Applications of the combined tanh function method with symmetry method to the nonlinear evolution equations. (English) Zbl 1114.65356
Summary: Based on the symbolic computation, we combine the tanh function method with the symmetry method to construct new type of solutions of the nonlinear evolution equations for the first time. With the combined method, some new types of solutions of the coupled (2+1)-dimensional nonlinear system of Schrödinger equations are obtained and their properties are studied by analyzing their figures.
65M70Spectral, collocation and related methods (IVP of PDE)
35Q55NLS-like (nonlinear Schrödinger) equations
68W30Symbolic computation and algebraic computation |
6b26be0dd2c23eb9 | Discover Interview: Roger Penrose Says Physics Is Wrong, From String Theory to Quantum Mechanics
One of the greatest thinkers in physics says the human brain—and the universe itself—must function according to some theory we haven't yet discovered.
By Susan Kruglinski, Oliver Chanarin|Tuesday, October 06, 2009
Roger Penrose could easily be excused for having a big ego. A theorist whose name will be forever linked with such giants as Hawking and Einstein, Penrose has made fundamental contributions to physics, mathematics, and geometry. He reinterpreted general relativity to prove that black holes can form from dying stars. He invented twistor theory—a novel way to look at the structure of space-time—and so led us to a deeper understanding of the nature of gravity. He discovered a remarkable family of geometric forms that came to be known as Penrose tiles. He even moonlighted as a brain researcher, coming up with a provocative theory that consciousness arises from quantum-mechanical processes. And he wrote a series of incredibly readable, best-selling science books to boot.
And yet the 78-year-old Penrose—now an emeritus professor at the Mathematical Institute, University of Oxford—seems to live the humble life of a researcher just getting started in his career. His small office is cramped with the belongings of the six other professors with whom he shares it, and at the end of the day you might find him rushing off to pick up his 9-year-old son from school. With the curiosity of a man still trying to make a name for himself, he cranks away on fundamental, wide-ranging questions: How did the universe begin? Are there higher dimensions of space and time? Does the current front-running theory in theoretical physics, string theory, actually make sense?
Because he has lived a lifetime of complicated calculations, though, Penrose has quite a bit more perspective than the average starting scientist. To get to the bottom of it all, he insists, physicists must force themselves to grapple with the greatest riddle of them all: the relationship between the rules that govern fundamental particles and the rules that govern the big things—like us—that those particles make up. In his powwow with DISCOVER contributing editor Susan Kruglinksi, Penrose did not flinch from questioning the central tenets of modern physics, including string theory and quantum mechanics. Physicists will never come to grips with the grand theories of the universe, Penrose holds, until they see past the blinding distractions of today’s half-baked theories to the deepest layer of the reality in which we live.
You come from a colorful family of overachievers, don’t you?
My older brother is a distinguished theoretical physicist, a fellow of the Royal Society. My younger brother ended up the British chess champion 10 times, a record. My father came from a Quaker family. His father was a professional artist who did portraits—very traditional, a lot of religious subjects. The family was very strict. I don’t think we were even allowed to read novels, certainly not on Sundays. My father was one of four brothers, all of whom were very good artists. One of them became well known in the art world, Sir Roland. He was cofounder of the Institute of Contemporary Arts in London. My father himself was a human geneticist who was recognized for demonstrating that older mothers tend to get more Down syndrome children, but he had lots of scientific interests.
How did your father influence your thinking?
The important thing about my father was that there wasn’t any boundary between his work and what he did for fun. That rubbed off on me. He would make puzzles and toys for his children and grandchildren. He used to have a little shed out back where he cut things from wood with his little pedal saw. I remember he once made a slide rule with about 12 different slides, with various characters that we could combine in complicated ways. Later in his life he spent a lot of time making wooden models that reproduced themselves—what people now refer to as artificial life. These were simple devices that, when linked together, would cause other bits to link together in the same way. He sat in his woodshed and cut these things out of wood in great, huge numbers.
So I assume your father helped spark your discovery of Penrose tiles, repeating shapes that fit together to form a solid surface with pentagonal symmetry.
It was silly in a way. I remember asking him—I was around 9 years old—about whether you could fit regular hexagons together and make it round like a sphere. And he said, “No, no, you can’t do that, but you can do it with pentagons,” which was a surprise to me. He showed me how to make polyhedra, and so I got started on that.
Are Penrose tiles useful or just beautiful?
My interest in the tiles has to do with the idea of a universe controlled by very simple forces, even though we see complications all over the place. The tilings follow conventional rules to make complicated patterns. It was an attempt to see how the complicated could be satisfied by very simple rules that reflect what we see in the world.
The artist M. C. Escher was influenced by your geometric inventions. What was the story there?
In my second year as a graduate student at Cambridge, I attended the International Congress of Mathematicians in Amsterdam. I remember seeing one of the lecturers there I knew quite well, and he had this catalog. On the front of it was the Escher picture Day and Night, the one with birds going in opposite directions. The scenery is nighttime on one side and daytime on the other. I remember being intrigued by this, and I asked him where he got it. He said, “Oh, well, there’s an exhibition you might be interested in of some artist called Escher.” So I went and was very taken by these very weird and wonderful things that I’d never seen anything like. I decided to try and draw some impossible scenes myself and came up with this thing that’s referred to as a tri-bar. It’s a triangle that looks like a three-dimensional object, but actually it’s impossible for it to be three-dimensional. I showed it to my father and he worked out some impossible buildings and things. Then we published an article in the British Journal of Psychology on this stuff and acknowledged Escher.
Escher saw the article and was inspired by it?
Is it true that you were bad at math as a kid?
I was unbelievably slow. I lived in Canada for a while, for about six years, during the war. When I was 8, sitting in class, we had to do this mental arithmetic very fast, or what seemed to me very fast. I always got lost. And the teacher, who didn’t like me very much, moved me down a class. There was one rather insightful teacher who decided, after I’d done so badly on these tests, that he would have timeless tests. You could just take as long as you’d like. We all had the same test. I was allowed to take the entire next period to continue, which was a play period. Everyone was always out and enjoying themselves, and I was struggling away to do these tests. And even then sometimes it would stretch into the period beyond that. So I was at least twice as slow as anybody else. Eventually I would do very well. You see, if I could do it that way, I would get very high marks.
You have called the real-world implications of quantum physics nonsensical. What is your objection?
Quantum mechanics is an incredible theory that explains all sorts of things that couldn’t be explained before, starting with the stability of atoms. But when you accept the weirdness of quantum mechanics [in the macro world], you have to give up the idea of space-time as we know it from Einstein. The greatest weirdness here is that it doesn’t make sense. If you follow the rules, you come up with something that just isn’t right.
In quantum mechanics an object can exist in many states at once, which sounds crazy. The quantum description of the world seems completely contrary to the world as we experience it.
It doesn’t make any sense, and there is a simple reason. You see, the mathematics of quantum mechanics has two parts to it. One is the evolution of a quantum system, which is described extremely precisely and accurately by the Schrödinger equation. That equation tells you this: If you know what the state of the system is now, you can calculate what it will be doing 10 minutes from now. However, there is the second part of quantum mechanics—the thing that happens when you want to make a measurement. Instead of getting a single answer, you use the equation to work out the probabilities of certain outcomes. The results don’t say, “This is what the world is doing.” Instead, they just describe the probability of its doing any one thing. The equation should describe the world in a completely deterministic way, but it doesn’t.
Erwin Schrödinger, who created that equation, was considered a genius. Surely he appreciated that conflict.
Schrödinger was as aware of this as anybody. He talks about his hypothetical cat and says, more or less, “Okay, if you believe what my equation says, you must believe that this cat is dead and alive at the same time.” He says, “That’s obviously nonsense, because it’s not like that. Therefore, my equation can’t be right for a cat. So there must be some other factor involved.”
So Schrödinger himself never believed that the cat analogy reflected the nature of reality?
Oh yes, I think he was pointing this out. I mean, look at three of the biggest figures in quantum mechanics, Schrödinger, Einstein, and Paul Dirac. They were all quantum skeptics in a sense. Dirac is the one whom people find most surprising, because he set up the whole foundation, the general framework of quantum mechanics. People think of him as this hard-liner, but he was very cautious in what he said. When he was asked, “What’s the answer to the measurement problem?” his response was, “Quantum mechanics is a provisional theory. Why should I look for an answer in quantum mechanics?” He didn’t believe that it was true. But he didn’t say this out loud much.
Yet the analogy of Schrödinger’s cat is always presented as a strange reality that we have to accept. Doesn’t the concept drive many of today’s ideas about theoretical physics?
That’s right. People don’t want to change the Schrödinger equation, leading them to what’s called the “many worlds” interpretation of quantum mechanics.
That interpretation says that all probabilities are playing out somewhere in parallel universes?
It says OK, the cat is somehow alive and dead at the same time. To look at that cat, you must become a superposition [two states existing at the same time] of you seeing the live cat and you seeing the dead cat. Of course, we don’t seem to experience that, so the physicists have to say, well, somehow your consciousness takes one route or the other route without your knowing it. You’re led to a completely crazy point of view. You’re led into this “many worlds” stuff, which has no relationship to what we actually perceive.
The idea of parallel universes—many worlds—is a very human-centered idea, as if everything has to be understood from the perspective of what we can detect with our five senses.
The trouble is, what can you do with it? Nothing. You want a physical theory that describes the world that we see around us. That’s what physics has always been: Explain what the world that we see does, and why or how it does it. Many worlds quantum mechanics doesn’t do that. Either you accept it and try to make sense of it, which is what a lot of people do, or, like me, you say no—that’s beyond the limits of what quantum mechanics can tell us. Which is, surprisingly, a very uncommon position to take. My own view is that quantum mechanics is not exactly right, and I think there’s a lot of evidence for that. It’s just not direct experimental evidence within the scope of current experiments.
In general, the ideas in theoretical physics seem increasingly fantastical. Take string theory. All that talk about 11 dimensions or our universe’s existing on a giant membrane seems surreal.
You’re absolutely right. And in a certain sense, I blame quantum mechanics, because people say, “Well, quantum mechanics is so nonintuitive; if you believe that, you can believe anything that’s nonintuitive.” But, you see, quantum mechanics has a lot of experimental support, so you’ve got to go along with a lot of it. Whereas string theory has no experimental support.
I understand you are setting out this critique of quantum mechanics in your new book.
The book is called Fashion, Faith and Fantasy in the New Physics of the Universe. Each of those words stands for a major theoretical physics idea. The fashion is string theory; the fantasy has to do with various cosmological schemes, mainly inflationary cosmology [which suggests that the universe inflated exponentially within a small fraction of a second after the Big Bang]. Big fish, those things are. It’s almost sacrilegious to attack them. And the other one, even more sacrilegious, is quantum mechanics at all levels—so that’s the faith. People somehow got the view that you really can’t question it.
A few years ago you suggested that gravity is what separates the classical world from the quantum one. Are there enough people out there putting quantum mechanics to this kind of test?
No, although it’s sort of encouraging that there are people working on it at all. It used to be thought of as a sort of crackpot, fringe activity that people could do when they were old and retired. Well, I am old and retired! But it’s not regarded as a central, as a mainstream activity, which is a shame.
After Newton, and again after Einstein, the way people thought about the world shifted. When the puzzle of quantum mechanics is solved, will there be another revolution in thinking?
It’s hard to make predictions. Ernest Rutherford said his model of the atom [which led to nuclear physics and the atomic bomb] would never be of any use. But yes, I would be pretty sure that it will have a huge influence. There are things like how quantum mechanics could be used in biology. It will eventually make a huge difference, probably in all sorts of unimaginable ways.
In your book The Emperor’s New Mind, you posited that consciousness emerges from quantum physical actions within the cells of the brain. Two decades later, do you stand by that?
In my view the conscious brain does not act according to classical physics. It doesn’t even act according to conventional quantum mechanics. It acts according to a theory we don’t yet have. This is being a bit big-headed, but I think it’s a little bit like William Harvey’s discovery of the circulation of blood. He worked out that it had to circulate, but the veins and arteries just peter out, so how could the blood get through from one to the other? And he said, “Well, it must be tiny little tubes there, and we can’t see them, but they must be there.” Nobody believed it for some time. So I’m still hoping to find something like that—some structure that preserves coherence, because I believe it ought to be there.
When physicists finally understand the core of quantum physics, what do you think the theory will look like?
I think it will be beautiful.
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516714e090eb9bff | Essential Computational Modeling in Chemistry book cover
Essential Computational Modeling in Chemistry
• Philippe Ciarlet, City University of Hong Kong, Kowloon
Essential Computational Modeling in Chemistry presents key contributions selected from the volume in the Handbook of Numerical Analysis: Computational Modeling in Chemistry Vol. 10(2005).
Computational Modeling is an active field of scientific computing at the crossroads between Physics, Chemistry, Applied Mathematics and Computer Science. Sophisticated mathematical models are increasingly complex and extensive computer simulations are on the rise. Numerical Analysis and scientific software have emerged as essential steps for validating mathematical models and simulations based on these models. This guide provides a quick reference of computational methods for use in understanding chemical reactions and how to control them. By demonstrating various computational methods in research, scientists can predict such things as molecular properties. The reference offers a number of techniques and the numerical analysis needed to perform rigorously founded computations.
Scientists, Engineers, Computational Biologists, Medical Researchers, Computational Physicists, Biophysicists, Bioinformatics Specialists, Computational Chemists and other Biological or Behavioral Science Researchers who need to understand numerical techniques for various systems and applications.
Paperback, 400 Pages
Published: December 2010
Imprint: North-holland
ISBN: 978-0-444-53754-6
• 1. The modeling and simulation of the liquid phase;
2. Computational approaches of relativistic models in quantum chemistry;
3. Quantum Monte Carlo methods for the solution of the Schrödinger equation for molecular systems;
4. Finite difference methods for ab initio electronic structure and quantum transport calculations of nanostructures;
5. Simulating chemical reactions in complex systems;
6. Biomolecular conformations can be identified as metastable sets of molecular dynamics;
7. Numerical methods for molecular time-dependent schrödinger equations - bridging the perturbative to nonperturbative regime;
8. Control of quantum dynamics: Concepts, procedures and future prospects;
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6a1c3a97de4ba485 |
Open Access Nano Express
Singly ionized double-donor complex in vertically coupled quantum dots
Ramón Manjarres-García1, Gene Elizabeth Escorcia-Salas1, Ilia D Mikhailov2 and José Sierra-Ortega1*
Author affiliations
1 Group of Investigation in Condensed Matter Theory, Universidad del Magdalena, Santa Marta, Colombia
2 Universidad Industrial de Santander, A. A. 678, Bucaramanga, Colombia
For all author emails, please log on.
Citation and License
Received:10 July 2012
Accepted:3 August 2012
Published:31 August 2012
© 2012 Manjarres-García et al.; licensee Springer.
The electronic states of a singly ionized on-axis double-donor complex (D2+) confined in two identical vertically coupled, axially symmetrical quantum dots in a threading magnetic field are calculated. The solutions of the Schrödinger equation are obtained by a variational separation of variables in the adiabatic limit. Numerical results are shown for bonding and antibonding lowest-lying artificial molecule states corresponding to different quantum dot morphologies, dimensions, separation between them, thicknesses of the wetting layers, and magnetic field strength.
Quantum dots; Adiabatic approximation; Artificial molecule; 78.67.-n; 78.67.Hc; 3.21.-b
Quantum dots (QDs) have opened the possibility to fabricate both artificial atoms and molecules with novel and fascinating optoelectronic properties which are not accessible in bulk semiconductor materials. An attractive route for nano-structuring semiconductor materials offers self-assembled quantum dots which are formed by the Stranski-Krastanow growth mode by depositing the material on a substrate with different lattice parameters [1-5]. The electrical and optical properties of these structures may be changed in a controlled form by doping the shallow impurities whose energy levels are defined by the interplay between the reductions of the physical dimension, the Coulomb attraction, and the inter-particle correlation.
Recently, it has been proposed to use the singly ionized double-donor system (D2+) confined in a single semiconductor QD [6] or ring [7] as an adequate functional part in a wide range of device applications, including spintronics, optoelectronics, photovoltaics, and quantum information technologies. This two-level system encodes logical information either on the spin or on the charge degrees of freedom of the single electron and allows us to manipulate conveniently its molecular properties, such as the energy splitting between the bonding and antibonding lowest-lying molecular-like states or the spatial distribution of carriers in the system [8-12]. One can expect that the singly ionized double-donor system (D2+) confined in vertically coupled QDs should have similar properties. In this paper, we analyze the electronic states of an artificial hydrogen molecular ion (D2+) compound by two positive ions that interchange their electron, which is constrained to exchange between two identical vertically coupled, axially symmetrical QDs in the presence of a threading magnetic field.
Below, we analyze the model of two separated on-axis singly ionized donors, confined in two coaxial, vertically stacked QDs, whose identical morphologies present axially symmetrical layers whose shape is given by the dependence of the layer thickness h on the distance ρ from the axis as follows: h(ρ) = db + d0fn(ρ)ϑ(R0ρ). Here, R0 is the base radius, db is the wetting layer thickness, d0 is the maximum height of the QD over this layer, ϑ(x) is the Heaviside step function, equal to 0 for x < 0 and to 1 for x > 0, and fn(ρ) = [1 − (ρ/R0)n]1/n. The morphology is controlled in this model by means of the integer shape-generating parameter n which is equal to 1, 2, or tends to infinity for conical pyramid-like, lens-like, and disk-like geometrical shapes, respectively. As an example, the 3D image of an artificial singly ionized molecule confined in lens-like QDs is presented in Figure 1.
thumbnailFigure 1. Image of the singly ionized molecule confined in lens-like QDs.
Besides, we assume that the external homogeneous magnetic field B = Bẑ is applied along the quantum dot's axis. The dimensionless Hamiltonian of the single electron in this D2+ complex in the effective-mass approximation can be written as
where Vc(ρ, z) is the confinement potential, equal to 0 and V0 inside and outside the QD, respectively. The last two terms in Equation 1 correspond to the attraction between electron and ions. The effective Bohr radius a0* = ℏ2ε/m*e2, the effective Rydberg Ry* = e2/2εa0*, and γ = eB/2m*cRy* have been taken above as units of length, energy, and the conventional dimensionless magnetic field strength, respectively.
As both donors are located at the axis, the potential is axially symmetrical, the angular momentum Lz commutes with the Hamiltonian, and the corresponding eigenvalues give us one good quantum number m. At this representation, the Hamiltonian (Equation 1) cylindrically coordinates only on two coordinates:
Taking into account that the thickness of QDs is typically much smaller than their lateral dimension and therefore the electron motion in the first direction is much faster than in-plane motion, one can use the advantage of the adiabatic approximation [13] in which the wave function is presented as a product of two functions:
where the first function f(ρ, z) describes the fast motion in z direction and satisfies the wave equation
with ‘frozen out’ radial coordinate ρ, while the radial part of the wave function is found in the second step from the equation
In our numerical procedure, we solve Equation 4 repeatedly for each value ρ by using the trigonometric sweep method [13] in order to restore the unknown function Ef(ρ). Once this function is found, then the energies Em of the molecular complex can be established by solving Equation 5.
As the potential V(ρ, z) for each fixed value of ρ presents an even function V(ρ, − z) = V(ρ, z) with respect to the variable z corresponding to a symmetrical (no-rectangle) quantum well, then all solutions of Equation 4 can be arranged in two sets: odd solutions f(ρ, − z) = − f(ρ, z) and even solutions f+(ρ, − z) = f+(ρ, z), called antibonding and bonding states, respectively. These sets of functions can be found as the solutions of the boundary value problems corresponding to the differential Equation 4 within the range 0 < z < ∞ with the frontier conditions <a onClick="popup('','MathML',630,470);return false;" target="_blank" href="">View MathML</a>.
Results and discussion
We have performed numerical calculations of two-electron renormalized energies Em as a function of the magnetic flux and for QDs with different morphologies, dimensions, and separation between layers in order to analyze the Aharonov-Bohm and the quantum size effects. We consider for our simulations the In0.55Al0.45As/Al0.35 Ga0.65As structures with the following values of physical parameters: dielectric constant ε = 12.71, the effective mass in the dot region and the region outside the dot for the electron m * = 0.076m0, the conduction and the valence band offset in junctions is V0 = 358meV, the effective Bohr radius a0* ≈ 10nm, and the effective Rydberg Ry* ≈ 5meV.
First, we calculate the energies of the molecular complex as functions of the magnetic field in disk-like, lens-like, and cone-like vertically coupled QDs and in a single one-electron QR with smooth non-homogeneity of the surface. Results for vertically coupled QDs with the heights d0 = 4nm, the wetting layer thicknesses db = 1nm, radii R0 = 20nm, and the separation between them d = 6nm are shown in Figure 2.
thumbnailFigure 2. Energies as functions of the magnetic field of a D2+ in vertically coupled quantum dots. (Heights 4 nm, wetting layer thicknesses 1 nm, radii 20 nm, and separation between them 6 nm).
It is seen that in all cases, the energy levels are very sensitive to the magnetic field and their dependencies on the magnetic field strength exhibit multiple crossovers and reordering. Comparing these dependencies for the disk, the lens, and the cone in Figure 2, one can also observe a successive increase of the number of crossovers and the lowering of the region energies where such crossovers occur. It is related to the variation of the electron probability distribution inside and around their InAs layers, which is similar to charge distribution in a metallic surface when its geometry varies from the flat to the spiked-type one. Such variation of the probability distribution is a consequence of the stronger confinement in structures with spiked-type QD geometry where the electron-ion separation is defined by interplays between the electrostatic interaction between them and the strong structural confinement, making it more stable with respect to the external magnetic field and the ring-like electron probability density distribution. Therefore, the energy dependencies for cone-like QDs have a shape similar to those that exhibit structures with ring-like geometry known as the Aharonov-Bohm effect.
The Aharonov-Bohm effect observed usually in ring-like heterostructures is a manifestation of the competition between the paramagnetic and diamagnetic terms in the Hamiltonian, resulting in the oscillation of the ground state energy. Such oscillations are impossible in the disk-like structures because of a significant decrease of the diamagnetic term contribution as the magnetic field increases and the electron probability distribution becomes more contracted. In QDs with a spike-like morphology, the electron probability density is already strongly confined, the external magnetic field can no longer decrease more the diamagnetic term contribution, and the energy dependencies on the increasing magnetic field become similar to those of ring-like structures.
In Figure 3, we present results of the calculation of the density of electronic states in the zero-magnetic field for QDs with three different morphologies on the left side case γ = 0 and on the right side for γ = 0.8. It is seen that the density of electronic states in the case of the zero-magnetic field for the disk-like structure has a larger value in the region of the low-lying energy levels and it decreases successively while the morphology becomes more and more spike-liked. It is due to the fact that the electron confinement in the disk is weaker than that in the lens and that in the lens is weaker than that in the cone.
thumbnailFigure 3. Density of the electronic states for a D2+ in vertically coupled quantum dots. (Heights 3 nm, wetting layer thicknesses 2 nm, radii 20 nm, and separation between them 6 nm for two different values of the magnetic field (γ = 0) and (γ = 0.8)).
Also, it is seen that the lowest peak corresponding to the ground bonding state in the cone-like structure is more significantly separated from other excited states than in two other structures. It is due to the stronger confinement of the electron in the cone-like structure where the electron is mainly located nearer to the donor than in disk-like and lens-like structures.
Comparing the densities of states presented on the left and right sides of Figure 3, one can see remarkable modifications that suffer the corresponding curves. Particularly, in the disk-like structure, the presence of the magnetic field provides a displacement of the peaks at the region of the low-lying energies. In the lens-like and cone-like structures, the modification is inversed; the peaks are reorganized in such a way that their distribution becomes almost homogeneous. Redistribution of the peaks' positions in the lens is defined mainly by the additional confinement that provides the external magnetic field, while analogous redistribution in other two spike-liked structures is mainly due to the Aharonov-Bohm effect.
In short, we propose a simple numerical procedure for calculating the energies and wave functions of a singly ionized molecular complex formed by two separated on-axis donors located at vertically coupled QDs in the presence of the external magnetic field. Our calculation includes some important characteristics of the heterostructure such as the presence of the wetting layer and the possibility of the variation of the QD morphology. The curves of the energy dependencies on the external magnetic field for the disk-like, lens-like, and cone-like structures are presented. We find that the effect of the in-plane confinement on the electron-ion separation is stronger in spike-shaped QDs and therefore the energy dependencies in such structures exhibit a behavior similar to that in ring-like structures. The analysis of the curves of the density of electronic states also confirms this result.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to this work. JSO created the analytic model with contributions from IM. RMG and GES performed the numerical calculations and wrote the manuscript. All authors discussed the results and implications and commented on the manuscript at all stages. All authors read and approved the final manuscript.
Authors’ information
JSO obtained his Ph.D. in 2004 at the Universidad Industrial de Santander, where IM was his advisor. His research interests include the theory of semiconductor nanostructures. JSO is the head of the research group ‘Condensed Matter Theory’ at the University of Magdalena. GES and RMG are master's degree and Ph.D. students, respectively, and teachers at the University of Magdalena.
This work was financed by the Universidad del Magdalena through the Vicerrectoría de Investigaciones (Código 01).
1. Jacak L, Hawrylak P, Wójs A: Quantum Dots. Berlin: Springer; 1997. OpenURL
2. Leonard D, Pond K, Petroff PM: Critical layer thickness for self-assembled InAs islands on GaAs.
Phys Rev B 1994, 50:11687-11692. Publisher Full Text OpenURL
3. Lorke A, Luyken RJ, Govorov AO, Kotthaus JP: Spectroscopy of nanoscopic semiconductor rings.
Phys Rev Lett 2000, 84:2223-2226. PubMed Abstract | Publisher Full Text OpenURL
4. Granados D, García JM: In(Ga)As self-assembled quantum ring formation by molecular beam epitaxy.
Appl Phys Lett 2003, 82:2401. Publisher Full Text OpenURL
5. Raz T, Ritter D, Bahir G: Formation of InAs self-assembled quantum rings on InP.
Appl Phys Lett 2003, 82:1706. Publisher Full Text OpenURL
6. Movilla JL, Ballester A, Planelles J: Coupled donors in quantum dots: quantum size and dielectric mismatch effects.
Phys Rev B 2009, 79:195319. OpenURL
7. Gutiérrez W, García LF, Mikhailov ID: Coupled donors in quantum ring in a threading magnetic field.
Physica E 2010, 43:559. Publisher Full Text OpenURL
8. Calderón MJ, Koiller B: External field control of donor electron exchange at the Si/SiO2 interface.
Phys Rev B 2007, 75:125311. OpenURL
9. Tsukanov AV: Single-qubit operations in the double-donor structure driven by optical and voltage pulses.
Phys Rev B 2007, 76:035328. OpenURL
10. Openov LA: Resonant pulse operations on the buried donor charge qubits in semiconductors.
Phys Rev B 2004, 70:233313. OpenURL
11. Koiller B, Hu X: Electric-field driven donor-based charge qubits in semiconductors.
Phys Rev B 2006, 73:045319. OpenURL
12. Barrett SD, Milburn GJ: Measuring the decoherence rate in a semiconductor charge qubit.
Phys Rev B 2003, 68:155307. OpenURL
13. Mikhailov ID, Marín JH, García LF: Off-axis donors in quasi-two-dimensional quantum dots with cylindrical symmetry.
Phys Stat Sol (b) 2005, 242:1636. Publisher Full Text OpenURL |
bc50291f86f1501b | KdV '95 - 100 Years Korteweg - De Vries Equation
by Henk Nieland
Hundred years ago an article appeared in the well-known British journal Philosophical Magazine, written by the Dutch mathematicians Korteweg and De Vries. It described the behaviour of certain types of waves occurring in a shallow canal in terms of a non-linear differential equation. This equation, now known as the Korteweg - De Vries equation (KdV), has played a central role in the study of non-linear phenomena, which has flourished so much since the 1970s. The world's leading experts in the field gathered 24-26 April 1995 at CWI to celebrate this centennial.
Diederik Johannes Korteweg at the age of 80 in 1928
The article was based on De Vries' Ph.D. thesis, in which he proved that the 'solitary wave', discovered and described by the Scottish engineer John Scott Russell about half a century earlier, really could keep its form. Many scientists, including Lord Rayleigh, had believed that this wave too would smooth out in the long run, as it should according to `linear science', such as the waves caused by a stone thrown in a pond. However, the KdV-equation contains a non-linear part which compensates this dispersive effect.
The KdV-equation remained in obscurity until 1965, when Zabusky and Kruskal (both speakers at the centennial symposium) discovered that two such solitary waves, or `solitons', emerge unchanged from a collision. The discovery of this remarkable stability property caused a - still ongoing - tide of research (sometimes called the `soliton revolution'). Two years later, Gardner, Greene, Kruskal and Miura invented the Inverse Spectral Transform to solve the Cauchy problem for KdV, and with this method Zakharov and Faddeev (both speakers in Amsterdam) proved in 1971 the complete integrability of KdV.
The Korteweg - De Vries equation, as published in the Philosophical Magazine, 1895, fifth series, Volume XXXIX, p. 422-433
These far-reaching breakthroughs had an enormous impact on the development of modern non-linear mathematical science. Whereas many applications of KdV theory are in fluid mechanics - not surprising because in this discipline several key non-linear structures, such as shocks, bifurcations, solitons, deterministic chaos, and hypercomplex systems were first discovered - , also vast areas of mathematics (ordinary differential equations, algebraic geometry, Lie group theory, differential geometry, asymptotics) and theoretical physics (quantum field theory, string and conformal field theory, quantum gravity, classical general relativity) opened up as a consequence of the basic research into the KdV-equation.
With KdV as a prototype, research into other famous integrable systems such as the Non-linear Schrödinger equation and the Sine-Gordon equation, has led to applications ranging from condensed matter and semi-conductor physics through non-linear optics and laser physics, hydrodynamics, meteorology and plasma physics to protein systems and neurophysiology. Examples highlighted at the symposium included internal solitons in the ocean, the transport of magma from the Earth's interior to the surface, non-linear acoustics of bubbly liquids, the Great Red Spot and other features in the Jovian atmosphere, and ion-acoustic plasma waves.
The most spectacular application so far was highlighted at the symposium by Hasegawa (Osaka): the use of optical solitons in fibres as an efficient (reliable and fast) means of long-distance communication. Proposed already more than twenty years ago by Tappert and Hasegawa (then at Bell Labs), these solitons are the best long-distance information carriers discovered so far. Recently 20 Gigabit/s soliton signals were transmitted error-free over a distance of 14.000 kilometers in a loop experiment. In a concrete project, 60 billion dollar is being invested in transatlantic optical fibre communication.
In the extensive obituary of Korteweg in the 1945/1946 Annals of the Royal Dutch Academy of Arts and Science, the KdV-equation is not even mentioned. Apparently at that time nobody could foresee that fifty years later articles with the acronym KdV would be counted by thousands.
Please contact
Michiel Hazewinkel - CWI
Tel: +31 20 592 4204
E-mail: mich@cwi.nl
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c5867afca0f8e08b | Orbital hybridisation
From Wikipedia, the free encyclopedia - View original article
(Redirected from Hybrid orbital)
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Not to be confused with s-p mixing in Molecular Orbital theory. See Molecular orbital diagram.
Historical development[edit]
Chemist Linus Pauling first developed the hybridisation theory in order to explain the structure of molecules such as methane (CH4) in 1931.[2] This concept was developed for such simple chemical systems, but the approach was later applied more widely, and today it is considered an effective heuristic for rationalising the structures of organic compounds.
Orbitals are a model representation of the behaviour of electrons within molecules. In the case of simple hybridisation, this approximation is based on atomic orbitals, similar to those obtained for the hydrogen atom, the only neutral atom for which the Schrödinger equation can be solved exactly. In heavier atoms, such as carbon, nitrogen, and oxygen, the atomic orbitals used are the 2s and 2p orbitals, similar to excited state orbitals for hydrogen. Hybrid orbitals are assumed to be mixtures of these atomic orbitals, superimposed on each other in various proportions. It provides a quantum mechanical insight to Lewis structures. Hybridisation theory finds its use mainly in organic chemistry.
spx and sdx terminology[edit]
For atoms forming equivalent hybrids with no lone pairs, there is a correspondence to the number and type of orbitals used. Thus sp3 hybrids in methane are formed from one s and three p orbitals. However, in other cases, there may be no such correspondence. For example, the two bond-forming hybrid orbitals of oxygen in water can be described as sp4, which means that they have 20% s character and 80% p character, but does not imply that they are formed from one s and four p orbitals. As a result, the amount of p-character is not restricted to integer values; i.e., hybridisations like sp2.5 are also readily described. For more information see variable hybridization.
An analogous notation is used to describe sdx hybrids. For example, the permanganate ion (MnO4) has sd3 hybridisation with orbitals that are 25% s and 75% d.
Types of hybridisation[edit]
sp3 hybrids[edit]
Four sp3 orbitals.
Carbon's ground state configuration is 1s2 2s2 2px1 2py1 or more easily read:
The carbon atom can utilize its two singly occupied p-type orbitals (the designations px py or pz are meaningless at this point, as they do not fill in any particular order), to form two covalent bonds with two hydrogen atoms, yielding the "free radical" methylene CH2, the simplest of the carbenes. The carbon atom can also bond to four hydrogen atoms by an excitation of an electron from the doubly occupied 2s orbital to the empty 2p orbital, so that there are four singly occupied orbitals.
As the energy released by formation of two additional bonds more than compensates for the excitation energy required, the formation of four C-H bonds is energetically favoured.
Quantum mechanically, the lowest energy is obtained if the four bonds are equivalent which requires that they be formed from equivalent orbitals on the carbon. A set of four equivalent orbitals can be obtained which are linear combinations of the valence-shell (core orbitals are almost never involved in bonding) s and p wave functions[3] which are the four sp3 hybrids.
sp2 hybrids[edit]
Three sp2 orbitals.
Ethene structure
Other carbon based compounds and other molecules may be explained in a similar way as methane. For example, ethene (C2H4) has a double bond between the carbons.
For this molecule, carbon will sp2 hybridise, because one π (pi) bond is required for the double bond between the carbons, and only three σ bonds are formed per carbon atom. In sp2 hybridisation the 2s orbital is mixed with only two of the three available 2p orbitals:
sp hybrids[edit]
Two sp orbitals
In this model, the 2s orbital mixes with only one of the three p orbitals resulting in two sp orbitals and two remaining unchanged p orbitals. The chemical bonding in acetylene (ethyne) (C2H2) consists of sp–sp overlap between the two carbon atoms forming a σ bond and two additional π bonds formed by p–p overlap. Each carbon also bonds to hydrogen in a σ s–sp overlap at 180° angles.
Hybridisation and molecule shape[edit]
ClassificationMain groupTransition metal[4]
• Linear (180°)
• sp hybridisation
• E.g., CO2
• Bent (90°)
• sd hybridisation
• E.g., VO2+
Main group compounds with lone pairs[edit]
For main group compounds with lone electron pairs, the s orbital lone pair can be hybridised to a certain extent with the bond pairs.[7] This is analogous to s-p mixing in molecular orbital theory, and maximizes energetic stability according to the Walsh diagram for the molecule.
Hybridisation of hypervalent molecules[edit]
Traditional description[edit]
In general chemistry courses and mainstream textbooks, hybridisation is often presented for main group AX5 and above, as well as for transition metal complexes, using the hybridisation scheme first proposed by Pauling.
ClassificationMain groupTransition metal
• Linear (180°)
• sp hybridisation
• E.g., Ag(NH3)2+
In this notation, d orbitals of main group atoms are listed after the s and p orbitals since they have the same principal quantum number (n), while d orbitals of transition metals are listed first since the s and p orbitals have a higher n. Thus for AX5 molecules, sp3d hybridisation in the P atom involves 3s, 3p and 3d orbitals, while dsp3 for Fe involves 3d, 4s and 4p orbitals.
However, hybridisation of s, p and d orbitals together is no longer accepted, as more recent calculations based on molecular orbital theory have shown that in main-group molecules the d component is insignificant, while in transition metal complexes the p component is insignificant (see below).
Resonance description[edit]
As shown by computational chemistry, hypervalent molecules can only be stable given strongly polar (and weakened) bonds with electronegative ligands such as fluorine or oxygen to reduce the valence electron occupancy of the central atom to a maximum of 8[8] (or 12 for transition metals). This requires an explanation that invokes sigma resonance in addition to hybridisation, which implies that each resonance structure has its own hybridisation scheme. As a guideline, all resonance structures have to obey the octet rule for main group compounds and the dodectet (12) rule for transition metal complexes.
ClassificationMain groupTransition metal
AX2-Linear (180°)
Di silv.svg
AX3-Trigonal planar (120°)
Tri copp.svg
AX4-Square planar (90°)
Tetra plat.svg
AX5Trigonal bipyramidal (90°, 120°)Trigonal bipyramidal (90°, 120°)
Penta phos.png
• Fractional hybridisation (s and d orbitals)
• E.g., Fe(CO)5
AX6Octahedral (90°)Octahedral (90°)
Hexa sulf.pngHexa moly.svg
AX7Pentagonal bipyramidal (90°, 72°)Pentagonal bipyramidal (90°, 72°)
Hepta iodi.svg
• Fractional hybridisation (s and three d orbitals)
• E.g., V(CN)74−
AX8Square antiprismaticSquare antiprismatic
• Fractional hybridisation (s and three p orbitals)
• E.g., IF8
• Fractional hybridisation (s and four d orbitals)
• E.g., Re(CN)83−
AX9-Tricapped trigonal prismatic
• Fractional hybridisation (s and five d orbitals)
• E.g., ReH92−
Main group compounds with lone pairs[edit]
Regular bonding component (marked in red)
Tetra sulf.svgTri chlo.svgDi xeno.svg
Penta chlo.svgTetra xeno.svg
Hexa xeno.svgPenta xeno.svg
Clarifying misconceptions[edit]
VSEPR electron domains and hybrid orbitals are different[edit]
The simplistic picture of hybridisation taught in conjunction with VSEPR theory does not agree with high-level theoretical calculations[7] despite its widespread usage in many textbooks. For example, following the guidelines of VSEPR, the hybridization of the oxygen in water is described with two equivalent lone electron-pairs.[9] However, molecular orbital calculations give orbitals that reflect the C2v symmetry of the molecule.[10] One of the two lone pairs is in a pure p-type orbital, with its electron density perpendicular to the H–O–H framework.[11] The other lone pair is in an approximately sp0.8 orbital that is in the same plane as the H–O–H bonding.[11] Photoelectron spectra confirm the presence of two different energies for the nonbonded electrons.[12]
Exclusion of d orbitals in main group compounds[edit]
Main article: Hypervalent molecule
Exclusion of p orbitals in transition metal complexes[edit]
Similarly, p orbitals have long been thought to be utilized by transition metal centers in bonding with ligands, hence the 18-electron description; however, recent molecular orbital calculations have found that such p orbital participation in bonding is insignificant,[14][15] even though the contribution of the p-function to the molecular wavefunction is calculated to be somewhat larger than that of the d-function in main group compounds.
Hybridization theory vs. Molecular Orbital theory[edit]
Hybridisation theory is an integral part of organic chemistry and in general discussed together with molecular orbital theory in advanced organic chemistry textbooks although for different reasons. One textbook notes that for drawing reaction mechanisms sometimes a classical bonding picture is needed with two atoms sharing two electrons.[16] It also comments that predicting bond angles in methane with MO theory is not straightforward. Another textbook treats hybridisation theory when explaining bonding in alkenes[17] and a third[18] uses MO theory to explain bonding in hydrogen but hybridisation theory for methane.
Bonding orbitals formed from hybrid atomic orbitals may be considered as localized molecular orbitals, which can be formed from the delocalized orbitals of molecular orbital theory by an appropriate mathematical transformation. For molecules with a closed electron shell in the ground state, this transformation of the orbitals leaves the total many-electron wave function unchanged. The hybrid orbital description of the ground state is therefore equivalent to the delocalized orbital description for explaining the ground state total energy and electron density, as well as the molecular geometry which corresponds to the minimum value of the total energy.
There is no such equivalence, however, for ionized or excited states with open electron shells. Hybrid orbitals cannot therefore be used to interpret photoelectron spectra, which measure the energies of ionized states, identified with delocalized orbital energies using Koopmans' theorem. Nor can they be used to interpret UV-visible spectra which correspond to electronic transitions between delocalized orbitals. From a pedagogical perspective, the hybridisation approach tends to over-emphasize localisation of bonding electrons and does not effectively embrace molecular symmetry as does MO theory.
See also[edit]
4. ^ Weinhold, Frank; Landis, Clark R. (2005). Valency and bonding: A Natural Bond Orbital Donor-Acceptor Perspective. Cambridge: Cambridge University Press. pp. 381–383. ISBN 978-0-521-83128-4.
6. ^ a b King, R. Bruce (2000). "Atomic orbitals, symmetry, and coordination polyhedra". Coordination Chemistry Reviews 197: 141–168.
7. ^ a b Weinhold, Frank. "Rabbit Ears Hybrids, VSEPR Sterics, and Other Orbital Absurdities". University of Wisconsin. Retrieved 2012-11-11.
9. ^ Petrucci R.H., Harwood W.S. and Herring F.G. "General Chemistry. Principles and Modern Applications" (Prentice-Hall 8th edn 2002) p. 441
10. ^ Levine I.N. “Quantum chemistry” (4th edn, Prentice-Hall) p. 470–2
11. ^ a b Laing, Michael J. Chem. Educ. (1987) 64, 124–128 "No rabbit ears on water. The structure of the water molecule: What should we tell the students?"
12. ^ Levine p. 475
13. ^ E. Magnusson. Hypercoordinate molecules of second-row elements: d functions or d orbitals? J. Am. Chem. Soc. 1990, 112, 7940-7951. doi:10.1021/ja00178a014
15. ^ O’Donnell, Mark (2012). "Investigating P-Orbital Character In Transition Metal-to-Ligand Bonding". Brunswick, ME: Bowdoin College. Retrieved 2012-09-16.
External links[edit] |
35bce3f7c91cef35 | Nonadiabatic Molecular Dynamics in Three Different Flavors
February 26, 2018 to March 2, 2018
Location : CECAM-HQ-EPFL, Lausanne, Switzerland
EPFL on iPhone
Visa requirements
• Basile Curchod (Durham University, United Kingdom)
• Ivano Tavernelli (IBM-Zurich Research, Switzerland)
• Graham A. Worth (University College London, United Kingdom)
• Todd Martinez (Stanford University, USA)
The main purpose of this school is to teach the participants different methods for performing excited-state molecular dynamics.
The first day of the school will be devoted to a general introduction on nonadiabatic molecular dynamics and potential energy surfaces.
Each of the three following days will discuss a particular nonadiabatic method, from a theoretical and a practical perspective, via dedicated lectures in the morning and tutorials on the computer during the afternoon. The three techniques that will be introduced during this school are Multi Configuration Time Dependent Hartree (MCTDH), Ab Initio Multiple Spawning (AIMS), and Trajectory Surface Hopping (TSH). TSH, AIMS, and MCTDH are currently the most popular nonadiabatic dynamics strategies for molecular applications. Furthermore, these three techniques form a hierarchy, from the most accurate quantum dynamics (MCTDH), passing by the approximate yet rigorous trajectory-guided technique AIMS, down to the mixed quantum/classical algorithm TSH.
This school will offer a unique opportunity to learn these methods in parallel, allowing the participants to gain a clear understanding of their differences, but also of their complementarity.
1. General introduction to excited-state dynamics. (Day 1)
a. Time-dependent Schrödinger equation
b. Representations and Ansätze for the time-dependent molecular wavefunction
c. Born-Oppenheimer approximation and beyond
2. Concept of potential energy surfaces. (Day 1)
a. Potential energy surfaces and conical intersections
b. Potential energy fitting procedures
3. Electronic structure properties required for nonadiabatic dynamics. (Day 1)
a. Electronic structure methods for excited states
b. Forces and nonadiabatic couplings
c. On-the-fly dynamics
4. MCTDH and its Gaussian-based versions. (Day 2)
5. Full and Ab Initio Multiple Spawning. (Day 3)
6. Mixed quantum/classical methods and Trajectory Surface Hopping. (Day 4) |
2ea5aca5a2025d88 | Quantum mechanics also rules astronomical processes
New research has discovered that quantum mechanics, which describes the world of the infinitely small, also serves to unveil the long-term evolution of the massive astrophysical objects that populate the Universe: they are governed by the same Schrödinger equation that rules the world of the elementary particles.
Quantum mechanics is the branch of physics that governs the sometimes weird behavior of the elementary particles that make up our universe. The elementary particles are those that are not made up of smaller particles, nor are they known to have internal structure. Your world is the one that describes quantum mechanics. The equations that describe the world of elementary particles are generally limited to the subatomic realm because relevant mathematics at very small scales are not relevant at larger scales, and vice versa.
However, new research suggests that the Schrödinger equation – the fundamental equation of quantum mechanics – is remarkably useful in describing the long-term evolution of certain astronomical structures. The results are published in the Monthly Notices of the Royal Astronomical Society. Massive astronomical objects are often surrounded by groups of smaller objects that revolve around them, like planets around the sun. For example, supermassive black holes are in orbit around swarms of stars, which in turn are orbited by huge amounts of rocks, ice and other space debris.
Due to gravitational forces, these huge volumes of material become flat and round discs. These disks, formed by innumerable individual particles that orbit in mass, can have a mass that varies from the size of the solar system, to a diameter of many light years (a light year is the distance that the light travels in the vacuum in the span of one year).
Astrophysical discs generally do not retain simple circular shapes throughout their lives. Instead, over millions of years, these discs evolve slowly to exhibit distortions on a large scale, bending and bending like waves in a pond.
However, science does not know very well how these deformations emerge and spread. Even computer simulations have not offered a definitive answer, since the process is as complex as it is prohibitively expensive, to be able to model it.
The solution has come from a well-known scientist, Konstantin Batyguine,has resorted to the so-called Perturbational Theory, typical of quantum mechanics, to formulate a simple mathematical representation of the evolution of astrophysical discs. The idea is brilliant, because that theory describes complicated quantum systems in terms of other, simpler systems.Batygin’s work suggests that the large-scale deformations that occur in astrophysical discs behave similarly to elementary particles, and that their propagation within the cosmic material of the astrophysical disc can be described by the same mathematics used to describe the behavior of a single quantum particle bouncing between the inner and outer edges of the astronomical disk.
Schrödinger evolution of self-gravitating discs. Konstantin Batygin. Monthly Notices of the Royal Astronomical Society, Volume 475, Issue 4, 21 April 2018, Pages 5070–5084. DOI:https://doi.org/10.1093/mnras/sty162
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3d03ac164dec7d1e | Welcome to Matt Kalinski'sHomepage
Matt Kalinski's homepage
strongly localized Starkstates for Zeeman Hamiltonian
While considered within the Mathieu theory and spanned by the circular eigenstates of Hydrogen it is a direct example of Poincare reccurence theorem in a simple system when the full exact quantum revival and Poincare reccurence time of zero-field evolution after the creation became one for the interval being devisible by the all squares of the quantum numbers in the expansion at once. Even for quantum numbers as low as 60 it becames hyperastronomical and therefore the Poincare time (10 to power 70 years).
When the relative dielectric susceptibility of matter X is -1 its relative dielectric constant epsilon = 1 + X is 0 (i.e. less than for the vacuum 1) and according to the screened Coulomb law the infinitely small external field from the point charge will cause the infinite Coulomb 1/epilon r^2 force and will induce the infinite internal electric field E=D/epsilon and the infinite opposite polarization causing it. Because of the NEGATIVE effective mass of the electron (anti-electricity in analogy to anti-gravity) this is the case of the Trojan matter. In practice because of the nonlinearity of response the polarization due to the infinitesimal field will saturate to finite. When considering the collective effects Trojan wave packets from their dipoles can generate together the internal electric field which consistently supports them so the external field does not need to be present. While the atomic dynamical response (Nonlinear unisotropic parallel resonant response theory of Hydrogen Circular States below ionization) and the polarizability of the Trojan state at the driving frequency is anomalous and negative (the dipole moment of the atom like a growing plant towards the light polarizes Trojan dipoles in the direction opposite to the electrostatic force from the capacitor plates charge surface density acting on the negative charge which normally orders dipoles to grow the capacitance) i.e. opposite to that of the normal dielectric at static response and so their electric susceptibility X_0(omega_0) <0 and they are both nonlinear and strongly depend on the external electric field under the resonant distance conditions between Trojan atoms the gas can self-lock on nonlinearity without any external field (While matching the outer empty space no normal field D=0 boundary condition for a Trojan electret stlab instantenously D = E - 4 Pi P(E) = 0 => E.ne.0, P(E).ne.0 i.e. the polarization is fully creating the internal electric field to cause it, above the critical Trojan dipole density and P(E) is fast-increasing and strongly nonlinear with "transistor Ic vs. Vc" saturation) and the native state of interacting Trojan gas is dynamically ferroelectric (superconducting with ethernal intra-atomic Meissner-like sigle-electron persistent currents). The two simplest three dimensional lattices supporting the native Trojan ferroelectric (and not anty-ferroelectric) order (all packets move in phase) on the microscopic Clausius-Mossotti cluster level (locked only on local field) are primitive tetragonal lattice (direction stretched (here compressed) simple cubic lattice) and parallel hexagonal lattice - Trojan hydrogen atoms are on honeycomb Graphene-like two-dimensional lattices parallel to each other and closer than the critical distance related to the intra-plane distances matching the atom positions projections perpendicularly and for both with packets move in planes. When the finite temperature is considered the temperature averaged quantum states being the thermal superposition of the circular Trojan manifold states will under the Einstein B coefficient (de)-excite towards low dipole moment rotational states and therefore new kind of the superradiant first order phase transition occures (nuclear-like shape transition) from non-ferroelectric (circular states with zero polarization) to ferroelectric (Trojan states) state when the temperature changes and the dipoles become thermally insufficient. Because once localized by the field the dipole moment of the Trojan wave packet saturates to its semiclassical value r_0 e (where r_0 is the Trojan Wavepacket orbit radius) and further almost does not depend on the field electric i.e. self-consistently E = E_sc/r_0^2 = - P = N/V e r_0 = 1/a^3 e r_0 (where a is the distance between the nearest Trojan atoms and Esc<0.11, a^3 is per-dipole volume) the value of the critical parameter N/V e^2/m w^2 around 1 (omega is the main oscillator frequency i.e. the Kepler frequency of the packet) remains the same that for the normal ground-state Lieb superradiance superradiance which is also the same order (1/3 of that) when the susceptibility becomes infinite on singularity due to the positive polarizaion feedback from the spherical dielectric hollow within the simple Clausius-Mossotti theory N/V e^2 / 3 m w^2 = 1 when the polarization centers are simple harmonic oscillators.
Classical Trojan Wave Packet Simulation using 1982 3.5MHz 48kB ZX Spectrum:
The Sinclair Basic code:
No CP field (spreading):
CP field ee=0.04 (non-spreading):
Simple man theory of Trojan wavepacket
Trojan wavepacket viewer
Quantum motor while it works:
Trojan motor as circulating current has the magnetic moment and is subjected to forces in the magnetic field gradients amd therefore is a microscopic Flying Gyro experincing the magnetic antygravity
Old US patents of electrostatic motors
Since is accelerated with 10^25 g there
Other small but not yet quantum motors
Enzyme powered rotating γ subunit
Metal droplet motors
Carbon nanotube bearing nanomotor powered by exchage of surface electrons angular momentum
Other quantum but not electric motors
Brownian motors - climbing with fluctuations
Carbon nanotube steam locomotive-like nanomotor powered by temperature gradient
American Physical Society Apple Cray Research
Optical Society of America Silicon Graphics Wolfram Research
My hot topics for today:
Quantum theory of dielectric constant of organic materials
Inertial confinement of solid hydrogen
Theory of dielectric constant of Bose condensate
Superradiance of Rydberg gases in quantum cavities
Car-Parinello simulations of egzotermic reactions
Cellular automata by quantum dots arrays
Physics of Magnetars
Fussion in atomic clusters in strong fields
Quantum monodromy in helium in crossed fields
Spontaneous light amplification through superradiant emission of radiation (SLASER)
Non-fixed node Monte Carlo with genetic algorithm
Bose-Hubbart theory of color Bose and mixed gases
Nonlinear acoustic subsceptibilities of Bose Gases
Beta electron beams from on-wave acceleration
Detection of heavy mases with mesoscopic SQUIDs
Unruh-Davies effect from quantum entanglement
Ionization Kondo effect in Rydberg gases
Mesoscopic superconductivity
Contraction-of-propagator methods for protein folding
Quantum computing with quantum dots
Trojan wavepackets with Wannier excitons
Spontaneous order in systems with negative mass
Superradiance with mouonic hydrogen with cyclotronic motion
Spacial control of dielectric constant
Optical phase solvers of Schrodinger equations
Quantum entropy ordering with Trojan wavepackets
Ticking of quantum states
Quantum entropy fluctuations in systems with anomalous spectra
Chaotic oscillations of quantum states in rotating systems
Channeling in multi-center scattering
Model Born-Infeld theories with electromagnetic vacuum collapse
Bohm Hydrodynamical Quantum Mechanics and real time Diffussion Monte Carlo
Cactoo fractals
Configuration existence theorem for N electrons in magnetic and circularly polarized fields: The maximum number of configurations may be the product of all differential foldings (maximum number of times the ZVS gradient manifold is cut by a straight line to the separate points): There may be at least 2^180=1532495540865888858358347027150309183618739122183602176 maximum number of configurations of electrons corresponding to carbon C60 (N=60) assuming the lowest nontrivial folding 2 (parabola-like). The same applies to the molecules consisting of atoms if the binding forces can be assumed to be conservative and have the potentials and their all components have notrivial folding at least 2: There may be in principle 2^(3 N) distinct molecules consisting of N atoms i.e. for example 2^180 allotropes of the C60 buckyball molecules (like 3 x 20, 5 x 12 or 6 x 10 sheet of Graphene or 60-atoms short Carbon nanotube etc.). To see this theorem working in 3D one may consider a rose and two planes as some differential ZVS gradient manifolds. The rose has clearly high folding and then when cut again perpendicularly is cuts to a lot of points. In 4 dimensions the line can cut the perpendicular or non-containing 3D space at 1 point (the system of 4 linear equations - 3D space equation in 4D + the line equation has only one unique solution) and therefore the 4 dimensional ellipsoid a x^2 + b y^2 + c z^2 + d t^2 = e in 2 points so there is maximum 2^4 = 16 solutions of the system of nonlinear equations consisting of 4 ellipsoid equations embadded 4 dimensional space or otherwise analytically when such system of equations is linear for x^2, y^2, z^2, t^2 it has only one nontrivial solution for them and if they are all positive it has 2*2*2*2=16 solutions for (x,y,z,t) because of the double sign of the square roots of the former.
Trojan Hydrogen as Josephson Junction through Quantum Phase Model (QPM)
Coherent tunneling between two Trojan atoms in two ultrashort delta laser pulses
Uncertainty relations for excited Trojan wave packets
Dynamic ferroelectricity and antyferroelectricity in the system of interacting Trojan atoms.
Quantum Hall effect and annihilation suppression in electron-positron gas
Cold fussion of magnetically stabilized two-nuclei Trojan states of Deuterium
Slaloming of Trojan wave packet of Deuterium nucleus in Palladium crystal lattice
Dynamic ferrolectricity of Trojan hydrogen atoms on 2D honeycomb lattice (Elok > 0).
Internal coordinate exitations of Trojan wave packet as the Hawking radiation
Attosecond Transient Absorption by Trojan Wave Packets
Microwave superconductivity of Trojan electron-positron pairs at finite temperatures
Hand writting with Landau states: Because there is a symmetric gauge A = (-B/2 y, B/2 x , 0) in which the potential of the quantum particle in the magnetic field is symmetrically harmonic with the diamagnetic term with the oscillator frequency proportional to the field it is possible to drive the Gaussian Landau state along the arbitrary trajectory using the exernal potentials (time-split the propagator as Landau and the external potential). First the paramagnetic omega_c/2 L_z term may be eliminated by the equaivalent counter-rotation of the coordinate system and the new time-rotated trajectory is the arbitrary for the Ehrenfest inverse dynamics problem for the ordinary potential for the Gausian packet. It is why the extra strong magnetic field is improving the spatial confinement of Trojan Wave Packet by x-reverting the stable rotating saddle point into the potential diamagnetic minimum and making it more orbiting Gaussian Landau state than the Trojan Wave Packet itself. In two dimensions because the quantum and the classical time evolution equation for the Wigner function (i.e. clasically the Liouville phase space distribution) is identical for the harmonic oscillator which as quantum can only have piecewise negative Wigner functions Trojan Wavepacket is stabilized to exist for arbitrary low resonant quantum number by the pendular quantum nonlinearity and Trojan-like wave packets exist without the nuclear hydrogen Coulomb field as simply C.P. field accelerated Gaussian Landau states when one turnes-off the field of the hydrogen nucleus. It is possible to find the time dependent CP field such that arbitrary trajectory including hand written messages can be achieved in time first found by the inverse problem. While the changable width of the Gaussian is pemitted the time dependent magnetic field may be also used in combination. Some kind of trasverse confining potentials like the grid of Coulomb potential exends the method to three dimensions.
Weak excitations of uniform Gross-Pitaevskii condensate with ultra strong self-focusing interaction
Stability of true quantum cubical atom of Oxygen in electromagnetic fields. While the exact static cubical atom with no fields in the vacuum is the ion with only the fractional nuclear charge Z=(1+3*3^0.5+6^0.5/2)/2=2.4676... (end always unstable according to Ernshaw´s theorem the physical integer nuclear charge maybe tuned either by the symmetric screening with the dielectric constant cubic inner zone (precise - not Menger Sierpinski sponge (fractal dimension Log(26)/Log(3)=2.96565... > Log(20)/log(3)=2.72683... ) 1-st interation (by removing (altering the dielectric constatnt in) only the central cube in lower partition)) or by the spherical harmonic quantum dot potential.
Collective"Hydrino" flakes: Storing chemical energy in Honeycomb self-sustained Trojan Atom clusters (Hydren) approximately 13.6 eV per excited Trojan atom (N*13.6 eV per N Trojan Hydrogen atoms)
Trojan wave packets in the quantum cavity as the ethernal, non-collapse and no-revival electron-photon superpositions immune to the spontaneous emission: Jaynes-Cummings-like model for the infinite number of slightly off-resonant quantum levels: Our Mathieu theory can be extended to the full electron-photon system with the quantum electromagnetic field. The states with the fixed deviation of circularity may be multiplied with the Fock states with the fixed number of photons and the augmented polarization in reversed quantum number order providing the proper Jaynes-Cummings ladder energy exchange conditions. The collective pendula in the electron-photon space are immune to spontaneous emission. Collective electron-photon Trojan wave packets with the discrete level of confinement as the new quantum number dependent on the photon-electron ladder offset in such space are fully ethernal and immune to the radiative decay and are the single electron Meissner effect for the photonic superconductivity. While the circular energy states interact with the resonant modes of the cylindrical cavity the harmonization of total Jaynes-Cummings spectrum occures. The field-electron superposition is the collective Schroedinger-Lorentz coherent Brown state for the Hydrogen-cavity with the non-spreading electron density and without the electromagmetic decay as only the phases of the circular states evolve but not the populations. While the dressed pendular states are stationary in the laboratory frame the coherent superpositions of states with the different photon number offset (for example with the coeeficients of the corresponding photon coherent states) with respect to the circular state running quantum number and with energes that now differ exacly by the harmonic hbar omega are nondispersing electron density wave packets. While they are photon-electron collective they are ethernal also with respect to the spontaneous emission. New method of no transient absorption spectroscopy detection of Trojan wave packet is implied with Positronium Trojan atom: Positronium Trojan atoms in presumably Trojan state are injected to quantum cavity with compatibile quantum C.P. field to maintain them in the Trojan states indefinitely. While the cavity has finite Q the Klystron power injection is nessesary. After Klystron power turn off the enhanced gamma emission from the recombination should be observed while the positronium was Trojan.
Ionization lifetimes of Trojan Wave Packets with hypergeometric-coordinate method appplied to normal circular or spherical coordinates as the high angular momentum Stark states. The absolute scaled critical field is approximately 1/3.
"Recursion formula" (theorem) to construct arbitrarily large primes: Let N be the primorial of the prime k (denoted by the following #) N = k# i.e. the product of all primes not larger than the prime k, then there is such integer s (also may be negatives and sometimes larger then 10) that N - 1 + s*10^l is prime, where l is the decimal length of N . For example 2*3*5*7*11*13*17*19*23*29*31*33*37*41*43 - 1 + 800000000000000000 = 431731123945110989 + 800000000000000000 is prime and there is such number s(=8) for arbitrary k(=43), l(=17) or 2*3*5*7*11*13*17*19*23*29*31*33*37*41*43*47 - 1+ 200000000000000000000 = 202913628254202165299 + 200000000000000000000 is prime k=47, s=2, k=20. It is Fermat-like theorem that the sum of N - 1 plus some existing multiple of its decimal length 10 power is prime. The first candidates are therefore k s near the exponent of 2 (e) in the largest known primes. While N can be arbitrary large by induction theorem it proves again there is no the largest prime.
For example 29*10^15 - 1, 2*10^27 - 1, 6*10^28 - 1, 9*10^29 - 1, 48*10^30 - 1, 8*10^31 - 1, 21*10^32 - 1, 5*10^33 - 1, 6*10^34 - 1, 44*10^35 - 1, 11*10^36 - 1, 11*10^37 - 1, 15*10^38 - 1, 18*10^39 - 1, 6*10^40 - 1, 33*10^41 - 1, 30*10^42 - 1, 77*10^43 - 1 6*10^61 - 1, 6*10^73 - 1 are primes alone but 109# + 23*10^45-1 and 179# + 11*10^70 -1 = 139819592777931214269172453467810429868925511217482600306406141434158089, 181# + 60*10^72 -1 = 65397346292805549782720214077673687806275517530364350655459511599582614289 are primes.
Similar case: Localization of primes near the decimal powers: (large primorial is replaced by the small even number) While the number 10 is only devisable by the primes 2 or 5 and so the 10^n it seems very unlikely that 10^n + s for small s (too small to be devisable by anything large) is devidable by many primes while is close to 10^n and s is not devidable by 5 or 2 so must be sometimes only by one and be the prime. So for the arbitrary n there is always a small number s with number of digits much less than 10^n (of the order of n in value) such that 10^n + s is prime: For example 10^123 + 3 is prime, 10^127 + 283, 10^1000 + 453, 10^1001 + 9337, 10^1002 + 1383, 10^1003 + 69, 10^1004+613, 10^1013+777, 10^1103 + 1693, 10^1203 + 597, 10^1303 + 729, 10^2000 + 4561, 10^3000 + 1027, 10^4000 + 16483, 10^5000+ 12123, 10^6000 + 9873, 10^7000 + 4981, 10^8000 + 5079 are primes. Further there is always such s that 10^n + s is the prime and the s is the prime. For example while 7 is the prime directly by the construction 10^100 000 000 + 7 may be the current 250 000 $ winning prime.
While the Mersenne number i.e of the form 2^n-1 has always the binary form 1111111.....111111 while 1 adds up to it to form that with one 1 and all zeros and n is its number of digigs while this number of digits is devisable by some number it is devisable by a shorter Mersenne number 111111 and the result is 1000001000001....1000001 By the binary multiplication that logically NAND-adds 1 from the left-shifted repetitions and AND-transfers 1 to th next column the length n is then devisable by the shorter Marsenne number binary length tha can be any. Therefore n must be the prime for 2^n -1 to be a prime. From the binary form form it is also the sum of the geometric series 1 + 2 + 2^2 + 2^3 + 2^(n-1) = (1-2^n)/(1-2). Therefore by the same argument in any base system and the numbers looking the same i.e. in the form 111111............11111 (1-k^n)/(1-k) may be good prime if n is the prime. For example in decimal (10^19-1)/9 = 1111111111111111111 and (10^23-1)/9 = 11111111111111111111111 are primes while from the form in the base 5 (5^47-1)/4 = 177635683940025046467781066894531, in the base 6 (6^71-1)/5 = 3546245297457217493590449191748546458005595187661976371 and in base 7 (7^13-1)/6 = 16148168401 is prime while in base 11 (11^73 -1)/10 = 1051153199500053598403188407217590190707671147285551702341089650185945215953 is prime.
Lucas-Lehmer primality test of Mersenne number using Trojan wave packet evolution after turn-off: Let M = 2^n - 1 is the Mersenne number to be checked if it is a prime and Lm is the lowest common multiple of squares of all of the quantum numbers of the states spanning the Trojan wave packet. Let us monitor the Trojan wave packet free (no-field) evolution at times proportional to Lm*s_n/M when s_n is the sequence such that s_0 = 4 and s_n=s_{n-1}^2 - 2 then M is the prime (if and only if) when the autocorrelation function < Psi(0)|Psi(t)>=1 for t = 2*Pi hbar/ R*Lm *s_{n-2}/M.
2^94897643-1 may be the largest Mersenne prime known. It takes about 100 days to check it up with Mathematica with the following code on multi-GHz processor: M=2^94897643-1; s=4; Do[temp = PrintTemporary[n]; s = Mod[s*s - 2, M]; NotebookDelete[temp], {n, 1, 94897643 - 2}]; Print[Mod[s, M]] (when the last s Mod[s, M] is 0 M is prime)
Breathing chains of self-sustained Trojan Hydrogen in Quantum Gear modes: When two hydrogen atoms are in the trojan state in the same plane at the on-line point between them they generate together the elliptically polarized electric field with the polarization ellipse with 2:1 axis ratio but with the main circular component opposite to their dipole rotation. Therefore the neighboring wave packets will self-consistently counter-rotate while breathing in shape.
Noninterating entanglement between two far Trojan atoms pairwise close to and interacting with others two: Let two very far atoms almost non-interating via dipole-dipole interation be in the entangled Trojan state, for example in entangled Trojan cat state: clockwise Trojan wave packet on one atom A times anti-clockwise on the other B plus the opposite |TA+> ox |TB-> + |TA-> ox |TB+> + terms to enforce fermion anti-symmetry (anti-symmetric entangled Hartree-Fock state). If now each of the atoms interacts with close Trojan atom (there are two far H2 semi-molecules) via the dipole-dipole interaction (There is totally 4 atoms) the level of entaglemnt between far Trojan atoms will change despite there is no interation between them but only they interact with close others (The close atoms perform semi-EPR measurement on the other by the dipole interaction).
Henon-Heiles dynamics and chaos suppression near Trojan equillibrium point: The nonlinear Hamiltonian for Trojan wave packet up to third order is H = px^2/2 + py^2/2 + a x^2/2 + b y^2/2 - (xpy-ypx) + c (x^3 - 3/2 y^2 x), with c approximately 1/N^2 a, b, and thereforeHenon-Heiles-like type and full of soft chaos but quantum forces surpress it even when the wave packet covers large regions of the classical phase-space for low quantum numbers N of the central component state.
Trojan wave packet generation from resonant circular states by sudden CP field rectangular turn-on/off, free sweep of quantum phase hypercube till space localization and suddent matching turn on of the supporting CP field: The new method of the Trojan wave packet generation seems to be possible in addition to adiabatic-rapid method to generate it from the circular resonant Rydberg state using quantum energy rails Zener switches. The suddent (growing much faster than the Kepler period) CP field turn-on projects the circular state onto the highest Trojan states manifold consisting mainly of the circular states with a different n number to superpose all the states. The circular state becomes the superposition of the angularly excited Trojan states freely evolving according to the phases proportional in time to the Trojan pendular energies meaning that the circular states populations themselves change in time. The field is than turn off abruptly to maximize the circularstates population closely matching the populations of the Trojan wave packet neglecting the phases. The spacially focused packet is rephased during the free no-CP field Rydberg evolution sweeping the quantum phases hypercube modulo 2Pi and the matching CP field is turned-on abruptly at the proper position phase to maintain it.
Quantum Mechanics as the Diffussion with Transmutation: A new Time Dependent Quantum Monte Carlo Method is possible: Two kind of diffussing particles are considered which can transmute one into the other. Each of the species undergoes the hyphotetical Einstein random walk progression with transmutation. The progressed particles transmute into the particles of the other king before contributing to or annihilating the other particles density. This fully emulates the Time Dependent Schrödinger equation for any number of quantum particles. The negative sign of the real and the imaginary parts of the wave function is handled by the "spinor" densities carrying the sign as the degree of freedom. The walkers densities are substructed by defining the critical distance of nearest neighbours epsilon within which the substructed walkers annihilate each other like the antyparticles or are removing (capturing) each other from the board like in the game of Chequers and even change sign if the other walker space islocally empty within epsilon.
Kosterlitz-Thouless motion phase dynamic phase transition in Trojan Wave Packetsas rotors system on regular lattices
Trojan Wave Packetsas artificial time-optical lattice(Amost)-Quantum Time Crystals Quantum Time Crystals are defined as "alive" ground states in which the internal motion cannot be stopped even in the absolute zero like the persistent current is the superconductor so there is spontaneous periodicity and therefore crystalization in time. Because the following simple theorem holds that: for abritrary positive numbers a_i adding to 1 i.e Sum a_i = 1 the sum Sum a_i E_i (the Hamiltonian expectation in superposition) is always larger than E_0 if E_i are ordered growing the time dependent solutions with the energy lower then the ground state are not possible for linear Schrödingers equation with the lowest eigenstate. It is however not immediately clear for the time-dependent nonlinear equations. But those are also very difficult to imagine since only moving exact many-body states (superpositions) build from the full space eigenstates could lead to moving mean fields through the density functionals but those would have to have also higher energy above the ground state because of the theorem. For the quantum rotor in the magnetic field 2*alpha with the Hamiltonian - hbar^2/2 m r0^2 d^2/d phi^2 + alpha (hbar/i) d/dphi + m*r0^2*alpha^2/2 two-soliton-like solution (two bumps in density rotating with Larmor (half of the cyclotron) frequency) exists as e^-i*E*t Cos(phi-alpha*t) with the "kinetic" energy E=hbar^2/2 m r0^2 + m*r0^2*alpha^2/2 arbitrarily close by hbar^2/2 m r0^2 (the best condition hbar^2/2 m r0^2 = hbar alpha = 1 quant of Larmor which may be made as (negligibly) small as possible in temperature kT but still large (acoustic) in frequency 7.637 nK/kHz to the constant ground state 1 (trivial in shape) as m or r0 grows to infinity. It therefore seems that it may generate similar solutions with the non-linear self-interaction but with the energy below the ground state. Similar states can be immediately constructed from three consequitive circular states i.e. from two without the cetral for two-bump states but they will lower energy in the magnetic field because of the effectively negative mass of radial electron. Similar states can be immediately constructed from three energy consecutive circular states spanning the Trojan wave packets with the Kepler orbital frequency resonant to the central state i.e. from two without the central one for moving two-bump states (standing in the Kepler rotating frame) but they will indeed lower energy at first look to the central state in the magnetic field because of the effectively negative mass of radial electron (negative kinetic energy on motion which has formally no botton but which is really the Hydrogen ground state) and all is spanned around the excited states. They for example may be Gaussons (As mean-field it corresponds to infinite series of N-body Psi+Psi+Psi+...PsiPsiPsi interactions but is easier to solve) put in the solid state Born-Karman boundary conditions in one dimension coupled to the one dimensional scalar magnetic field alpha as alpha*p=alpha/i d/dx (in one dimension here the magnetic field and the potential is one). While the Gausson momentum may be only discrete due to the B-K conditions and cannot match the arbitrary magnetic field alpha precisely while it is descrete the Gausson center velocity (alpha-k) cannot match 0 for abitrary alpha, the Gausson center must move at the lowest energy (as it stays steady in the comoving (rotating) frame canceling the rest (alpha-k) i d/dx term). Similar states can be constructed from three energy consecutive circular states which constitute mostly to Trojan wave packets but because of the negtive mass they will lower the absolute energy in the magnetic field. Similar states can be constructed from three energy consecutive circular states which constitute mostly to Trojan wave packets but because of the negtive mass they will lower the absolute energy in the magnetic field. Even if they are disputed as true normal ground states with the lowest energy to exist rigorously (the self-consistent stationary Bloch e^i{alpha-k)x* u(x) function with the Bloch vector (alpha-k) clearly exists for the above problem while log |u(x)|^2 stays consistently periodic which in principle may have the lower energy (the leading correction from the term (alpha-k)/i d/dx within the first order perturbation theory for example from the tight-binding Bloch function structure (simply average in this state) is only -2 (k-alpha)^2 Cos L (k-alpha)) <Psi0(0)|Psi0(x-L)> where the last factor is the overlap between infinite line shifted perfectly Gaussian Gaussons with the energy lower then "free band" kinetic (alpha-k)^2/2 and even negative (below hbar omega0/2) for small (k-alpha) and it later grows above 0 but is very small because the overlap is exponential in -L for confined gausson and can only exeed formally the kinetic for overlaps of the order 1 (very weak interaction) and (alpha-k) = Pi/L) and which periodic solitonic part is the solution and it can move on weak excitation as the interference pattern when unfolded above L = 2 Pi while there is no fixed 0 - the "wrong" Bloch function will creep under extra flux turned-on abruptly for which it would be the (ground) eigenstate otherwise but was for different) one may imagine interactios leading to Gausson broadening (self-energy lowering e.g. for relativistic) with motion and there are clearly infinitesimaly excited states which approximate them very well and will be excited by the infinitesemal temperature or fluctuating stray fields. It is a matter of the energy difference between the moving Gausson which is approximately hbar omega0/2 + (k-alpha)^2/2 and even looks conduction Bloch-band-like despite the state moving shape time-dependence (exact value on the infinite line) and the energy of the Bloch function with the Bloch vector (k-alpha) (which on the closed boundaries can be found precisely only numerically) which periodic part is the solution which appears to be not significant since for the symetric part it will be only from the broadening or narrowing Gaussian shape to the moving Gausson so both are semi-degenerated in energy. Trojan wave packets are quite similar even if it is difficult to talk about their lowest energy but better about their best topological simplicity. Especially the Trojan Wave Packet consisting of two levels in extreme, the ground state and the first exited state with l=1 will move periodically in time even without the C.P. field and with the energy near the ground state which is the direct equivalent of the Zeeman creeping of the periodic Gausson with momentum k0: alpha= alpha-k0 +k0, in the smallest (k0-alpha) (k0,alpha>0) Zeeman field acting on its components on closed boundaries splitting the propagation operator ((E1-E0) -> (k0-alpha)). For the exited Trojan Wave Packet being the pendular "ground state" in the sense of nodal simplicity (Trojan wave packet the the maximal energy state because of the negative mass) the time crystalization as exactly defined originally would be the small Trojan packet oscillations or even oscillations-modulated rotational motion around the electric field vector after the sudden small off-resonance detuning 1/n^3-omega turn-on from the formal discrete Kepler resonance equivalent to the magnetic flux i.e. the attempt of the cirular motion in the pendular potential around the C.P. vector static in the rotating frame rather than the Trojan circular motion itself or simply magnetic field can be turned-on while the electric off or lowered on generated Trojan wave packets for high Rydberg states to generate similar motion relatively in the rotating frame on slowly spreading or width-oscillating wave packets.
Almost Quantum Time Crystals in the model nonlinear system with "Coulomb" potential in mean field modulus: The nolinear Gausson-like Hamiltonian H = -(1/2) d^2/d phi^2 + (A/2) 1/|Psi| (Singular repulsive 1/|Psi| v.s. singular a*Log|Psi|^2 leading to Gaussons) has the "Solitonic" (+1 elevated cosine looking Gaussian) solutions with the periodic boundary condition Psi(phi) = Psi(phi + 2 Pi) Psi(phi) =A (1 + Cos(phi)). Those solutions have energy E=1/2. When the system is put in the magnetic field alpha then H = (1/2) (1/i d/ d phi - alpha)^2 + A/2 1/ |Psi| and the time dependent solution A (1 + Cos(phi-alpha*t)) has the energy only E = 1/2 + alpha^2/2 while the constant solution Psi= (3/2) A (the same particles normalization) the energy E = alpha^2/2 + A/2 (2/3)^0.5. It can be higher than for the time-dependent if A > (3/2)^0.5 The approximate static solution (without caring about closed boundaries for small alpha) A e^{-i alpha phi} (1 + Cos(Psi)) has still lower energy 1/2 (exact for alpha = 1) but the nodally simplest eigenstate can have higher. The first order perturbation theory with respect to 1/i alpha d/dphi (zero contribution from symmetry of the solution without it) gives on the other hand exactly the energy of the moving state 1/2 + alpha^2/2.
Quantum Mechanics as the perturbation to Classical Mechanics: The nonlinear Schrödinger equation proposed by Shay propagates the wave function which is exactly equivalent to the density of walkers moving according to the Newtonian equations and with the velocities contained it the gradient of its phase. Therefore the exact quantum Schrödinger equation can be considered as the perturbed to the Shay equation. The time-dependent perturbation calculus can be performed with respect to the quantum potential back added while it is expressed is some scaled coordinates in which it is small (1/N)(for example N=omega^-(1/3) where omega is the characteristic frequency of the quantum system) correction and the zero quantum perturbation wave function is reconstructed from the purely classical simulations of the ensamble.
Inverse classical harmonic generation problem for the one dimensional potential: The time dependent solution for the motion of the classical charged particle is assumed as the high terms desired Fourier series expansion up to the very high term enforcing the n-th harmonic. The external harmonic force is added to the classical uknown potential to produce the proposed highly anharmonic solution. The potential is Taylor expanded also to the very high power of the space coordinate (time dependent polarization). While the high powers of the proposed Fourier series are calculated the terms multiplying the n-th order Fourier oscillatory terms are collected containing also the potential Taylor series coefficients as uknowns which can further be obtained from the resulting linear system of equations to reconstruct the desired potential. The problem effectively reduces to finding the desired trunaction of the n-th power of the time Fourier series in the Exp[i n omega t] which may me obtained by the recursive tensor multiplication of the vector of the lenght N by the another vector of the same length to costruct the NxN matrix containg the all products of the original vectors elements and then the summation of all anty-diagonals of the the smaller submatrices of the resulting product matrix with the common left upper corner to contruct the next vector to multiply it by the original till the desired power is obtained.
The nonlinear quantum mechanics of Shay with the Schrödinger equation - d^2/2 d x^2 + V(x) + d^2/d x^2 rho^0.5/2 rho^0.5 = i d/dt (rho = |Psi|^2) immedialy predicts the approximate short time solutions with a stationary density but with the internal motion: Let the wave function Phi was real at the t=0 i.e. Phi=rho^0.5. Then assuming it did no become significantly complex during the initial evolution the kinetic energy term cancels exacly with the action of the anti-quantum potential and the Shay Schrödinger equation simplifies to V(x) = i d/ dt. Then the solution is Psi= rho^0.5 e^-i V(x)t and the velocity field of Bohmian trajectories is vel(x)= - d/dx V(x) t which is a solution of the Newtons equation for short time with the local acceleration field a(x) = - d/dt V(x). Therefore it appears long time solutions are possible with the internal motion but the steady density. For the harmonic oscillator the approximate solutions are possible in the form of the constant density eigenstates but full of the internal motion in the form Psi= e^-a*x^2/2 e^i*Integral(2(E - V(x))^0.5 (a is related to the turning point for the energy E as 1/x_turn^2 =a). Those states are full of the internal natural velocity field v(x)= (2(E-V(x))^0.5 with the Bohm trajectories obviously executing the harmonic oscillatory motion but not changing their density in time. In general there is no reason why this motion would not be chaotic but the desity still stationary. Chaos drag and steady flows of Bohmian Trajectories within the Trojan states in Shay Quantum Mechanics: The Bohm Trajectories for the Quantum mechanics of Shay are exactly the classical trajectories. The stable shape invariant Trojan Wave Packets within the nonlinear Shay Quantum Mechanics therefore contain the steady chaotic flows around the Lagrange equillibrium points leading to non-changing particle densities as the dynamics is locally Henon-Heiles-like.
Chaos in Bohmian trajectories in quantum extended two-dimentional rotating continuous Feigenbaum map : It is known that the functional powers of the Feigenbaum (logistic) function f(x) = r x (1-x) f^n(x) = f(f(f(f(...(x)))) as higher and higer order polynomials change the number of solutions of the polynomial equation f^n(x) - x = 0 upon the change of the parameter r from the powers of 2 to infinity which leads to chaos of the interations trajectories x_n+1 = f(x_n). The discrete map can be readily extended to 2 dimensions as x_n+1 = f(y_n), y_n+1=f(x_n). One may rewrite it as x_n+1 = f(y_n) - x_n + x_n, y_n+1 = f(x_n) - y_n + y_n or x_n+1 - x_n = f(y_n) - x_n, y_n+1 - y_n = f(x_n) - y_n. While mapping 1 -> dt the continuos version is dx/dt = f(y) - x, dy/dt = f(x) - y and the Newtonian (Hamilton) version is d^2x/dt^2 = df/dy dy/dt - dx/dt, d^2/dt^2 = df/dx dx/dt - dy/dt. The Hamilton function can be contructed and quantized assuming px = dx/dt, py = dy/dt. While the rotation is imposed by adding the term L_z = x py - y px the original Hamiltonian Bohmian trajectories of the eigenstates will exhibit chaotic behaviour.
Origin of classical chaos around the Trojan and anti-Trojan Lagrange equilibrium Point: When the Coulomb potential in the rotating frame which is the same Coulomb potential is expanded around the Trojan or anti-Trojan Lagrange equilibrium point till the forth order and the discrete Euler integration method is applied the resulting map resambles the two dimensional Feigenbaum map. Therefore the origin of chaos around that points is that of Feigenbaim. It is a result of the periodic alternating reflection from the line inclined at the 45 degree angle and the 90 degree refraction from the smooth polynomial curve which leads to the random distribution of the refraction and the reflection points upon exceeding the critical values by the curve coefficients.
Sligthly wiggling but a very good approximation for the error function Erf(x)=(2/Pi)*ArcTan((x*(2+x^4*2)) reversible to quintic equation solvable by Banach-Newton operator iteration gn(x)=(4*f1(x)^5 + f1(x))/(5*f1(x)^4 + 1), f1(x) = 0.5*Tan(x*Pi/2) and another one reversible by Ferrari's formulas from quartic equation Erf(x)=Sign(x)*Tanh(|x|*1.152 + 0.064*|x|^4)
Einstein equations as the intrinsic property of the differentiable manifold: While the Lentz-Faraday induction law and the Maxwell-Faraday equation can be derived from the existence of the (x) Lorentz force by a vector division of the induced electric-like force field and the magnetic field sweep and the coefficients of the affine connection are forming the gravitomagetic field vector generating the gravitomagetic Lorentz force through the geodesics equation the Einstein equations appear to be more the intrinsic geometric property of the manifold than externally imposed as the augmented gravity induction laws are therefore implied for the affine connection and the components of the energy-momentum-stress tensor with the main Gauss law component are not independent.
Critical condition for the superradiant phase transition from the electrostatic and classical considerations only: consider the cubic lattice of the (locally) harmonic oscillators with a frequency omega with the lattice constant a and each filled with one electron. Consider one electron with 4 in-plane nearest neighbors which when assumed rigid create the anti-harmonic oscillator potential in the plane perpendicular direction. The 5-th central electron motion becomes unstable in the Coulomb field of 4 when the net harmonicity of the oscillator is negative i.e m omega^2/2 - 4 *e^2/ 4Pi /epsilon 0 (1/a^3) < 0 or e^2/Pi/epsilon 0 /m/omega^2 >1 . Making it artificially quantum by multiplying the fraction up and down by hbar we have e^2 hbar /Pi/epsilon 0/m/hbar/omega^2 > 1 or 2/Pi |D12|^2/E12/epsilon0 (N/V) >1 where |D12|^2 = e^2 hbar/2 m / omega is the transition strength between first two quantum harmonic oscillator levels, E12= hbar omega is the energy gap between them and (1/a^3) = N/V is the spatial 3D harmonic oscillator density. This is identical to this of the original superradiance by the Hepp and Lieb if the two levels are approximated by the harmonic oscillator (Holstein-Primakoff approximation).
Single atom Quantum Hall Effect with the Trojan Wavepacket. While the motion of the Trojan Wave packet is a single electron current I =e/T = e f = e omega / 2Pi and defining the Hall Voltage as the Coulomb Voltage (the difference between the Coulomb potentials) (Coulomb force is acting as the Lorentz force balancing the centrifugal like for the free cyclotron motion) between the infinity and the point on the Trojan wave packet orbit U = e/2 Pi/Epsilon_0/r the sudden jumps of r as the function of omega taking the Bohr values r_n = n^2 a_0 for omega_n=1/n^3 (a_0 is the Bohr) radius lead to the quantization of the Hall resistance R=U/I around the jump points, R_n=n (h/e^2) (here linear but not inverse in n).
Trojan Wave Packets above the classical stability threshold. While the anti-Trojan wave Packets exist as an approximate product of the inverted pedulum state which means that the classical trajectories indeed fall and spill from the unstable radial maximum and a Gaussian circular state and are localized around the always ustable equillibrium point it appears that also states of such type should localize around the Trojan Wave Packet equillibrium point when it is unstable above the critical field but when the rotational stabilization or the destabilization of the saddle potential becomes insignificant to the electric field potential. Those will no longer have Gaussian radialfunctions but some of the Bessel J_0 like inverted pendulum type. The destruction of the Coulomb spectrum in the rotating frame is so strong that the Mathieu theory in the Hydrogen eignestates basis is no longer valid because of large energy coupling between the states.
Finding the perfect Trojan Wave Packets directly with the Power Method. The Power Method http://ergodic.ugr.es/cphys/LECCIONES/FORTRAN/power_method.pdf for the eigenvalue problem does not require the matrix storage to solve the eigenvalue equation. In the essential two space dimensions the method may be used for sparse Hamiltonian of the Hydrogen atom in the C.P. field in the finite difference approximation on either x-y grid or r-phi (r - angular momentu m) grid. Each recursion of the for example Psi(x_i, y_j) 512 x 512 numerical vector is equivalent the time dependent numerical integration of the Schrödingers equation using the FFT method i.e. one integration for one consequitive eigenvector to project it out for the next dominant eigenvalue.
Cold fusion in the Trojan dynamic ferroelectric clusters: While the regular ultra-cold 2D lattices (flakes) consisting of the Hydrogen atoms in the Trojan wave packet states coupled to each other the volumetric dynamic ferroelectric is possible due to the dipole-dipole interaction between the Trojan electrons. Each atom while highly excited to the Trojan state carries approximately 13.6 eV of the chemical energy. While such matter suddenly collapses due to the thermal decoherence to the hydrogenic ground state 13.6 eV of the energy is released from each atom through radiation. When the Hydrogen is replaced by the Deuterium 4 keV of the fussion activation energy can be obtained for the primary ignition from 300 atoms for one secondary fussion event and the further chain reaction.
Self-amplyfying high-energy solitonic phononic waves in the Deuterium sublattice of the fully packed Palladium Hydride as initially displaced phase-triggered system of Coulomb-interacing quartic oscillators
Trojan Wave Packets in Helium atom in configuration space: While the 1D model of the Helium atom is similar to 2D model of the Hydrogen atom and the hyper-linear polarized electromagnetic field has two counter-rotating hyper-Circularly Polarized components the semi-Trojan Wave Packets are possible in the configuration space of Helium in the linearly polarized electromagnetic field. For the true physical 6D helium atom they correspond to highly correlated low angular momentum wave packet motions along the electromagnetic field linear polarization phased withfield when the electrons periodically avoid each other approaching the nucleus and later tunnel through each other.
American Physical Society Apple Cray Research
Optical Society of America Silicon Graphics Wolfram Research
Quantum kaleidoscops with BEC in 1D optical lattice with oscillatory interaction
Ergodicity of quantum phase in three state model
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Arbitrary multipulse fieldcan generate arbitrary quantum state from other arbitrary state within the Hydrogen all bound states manifold but can one generate high angular momentum Trojan Wavepacket directly with one "pulse" from easy to prepare zero angular momentum static field Stark State ?:
Barry Dunning lab to generate Trojan Wavepackets directly by sudden normal intra-n Stark upside-down/strongly swung pendulum "gravity" turn-on and off to angle-accelerate and cause inter n-manifold mixing to space-focus 1)
Barry Dunning lab to generate Trojan Wavepackets directly by sudden normal intra-n manifold Stark upside-down/strongly swung pendulum "gravity" turn-on and off to angle-accelerate and cause inter n-manifold mixing to space focus 6)
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02f4a077744ce8ea | Friday, June 08, 2018
Myths of Copenhagen
Discussing the Copenhagen interpretation of quantum mechanics with Adam Becker and Jim Baggott makes me think it would be worthwhile setting down how I see it. I don’t claim that this is necessarily the “right” way to look at Copenhagen (there probably isn’t a right way), and I’m conscious that what Bohr wrote and said is often hard to fathom – not, I think, because his thinking was vague, but because he struggled to express it through the limited medium of language. Many people have pored over Bohr’s words more closely than I have, and they might find different interpretations. So if anyone takes issue with what I say here, please do tell me.
Part of the problem too, as Adam said (and reiterates in his excellent new book What is Real?, is that there isn’t really a “Copenhagen interpretation”. I think James Cushing makes a good case that it was largely a retrospective invention of Heisenberg’s, quite possibly as an attempt to rehabilitate himself into the physics community after the war. As I say in Beyond Weird, my feeling is that when we talk about “Copenhagen”, we ought really to stick as close as we can to Bohr – not just for consistency but also because he was the most careful of the Copenhagenist thinkers.
It’s perhaps for this reason too that I think there are misconceptions about the Copenhagen interpretation. The first is that it denies any reality beyond what we can measure: that it is anti-realist. I see no reason to think this. People might read that into Bohr’s famous words: “There is no quantum world. There is only an abstract quantum physical description.” But it seems to me that the meaning here is quite clear: quantum mechanics does not describe a physical reality. We cannot mine it to discover “bits of the world”, nor “histories of the world”. Quantum mechanics is the formal apparatus that allows us to make predictions about the world. There is nothing in that formulation, however, that denies the existence of some underlying stratum in which phenomena take place that produce the outcomes quantum mechanics enables us to predict.
Indeed, what Bohr goes on to say makes this perfectly clear: “It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature.” (Here you can see the influence of Kant on Bohr, who read him.) Here Bohr explicitly acknowledges the existence of “nature” – an underlying reality – but doesn’t think we can get at it, beyond what we can observe.
This is what I like about Copenhagen. I don’t think that Bohr is necessarily right to abandon a quest to probe beneath the theory’s capacity to predict, but I think he is right to caution that nothing in quantum mechanics obviously permits us to make assumptions about that. Once we accept the Born rule, which makes the wavefunction a probability density distribution, we are forced to recognize that.
Here’s the next fallacy about the Copenhagen interpretation: that it insists classical physics, such as governs measuring apparatus, works according to fundamentally different rules from quantum physics, and we just have to accept that sharp division.
Again, I understand why it looks as though Bohr might be saying that. But what he’s really saying is that measurements exist only in the classical realm. Only there can we claim definitive knowledge of some quantum state of affairs – what the position of an electron “is”, say. This split, then, is epistemic: knowledge is classical (because we are).
Bohr didn’t see any prospect of that ever being otherwise. What’s often forgotten is how absolute the distinction seemed in Bohr’s day between the atomic/microscopic and the macroscopic. Schrödinger, who was of course no Copenhagenist, made that clear in What Is Life?, which expresses not the slightest notion that we could ever see individual molecules and follow their behaviour. To him, as to Bohr, we must describe the microscopic world in necessarily statistical terms, and it would have seemed absurd to imagine we would ever point to this or that molecule.
Bohr’s comments about the quantum/classical divide reflect this mindset. It’s a great shame he hasn’t been around to see it dissolve – to see us probe the mesoscale and even manipulate single atoms and photons. It would have been great to know what he would have made of it.
But I don’t believe there is any reason to suppose that, as is sometimes said, he felt that quantum mechanics just had to “stop working” at some particular scale, and classical physics take over. And of course today we have absolutely no reason to suppose that happens. On the contrary, the theory of decoherence (pioneered by the late Dieter Zeh) can go an awfully long way to deconstructing and demystifying measurement. It’s enabled us to chip away at Bohr’s overly pessimistic epistemological quantum-classical divide, both theoretically and experimentally, and understand a great deal about how classical rules emerge from quantum. Some think it has in fact pretty much solved the “measurement problem”, but I think that’s too optimistic, for the reasons below.
But I don’t see anything in those developments that conflicts with Copenhagen. After all, one of the pioneers of such developments, Anton Zeilinger, would describe himself (I’m reliably told) as basically a Copenhagenist. Some will object to this that Bohr was so vague that his ideas can be made to fit anything. But I believe that, in this much at least, apparent conflicts with work on decoherence come from not attending carefully enough to what Bohr said. (I think Henrik Zinkernagel’s discussions of “what Bohr said” are useful here and here.)
I think that in fact these recent developments have helped to refine Bohr’s picture until we can see more clearly what it really boils down to. Bohr saw measurement as an irreversible process, in the sense that once you had classical knowledge about an outcome, that outcome could not be undone. From the perspective of decoherence, this is now viewed in terms that sound a little like the Second Law: measurement entails the entanglement of quantum object and environment, which, as it proceeds and spreads, becomes for all practical purposes irreversible because you can’t hope to untangle it again. (We know that in some special cases where you can keep track, recoherence is possible, much as it is possible in principle to “undo” the Second Law if you keep track of all the interactions and collisions.)
This decoherence remains a “fully quantum” process, even while we can see how it gives rise to classical-like behaviour (via Zurek’s quantum Darwinism, for example). But what the theory can’t then do, as Roland Omnès has pointed out, is explain uniqueness of outcomes: why only one particular outcome is (classically) observed. In my view, that is the right way to put into more specific and updated language what Bohr was driving at with his insistence on the classicality of measurement. Omnès is content to posit uniqueness of outcomes as an axiom: he thinks we have a complete theory of measurement that amounts to “decoherence + uniqueness”. The Everett interpretation, of course, ditches uniqueness, on the grounds of “why add an extra, arbitrary axiom?” To my mind, and for the reasons explained in my book, I think this leads to a “cognitive instability”, to purloin Sean Carroll’s useful phrase, in our ability to explain the world. So the incoherence that Adam sees in Copenhagen, I see in the Everett view (albeit for different reasons).
But this then is the value I see in Copenhagen: if we stick with it through the theory of decoherence, it takes us to the crux of the matter: the part it just can’t explain, which is uniqueness of outcomes. And by that I mean (irreversible) uniqueness of our knowledge – better known as facts. What the Copenhagenists called collapse or reduction of the wavefunction boils down to the emergence of facts about the world. And because I think they – at least, Bohr – always saw wavefunction collapse in epistemic terms, there is a consistency to this. So Copenhagen doesn’t solve the problem, but it leads us to the right question (indeed, the question that confronts the Everettian view too).
One might say that the Bohmian interpretation solves that issue, because it is a realist model: the facts are there all along, albeit hidden from us. I can see the attraction of that. My problem with it is that the solution comes by fiat – one puts in the hidden facts from the outset, and then explains all the potential problems with that by fiat too: by devising a form of nonlocality that does everything you need it to, without any real physical basis, and insisting that this type of nonlocality just – well, just is. It is ingenious, and sometimes useful, but it doesn’t seem to me that you satisfactorily solve a problem by building the solution into the axioms. I don’t understand the Bohmian model well enough to know how it deals with issues of contextuality and the apparent “non-universality of facts” (as this paper by Caslav Brukner points out), but on the face of it those seem to pose problems for a realist viewpoint too.
It seems to me that a currently very fruitful way to approach quantum mechanics is to think about the issue of why the answers the world gives us seem to depend on the questions we ask (à la John Wheeler’s “20 Questions” analogy). And I feel that Bohr helps point us in that direction, and without any need to suppose some mystical “effect of consciousness on physical reality”. He didn’t have all the answers – but we do him no favours by misrepresenting his questions. A tyrannical imposition of the Copenhagen position is bad for quantum mechanics, but Copenhagen itself is not the problem.
Adam Becker said...
Hi Philip,
I like this, but I don't agree with everything you've said here.
On the issue of contextuality, which you're right to emphasize, I think that Bohr himself gives one possible good answer: he talked about "the impossibility of any sharp distinction between the behaviour of atomic objects and the interaction with the measuring instruments which serve to define the conditions under which the phenomena appear." When dealing with very small objects, our large measurement devices are necessarily clumsy, by virtue of their largeness. So contextuality can be seen as a purely mechanical effect. This is, for example, how it works in the Bohmian interpretation: there, contextuality is guaranteed by the interaction between a measurement device and the thing it's measuring.
But more generally, I am loath to ascribe positions to Bohr. He really was unclear. His students said he spoke of a complementarity between clarity and truth, and thus Bohr's seeming incomprehensibility was merely the result of his concern for the truth. I think that you're giving one possible reading of Bohr, but it's certainly not clear that this is the single best way to read Bohr. Another possible reading is that he really did see a divide between the world of the classical and the world of the quantum, and was simply unclear about where that divide might lie. And another possibility is that he changed his mind a lot, or was simply (and understandably) confused. As Jim said on Twitter, Mara Beller's Quantum Dialogue is particularly good on this subject.
I also don't think it's right to say that knowledge is classical. (I see the connection to Kant, but I don't think that helps much.) It's simply not true that human experience of the everyday world is necessarily classical, any more than it's necessarily Aristotelian or necessarily astrological. Classical physics has plenty of profoundly counterintuitive consequences. Think of the first time you held a spinning bicycle wheel and tried to move its axis, the way it kicked back at you in an unexpected way. Or, even more fundamental, the idea that an object in motion tends to stay in motion -- certainly not an idea that lines up with everyday experience on Earth! If there's a way that all human minds universally organize perceptions (a thesis I'm somewhat skeptical of to begin with), it sure ain't classical. This is a great deal of what was at stake in the debates between Einstein and Bohr: Bohr (the conservative) insisted that classical concepts like energy and momentum were required for thinking about the outcomes of experiments, whereas Einstein (the radical, as always) insisted that we could develop new concepts that would give a greater understanding of what was actually happening in the quantum realm, just as spacetime replaced the concepts of individual space and time.
(to be continued, I hit the character limit for comments...)
Adam Becker said...
(Continuing where I left off in my previous comment.)
I'll end with a question: it sounds to me like what you're really defending here, aside from a particular reading of Bohr, is the idea that a psi-epistemic viewpoint (the wave function is knowledge about something, rather than a real thing in the world) is not incompatible with a broadly realist stance about the world, including the world of very small things. Is that your position? If so, I agree with you! But I am somewhat more sympathetic to psi-ontic views (the wave function is something real, be it physical or lawlike). This is, in part, because the PBR theorem is a problem for the most straightforward kinds of psi-epistemic positions. (Matt Leifer, a psi-epistemicist and realist, has a good post on this here.) Furthermore, being psi-epistemic doesn't automatically give you a way out of the kind of nonlocality that Bell's theorem demands, especially if you still want to be a realist of some stripe. So given the choice between "the wave function is my information about something, I don't know what that something is, but that something is nonlocal" and "the wave function is a thing, I know what it is, and it's nonlocal," I'll probably choose the latter. (That's not the choice, and there are both psi-epistemic and psi-ontic ways to avoid nonlocality. But those ways out have other unpleasant consequences of their own.)
Philip Ball said...
Thanks so much for these comments Adam.
Clearly it's not possible to say for sure which of us is right about Bohr; mine is just the generous interpretation. I do agree that it would seem unwise to regard his writings as monolithic and consistent.
I'm not sure what you mean by saying that human experience is not necessarily classical. By that I certainly didn't mean that human intuitions must fit with what classical physics tells us, because as you say there can be plenty that is counter-intuitive about classical physics too. I mean that all perception, and thus measurement in Bohr's sense, ultimately takes place at the classical, macroscopic limit, where decoherence has kicked in. This is, I think, what Bohr was saying, though now our understanding of decoherence allows us to express it in clearer terms.
I'm surprised to find that there are notions that quantum contextuality can be explained as a purely mechanical effect. To me that smacks of Heisenberg's (misconceived) gamma-ray microscope. My understanding is that Kochen and Specker (and indeed Bell, though he published it later than them) established that contextuality is as fundamental as nonlocality: just as we can be confident that any "deeper" theory below QM will have to be nonlocal, it will have to be contextual.
Your phrasing "the wave function is my information about something, I don't know what that something is, but that something is nonlocal" actually sums up very nicely the way I tend to lean, though I wouldn't be dogmatic about it. To my mind, that encapsulates what we can currently say with confidence about QM (and you can probably see why I think Copenhagen at least starts us, imperfectly, down that road). In contrast, to say "The wave function is a thing, I know what it is" strikes me as an article of faith right now - it may turn out to be true, but we can't be sure about that right now. Perhaps I'm just more conservative!
I do like it that you make Einstein the radical! It's so tiresome how he is so often portrayed as the stick-in-the-mud about QM.
C Adams said...
Thanks for your interesting piece and opening a discussion. A couple of comments.
“to “undo” the Second Law if you keep track of all the interactions and collisions”
You do not undo the 2nd law. The second law (increase in entropy) only kicks in if you throw the information away, but if you can recover it, then you have not really thrown it away.
“But what the theory can’t then do, as Roland Omnès has pointed out, is explain uniqueness of outcomes”
One way to view this is that there was only ever going to be one outcome, it is just that we did not know which. Classical physics says we do not know if because we do not have sufficient information. Quantum mechanics says that we still do not know, even if we have all the information that it is possible to have.
In summary, quantum mechanics is simply a formulation of what we can say about the World, it does make any claims on whether the World exists.
Clara, once known as Nemo said...
Philip, Adam,
you speculate about a possible deeper theory for the wave function, one that is nonlocal and contextual. For a number of years, I have followed what Schiller proposes on this topic at . In his talk slides, he indeed presents a deeper theory, where psi is an average of a crossing density due to fluctuating strands. That model for psi is nonlocal and it is contextual, following Bohr, Copenhagen and decoherence rather closely. My own view is somewhat different - I do not think that psi is a "thing" - but the proposal keeps me wondering whether Bohr might have been right after all.
Jim said...
If it's not too late... I think it's important to be clear on what the PBR theorem is saying (and, indeed, Matt Leifer's blog post and subsequent paper on this are models of clarity). PBR's no-go theorem does not (it cannot) rule out 'pure' psi-epistemic interpretations of the wavefunction. It does rule out epistemic interpretations in which what we see is presumed to result from the statistical behaviour of underlying real (ontic) physical states. And, whilst a pure psi-epistemic interpretation is anti-realist, this shouldn't be taken to imply that advocates of such an interpretation are out-and-out empiricists or deny at least some aspects of scientific realism. I believe it is possible to hold to a position which accepts objective reality (the Moon is still there when nobody looks); entity realism ("if you can spray them then they are real"); and yet still question whether the QM *representation* and particularly the wavefunction corresponds (directly or statistically) with the real physical states of such entities. This, I believe, is essentially Carlo Rovelli's position. Then, as I said in a tweet, once you start doubting the QM representation, you can't help but wonder if we've been kidding ourselves all these years with classical mechanics...
Chris Dewdney said...
I don’t agree that the “solution comes by fiat”. In deBB theory, as Bohm presented it in 1952, one reformulates the quantum theory, in a form as close as possible to classical theory, by simply rewriting the complex Schrödinger equation as two, real equations. One of these equations is similar to a classical Hamilton-Jacobi equation (with an extra quantum potential) and the other to a continuity equation for the probability density. The pair of equations is just quantum theory rewritten in a pseudo-classical form, a form that allows one to maintain a very natural definition of particle trajectories. Every particle has a definite (but unknown and uncontrollable) trajectory and the trajectories accounts for the definite results of measurement.
Regarding nonlocality, the essential point is that it concerns many particles (there is no non locality for a single particle) - for which the Schrödinger equation determines the evolution of a configuration space wave function. This is often overlooked when the emphasis of discussion is on single particle quantum mechanics (it might be said to be deceptively spacious). In deBB theory the velocity of the individual particles depends on the multi-particle configuration space wave function and hence on all of the particle coordinates at once. In the Hamilton-Jacobi form of this configuration space description one naturally finds nonlocal quantum forces - without adding them artificially.
So, in deBB theory one does not proceed by “devising a form of nonlocality ….without any real physical basis…” instead, nonlocality arises naturally within the simple mathematical reformulation of the theory. I saw this amazingly clearly when, working in JP Vigier’s lab in the Institut Henri Poincaré in Paris and sitting at what had been de Broglie’s desk, I first calculated the trajectories of a pair of spin one-half particles, in a spin zero singlet state, undergoing spin measurements in Bohm’s version of the EPR experiment (see the 1986 paper - Spin and non-locality in quantum mechanics C Dewdney, PR Holland, A Kyprianidis, JP Vigier Nature 336 (6199), 536). It became very clear from these calculations that what happened to one of the particles depended not only on where the other particle was, but also on which measurements were carried out at the location of the distant particle. Contextuality is also a natural part of deBB theory. In deBB theory the value assigned to an individual physical observable depends not only on the set of hidden variables but also on the wave function. Consequently, the value revealed by a measurement depends on the hidden variables, the initial wave function and the measurement Hamiltonian (describing all of the measurements taking place). (see Constraints on quantum hidden-variables and the Bohm theory. C Dewdney 1992 J. Phys. A: Math. Gen. 25 3615).
When I was a PhD student at Birkbeck in the late 70’s, I remember Bohm saying to me in discussion, that one could imagine the counterfactual historical scenario in which de Broglie’s theory had been accepted at the outset (there were no conclusive arguments against it). Nonlocally correlated particle trajectories would then have been recognised as a natural and irreducible aspect of quantum theory from the beginning and there would have been no measurement problem relying on observers for its “resolution”. Further imagine, he said, that after some 25 years it was suggested that one should remove “by fiat” the idea of particle trajectories from quantum theory, this would have looked very strange indeed, and would have been rejected by the physics community, as it would immediately have given rise to the host of interpretational difficulties with which we all too familiar today.
From this point of view, one could argue that all of the interpretational difficulties within quantum theory arise as a result of the removal from physics of the idea of particle trajectories – “by fiat”.
Jayarava Attwood said...
Understanding quantum is a two-sided problem. Firstly there is the weirdness of quantum mechanics and secondly the weirdness of "understanding". Sometimes we focus on the quantum side of things without considering what knowledge even is. As you say Kant was entirely pessimistic about knowledge of reality. But Kant was writing before the development of modern science and I think we can safely say that he was not entirely right. We can infer a great deal about reality from comparing notes about how we experience it. On the scales of mass, length, and energy where classical physics is a good description, we understand reality quite well.
It so happens that we are somewhere in the middle of the scales of mass, length, and energy spanning 60-100 orders of magnitude. We experience about as much of reality as we see of the EM spectrum.
The quantum problem is that we cannot *experience* quantum phenomena. Thus knowledge about reality at the quantum level is always going to be abstract. The same is, less obviously, true on the largest scales.
I may grasp that there is EM radiation I cannot see or feel, but do I really understand the *reality* of radio waves or X-rays? I can probably with some revision cope with Maxwell's equations. But so what? I still have no experience of radio waves because none of my senses can detect them. When they X-rayed my broken wrist last year, it gave me no sense impressions that I might develop into knowledge.
Along with the breakdown of classical physics, there is a breakdown of the classical concept of knowledge at the scales where quantum descriptions rule. We talk about having images of atoms, for example, but there are many layers of technology between us and the object. If I see a static image of an atom, for example, I tacitly "translate" that into a classical object and believe I understand what I am seeing. But in many ways this picture is false. It tells me nothing about atoms generally if I measure the intensity of electric fields around an atom frozen close to absolute zero and plot them on a graph. The map is not the territory, let alone the pixelated image of the map.
Philip Ball said...
Thanks very much Chris for those very helpful comments. I don't by any means reject the deBB formalism, any more than I reject most of the other interpretations. Indeed, I can see that it has virtues. My impression is that many quantum physicists don't engage with it simply because it seems like a lot of effort for no real gain - they end up with exactly the same predictions as the standard quantum formalism (by design, of course!). I think that the two state vector formalism of Aharonov suffers from neglect for the same reasons, though I appreciate that both can, in certain circumstances, offer a useful way of looking at quantum problems that is not easily evident in other viewpoints. The question, of course, is whether one should deduce any actual ontology from these reformulations of quantum theory.
I don't understand the formalism well enough to be sure, but my understanding is that to connect with the standard quantum formalism you do need at least one extra assumption in the deBB approach (aside from those hidden variables) - the quantum equilibrium hypothesis. And the fact that nonlocality comes out quite naturally doesn't in itself seem an obvious gain over standard QM. I shouldn't say that this nonlocality is "put in by hand", but rather, that it seems to me the deBB formalism just ushers the nonlocality of standard QM into a particular place that then allows one to create a realist, deterministic description of the rest. That's an interesting way to do things, but I'm not convinced it is obviously an advance.
All the same, the counterfactual history you suggest is indeed an interesting one, and I fully buy James Cushing's argument that things could look very different now if the Copenhagen interpretation had not, for what ever reason, got in first. I suspect people would then just be railing against the "absurd Bohmian tyranny" and demanding that a Copenhagenist view be admitted to the textbooks too...
Nicophil said...
E.T. Jaynes wrote :
"" Although Bohr's whole way of thinking was very different from Einstein's, it does not follow that either was wrong.
Einstein's thinking is always on the ontological level traditional in physics; trying to describe the realities of Nature. Bohr's thinking is always on the epistemological level, describing not reality but only our information about reality.
The peculiar flavor of his language arises from the absence of all words with any ontological import. Those who, like Einstein, tried to read ontological meaning into Bohr's statements, were quite unable to comprehend his message. This applies not only to his critics but equally to his disciples, who undoubtedly embarrassed Bohr considerably by offering such ontological explanations as [...] the remark of Pauli quoted above, which might be rendered loosely as "Not only are you and I ignorant of x and p ; Nature herself does not know what they are" [or "Eine prinzipielle Unbestimmtheit, nicht nur Unbekanntheit"].
We routinely commit the Mind Projection Fallacy: supposing that creations of our own imagination are real properties of Nature, or that our own ignorance signifies some indecision on the part of Nature. It is then impossible to agree on the proper place of information in physics. This muddying up of the distinction between reality and our knowledge of reality is carried to the point where we find some otherwise rational physicists, on the basis of the Bell inequality experiments, asserting the objective reality of probabilities, while denying the objective reality of atoms !""
Adam Becker said...
Belatedly throwing in a few final comments:
Philip, I believe you're correct about dBB needing a quantum equilibrium hypothesis regarding the initial conditions. But basically every cosmological theory requires an initial condition that's somehow special, so that's not a problem that's unique to dBB (though of course it doesn't mean it's not a problem at all).
To clarify what I was saying about contextuality in dBB: in that theory, position is a privileged observable. Measurements of all other observables boil down to measurements of position in dBB, and the outcome of position measurements depend not only on the hidden variables, but on the wave function and the interaction Hamiltonian between the measurement apparatus and the thing being measured, just as Chris said. That's how contextuality works in dBB, or at least that's my understanding of it. So in dBB, contextuality really does come down to a mechanical disturbance of a particle's position by the measurement device — even when position isn't one of the observables being measured.
Also, a historical note: Bell published his proof of contextuality before Kochen and Specker. His paper was written in 1964 and published in 1966 (the two year delay was due to an editorial snafu). Kochen and Specker's result was published in 1967. So it should really be called the Bell-Kochen-Specker theorem.
Finally, regarding PBR: you can definitely hold the view that the wave function is our information about some underlying reality, you just need to give up on one of the assumptions of the PBR theorem to do that. Leifer, for example, lifts the ban on retrocausality, which seems like a reasonable move to me in light of the difficulties here. But as usual, just because I'm saying Leifer's view is reasonable doesn't mean I subscribe to it (or to dBB, or MWI).
PS. Jim, I don't understand how Rovelli's interpretation is realist. But that's probably my fault for not reading enough of his work.
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Steve said...
There is an interesting new take on this. Maybe just maybe there really is a quantum/classical divide. I know this is a comment on an article from a while ago but I would love to hear some feedback on this.
Best of all it would end this MWI nonsense for good as would GOC
Dr. Ball?
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cec7fe49de290902 | Neutron Optics
Neutron optics
The general class of experiments designed to emphasize the wavelike character of neutrons. Like all elementary particles, neutrons can be made to display wavelike, as well as particlelike, behavior. They can be reflected and refracted, and they can scatter, diffract, and interfere, like light or any other type of wave. Many classical optical effects, such as Fresnel diffraction, have been performed with neutrons, including even those involving the construction of Fresnel zone plates. See Diffraction, Interference of waves, Reflection of electromagnetic radiation, Refraction of waves, Scattering of electromagnetic radiation, Wave (physics)
The typical energy of a neutron produced by a moderated nuclear reactor is about 0.02 eV, which is approximately equal to the kinetic energy of a particle at about room temperature (80°F or 300 K), and which corresponds to a wavelength of about 10-10 m. This is also the typical spacing of atoms in a crystal, so that solids form natural diffraction gratings for the scattering of neutrons, and much information about crystal structure can be obtained in this way. However, the wavelike properties of neutrons have been confirmed over a vast energy range from 10-7 eV to over 100 MeV. See Neutron diffraction
Neutrons, being uncharged, can be made to interfere over large spatial distances, since they are relatively unaffected by the stray fields in the laboratory that deflect charged particles. This property has been exploited by using the neutron interferometer. This device is made possible by the ability to grow essentially perfect crystals of up to 4 in. (10 cm). The typical interferometer is made from a single perfect crystal cut so that three parallel “ears” are presented to the neutron beam. This allows the incident beam to be split and subsequently recombined coherently. See Coherence, Interferometry, Single crystal
One of the most significant experiments performed with the interferometer involved rotating the interferometer about the incident beam so that one neutron path was higher than the other, creating a minute gravitational potential difference (of 10-9 eV) between the paths. This was sufficient to cause a path difference of 20 or so wavelengths between the beams. This remains the only type of experiment that has ever seen a quantum-mechanical interference effect due to gravity. It also verifies the extension of the equivalence principle to quantum theory (although in a form more subtle than its classical counterpart). See Gravitation, Relativity
Many noninterferometer experiments have also been done with neutrons. In one experiment, resonances were produced in transmitting ultracold neutrons (energy about 10-7 eV) through several sheets of material. This is theoretically similar to seeing the few lowest states in a square-well potential in the Schrödinger equation. See Neutron, Quantum mechanics
Neutron Optics
a branch of neutron physics that deals with a number of phenomena that have optical analogues and that arise during the interaction of neutron beams with matter or with fields (magnetic, gravitational). These phenomena are characteristic of slow neutrons and include the refraction and reflection of neutron beams at the boundary between two media, the total reflection of a neutron beam from an interface (which is observed under certain conditions), the diffraction of neutrons by individual heterogeneities in a medium (small-angle neutron scattering), and the diffraction of neutrons by periodic structures. The polarization of neutrons, which, in the first approximation, may be compared with the circular polarization of light, arises for certain substances upon reflection and refraction. The inelastic scattering of neutrons in gases, liquids, and solids is analogous to the Raman effect.
In a number of phenomena in neutron optics, the wave properties of neutrons predominate. The neutron wavelength λ is defined by the neutron mass m = 1.67 × 10-24 g and velocity v:
(1) λ = h/mv
where h is Planck’s constant. The mean velocity of thermal neutrons is ν = 2.2 × 105 cm/sec. For such neutrons, the wavelength is λ = 1.8 × 10-8 cm; that is, the wavelength is of the same order as the X-ray wavelength. The wavelengths of the slowest, or ultracold, neutrons (see below) are the same as the wavelengths of ultraviolet and visible light. The analogy between neutron beams and electromagnetic waves is also underscored by the fact that neutrons, like photons, do not have an electric charge. At the same time, neutron waves and electromagnetic waves are different in nature. Photons interact with the electron shell of the atom, whereas neutrons interact chiefly with atomic nuclei. The neutron has a rest mass, which makes it possible to use nonoptical methods for neutron studies. The presence of a magnetic moment for the neutron accounts for the magnetic interaction of neutrons with magnetic materials and magnetic fields. Photons do not interact in this manner.
The development of neutron optics began in the 1940’s, after the appearance of nuclear reactors. E. Fermi introduced the concept of the refractive index n to describe the interaction of neutrons with condensed media. When neutrons pass through a medium, they are scattered by atomic nuclei. In wave terminology, this means that an incident neutron wave produces secondary waves, and the coherent combination of these secondary waves determines the refracted and reflected waves. As a result of the interaction of neutrons with nuclei, there is a change in the velocity and, consequently, in the wavelength λ1 of neutrons in a medium, as compared with the wavelength λ in a vacuum. Under ordinary conditions, when the absorption of neutrons over a path of the order of λ1 can be neglected (just as in optics), we have n = λ/λ1. It follows from the de Broglie relation that n = λ/λ1=v1/v.
If U is the interaction potential between neutrons and nuclei, averaged over the volume of the medium, then a neutron should perform work when it strikes the medium. The kinetic energy of the neutron in the medium decreases from its initial value ε0 = mv2/2 to the value ℰ1 = ℰ – U. When U > 0, the neutron velocityin the medium decreases (ν1 < ν); in this case, we also have λ1 > λ and n < 1. When U < 0, the velocity increases and n > 1. If a quantity analogous to the dielectric constant (∊ = n2) is introduced for neutrons, then ∊ = λ212 = v12/v2 = ℰ1/ℰ.. The potential is U = h2Nb/2πm, from which
(2) ∊ = n2 = 1 - h2Nb/πm2v2
Here, b is the coherent scattering length for the scattering of neutrons by nuclei and N is the number of nuclei per unit volume of the medium. For most substances, b > 0, and equation (2) may be put in the form
(3) ∊ = n2 = 1 - v02/v2
Neutrons with velocity ν < vo have energy ε0 > < U; for such neutrons, n2 < 0, that is, the refractive index is imaginary. Such neutrons cannot overcome the repulsive forces of the medium and are completely reflected from the surface of the medium. They are called ultracold neutrons. For metals, vo is about several m/sec (for example, for copper, vo = 5.7 m/sec).
The velocity of thermal neutrons is several hundred times greater than the velocity of ultracold neutrons, and n is close to unity (1 —n ≈ 10-5). When a beam of thermal neutrons is incident upon the surface of a dense substance at a glancing angle, the beam undergoes total reflection analogous to the total internal reflection of light. This happens at glancing angles ɸ < Φcr, that is, at angles of incidence θ ≥ θcr = (π/2) —ɸcr. The critical angle is determined from the condition
For example, for copper, ɸcr = 9.5’. It can be shown that the condition of total reflection (4) is equivalent to the requirement ν = ≤ νo, where vz. is the component of the neutron’s velocity that is normal to the reflecting surface. The velocity of cold neutrons is several times less than the velocity of thermal neutrons, and the angle Φcr, is correspondingly greater.
Total reflection is used to transport, with minimal losses, thermal and cold neutrons from a nuclear reactor to the experimental facilities (distances of about 100 m). This is accomplished by means of mirror neutron guides—evacuated tubes whose inner surface reflects neutrons. Mirror neutron guides are made of copper or glass and may or may not have a metal coating.
The reflection coefficient of neutrons is actually always somewhat less than unity because nuclei not only scatter neutrons but also absorb them. Taking absorption into account leads to a refinement of equation (3):
Here, σ is the effective cross section of all processes that lead to attenuation of the neutron beam. The sum of the capture cross section and inelastic-scattering cross section is significant for cold and ultracold neutrons and is inversely proportional to the velocity ν. The product σv is therefore independent of ν. This means that for neutrons, just as in optics, ∊ and n are complex quantities: ∊ = ∊’ + i∊” and n = η” + iη’. For ultracold neutrons, the real part is ∊, that is, ∊’ < 0 and n” > n’. This is characteristic of metals in the case of light, and the reflection of ultracold neutrons from many substances is analogous to the reflection of light from metals with an extremely high reflectance (seeMETAL OPTICS). If b < 0, then a plus sign precedes the term νo22 in equation (5) and ∊ > 1 (the dielectric constant increases with decreasing ν). Such substances reflect and refract very slow neutrons, just as dielectrics reflect and refract light.
Equation (2) can be easily generalized to the case in which a magnetic field is present in the medium if we add, to the energy U of the interaction of neutrons with the medium, the energy of magnetic interaction ± μB, where μ is the magnetic moment of the neutron and B is the magnetic induction; here, the ± sign refers to the two possible orientations of the neutron’s magnetic moment with respect to the vector B, that is, the two polarizations of the neutron beam. The generalized equation is
(6) n2 = 1 - h2Nb/πm2v2 ± 2μB/mν2
By selecting the material for the reflecting mirror, the magnetic field, and the glancing angle, it is possible to ensure that the neutrons of one of the two polarizations undergo total reflection, whereas neutrons of the other polarization do not. A device making use of this principle is used to produce beams of polarized neutrons and to determine the degree of polarization of the beam.
A number of devices that are used both in experiments and in the solution of practical problems are based on the principles of neutron optics. These include neutron mirrors, straight and curved neutron guides with total internal reflection, neutron crystal monochromators, mirror and crystal neutron polarizers and analyzers, devices for focusing neutron beams, refracting prisms, and neutron interferometers. Neutron diffraction is extensively used to investigate submicroscopic properties of matter: atomic-crystal structure, vibrations of the crystal lattice, magnetic structure, and the dynamics of the crystal lattice.
Fermi, E. Lektsii po atomnoi fizike. Moscow, 1952. (Translated from English.)
Hughes, D. J. Neitronnaia optika. Moscow, 1955. (Translated from English.)
Frank, I. M. “Nekotorye novye aspekty neitronnoi optiki.” Priroda, 1972, no. 9.
neutron optics
[′nü‚trän ′äp·tiks]
The study of certain phenomena, for example, crystal diffraction, in which the wave character of neutrons dominates and leads to behavior similar to that of light.
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References in periodicals archive ?
They highlight new results and developments in such topics as neutron electric dipole moment searches, neutron optics and interferometry, Standard Model tests using neutron beta decay, neutron facilities, neutron polarimetry, and nucleon-nucleon interactions.
With its treatment of the theories, experiments and applications involved in neutron optics, this relevant reading for nuclear physicists and materials scientists alike.
He devoted most of his professional life to neutron optics research as the head professor for neutron science at the Research Reactor Institute.
Bowman, Los Alamos National Laboratory Neutron Optics II -- Chair: David Jacobson, National Institute of Standards and Technology 2:00-2:15 Experimental test of Laue diffraction method of a search for neutron EDM V.
In my view, the PNC neutron optics experiments are most beautiful manifestations of parity-violation properties of ordinary matter.
Monte Carlo simulations are being performed to design and characterize the neutron optics components for the two fundamental neutron physics beamlines at the Spallation Neutron Source.
Hughes, Neutron Optics, Interscience Publishers, Inc.
Sears, Fundamental Aspects of Neutron Optics, Phys.
The basis of neutron optics is the Schrodinger equation for the coherent interaction of the neutron with matter using an optical potential.
Valsky, Neutron optics P-violation effects near p-wave resonance, Physica B 267-268, 289-293 (1999).
Glinka (1975-present) Small-angle neutron scattering, mesoporous materials, neutron optics and scattering instrumentation. |
a94a482acb847e0e | Multiverse January 2017
The Many Mice Theory of Quantum Mechanics
The father of the quantum multiverse didn’t actually think of it as a multiverse.
By Peter Byrne
On April 14, 1954, Albert Einstein gave the last lecture of his life. Speaking to physics students at Princeton University, he remarked that although quantum mechanics works, “it is difficult to believe that this description is complete. It seems to make the world quite nebulous unless somebody, like a mouse, is looking at it.”
Einstein was referring to the central paradox that bedevils quantum physics, then as now: the measurement problem. Before an atom is detected by a scientific instrument—a Geiger counter, the measuring devices of the Large Hadron Collider, arrays of half-silvered mirrors, or what have you—it can inhabit multiple locations simultaneously. This phenomenon, called “superposition,” seems absurd because, upon detection, the atom is found at a single position.
Each mouse state believes there is but one mouse-self making merry in one world.
Einstein’s mouse simile was a jab at the widely accepted “collapse” postulate, which declares that the act of measuring or observing an atomic particle forces it into one location, somehow. The postulate arbitrarily discards all of the atom’s positions but one, contradicting the fundamental rule of quantum mechanics, the Schrödinger equation, which tracks the atom’s evolution through time without losing any information about its superposed properties.
Taking notice of Einstein’s rodential remark was a second-year grad student, Hugh Everett III. In his soon-to-be-written doctoral thesis, Everett would explain away the measurement paradox. His solution treated the universe as completely quantum mechanical, obeying the Schrödinger equation without recourse to the illogical (and never observed) collapse mechanism. Everything that ever was or ever will be is a giant quantum superposition, a completely determinate universe of universes.
The penalty for theoretical simplicity? Mouse becomes mice.
In September 1970, cosmologist Bryce Dewitt popularized Everett’s model as the many-worlds interpretation of quantum mechanics. Writing in Physics Today magazine, DeWitt was deliberately sensational, insisting on metaphysical realism for the possibilities inherent in the superposition: “[Everett’s] universe is constantly splitting into a stupendous number of branches, all resulting from the measurement-like interactions between its myriad of components. Moreover, every quantum transition taking place on every star, in every galaxy, in every remote corner of the universe is splitting our local world on earth into myriads of copies of itself.”
An evocative image, indeed, but not Everett’s. Although he is widely credited with inventing the quantum multiverse model, his language was less dramatic.
As philosopher Jeffrey A. Barrett at U.C. Irvine has noted, it is important to realize that Everett was not a metaphysician, but an empiricist. His argument was simply that vanilla quantum theory can already explain our observations without any need for collapse. The reason is that observers are every bit as quantum as atoms. They, too, can exist in superpositions, the elements of which are encoded in a mathematical device called a “wavefunction.” Each time a mouse-observer measures a superposed atom—a process governed by quantum mechanics—the observer’s wavefunction splits repeatedly until it connects to each of the many locations of the atom. If the atom is superposed at 100 positions, or states, the mouse who interacts with it is superposed in 100 states, each with a memory of finding the atom at one location. Each of those mouse states proceeds to evolve independently, never feeling the splits, believing there is but one mouse-self making merry in one world, blind to her copies.
Everett explained: “[I]t is not so much the system which is affected by the observation as the observer, who becomes correlated to the system.… From the present viewpoint all elements of the superposition are equally ‘real.’”
"At this point we encounter a language difficulty.”
Without a collapse to select one possibility out of the full range, the theory implies that all physically possible events occur, somewhere. But “somewhere,” to Everett, merely meant a term in a mathematical superposition. He did not associate it with a literal place, as DeWitt and later interpreters did. He believed that science should not make ontological claims concerning the unobservable. We can infer the full superposition by performing repeated measurements on identically prepared atoms and observing the variety of outcomes, but we do not ever observe it directly.
Everett wrote in his dissertation: “Once we have granted that any physical theory is essentially only a model for the world of experience, we must renounce all hope of finding anything like ‘the correct theory.’ There is nothing which prevents any number of quite distinct models from being in correspondence with experience (i.e., all ‘correct’). And furthermore no way of verifying that any model is completely correct, simply because the totality of all experience is never accessible to us.” This inability to see the full range of possible experimental outcomes is not a limitation of the theory, but an essential feature, because it explains the appearance of indeterminism in a deterministic theory. The indeterminism reflects our imperfect knowledge.
That is not to say that Everett did not engage in metaphysical speculation, as long as it was kept separated from science. He knew that his interpretation could lead you down a mouse hole. “The price,” he wrote, “is the abandonment of the concept of the uniqueness of the observer, with its somewhat disconcerting philosophical implications.” He added: "At this point we encounter a language difficulty.”
Everett did not dismiss alternative interpretations that supplemented quantum theory with new elements, going by the name of hidden variables and ensembles. As if to confirm his ontological agnosticism, different physicists have interpreted his formalism in vastly different ways: many worlds, many minds, decoherent histories, relational quantum mechanics. But the core of Everett’s insight—accepting the ruthlessly linear logic of quantum mechanics—has lost none of its power in more than half a century.
After his dissertation was published in 1957, Everett abandoned quantum mechanics. He made a living designing top-secret weapons systems for the Pentagon and died of a heart attack in 1982, even as ever-widening acceptance of the many worlds was making him famous.
The fate of the mouse is unknown. |
8f46c0843e4e7ec0 | Learning About Atoms
The Science PLC at our school is considering what students should know about atoms in 8th and 9th grade science classes (including Physics First). Just recently, Amber (Strunk) Henry posted on Twitter:
This is my attempt to arrange the ideas.
Map of the Territory of Things to Know
Next Generation Science Standards (NGSS)
Here are the progressions found in Appendix E of the standards. I do digress into talk of matter and substance when it supports later understanding of atoms. I’ve expanded these to list ideas explicitly and separately.
ESS1.A The universe and its stars
• (Grades 9-12) Light spectra from stars are used to determine their characteristics, processes, and lifecycles. Solar activity creates the elements through nuclear fusion. The development of technologies has provided the astronomical data that provide the empirical evidence for the Big Bang theory.
• Excited atoms/molecules emit light of particular frequencies and wavelengths (collectively called the emission spectrum of an atom/molecule).
• The frequencies and wavelengths of light emitted by atoms/molecules depend on the structure of the atom/molecules.
• Atoms/molecules absorb light at particular frequencies and wavelengths (collectively called the absorption spectrum of an atom/molecule).
PS1.A Structure of matter (includes PS1.C Nuclear processes)
• (Grades K-2) Matter exists as different substances that have observable different properties. Different properties are suited to different purposes. Objects can be built up from smaller parts.
• Matter can be made of different substances.
• Substances have many properties, each with their own uses.
• Objects are made of smaller parts.
• Indivisible particles of matter are too small to see.
• Measurements of properties characterize substances.
• Molecules are made of atoms.
• Matter is made of atoms and molecules.
• Different atoms and molecules explain different substances.
• Atoms and molecules behave differently in different states of matter.
• Atoms and molecules change their qualitative behavior at phase transitions.
• Matter is conserved because atoms are not destroyed in physical and chemical processes.
• (Grades 9-12) The sub-atomic structural model and interactions between electric charges at the atomic scale can be used to explain the structure and interactions of matter, including chemical reactions and nuclear processes. Repeating patterns of the periodic table reflect patterns of outer electrons. A stable molecule has less energy than the same set of atoms separated; one must provide at least this energy to take the molecule apart.
• An individual atom has structure explained by electromagnetic and nuclear interactions.
• The structure of the atom explains:
• arrangement of atoms into molecules
• chemical reactions
• nuclear processes
• trends in periodic table
• Energy is required to remove electrons from an atom.
• Energy is required to break molecular bonds.
PS1.B Chemical reactions
• (Grades K-2) Heating and cooling substances cause changes that are sometimes reversible and sometimes not.
• (Grades 3-5) Chemical reactions that occur when substances are mixed can be identified by the emergence of substances with different properties; the total mass remains the same.
• Mass is conserved in chemical reactions.
• Measurement of properties of substances identifies when chemical reactions have taken place.
• (Grades 6-8) Reacting substances rearrange to form different molecules, but the number of atoms is conserved. Some reactions release energy and others absorb energy.
• Chemical reactions result in different molecular arrangements of atoms.
• (Grades 9-12) Chemical processes are understood in terms of collisions of molecules, rearrangement of atoms, and changes in energy as determined by properties of elements involved.
• Chemical reactions occur when molecules collide and atoms rearrange.
• Changes in energy during a chemical reaction depend on properties of the atoms involved.
Let me know if you think I’ve forgotten anything here!
AAAS Science Assessment
The AAAS has a great website under the auspices of Project 2061 that lists ideas and misconceptions related to Atoms, Molecules, and States of Matter.
Arnold B. Arons. Teaching Introductory Physics
Arons identifies four lines of evidence necessary to build an early quantum model of the atom:
1. Bright line spectra of gases. This requires understanding of how accelerated charged particles can emit light, how charged particles can absorb light. It should include the Balmer-Rydberg formulae for hydrogen.
2. Radioactivity
3. Size of atoms (electron cloud and nuclear). Evidence from multiple sources.
4. Photoelectric effect and photon concept
How should this knowledge be arranged?
TODO: I’d like to work on a Learning Landscape, Knowledge Packet, or Learning Progression synthesizing these sources, but that will have to be added later.
Models of Atoms
1. BB Model of Atoms and Molecules (hard, indivisible balls)
• Needed to explain phases of matter.
2. Dalton Model of Atoms (hard, indivisible balls that can combine)
• Needed to explain chemical reactions in integer ratios.
3. Plum Pudding Model / Thompson Model (negatively charged electrons embedded in a positively charged medium)
• Needed to explain static electricity.
4. Planetary Model / Rutherford Model
• Needed to explain the Geiger-Marsden gold foil experiments.
5. Bohr Model / Rutherford-Bohr Model
• Needed to explain why electrons don’t fall into the nucleus after radiating EM waves.
• Needed to explain the Rydberg formula.
6. Bohr-Summerfeld Model
• Needed to allow elliptical orbits
7. Schrödinger Model / Electron Cloud Model
• Needed to explain more satisfactorily why electrons don’t fall into the nucleus after radiating EM waves
• Needed to explain atoms with more than one electron
• Needed to explain periodic table trends
• Needed to explain spectra of large Z atoms
• Needed to explain intensities of spectral lines
• Needed to explain Zeeman effect from magnetic fields
• Needed to explain spectral splittings (fine, although this could be done with Klein-Gordan equation and is really a hack onto the non-relativisticSchrödinger equation, and hyperfine) Note: I need to go back to my QM books on this one.
8. Swirles/Dirac Model
• Needed to explain spectra of large Z atoms better
• Needed to explain the color of gold and cesium
• Needed to explain chemical and physical property differences between the 5th and 6th periods
9. Quantum Field Theory Model
• Needed for ???
10. Nuclear Shell Model / Goeppert-Mayer et al. Model
• Needed to explain radioactivity
About the biggest controversy is disagreement over the need to teach pseudo-historically these models. This is leaving out all the really bad ones. However, the terrible picture that society has adopted as the meme for atom (see below) affects student perceptions of the atom.
Stylised Lithium Atom by Indolences, Rainer Klute on Wikimedia Commons. Note that this is only a model, based loosely on the Bohr model. Also, these 3 electrons couldn’t all occupy the same circular orbit.
It would be nicer if students came into classrooms with the following conception of an atom.
Helium Atom QM by Yzmo from Wikimedia Commons. This is a much better rendition of the electron cloud but might be as bad for the nucleus. However, it is nice that it shows scale.
Physics teachers tend to like the Bohr model in that it can quickly (although magically) explain the Rydberg formula. However, there are many reasons to dislike the Bohr model.
Classroom Experiments
TODO: What classroom experiments or simulations could help students to progress in their knowledge of atoms? |
dd3cad685a23b04b | Acta Physica Sinica
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Experimental determination of scattering matrix of fire smoke particles at 532 nm
Zhang Qi-Xing, Li Yao-Dong, Deng Xiao-Jiu, Zhang Yong-Ming
Acta Physica Sinica, 2011, 60 (8): 084216
Transport characteristic of photoelectrons in uniform-doping GaAs photocathode
Ren Ling, Chang Ben-Kang, Hou Rui-Li, Wang Yong
Acta Physica Sinica, 2011, 60 (8): 087202
Effect of palladium adsorption on the electrical transport of semiconducting carbon nanotubes
Zhao Hua-Bo, Wang Liang, Zhang Zhao-Hui
Acta Physica Sinica, 2011, 60 (8): 087302
Acta Physica Sinica
Acta Physica Sinica--2011, 60 (8) Published: 15 August 2011
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Progress of white organic light-emitting device
Wang Xu-Peng, Mi Bao-Xiu, Gao Zhi-Qiang, Guo Qing, Huang Wei
Acta Physica Sinica. 2011, 60 (8): 087808 doi: 10.7498/aps.60.087808
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White organic light-emitting device (OLED) has potential of producing highly efficient saturated white light with the advantages of low-driving voltage, large area available, and flexible display, hence presenting many potential applications in solid state lighting and display industry. At present, the architecture of white OLED device includes mainly single emission layer, multilayer, down-conversion, stacked OLED, etc. which possess their own benefits, and it has attracted much attention respectively. In this paper after introducing the performance standards of white light, we review the development of white OLED in the aspects of architecture and performance. After that, we summarize the approaches to obtaining the high-performance white OLED. Meanwhile, we discuss the challenges to improving white OLED performance. Finally we look forward to the development of white OLED in the future.
Research and development of GaN photocathode
Li Biao, Chang Ben-Kang, Xu Yuan, Du Xiao-Qing, Du Yu-Jie, Wang Xiao-Hui, Zhang Jun-Ju
Acta Physica Sinica. 2011, 60 (8): 088503 doi: 10.7498/aps.60.088503
Full Text: [PDF 1183 KB] Download:(1198)
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Negative electron affinity GaN photocathode with greatly advanced photoelectricity performance is described. The research of GaN photocathode focuses on the three points, i. e. , quantum yield, electron energy distribution and surface model, in the last decade. The domestic research of GaN photocathode is still in its infancy, the basic theory is not established, and preparation technology is not mature. In this paper we review emission mechanism, material growth, surface cleaning, activation process optimization, varied-doping structure design and stability of GaN photocathode. The latest experimental results confirm that the fabrication technology of GaN photocathode is feasible.
Experimental study on X-band repetitively oversized backward wave oscillator
Wu Yang, Jin Xiao, Ma Qiao-Sheng, Li Zheng-Hong, Ju Bing-Quan, Su Chang, Xu Zhou, Tang Chuan-Xiang
Acta Physica Sinica. 2011, 60 (8): 084101 doi: 10.7498/aps.60.084101
Full Text: [PDF 2272 KB] Download:(725)
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A new type of high power microwave device is developed based on bitron and backward wave oscillator. The device is composed of two parts: the modulation cavity and the extraction cavity (which is similar to slow wave structure). The modulation cavity acts as electron beam modulator and microwave reflector, which forms a microwave resonator in combination of the extraction cavity. The electron is modulated when it passes through the modulation cavity, and the high power microwave is generated when the modulated beam passes through the extraction cavity. An X-band high power microwave device is designed for a 20 GW accelerator, and the simulation results are frequency 8.25 GHz and output power 5.70 GW. Using superconducting magnet as guiding magnet, a microwave power of 5.20 GW at X-band (frequency (8.25±0.01)GHz) is obtained in single pulse mode. The radiation power is 5.06 GW when the repetition rate is 30 Hz, and the pulse length is 13.8 ns.
Three-dimensional particle-in-cell simulation studies on a new radial three-cavity coaxial virtual cathode oscillator
Yang Chao, Liu Da-Gang, Zhou Jun, Liao Chen, Peng Kai, Liu Sheng-Gang
Acta Physica Sinica. 2011, 60 (8): 084102 doi: 10.7498/aps.60.084102
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A new radial three-cavity structure of the coaxial virtual cathode oscillator is proposed and studied numerically in this paper. Using the radial three-cavity structure, the beam-wave conversion efficiency is enhanced by modulating the electric field in the beam-wave interaction area, while the resonator composed of the radial three-cavity configuration and the mesh anode helps restrain mode competition effectively . And then the coaxial extraction structure benefits the energy extraction, and it can also absorb the used electrons entering into the drifting tube. Therefore, this new kind of virtual cathode oscillator can achieve a high output power. With an electron beam of 50 kA at 400 kV, a peak power of about 6 GW is achieved by simulation at 4.5 GHz. The mean power reaches 3.1 GW and the beam-wave conversion efficiency is about 15%.
Influence of longitudinal radio frequency electric field on multipactor effect on a dielectric surface
Zhu Fang, Zhang Zhao-Chuan, Dai Shun, Luo Ji-Run
Acta Physica Sinica. 2011, 60 (8): 084103 doi: 10.7498/aps.60.084103
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Based on the multipactor dynamic model and the secondary electron emission yield curves, the multipactor phenomenon of the secondary electron emission subjected to the longitudinal radio frequency (RF) electric field existing on a dielectric surface is simulated using the Monte Carlo method. The susceptibility curve of the electric field on the surface and the temporal evolution image of the multipactor discharge are investigated. The power deposited on the dielectric surface by the multipactor is also obtained in terms of an S-band RF dielectric window. The results show that the longitudinal RF electric field may intensify the single-surface multipactor effect, which is likely to result in dielectric crack and detrimental to RF transmission.
Magnetic metamaterial based on connected split and closed rings
Chen Chun-Hui, Qu Shao-Bo, Wang Jia-Fu, Ma Hua, Xu Zhuo, He Hua
Acta Physica Sinica. 2011, 60 (8): 084104 doi: 10.7498/aps.60.084104
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Through connecting split and closed rings together, a three-dimensional magnetic metamaterial is proposed in this paper. When the electric field of the induced electromagnetic wave is perpendicular to the dielectric board, this construction exhibits negative effective permeability. Its magnetic resonant frequency is insensitive to the variation of the width of the metallic wires, which facilitates practical fabrications and applications. Meanwhile this construction is of significance for designing polarization-independent and isotropic magnetic and left-handed metamaterials.
Investigation on focus and transport characteristics of high transmission rate sheet electron beam
Ruan Cun-Jun, Wang Shu-Zhong, Han Ying, Li Qing-Sheng
Acta Physica Sinica. 2011, 60 (8): 084105 doi: 10.7498/aps.60.084105
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The investigation on focus and transport characteristics of sheet electron beam has been a key technique for the development of high-power microwave and millimeter-wave vacuum electronic devices. Compared with the period permanent magnetic system to transport the sheet electron beam, the uniform magnetic focusing system has many advantages, such as easily adjusting and matching the magnet with the beam, focusing the intensity electron beam, no cut off beam voltage restriction, etc. However, the Diocotron instability of the sheet electron beam in the uniform magnetic field can produce the distortion, deformation, vortex and oscillation to destroy the beam transportation. In this paper, the single-particle model and the cold-fluid model theory and calculation are used to indicate that if the electron optics system parameters of the sheet beam are designed more carefully, the magnitude of uniform magnetic field and the filling factor of the beam in transport tunnel are increased appropriately, the Diocotron instability can be reduced, even vanished completely to transport the sheet beam effectively in a long distance. To verify the above conclusion, the electron gun with the ellipse cathode and the electron optics system are designed and optimized with the three-dimensinal simulation software in detail. After the complex assembly and weld process with the small geometry and high precision, the W-band sheet electron beam tube is manufactured and tested. The sheet beam cross section of 10 mm×0.7 mm is achieved experimentally with the one-dimensional compression and formation of electron gun. Also, with a beam voltage of 20—80 kV, and beam current of 0.64—4.60 A,the experimental transmission rate of sheet beam electron tube manufactured is more than 95% with a drift length of 90 mm, which is higher than the periodic cusp magnetic field transport experiment result of 92% obtained recently.
Diffraction characteristic of a misaligned vortex beam through a phase-hologram grating
Li Fang, Jiang Yue-Song, Ou Jun, Tang Hua
Acta Physica Sinica. 2011, 60 (8): 084201 doi: 10.7498/aps.60.084201
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We study the analytical features of the output beam diffracted from a phase-hologram grating when the incident vortex beam is misaligned with respect to the grating. The analytical representation describing the diffracted beam of the 1st order is derived theoretically. Based on the representation, the central of gravity and the central intensity of the diffracted beam are investigated in the cases of the alignment, lateral displacement, angular tilt and simultaneous lateral misalignment and angular tilt, separately. It is shown that the diffracted beam is described through confluent hypergeometrical function. The misalignment of the incident vortex beam can give rise to the displacement of the beam center of gravity, which is independent of the misalignment direction and azimuth angle. The displacement is more obvious for the lager misalignment. In the case of angular tilt, the direction of beam center of gravity is nearly identical to the misalignment direction, whatever the topological charge of the incident beam and the fork number of the grating are. Besides, if the sum of the topological charge of the incident beam and the fork number of the grating is zero, with radial displacement and deflection angule increasing, the central intensity of the diffracted beam decreases gradually, otherwise, the central intensity is non-zero value any more. That means the misalignment between the phase-hologram grating and the incident vortex beam can influence the measurement of the topological charge of the vortex beam.
Analysis and measurement for passive tag modulationperformance of backscatter link
Li Bing, He Yi-Gang, Hou Zhou-Guo, She Kai, Zuo Lei
Acta Physica Sinica. 2011, 60 (8): 084202 doi: 10.7498/aps.60.084202
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Optimal conditions for maximizing the effectively received power of reader receiver in passive ultra high frequency radio-frequency identification system are analyzed in this paper. The mismatched impendence affecting modulation index of tag backscatter link is discussed. Expressions of backscattered modulation indexes for calculating normalized effective received power of reader receiver, lower boundary of signal-to-noise ratio for demodulated output signal, and bit error rate are derived. Modulation indexes of backscatter link under different parameters are measured in open indoor environment. The measurement results show that the tag can be detected successfully when the modulation index of backscatter link is in a range from 5% to 10%.
Phase noise detection method in fiber lasers based onphase modulation and demodulation
Wang Xiao-Lin, Zhou Pu, Ma Yan-Xing, Ma Hao-Tong, Li Xiao, Xu Xiao-Jun, Zhao Yi-Jun
Acta Physica Sinica. 2011, 60 (8): 084203 doi: 10.7498/aps.60.084203
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In coherent combination of active phase control, there have existed three main phase detection techniques until now, they are heterodyne phase detection technique, multi-dithering technique and stochastic parallel gradient descent algorithm. A new phase detection method based on phase modulation and demodulation is proposed according to the principle of heterodyne phase detection and multi-dithering technique. Periodic phase modulation signal is implemented on a reference laser, and the coherent detection is carried on between the reference laser and the signal laser. With some processing of the modulation signal and the coherent detected optoelectronic signal, the phase noise is detected and the noise compensation can be realized. Numerical simulation and experimental studies are conducated. Experimental results show that the phase detection accuracy is higher than 1/50 wavelength and the average phase compensation residual error is less than 1/50 wavelength in the case of a 2 kHz sine wave phase noise with a phase region of
Experiment and simulation of time lens using electro-opticphase modulation and cross phase modulation
Li Bo, Tan Zhong-Wei, Zhang Xiao-Xing
Acta Physica Sinica. 2011, 60 (8): 084204 doi: 10.7498/aps.60.084204
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Temporal imaging is one of the important research issues of time lens. The theory of temporal imaging is investigated briefly. The experiment using electro-optic phase modulation to perform optical pulses compression is demonstrated. And the simulation and the discussion using electro-optic phase modulation and cross phase modulation to perform temporal imaging system are presented. The experimental results show that the temporal imaging system which consisits of time lens basing electro-optic phase modulation can effectively compress the optical pulses. However, the compression coefficient is restricted by the aperture of the time lens, and the resolution of the temporal imaging system is low. Further more, the simulation results and the analysis results indicate that the temporal imaging system which consists of time lens basing cross phase modulation has a bigger compression coefficient and a higher resolution. However, this temporal imaging system is difficult to realize.
Stereoscopic visualization based on splitting of two-way polarized image beams
Dai Yu, Zhang Jian-Xun
Acta Physica Sinica. 2011, 60 (8): 084205 doi: 10.7498/aps.60.084205
Full Text: [PDF 2012 KB] Download:(888)
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A thin metal film is introduced to split two-way polarized image beams, thereby accomplishing a stereoscopic three-dimensional display. Linearly polarized beams emitted by two uniform liquid crystal displays (LCDs) are reflected and transmitted by ultrathin aluminum film separately. Since optical constant of ultrathin metal film depends on film thickness, a piecewise linear function is used to obtain the relationship between the volume fraction of aluminum and the thickness of the film, and then the optical constant can be estimated by the Sheng Model. Following this procedure it is proved that both the reflected and the transmitted beams become elliptically polarized light beams and their principal axes are approximately vertical. Thus two orthogonal polarizers can be used to separate them and then make the observer have a perception of depth in LCD image. In order to keep the quantities of light entering the observer’s two eyes balanced, the thickness of the aluminum film is optimized. Experimental results are in agreement with the theoretical analyses, so the correctness of the method is verified.
Analysis and correction of object wavefront reconstruction errorsin two-step phase-shifting interferometry
Xu Xian-Feng, Han Li-Li, Yuan Hong-Guang
Acta Physica Sinica. 2011, 60 (8): 084206 doi: 10.7498/aps.60.084206
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The method of calculating and correcting object wave reconstruction errors caused by phase shift errors in two-step phase-shifting interferometry is studied systematically. Based on the principle of random distribution and the amplitude-phase independence of diffractive object wave, the expression of objective wave reconstruction error is introduced and the formula for that in the two-step standard algorithm is deduced. The automatic error correction method is suggested by further analyzing the structures, the characters of those errors caused by phase shift errors, and the objective expression. By the proposed method, the reconstructive amplitude and phase errors can be corrected at the same time through simple operation on the objective complex amplitude reconstructed by the standard two-step method without the additional measurement or the acknowledge of phase shift. The computer simulations are carried out to verify the effectiveness of this method, and the results show that the method is robust and reduces the effect of phase shift error on object wave-front reconstruction by about 2 orders of magnitude. Optical experiments also indicate that this method is effective and efficient.
Statistics property of polarized photon emission driven bya pair of pulses in single quantum dot
Gu Li-Shan, Wang Dong-Sheng, Peng Yong-Gang, Zheng Yu-Ju
Acta Physica Sinica. 2011, 60 (8): 084207 doi: 10.7498/aps.60.084207
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We study the properties of the x- and the y-polarized photons of the single quantum dot system driven by a pair of pulses using the generating function approach. Our results show that the quantum interference effects on the line shapes and the Mandels parameter Q of the x- and y-polarized photons and the linear and the nonlinear cross correlations are important.
Wavelet transform of odd- and even-binomial states
Song Jun, Xu Ye-Jun, Fan Hong-Yi
Acta Physica Sinica. 2011, 60 (8): 084208 doi: 10.7498/aps.60.084208
Full Text: [PDF 1786 KB] Download:(627)
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In the context of quantum mechanics the classical wavelet transform of a function f with the mother wavelet ψ can be recast into a matrix element of the squeezing-displacing operator U(μ,s) as 〈ψ|U(μ,s) |f〉. The technique of integral within an ordered product of operators is used to support this theory. Based on this, wavelet transforms are done for even- and odd-binomial states, and the corresponding numerical calculation leads to the spectrum of wavelet transform, which is helpful for recognizing the difference between even- and odd-states.
Mapping between multi-photon polarization state and single-photon spatial qudit and its applications
Lin Qing
Acta Physica Sinica. 2011, 60 (8): 084209 doi: 10.7498/aps.60.084209
Full Text: [PDF 1787 KB] Download:(684)
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Based on a special controlled-NOT gate, a multi-photon state encoded in polarizations of photons could be transformed into the corresponding single photon qudit encoded in spatial mode. It will make the processing on multi-photon change into the operation on a single-photon, if the inverse transformation from a single photon qudit back to a multi-photon state could be realized also. Associated with linear optical multi-port interferometer for single-photon unitary operation, the positive-operator value measurement and the universal unitary operation for multi-photon state are realized. This approach is more efficient than the previous one with decomposition into two-qubit gates in the circuit-based quantum computation, and it is feasible with using the current experimental technology.
Theoretical analysis of two-stage pumping technology for high power fiber lasers
Yang Wei-Qiang, Hou Jing, Song Rui, Liu Ze-Jin
Acta Physica Sinica. 2011, 60 (8): 084210 doi: 10.7498/aps.60.084210
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The problems of direct pumping and two-stage pumping for high power fiber lasers are theoretically analyzed. The simulation results show that 1070 nm laser directly pumped by 975 nm laser provides a theoretical slope efficiency up to 80%. But it is hard to make the highest temperature of the fiber core reduce below 150 ℃ by forced water cooling when pump power is 10 kW. For the two-stage pumping technology, if the conventional cladding pumping technology is used the slope efficiency is less than 20% when a 1070 nm laser is pumped by a 1018 nm laser. With the pump power filling factor increasing from 0.0025 to 0.1, the slope efficiency can be raised from 18.5% to 80.9%, thus the total slope efficiency is raised from 15.5% to 68%. Two-stage pumping is inferior to direct pumping in terms of slope efficiency, but it has the obvious advantage of thermal management. How to raise the pump power filling factor to achieve high slope efficiency and good temperature characteristics at the same time is the key technology of the two-stage pumping.
Enhancing stimulated Raman scattering of benzene high-order Stokes lines by fluorescence of 4-sulfphenyl porphyin in liquid-core optical fiber
Li Zhan-Long, Lu Guo-Hui, Sun Cheng-Lin, Men Zhi-Wei, Li Zuo-Wei, Gao Shu-Qin
Acta Physica Sinica. 2011, 60 (8): 084211 doi: 10.7498/aps.60.084211
Full Text: [PDF 1243 KB] Download:(805)
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Stimulated Raman scattering of benzene is studied in liquid-core fiber. Owing to fluorescence and third-order nonlinear 4-sulfphenyl porphyin, the high-order stimulated Raman scattering, such as sixth-order Stokes line of benzene, can be observed at a relatively low input-laser power. The threshold of high Stokes line is lowered with the addition of the 4-sulfphenyl porphyin when the concentration of solution is within 10-6 and 10-9 mol/L. At the same time, the Stokes line width becomes narrow with Stokes line order increasing. These results are expected to be applied to tunable laser, seeding laser, biological molecular structures and biological molecule used in none biological area.
Interactions of Laguerre-Gaussian solitons instrongly nonlocal nonlinear media
Zhang Xia-Ping, Liu You-Wen
Acta Physica Sinica. 2011, 60 (8): 084212 doi: 10.7498/aps.60.084212
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Based on the modified Snyder-Mitchell model, the optical fields that are produced by two collinear Laguerre-Gaussian solitons (LGS) in a strongly nonlocal nonlinear medium are studied. Various novel kinds of solitons on the profiles which depend on the model-index and the relative amplitude of LGS are shown. It is the phase vortices of the LGS that lead to the optical singularities. The many-ring soliton is produced first with the collinear component LGS. The optical field may rotate in propagation, and the angular velocity of the spiral soliton is given.
Influence of nonlocalization degree on the interaction between spatial dark solitons
Gao Xing-Hui, Yang Zhen-Jun, Zhou Luo-Hong, Zheng Yi-Zhou, Lu Da-Quan, Hu Wei
Acta Physica Sinica. 2011, 60 (8): 084213 doi: 10.7498/aps.60.084213
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The interaction between dark solitons in nonlocal self-defocusing media is investigated. Numerical results show that there is a critical condition for interaction between dark solitons in nonlocal self-defocusing medium. Under the critical condition, dark solitons will neither attract nor repell each other because the attractive force and repulsive force between them are identical. Beyound the critical condition, dark solitons may attract or repell each other depending on the nonlocalization degree and distance between them. The value of the critical condition is found.
Mutual-induced fractional Fourier transform in strongly nonlocal nonlinear medium
Zhao Bao-Ping, Yang Zhen-Jun, Lu Da-Quan, Hu Wei
Acta Physica Sinica. 2011, 60 (8): 084214 doi: 10.7498/aps.60.084214
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The effect of mutual-induced fractional Fourier transform (FRFT) between a weak signal beam and a strong pump beam in a strongly nonlocal nonlinear medium is described. The signal beam is an FRFT during propagation and the order of FRFT is dependent on the propagation distance and the power of the pump beam. When the propagation distance is fixed, the order of FRFT of the signal beam is proportional to the square root of the power of the pump beam. Mutual-induced FRFT is another method of light controlling light, and its properties contribute to the development of new type FRFT devices, optical information processing and optical imaging.
Theoretical and experimental studies on ultra-broad-bandwavelength tunableness by optical soliton mechanism
Zhu Qi-Hua, Zhou Shou-Huan, Zhao Lei, Zeng Xiao-Ming, Huang Zheng, Zhou Kai-Nan, Wang Xiao, Huang Xiao-Jun, Feng Guo-Ying
Acta Physica Sinica. 2011, 60 (8): 084215 doi: 10.7498/aps.60.084215
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It is crucial but challengeable that the generation of optical exact by synchronized seed pulses both for the main amplifier chain and for the pump-laser chain of an optical parametric chirped pulse amplification system, which are tried to be developed by the soliton mechanism. Detailed numerical simulation of the soliton propagation mechanism are accomplished. So the evolutions of solitons in time-domain and frequency-domain as well as the reciprocity characteristic with other nonlinear effects are clarified. Experiments are carried out, for validating the method of using soliton mechanism to generate ultra-broad band tunable ultra-short laser pulses. Forming, breaking up and self-frequency shift of a soliton are observed. The favorable tunablenesses of the wavelength between the visible and the near-infrared regions are exhibited. All these experimental results are well consistent with the theoretical analyses.
Experimental determination of scattering matrix of fire smoke particles at 532 nm Hot!
Acta Physica Sinica. 2011, 60 (8): 084216 doi: 10.7498/aps.60.084216
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Based on polarization modulation and lock-in detection, an experimental apparatus is built to determine several important angular dependent scattering matrix elements at 532 nm. The apparatus is tested by water droplets through comparing measurement results with Mie calculations. Measurement results of scattering matrix elements and element ratios between smoke particles produced by smoldering cotton test fire and those produced from flaming n-heptane test fire are presented. We find that results of Mie calculations are able to describe the experimental data of smoldering cotton test fire smoke, which indicates that the particles generated by smoldering cotton test fire are mostly spherical in shape with considering the particle size relative to the wavelength. Using the optimization method, we estimate the refractive index (m=1.49+i0.01) and size distribution (lognormal distribution, σg=2.335 and dg=0.17 μm) of smoldering cotton test fire smoke. Contrarily, the experimental data of flaming n-heptane fire smoke cannot be described by Mie scattering, which is interpreted by the nonspherical, fractal aggregate morphology of the particulates.
Model of quench cooling and experimental analysis of cylindrical infrared chalcogenide glass
Song Bao-An, Dai Shi-Xun, Xu Tie-Feng, Nie Qiu-Hua, Shen Xiang, Wang Xun-Si, Lin Chang-Gui
Acta Physica Sinica. 2011, 60 (8): 084217 doi: 10.7498/aps.60.084217
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The tapping temperature and the cooling rate are the key parameters during the development process of chalcogenide glass. Based on the theory of heat conduction equation, a model for calculating the temperature distribution of cylindrical chalcogenide glass is established using the least square fitting method in this paper. The tapping temperature, the temperature distribution and the cooling rate are simulated by using the model. The simulation results are compared with experimental data. The results show that the glass temperature stays in a non-steady non-uniform distribution, the surface cooling is the fastest, and the temperature decreases exponentially with time when the glass is tapped off from the furnace; the temperature of glass rod from the center to the edge is approximately of parabola distribution; the crystallization is the most difficult when the glass is tapped off at 50—100 ℃ higher than crystallization temperature and a surface heat exchange coefficient of 180 W ·m-2K-1. Under the guidance of the theoretical model the uniform and transparent chalcogenide glass with a diameter of 110 mm and height of 80 mm is obtained. The glass transmission spectrum range is 0.8—17 μm. The 2 mm thick flat sheet has the average transmittance higher than 65% in a 8—12 μm range.
Frequency properties of the defect mode inside a photonic crystal band-gap with zero average refractive index
Liu Li-Xiang, Dong Li-Juan, Liu Yan-Hong, Yang Chun-Hua, Yang Cheng-Quan, Shi Yun-Long
Acta Physica Sinica. 2011, 60 (8): 084218 doi: 10.7498/aps.60.084218
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Left-handed metamaterial has been designed based on transmission line technology. A left-handed and a normal materials are studied alternately to form a one-dimensional photonic crystal with an average refractive index of zero but a special band-gap emerges. The two band-edge frequencies of this gap are insensitive to the incident angle and lattice constant. These characteristics can be used in the miniaturization of filters and high-quality coupling. Studies show that we can regulate the mode frequency by controlling the thickness of the defect layer, which can provide a method for the adjusting frequency in applications. The experimental results are consistent with numerical simulation.
Magnetically tunable magneto-photonic crystals for multifunctional terahertz polarization controller
Fan Fei, Guo Zhan, Bai Jin-Jun, Wang Xiang-Hui, Chang Sheng-Jiang
Acta Physica Sinica. 2011, 60 (8): 084219 doi: 10.7498/aps.60.084219
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A multifunctional terahertz polarization controller is designed based on the two-dimensional photonic crystal structure and the ferrite material. The different working devices including a controllable polarizer, a polarization beam splitter and a tunable phase retarder with continuous phase retardations of -π—π at 1 THz are controlled by the shift of photonic band gap with different external magnetic fields. By using the plane wave expansion method and the rigorous coupled wave analysis, we calculate the band gap positions and transmittances of device with the variation of magnetic field. The field distribution and phase are simulated by the finite difference time domain method.
Core-shape transitions in photonic crystal fiber
Sun Gui-Lin, Chen Zi-Lun, Xi Xiao-Ming, Chen Hong-Wei, Hou Jing, Jiang Zong-Fu
Acta Physica Sinica. 2011, 60 (8): 084220 doi: 10.7498/aps.60.084220
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The mode field and the translational efficiency are studied by the finite difference beam propagation method in an anamorphic photonic crystal fiber. Selective hole collapse is realized in laboratory and anamorphic fibers with circular to rectangular cores and small to large mode fields are made. The loss at wavelength 1550 nm is below 0.05 dB during each translation. The calculated and the experimental results are consistent with each other.
Effect of finite-amplitude acoustic wave nonlinear interaction on farfield directivity of sound source
Lü Jun, Zhao Zheng-Yu, Zhou Chen, Zhang Yuan-Nong
Acta Physica Sinica. 2011, 60 (8): 084301 doi: 10.7498/aps.60.084301
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The acoustic radiation pressure expression of a multi-frequency sound source is obtained on the basis of the Fenlon theory. Then using the solution method of a monochromatic source harmonic wave directivity, farfield directivity of second-order approximate is obtained in the case of the dual-frequency sound source interaction. Subsequently, the effect of two-wave interaction on the farfield of first-order and second-order wave of either wave is studied and discussed when the initial radiation pressures and frequencies are different. The conclusion is that there are some different effects of sound wave interaction on the farfield directivity of a sound source, when the relative initial radiation pressure and frequency between the sound waves is changed.
Dynamical behaviors of hydrodynamic cavitation bubble under ultrasound field
Shen Zhuang-Zhi, Lin Shu-Yu
Acta Physica Sinica. 2011, 60 (8): 084302 doi: 10.7498/aps.60.084302
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Considering liquid viscosity, surface tension, liquid compressibility and turbulence, the dynamical behaviors of cavitation bubble in venturi cavitation reactor are numerically investigated with using acoustic field regarding water as a work medium. The effects of acoustic frequency, acoustic pressure and ratio of throat to pipe diameter on cavitation bubble dynamics, bubble temperature and pressure pulse by rapid collapse of cavitation bubble are analysed. The results show that bubble motion of hydrodynamic cavitation modulated by ultrasound, becomes the high energy stable cavitation. It is favorable for enhancing the cavitation effect.
Symplectic symmetry feature of thermoacoustic network
Yang Zhi-Chun, Wu Feng, Guo Fang-Zhong, Zhang Chun-Ping
Acta Physica Sinica. 2011, 60 (8): 084303 doi: 10.7498/aps.60.084303
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Symplectic mathematics is introduced into the thermoacoustic network model. The transferring matrix of thermoacoustic system is analyzed, and the transferring matrix of working gas in isothermal fluid pipe of thermoacoustic system is a symplectic matrix. The transferring matrix of working gas in regenerator of thermoacoustic system is not a symplectic matrix, but it can be converted into a symplectic matrix by variable transformation. With variable transformation, the whole transferring matrix of thermoacoustic system can be represented by a symplectic matrix. The form of symplectic matrix is conducible to analyzing and calculating the thermoacoutic network model.
A type of new conserved quantity of Mei symmetry for Nielsen equations
Jia Li-Qun, Sun Xian-Ting, Zhang Mei-Ling, Wang Xiao-Xiao, Xie Yin-Li
Acta Physica Sinica. 2011, 60 (8): 084501 doi: 10.7498/aps.60.084501
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A type of new conserved quantity of Mei symmetry of Nielsen equations for a holonomic system is studied. Under the infinitesimal transformation of groups, new structural equation and new conserved quantity of Mei symmetry of Nielsen equations for a holonomic system are obtained from the definition and the criterion of Mei symmetry of Nielsen equations. Finally, an example is given to illustrate the application of the results.
A two-lane cellular automaton traffic flow model with the influence of driving psychology
Hua Xue-Dong, Wang Wei, Wang Hao
Acta Physica Sinica. 2011, 60 (8): 084502 doi: 10.7498/aps.60.084502
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A two-lane cellular automaton model is developed to analyze the urban traffic flow with considering the influence of driving psychology. In order to show the different psychological characters of drivers when changing lane and braking, lane-changing choice probability and safety parameter are introduced. By computer simulation, the relationships among speed, density and traffic volume are given to show the influence of driving psychology on the traffic flow. The simulation results indicate that the lane-changing choice probability has little effect on the average speed, but it makes the variance of speed larger. And the safety parameter can increase the average speed and the traffic volume.
The analysis of effects and theories for electromagnetic hydrodynamics propulsion by surface
Liu Zong-Kai, Zhou Ben-Mou, Liu Hui-Xing, Liu Zhi-Gang, Huang Yi-Fei
Acta Physica Sinica. 2011, 60 (8): 084701 doi: 10.7498/aps.60.084701
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The electromagnetic hydrodynamics(EMHD) propulsion by surface is performed through the reaction of electromagnetic body force, which is induced in conductive flow fluid (such as seawater, plasma and so on) around the propulsion unit. Based on the basic governing equations of electromagnetic field and hydrodynamics, by numerical simulations obtained by the finite volume method, the characteristics of flow field structures near the navigating and the strength variation of propulsion force are investigated at varying positions (the angle of attack). The results show that surface electromagnetic body force can modify the structure and the input energy of flow boundary layer, which enables the navigation to obtain the thrust. With the increase of interaction parameter the effect of viscous resistance and pressure drag to navigating decrease and the nonlinear relationship between propulsion coefficient and interaction parameter tends to be linear gradually. The strength of propulsion force depends mainly on the electromagnetic body force. The lift force can be improved effectively through the EMHD propulsion by surface at an angle of attack for navigating. The navigating surface can be designed as working space of propulsion units, which is of certain significance for optimizing the whole struction and improving the efficiency.
Direct method of determining the lattice parameters of a phase from X-ray diffraction pattern of multi-phase
Xu Xiao-Ming, Miao Wei, Tao Kun
Acta Physica Sinica. 2011, 60 (8): 086101 doi: 10.7498/aps.60.086101
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A new method of determining lattice parameters by peak fitting of X-ray diffraction pattern, without involving structure parameters, is introduced. The method can be applied to a single phase and one phase in multi-phase diffraction patterns. It can avoid getting different fitting results caused by using different extrapolation functions, and can get a more accurate result in a short time. The application program of the method has been used in the practical work. For improving the fitting accuracy, the program can also adjust off-axis deviation of the sample surface and the goniometric mechanical zero.
Neutron diffraction study of aging process effect on phase structure of single crystal superalloy
Sun Guang-Ai, Chen Bo, Wu Er-Dong, Li Wu-Hui, Zhang Gong, Wang Xiao-Lin, Vincent Ji, Thilo Pirling, Darren Hughes
Acta Physica Sinica. 2011, 60 (8): 086102 doi: 10.7498/aps.60.086102
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The neutron diffraction results obtained by oscillating and fixing φ during data collection show that the high ageing temperature is effective to eliminate dendrite crystals and the microstrain exists mainly in the γ' phase. Based on the microstructure obtained by neutron diffraction and scanning electron microscope, the influence of ageing temperature and time on γ' phase are evaluated. The unique misorientations among γ' phase grains are observed from superlattice measurements. According to the neutron diffractions of different crystal planes, the crystal symmetry is slightly changed from cubic to quartet (a < c) due mainly to the γ matrix phase and the experimental results also prove the strain deviation in difference orientations, thus providing the basis for the existence of driving force for the raft model. The calculation based on the superlattice diffraction shows that in the interfaces between the γ and γ' phases exists a complex distortion: the mismatch varies from -0.1% to -0.3% and the mismatch value can be reduced by high temperature during the first ageing and long time during the second ageing.
All-organic two-dimensional photonic crystal laser based on holographic polymer dispersed liquid crystals
Deng Shu-Peng, Li Wen-Cui, Huang Wen-Bin, Liu Yong-Gang, Peng Zeng-Hui, Lu Xing-Hai, Xuan Li
Acta Physica Sinica. 2011, 60 (8): 086103 doi: 10.7498/aps.60.086103
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In this paper, we report a kind of dye-doped two-dimensional photonic crystal based on holographic polymer dispersed liquid crystals (HPDLC) with a lattice constant of 582nm, which is prepared conveniently with a single step holographic exposure. Under the excitation of a frequency-doubled Nd:yttrium-aluminum-garnet laser operating at a wavelength of 532nm, optically pumped lasing with narrow bandwidth and low threshold is observed from a 4-(dicyanomethylene)-2-methyl-6-(4-dimethylaminostyryl)-4H-pyran dye-doped two-dimensional photonic crystal. The results show that the emitted lasing peak is centered at about 603nm with a full width at half maximum of only 0.4nm, and the threshold energy is about 22.7 μJ, which is evidently lower than the reported previously. The laser bandwidth decreases by a factor of three from 1.4nm to 0.4nm compared with that of the dye-doped HPDLC transmission grating. This result exhibits a bright prospect in application of tunable photonic crystal laser.
Effects of thermal evaporation and electron beam evaporation on two-dimensional patterned Ag nanostructure during nanosphere lithography
Luo Yin-Yan, Zhu Xian-Fang
Acta Physica Sinica. 2011, 60 (8): 086104 doi: 10.7498/aps.60.086104
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In the process of fabricating two-dimensional Ag nano-arrays via nanosphere lithography, different deposition methods present different lattice point shapes of Ag nanostructure. Nano-triangles are produced by thermal evaporation whereas nano-rings are obtained by electron beam evaporation although they are both of hexagonal lattice. It is indicated that the sizes, the surface nanocurvature, the thermal and the kinetic energies of the particles deposited are key factors controlling the formation of the shape of Ag lattice point.
Long-range Finnis-Sinclair potential for Zn-Mg alloy
Wang Zhao-Ke, Wu Yong-Quan, Shen Tong, Liu Yi-Hu, Jiang Guo-Chang
Acta Physica Sinica. 2011, 60 (8): 086105 doi: 10.7498/aps.60.086105
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A set of optimal long-range Finnis-Sinclair (F-S) potential parameters of single Mg are achieved by fitting the lattice energy, lattice constants, and elastic constants to experimental results. With the same method, the set of the F-S potential parameters of single Zn are obtained through the introduction of modifying factor to the repulsive term. Finally, the lattice energy and lattice constants of Mg21 Zn25, MgZn2 and Mg2Zn11 alloys are further fitted to achieve the F-S potential parameters of Zn-Mg based on the previous F-S potential parameters of Mg-Mg and Zn-Zn. After that, a series of molecular dynamics simulations of single Mg, Zn, and Mg21 Zn25, MgZn2, Mg2Zn11 alloys is performed at 300 K with the achieved F-S potential parameters, thereby proving the F-S potential parameters to be appropriate for the description of Zn-Mg alloys. The long-range F-S potential parameters of Zn and Zn, Mg and Mg, Zn and Mg are obtained.
Microstructures in polycrystalline pure copper induced by high-current pulsed electron beam——deformation structures
Guan Qing-Feng, Gu Qian-Qian, Li Yan, Qiu Dong-Hua, Peng Dong-Jin, Wang Xue-Tao
Acta Physica Sinica. 2011, 60 (8): 086106 doi: 10.7498/aps.60.086106
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In order to investigate the superfast deformation mechanism of metal, the high-current pulsed electron beam (HCPEB) technique is employed to irradiate the polycrystalline pure copper. The microstructure of the irradiated sublayer is investigated by using transmission electron microscopy. It is suggested that the stress with very high value and strain rate is introduced within the sublayer after HCPEB irradiation. The dislocation cell and the tangle dislocation formed by cross slip are the dominant defects after one-pulse HCPEB irradiation, whereas, dense dislocation walls and twins are the central microstructures after five- and ten-pulse irradiation. The diffusion and the climb of the atomic plane can cause the formation of the steps at the grain boundary and (or) the twin boundary. Based on the structure characteristics of the irradiated surface, the possible deformation mechanism induced by HCPEB irradiation is discussed.
Influence of interface traps of p-type metal-oxide-semiconductor field effect transistor on single event charge sharing collection
Chen Jian-Jun, Chen Shu-Ming, Liang Bin, Liu Bi-Wei, Chi Ya-Qing, Qin Jun-Rui, He Yi-Bai
Acta Physica Sinica. 2011, 60 (8): 086107 doi: 10.7498/aps.60.086107
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Due to negative bias temperature instability and hot carrier injection, p-type metal-oxide-semiconductor field effect transistor (MOSFET) will degrade with time, and the accumulation of interface traps is one major reason for the degradation. In this paper, the influence of the accumulation of pMOSFET interface traps on single event charge sharing collection between two adjacent pMOSFET is studied based on three-dimensional numerical simulations on a 130 nm bulk silicon complementary metal-oxide-semiconductor process, the results show that with the accumulated interface traps increasing, the charge sharing collection reducs for both the two pMOSFETs. The influence of the accumulation of pMOSFET interface traps on single event charge sharing induced multiple transient pulses between two adjacent inverters is also studied, the results show that the multiple transient pulses induced by the two pMOSFET charge sharings will be compressed, while multiple transient pulses induced by the two nMOSFET charge sharing will be broadened.
Research on the conductivity of KAg4 I5-AgI composite
Gao Shao-Hua, Wang Yu-Xia, Wang Hong-Wei, Yuan Shuai
Acta Physica Sinica. 2011, 60 (8): 086601 doi: 10.7498/aps.60.086601
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KAg4I5(10%AgI) composite is prepared by the solid-state reaction method in the dark and dry conditions, and its structure, morphology, ion conductivity properties, and phase transition temperature are studied by X-ray diffraction, scanning electron microscope, impedance spectroscopy, differential scanning calorimetry and other analytical tools. The results show that when the two phases (AgI and KAg4I5 phases) in the composite are both fast ionic conductor phases, ionic conductivity of composite is higher than that of the single phase, and the heating and cooling of the conductivity curves form a hysteresis loop.During heating and cooling, AgI phase transition temperature lags 5 and 10 ℃ respectively.We use the interaction interface, the interface stress phase and Gouy-Chapman model to analyze the mechanism for the improved conductivity of this composite and phase transition temperature change when the two phases are both the fast ionic conductive.
First principles study of H2 molecule adsorption on Li3 N(110) surfaces
Chen Yu-Hong, Du Rui, Zhang Zhi-Long, Wang Wei-Chao, Zhang Cai-Rong, Kang Long, Luo Yong-Chun
Acta Physica Sinica. 2011, 60 (8): 086801 doi: 10.7498/aps.60.086801
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The adsorption of H2 on a Li3N(110) crystal surface is studied by first principles. Preferred adsorption sites, adsorption energy, dissociation energy and electronic structure of the H2/Li3N(110) systems are calculated separately. It is found that H2 is adsorbed on the N bridge site more favorably than on the other sites, while two —NH radicles are formed on the Li3N(110) crystal surface. The calculated adsorption energy on the N bridge site is 1.909 eV, belonging to a strong chemical adsorption. The interaction between H2 and Li3N(110) surface is due mainly to the overlapping among H 1s, N 2s and N 2p states, through which covalent bonds are formed between N and H atoms. An activation barrier of 1.63 eV is found for the dissociation of H2 molecule in N bridge configuration, which indicates that the dissociative adsorption of H2 on Li3N(110) surface is favorable under the certain heat activation condition; —NH2 radicle is formed after the optimization of H2 adsorbed on the N top site. The adsorption energy on the N top site is negative. In other words, this adsorption is unstable. So it is concluded that it is not easy to produce the LiNH2 between Li3N(110) face and H2 directly.
Microstructure and mechanical properties of Ti-B-C-N nanocomposite coatings
Luo Qing-Hong, Lu Yong-Hao, Lou Yan-Zhi
Acta Physica Sinica. 2011, 60 (8): 086802 doi: 10.7498/aps.60.086802
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Ti-B-C-N nanocomposite coatings with different C quantities are deposited on Si(100) and high speed steel (W18Cr4V) substrates by the closed-field unbalanced reactive magnetron sputtering in the mixture of argon, nitrogen and acetylene gases. The microstructures of Ti-B-C-N nanocomposite coatings are characterized by X-ray diffraction and high-resolution transmission electron microscopy; while the nanohardness and elastic modulus values are measured by the nano-indention method. The results indicate that in the studied composition range, the deposited Ti-B-C-N nanocomposite coatings are found still only in the TiN base nanocrystalline. When the C2H2 flux is small, adding C can promote crystallization of Ti-B-C-N nanocomposite coatings, and the grain can be increased to improve the mechanical properties, when the grain size of about 6 nm (C2H2 flux rate 2 cm3/min), hardness, elastic modulus and fracture toughness of Ti-B-C-N nanocomposite coatings achieve the maximum, respectively, 35.7 GPa, 363.1 GPa and 2.46 MPa ·m1/2; the further increase of the C content of Ti-B-C-N nanocomposite coating can reduce mechanical properties of coating dramatically.
Structure and mechanical property of glow discharge polymer
He Zhi-Bing, Yang Zhi-Lin, Yan Jian-Cheng, Song Zhi-Min, Lu Tie-Cheng
Acta Physica Sinica. 2011, 60 (8): 086803 doi: 10.7498/aps.60.086803
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Different glow discharge polymer (GDP) thin-films are prepared at various working pressures and flux ratios of trans-2-butune (T2B) to H2 by using low-pressure plasma enhanced chemical vapor deposition technology. Hydrogen atomic content and network structure of GDP thin-film are characterized by element analysis and Fourier transform infrared spectrum. The hardness and modulus of GDP coating are measured by nanoindentation. It is found that when working pressure and flux ratio between T2B and H2 gradually decrease, hydrogen content and sp3 CH3 groups decrease, sp2 CH2 and sp3 CH1,2 groups increase and hardness H and Young’s modulus E of GDP coating and the cross-linking degree of carbon network inerease.
Calculation of valence band structure of uniaxial 〈111〉 stressed silicon
Ma Jian-Li, Zhang He-Ming, Song Jian-Jun, Wang Xiao-Yan, Wang Guan-Yu, Xu Xiao-Bo
Acta Physica Sinica. 2011, 60 (8): 087101 doi: 10.7498/aps.60.087101
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The valence band structure of uniaxial 〈111〉 stressed silicon is calculated in the frame of k·p perturbation method and compared with that of unstressed silicon. The valence band energy level shifting, splitting, and variation of the effective mass in the vicinity of the Γ point are presented for different uniaxail 〈111〉 stresses. The effective masses for the heavy and light hole bands in unstressed, our calculation results are in good agreement with the obtained published results of bulk silicon. The study extends the selective range of optimum stresses and crystal direction configuration of conduction channels for uniaxial stressed silicon devices. The obtained results of splitting energy and effective mass may serve as the reference for the calculation of other physical parameters of uniaxial 〈111〉 stressed silicon.
Helicity effects on Rh adsorption behavior inside and outside the single-wall carbon nanotubes
Liu Sha, Wu Feng-Min, Teng Bo-Tao, Yang Pei-Fang
Acta Physica Sinica. 2011, 60 (8): 087102 doi: 10.7498/aps.60.087102
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The curvature and the helicity of single-wall carbon nanotube (SWCNT) are the important factors which influence the adsorption behaviors of metal atoms inside and outside carbon tubes. However, it is difficult to investigate the separate effects of SWCNT helicity on the adsorption behaviors of metal atoms. In the present work, the armchair (6, 6), zigzag (10, 0), and chiral (8, 4) tubes with similar curvature are selected, then the Rh adsorption behaviors inside and outside the tubes are systematically investigated using the density functional theroy. Due to the different SWCNT helicities, the stable configurations of Rh atoms on tubes are different. The neighbor carbon atoms interacting with Rh atoms vary with tube helicity, therefore, the Rh adsorption energies for a similar configuration are also different. It indicates that the outer charge density of SWCNT is higher than the inner one. Different helicities lead to different charge density variations along the radial direction. Charge density difference shows that the orbital orientations of Rh adatom and the electrons obtained and lost are slightly different due to the different helicities. The bandstructure indicates that the doping band appears near the Fermi energy level. The (6, 6) tube with Rh adatom still exhibits metallicity. When Rh atoms are adsorbed inside the (10, 0) tube, the nanotube transforms from the semiconducting into the metallic one. However, the band gap reduces when Rh atoms adsorbed outside the tube. After the Rh adsorption, the (8, 4) tube band gap reduces.
Effects of Mn and N codoping on microstructure and performance of anatase TiO2
Zhang Xue-Jun, Liu Qing-Ju, Deng Shu-Guang, Chen Juan, Gao Pan
Acta Physica Sinica. 2011, 60 (8): 087103 doi: 10.7498/aps.60.087103
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The effects of Mn and N codoping on the crystal structure, defect formation energy, electronic structure, optical property and redox ability of anatase TiO2 are investigated by first-principles calculations of plane-wave ultrasoft pseudopotential. The calculation results show that the octahedral dipole moment of anatase TiO2 increases due to its lattice distortion after Mn, N codoping, which is favorable for effective separation of photogenerated electron-hole pairs. Some impurity bands appear in the band gap, which leads to the red-shift of optical absorption edge and to the increase in coefficient of light absorption, thereby facilitating the enhancement of the photocatalytic efficiency. If the impurity band is not taken into account, the band edge redox potential of codoped TiO2 is only slightly changed compared with that of pure TiO2 . All of these results can explain the better photocatalytic performances of Mn, N codoped anatase TiO2 under visible-light irradiation.
Lattice dynamical, dielectric and thermodynamic properties of LiNH2 from first principles
Li Xue-Mei, Han Hui-Lei, He Guang-Pu
Acta Physica Sinica. 2011, 60 (8): 087104 doi: 10.7498/aps.60.087104
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The lattice dynamical, dielectric properties and thermodynamic properties of LiNH2 are investigated by first principles calculations. Based on the density functional perturbation theory within the framework of linear response theory, the phonon dispersion curves and the phonon density of phonon states throughout the Brillouin zone are obtained. The calculated frequencies of the Raman active and infrared active modes are compared with previous experimental and theoretical results, and Born effective charge tensor as well as electronic dielectric permittivity tensor is presented. We find that the Born effective charge tensor of LiNH2 has quite small anisotropy. These calculated results are in good agreement with available experimental and theoretical values. Furthermore, the thermodynamic functions are predicted using the phonon density of states.
First principles study of structural, electronic and elastic properties of Mg2 Si polymorphs
Yu Ben-Hai, Liu Mo-Lin, Chen Dong
Acta Physica Sinica. 2011, 60 (8): 087105 doi: 10.7498/aps.60.087105
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The structural and the elastic properties of the Mg2Si polymorphs are calculated. The calculations are performed by using the plane-wave pseudo-potential method within the framework of first principles. The anti-fluorite structure, the anti-cotunnite structure and the Ni2In-type structure of Mg2Si can retain their mechanical stability in the pressure intervals 0—7 GPa,7.5—20.2 GPa and 21.9—40 GPa, separately. The relationships between pressure and the elastic moduli (elastic constant, bulk modulus, shear modulus, Young’s modulus, Poisson ratio and anisotropy factor) are discussed. The electron density distribution, the density of states, the bond length and the Mulliken population of these polymorphs are systemically investigated. Our results show that the anti-fluorite Mg2Si is a semiconductor and the other two polymorphs are metallic materials. The interaction between Mg 2p, 3s and Si 3p plays a dominant role in the stability of the Mg2Si polymorphs. The strongest interactions in the anti-fluorite Mg2Si and the Ni2In-type Mg2Si are Mg-Mg and Mg-Si interactions, respectively. Our results are concordant with the experimental data and the previous results.
Investigation on the dynamic conductance of mesoscopic system based on the self-consistent transport theory
Quan Jun, T. C. Au Yeung, Shao Le-Xi
Acta Physica Sinica. 2011, 60 (8): 087201 doi: 10.7498/aps.60.087201
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According to the self-consistent electronic dynamic transport theory of mesoscopic system, we present the dynamic conductance of mesoscopic structure. As an application of this theory, we employ a coherent mesoscopic parallel-plate capacitor model in the present study. The results show that the dynamic conductance of system depends on the frequency of external field and Fermi energy of system, and is a complex with a finite imaginary part. For a smaller frequency, the conductance shows a similar feature to dc case, but with the increase of the frequency of external fields, substantial deviations between dc case and ac case are observed, the dynamic conductance of system presents a peak structure with Fermi energy varying. For a given Fermi energy, the dynamic conductance is oscillatory with frequency varying, moreover some negative imaginary parts of conductance are observed. The negative imaginary part implies the capacitive behavior, and positive imaginary part refers to the inductive behavior.
Transport characteristic of photoelectrons in uniform-doping GaAs photocathode Hot!
Acta Physica Sinica. 2011, 60 (8): 087202 doi: 10.7498/aps.60.087202
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The transport of photoelectrons in a uniform-doping transmission-mode GaAs photocathode is calculated by establishing the models of atomic configuration and ionized impurity scattering. And the influence of the doping concentration of photocathode, the photocathode thickness, the electron diffusion length on the diffused circle and the ratio of the number of photoelectrons reaching the emit-surface to the number of exited photoelectrons at the back-interface of GaAs photocathode are analyzed. The calculated results show that the limiting linear resolution is 769 mm-1 with the cathode thickness being 2 μm, the electron diffusion length 3.6 μm and the uniform-doping concentration 1×1019 cm-3. The research on the transport of photoelectrons is worthwhile for preparing the high-performance GaAs cathode and improving the resolution of intensifier image.
Synthesis and electrical properties of dual doped CaMnO3 based ceramics
Wang Hong-Chao, Wang Chun-Lei, Su Wen-Bin, Liu Jian, Sun Yi, Peng Hua, Zhang Jia-Liang, Zhao Ming-Lei, Li Ji-Chao, Yin Na, Mei Liang-Mo
Acta Physica Sinica. 2011, 60 (8): 087203 doi: 10.7498/aps.60.087203
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Different Nb doped Ca0.9Yb0.1Mn1-xNbxO3 ceramics are successfully synthesized by the conventional solid state reaction technique. The crystal structures are of orthorhombic phase, belonging to the Pnma space group. The lattice constant and the volume increase with the increase of Nb content. Relatively high density is around 97%. Scanning electron microscope (SEM) images show that samples are well crystallized. The electrical resistivity and the Seebeck coefficient are measured in a temperature range between 300 and 1100 K. At low temperatures, the electrical resistivity shows a semiconductive-like behavior. At high temperatures, the electrical resistivity exhibits a typical metallic conductive behavior. The semiconductor-metal transition temperature shifts toward a higher temperature with the increase of Nb content. The electrical resistivity increases with Nb dopant, except that the electrical resistivity for x=0.03 is slight lower than that fox x=0.00 sample at high temperature range. This conductivity behavior can be understood as the fact that though Nb doping can introduce more carriers, it also distorts the MnO6 octahedra, and causes the carrier localization. The values of Seebeck coefficient are all negative, indicative of an n-type electrical conduction. The absolute value of Seebeck coefficient increases with temperature increasing, but decreases with the increase of Nb content. The highest power factor is obtained to be 297 μW/K2m at 497 K in the x=0.00 sample, and the power factor of this sample is less independent of temperature in the whole measured temperature range.
Influence of niobium doping on crystal structure and thermoelectric property of reduced titanium dioxide ceramics
Liu Jian, Wang Chun-Lei, Su Wen-Bin, Wang Hong-Chao, Zhang Jia-Liang, Mei Liang-Mo
Acta Physica Sinica. 2011, 60 (8): 087204 doi: 10.7498/aps.60.087204
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Titanium oxide ceramics doped with niobium is synthesized in reduced atmosphere at 1200 ℃ by conventional solid-state reaction technique. From their crystal structures determined by the powder X-ray diffraction(XRD), the samples have multiple-phase with low Nb concentration, but they have single tetragonal rutile phase when Nb content is larger than 0.02. The electrical conductivities, the Seebeck coefficients and the thermal conductivities of the samples with single phase are measured at a temperature range between room temperature and 900 K. The electrical conductivity and the Seebeck coefficient show non-metallic behaviors. According to the fitting, it is found that the samples show thermal-activation mechanism at low temperatures and small-polaron hopping conduction mechanism at high temperatures. Moreover, the analyses of XRD, electrical conductivity and Seebeck coefficient show that the concentration of oxygen vacancy decreases with Nb content increasing. Thermal conductivity decreases with temperature increasing, dominating by lattice thermal conductivity. In the measurement region, the figure of merit (ZT) reaches a highest value of 0.19 at 873 K in the Ti0.98Nb0.02O2-δ sample.
Preparation and electrical transport properties of Fe doped Ca1-xFexMnO3(x=0—0.12) oxide
Zhang Fei-Peng, Zhang Xin, Lu Qing-Mei, Liu Yan-Qin, Zhang Jiu-Xing
Acta Physica Sinica. 2011, 60 (8): 087205 doi: 10.7498/aps.60.087205
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The Fe doped Ca1-xFexMnO3(x=0—0.12) powder and bulk samples are fabricated by citric acid sol-gel and ceramic preparation process, the samples are analzed by X-ray diffraction pattern and electrical constant measurement. The results show that all samples are of single phase, the lattice constants are gradually lowered by Fe doping for Ca site, and the crystalline grain growth is restrained. All the bulk samples have semiconductor transporting characteristics in the whole temperature range of measurement. The transportation mechanism is not changed. The energy for polarons to hop is increased for doped samples and thus the electrical resistivity is increased by increasing Fe doping concentration.
Relationship between light efficiency and juction temperature of high power AlGaInP light-emitting diode
Chen Yi-Xin, Shen Guang-Di, Gao Zhi-Yuan, Guo Wei-Ling, Zhang Guang-Chen, Han Jun, Zhu Yan-Xu
Acta Physica Sinica. 2011, 60 (8): 087206 doi: 10.7498/aps.60.087206
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One of the main problems of high power AlGaInP light-emitting diode (LED) is the heat generated seriously at large working current, which is caused by the weak current spreading, the photon blocking and absorbing of p-type or n-type electrode, and the critical reflection at the interface between the device and air. The heat inside can lead to the restriction on light output, and gives rise to low light efficiency and the low luminous intensity. In this paper, we introduce a new LED structure which is composed of compound current spreading layers and compound distribute Bragg reflector (DBR) layers. For the new structure LED, the injected current spreads adequately and the reflectivity is improved by the compound DBR layers. The testing results show that the performance of new structure LED is much better than that of the conventional LED, and that at a working current of 350 mA, the output powers of the two kinds of LEDs (which are unpackaged) are 17 and 49.48 mW respectively. At the same time, the heat testing results show the relationship between LED light efficiency and juction temperature, and the consistence between the juction temperature ratio and the ratio of light efficiency for the two kinds LEDs, which implies that LED light efficiency can be improved by reducing heat generated inside and reducing the juction temperature.
Surface-plasmon-mediated emission enhancement from Ag-capped ZnO thin films
Qiu Dong-Jiang, Fan Wen-Zhi, Weng Sheng, Wu Hui-Zhen, Wang Jun
Acta Physica Sinica. 2011, 60 (8): 087301 doi: 10.7498/aps.60.087301
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Ag/ZnO bilayer thin films are fabricated on Si substrates via two-step approach of "ZnO sputtering + Ag evaporation". The enhancement of the near band edge (NBE) emission of the ZnO film is realized through coupling between the surface plasmon resonating energy at Ag/ZnO interface and the photonic energy of ZnO NBE emission. The dependence of the emission enhancement ratio η of ZnO on the thickness and the growth temperature T of Ag cap-layers are investigated. By evaporating Ag(8 nm) cap-layer onto ZnO(100 nm) film at high substrate temperatures (T≥300 ℃), the η value reaches about 18,i.e., η≈18, which is more than twice that of Ag(8 nm)/ZnO(100 nm) bilayer films grown at low temperatures (T≤200 ℃). It is found that the realization of the larger η can be ascribed to the bigger surface roughness of Ag/ZnO bilayer samples prepared under higher growth temperatures.
Effect of palladium adsorption on the electrical transport of semiconducting carbon nanotubes Hot!
Zhao Hua-Bo, Wang Liang, Zhang Zhao-Hui
Acta Physica Sinica. 2011, 60 (8): 087302 doi: 10.7498/aps.60.087302
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The metal Pd is deposited on semiconducting single-walled carbon nanotubes (SWNTs) by physical vapor deposition. The image of scanning electron microscopy shows that the Pd nanoparticles (10—30 nm) are formed on the carbon nanotubes. It is found by the conductive atomic force microscopy that with the increase of Pd nanoparticles, the semiconducting carbon nanotube is changed gradually into a metallic one. Furthermore, our density functional theory calculation demonstrates that with the Pd adsorption increasing the band gap of the SWNT becomes smaller, and eventually disappears, which is in good agreement with the experimental result.
Properties of MgB2 ultra-thin films grown by hybrid physical-chemical vapor deposition
Sun Xuan, Huang Xu, Wang Ya-Zhou, Feng Qing-Rong
Acta Physica Sinica. 2011, 60 (8): 087401 doi: 10.7498/aps.60.087401
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We fabricate MgB2 ultra-thin films via hybrid physical-chemical vapor deposition technique. Under the same background pressure, the same H2 flow rate, by changing B2H6 flow rate and deposition time, we fabricate a series of ultra-thin films with thickness ranging from 5 nm to 80 nm. These films grow on SiC substrate, and are all c-axis epitaxial. We study the Volmer-Weber mode in the film formation. As the thickness increases, critical transition temperature Tc(0) also increases and the residual resistivity decreases. Especially, a very high Tc(0) ≈ 32.8 K for the 7.5 nm film, and Tc(0) ≈ 36.5 K, low residual resistivity ρ(42 K)≈ 17.7 μΩcm, and extremely high critical current density Jc (0 T,4 K) ≈ 107 A/cm2, upper critical field Hc2(0) for 10 nm film are achieved. Moreover, by optimizing the H2 flow rate, we obtain relatively smooth surface of the 10 nm epitaxial film, with a root-mean-square roughness of 0.731 nm, which makes them well qualified for device applications.
Preparation and electric field driven magnetoelectric effect for multiferroic La2/3Sr1/3MnO3/BaTiO3 composite films
Li Ting-Xian, Zhang Ming, Wang Guang-Ming, Guo Hong-Rui, Li Kuo-She, Yan Hui
Acta Physica Sinica. 2011, 60 (8): 087501 doi: 10.7498/aps.60.087501
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Using pulsed laser deposition, multiferroic La2/3Sr1/3MnO3(LSMO)/BaTiO3(BTO) composite films are deposited on LaAlO3 (LAO)(001) substrate. X-ray diffraction results show that LSMO and BTO films exhibit only (001) orientation. Film smoothness is verified by their low root-mean-square surface roughness values as 1.4 nm from atomic force microscope study. The magnetic and the electric properties of these composite films are investigated. Furthermore, the variations of resistivity and metal-insulator transition temperature TMI of LSMO, induced by the external electric field, are studied. The resisitivity is reduced while the TMI is enhanced for hole accumulation state which is induced by negative electric field across BTO layer. In contrast, the resistivity is enhanced while the TMI is reduced for hole depletion state, which shows coupling between magnetic and electric order parameters, i.e., there is a magnetoelectric effect induced by electric field.
Local lattice structure and spin singlet contribution to zero-field splitting of ZnS:Cr2+
Lu Cheng, Wang Li, Lu Zhi-Wen, Song Hai-Zhen, Li Gen-Quan
Acta Physica Sinica. 2011, 60 (8): 087601 doi: 10.7498/aps.60.087601
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Using the unified ligand-field-coupling scheme, the 210×210 complete energy matrices including all the spin states for d4 configuration transition metal ions are constructed within a strong field representation. By diagonalizing the complete energy matrices, the local lattice structure and the Jahn-Teller energy of Cr2+ ions doped into ZnS are investigated. It is found that the theoretical results are in good agreement with the experimental data. Moreover, the contribution of the spin singlet to the zero-field splitting (ZFS) parameter of Cr2+ ions doped into ZnS is also investigated. The results indicate that the spin singlet contribution to ZFS parameter D is negligible, but the contribution to ZFS parameters a and F may not be neglected.
Design of a wide-band metamaterial absorber based on loaded magnetic resonators
Gu Chao, Qu Shao-Bo, Pei Zhi-Bin, Xu Zhuo, Bai Peng, Peng Wei-Dong, Lin Bao-Qin
Acta Physica Sinica. 2011, 60 (8): 087801 doi: 10.7498/aps.60.087801
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A wide-band, polarization-insensitive and wide-angle metamaterial absorber based on loaded magnetic resonator is proposed. A single unit cell of the absorber is comprised of a magnetic resonator loaded with lumped elements, a substrate and a back metal board. Simulated absorbances of the one-dimensional-array absorber under loading and unloading conditions indicate that compared with under the unloading condition, the one-dimensional absorber under the loading condition can realize a wide-band absorption. Simulated absorbances of the one-dimensional-array absorber with lossy and loss-free substrates indicate that the power loss in the absorber results from lumped resistances in magnetic resonators, and is insensitive to the loss of the substrate. Simulated absorbances of the one-dimensional-array absorber with different lumped resistances and capacitances indicate that there exist optimal values for lumped resistances and capacitances, where the absorbance is highest and the bandwidth is widest. Simulated absorbances of the two-dimensional-array absorber under different polarization angles and different incident angles indicate that the absorber is polarization-insensitive and angle-wide.
Design of a wide-band metamaterial absorber based on resistance films
Gu Chao, Qu Shao-Bo, Pei Zhi-Bin, Xu Zhuo, Lin Bao-Qin, Zhou Hang, Bai Peng, Gu Wei, Peng Wei-Dong, Ma Hua
Acta Physica Sinica. 2011, 60 (8): 087802 doi: 10.7498/aps.60.087802
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A wide-band, polarization-insensitive and wide-angle metamaterial absorber is presented, which is based on resistance films. A unit cell of the absorber consists of a hexagonal resistance film, a substrate and a metal backboard. Simulated reflectances and absorbances indicate that this absorber has a wide-band strong absorption for the incedent wave from 7.0 GHz to 27.5 GHz, indicating that electrocircuit resonances are more suited to realize a wide-band strong absorption than electromagnetic resonances. Simulated absorbances under different polarization angles and different incident angles show that this absorber is polarization-insensitive and angle-wide. Simulated influence of substrate and resistance film on the absorbance of the absorber indicates that there exist optimal values for the capacitance between the resistance film and the metal backboard and for the resistance of the resistance film, where electrocircuit resonances are strongest and the absorption band is widest.
Femtosecond photoinduced magnetization of terbium gallium garnet crystal
Jin Zuan-Ming, Guo Fei-Yun, Ma Hong, Wang Li-Hua, Ma Guo-Hong, Chen Jian-Zhong
Acta Physica Sinica. 2011, 60 (8): 087803 doi: 10.7498/aps.60.087803
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The photoinduced magnetization in magneto-optical crystal terbium gallium garnet (TGG) is investigated by time-resolved pump-probe spectroscopy. When the pump pulse is elliptically polarized, the rotation signal and the ellipticity signal of the probe pulse are observed at zero time delay, resulting from the optical Kerr effect and the inverse Faraday effect. The direction of the effective magnetic field is dominated by the helicity of the pump pulse, so the rotation signal and the ellipticity signal of the probe pulse can be triggered selectively by modifying the helicity of the pump pulse. The full widths at half maximum of the rotation signal and the ellipticity signal both can be as fast as about 500 fs, which indicates that TGG crystal is expected to be a candidate material of ultrafast all-optical magnetic switching.
Excitation spectrum intensity adjustment of SrWO4:Eu3+ red phosphors for light-emitting diode
Ren Yan-Dong, Lü Shu-Chen
Acta Physica Sinica. 2011, 60 (8): 087804 doi: 10.7498/aps.60.087804
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The SrWO4:Eu3+ red phosphors with different Eu3+ doping concentrations and different sintering temperatures are prepared by the co-precipitation method. The powders as-prepared exhibit sharply red characteristic emissions of Eu3+ ions at room temperature. The near-ultraviolet and blue light absorption intensities are controlled by adjusting sintering temperature and doping concentration and so the red emission intensities under the 395 or 465 nm excitation can be adjusted. Our results show that the SrWO4:Eu3+ red phosphors can be effectively excited by the ultraviolet light, the near-ultraviolet (395 nm) light and 465 nm blue light. Therefore the SrWO4:Eu3+ red phosphors may have a potential application to white light-emitting diodes.
Effect of spin-coating process on the performance of passive-matrix organic light-emitting display
Liu Nan-Liu, Ai Na, Hu Dian-Gang, Yu Shu-Fu, Peng Jun-Biao, Cao Yong, Wang Jian
Acta Physica Sinica. 2011, 60 (8): 087805 doi: 10.7498/aps.60.087805
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By improving the spin-coating process during the deposition of hole transport layer poly(3,4-ethylenedioxythiophene)-poly(styrene-sulfonate)(PEDOT:PSS), high efficient monochrome passive-matrix organic light-emitting display is fabricated. The PEDOT:PSS film is spin-coated with a two-step spin-coating process, in which the substrate is turned 180° during the spin-coating, forcing the piled-up materials to move reversely. By introducing the second spin-coating step, the film thickness difference between single and double lines significantly reduces, leading to a more uniform light emission and a larger fill factor for the 3.81 cm monochrome polymer light-emitting diode display. Green light-emitting display with a peak current efficiency of 17 cd/A is successfully fabricated, and the efficiency is improved by 40 percent compared with that made by traditional spin-coating method. A 3.81 cm 96×64 full color display with a current efficiency of 1.25 cd/A is also successfully made.
Effects of incident polarization and electric field coupling on the surface plasmon properties of square hollow Ag nanostructures
Li Shan, Zhong Ming-Liang, Zhang Li-Jie, Xiong Zu-Hong, Zhang Zhong-Yue
Acta Physica Sinica. 2011, 60 (8): 087806 doi: 10.7498/aps.60.087806
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Square hollow nanostructure can induce a large-area enhanced electric field at the main plasmon peak. Therefore, it can be used as a substrate for the surface enhanced Raman scattering. The effects of the incident polarization on the extinction spectrum and the electric field distribution of the square Ag nanostructure are studied by the discrete dipole approximation method. The results show that the plasmon peaks do not shift with the variation of incident polarization. However, the electric field distribution is strongly dependent on the direction of incident polarization. Additionally, the effect of the electric field coupling between adjacent square Ag nanostructures on the plasmon mode is also studied. It is found that the plasmon resonance can be tuned by varying the separation between adjacent squares. These results could be used to guide the preparation of such closed nanostructures for specific plasmonic applications.
Investigation on the band structures of AlN/InN and AlN/GaN superlattices
Lu Wei, Xu Ming, Wei Yi, He Lin
Acta Physica Sinica. 2011, 60 (8): 087807 doi: 10.7498/aps.60.087807
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The band structures of wurtzite-AlN/InN and AlN/GaN superlattices are calculated by the Krönig-Penney model and the deformation potential theory under considering the lattice strain. Our calculations include the variation of band structure with the parameters for the sublayers, and the energy dispersion relations. It is found that by varying the sublayer thickness, the band structures can be well designed in different ways. The strain will change the bandgaps, reduce the band offsets and the sub-bands obviously, and make the valence band more complex. In comparison with the experimental results, our model is rather suited for simulating the narrow-quantum-well structures, while for the wide-quantum-well structures, the build-in field should be considered.
Preparation of lamina-shape TiO2 nanoarray electrode and its electron transport in dye-sensitized solar cells
Hari Bala, Shi Lan, Jiang Lei, Guo Jin-Yu, Yuan Guang-Yu, Wang Li-Bo, Liu Zong-Rui
Acta Physica Sinica. 2011, 60 (8): 088101 doi: 10.7498/aps.60.088101
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Lamina-shape TiO2 nanoarrays (LTNA) film electrode which is vertically grown on the surface of a Ti sheet by the means of hydrogen peroxide oriented etching at low temperature. X-ray diffraction shows that amorphous phase transforms to highly-crystalline anatase phase of the LTNA film after having been calcined at 500 ℃ for 1 h. Field emission scanning electron microscope exhibits a vertically oriented lamina-shape array with the morphology uniformly distributed and perfectly coated on the surface of Ti sheet, and the average height (film thickness), width and thickness of the leave are 1.35 μm, 30—80 nm and 10—15 nm respectively, after 1 d etching in hydrogen peroxide at 80 ℃. The LTNA electrodes exhibit similar morphologies except for the film with a thickness of 2.12 μm by hydrogen peroxide etching for 2 d. Using the LTNA film electrode as photoanode based on dye C106 fabricate back-illumination type dye-sensitized solar cell (DSC), a power conversion efficiency can reach 3.2% under an irradiation of air mass 1.5 global (100 mW·cm-2) simulated sunlight. Mesoporous TiO2 films are also used in the fabrication of DSC under similar conditions. The devices are compared with each other by transient photoelectric attenuation and electrical impedance technique. The results demonstrate that the LTNA-electrode DSC has a much lower recombination rate and a longer electron life time.
Growth of Ge quantum dot at the mix-crystal interface self-induced on the ion beam sputtering deposition
Xiong Fei, Pan Hong-Xing, Zhang Hui, Yang Yu
Acta Physica Sinica. 2011, 60 (8): 088102 doi: 10.7498/aps.60.088102
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The dense domes of Ge quantum dots on Si (001) substrate with a monomodal morphology distribution are deposited at different temperatures by ion beam sputtering (IBS). The areal density of the Ge quantum dots is observed to increase with elevating temperature, but the dots size to decrease. As the deposition temperature increases to 750 ℃, the smaller Ge quantum dots each with a height of 14.5 nm and base width of 52.7 nm are obtained by sputtering 15 monolayer Ge coverage, and the dots areal density is up to 1.68×1010 cm-2 at the same time. Thus the evolution of Ge quantum dot prepared by IBS is very different from that by vapor deposition at thermal equilibrium condition. The stable shape and the size distribution are demonstrated to result from the kinetic behavior of the surface atoms which is restricted by the thermodynamic limitations. A mix-crystal interface including amorphous and crystal components is revealed by Raman spectrum, and this special interface is demonstrated to contribute to the high density of Ge quantum dots, since the boundaries between the two different components can provide more preferential centers for the nucleation. As the density increases at high deposition temperature, the elastic repulsion between islands is enhanced, resulting in the surface atoms growing along the orientation of high index during the IBS deposition, and inducing the increase in aspect ratio and the reduction in island size.
Preparation and structure characterization of Pd thin films by supercritical fluid deposition
Wang Yan-Lei, Zhang Zhan-Wen, Li Bo, Jiang Bo
Acta Physica Sinica. 2011, 60 (8): 088103 doi: 10.7498/aps.60.088103
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Pd films are deposited on the Si wafers by the reduction of palladium(Ⅱ) hexafluoroacetylacetonate, which is used as the precursor, in the supercritical CO2 solution at temperature 100 ℃ and pressures between 12 and 18 MPa, and with reaction for 10—20 h. The films are continuous, uniform and 0.3—1.5 μm thick. The analyses of the Pd films by X-ray photoelectron spectroscopy and X-ray diffraction indicate that the structures of the deposited films are of single matter and nanocrystalline. The scanning electron microscope images show that pressure is a factor of affecting the size of the grain of the deposited film. At a pressure of 12 MPa, the size of grain is between 30 and 60 nm, at a pressure of 15 MPa, it is between 90 and 120 nm. Moreover, at a pressure of 18 MPa, it is between 150 and 180 nm. At the same temperature, with higher pressures, the size of the grain is bigger. On the same conditions, Pd thin films are deposited on the inner and the outer surfaces of cylindrical cavity.
Simulation of multi-grain solidification and subsequent spinodal decomposition by using phase field crystal model
Zhang Qi, Wang Jin-Cheng, Zhang Ya-Cong, Yang Gen-Cang
Acta Physica Sinica. 2011, 60 (8): 088104 doi: 10.7498/aps.60.088104
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The phase field crystal model (PFCM) is employed to simulate the process of multi-grain solidification and its subsequent spinodal decomposition in a binary alloy system. Simulation results show that the PFCM can reproduce the whole process of the important material processing phenomena of multi-grain solidification, including nucleation, growth, coarsening and grain boundary formation. Furthermore, the PFCM can also successfully simulate the full course of the multi phase transformation process from solidification to spinodal decomposition.
Detection of seam tracking offset based on infrared image during high-power fiber laser welding
Gao Xiang-Dong, Mo Ling, Zhong Xun-Gao, You De-Yong, Katayama Seiji
Acta Physica Sinica. 2011, 60 (8): 088105 doi: 10.7498/aps.60.088105
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Seam tracking is a significant precondition to obtain good welding quality. During the laser welding, the laser beam focus must be controlled to follow the welding seam accurately. A novel approach to detecting the offset between the laser beam focus and the welding seam based on infrared image processing is investigated during high-power fiber laser butt-joint welding of type 304 austenitic stainless steel plates at a continuous wave fiber laser power of 10 kW. The joint gap width is less than 0.1 mm. An infrared sensitive high speed camera arranged in off-axis direction of laser beam is used to capture the dynamic thermal images of a molten pool. The characteristics of thermal distribution and infrared radiation of the molten pool, when the laser beam focus is deviated from the welding seam center, are analyzed. Two parameters called the keyhole morphological parameter and the heat accumulation effect parameter are defined as the characteristic values of seam tracking offset to determine the offset between the laser beam focus and the desired welding seam. Also, the image processing technique is used to analyze the infrared images of the molten pool, which indicates the presence of mathematic correlation between the defined two parameters and the seam tracking offset. The welding experiments confirm that the offset between the laser beam focus and the welding seam can be estimated by the keyhole morphological parameter and the heat accumulation effect parameter effectively.
Influence of annealing on thermal stability of IrMn-based magnetic tunnel juctions
Yan Jing, Qi Xian-Jin, Wang Yin-Gang
Acta Physica Sinica. 2011, 60 (8): 088106 doi: 10.7498/aps.60.088106
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The magnetic tunnel junction with a structure of IrMn/CoFe/AlOx/CoFe is deposited by magnetron sputtering and annealed at different temperatures in a magnetic field of parallel to the orienting field. Vibrating sample magnetometer is used to record the magnetic hysteresis loop at room temperature, and scanning probe microscope is used to record the interface morphology. The influence of annealing on thermal stability of the magnetic tunnel junction is investigated by holding the film in its negative saturation field. After annealing, the exchange bias increases due to the enhancement of unidirectional anisotropy of antiferromagnetic layer. The recoil loop of the pinned ferromagnetic layer shifts towards the positive field, and the exchange bias field decreases monotonically, with the film held in a negative saturation field, whereas annealing reduces the reduction speed of Hex.
Molecular dynamics simulation of low-energy sputtering of Pt (111) surface by oblique Ni atom bombardment
Yan Chao, Duan Jun-Hong, He Xing-Dao
Acta Physica Sinica. 2011, 60 (8): 088301 doi: 10.7498/aps.60.088301
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The low-energy sputtering on Pt (111) surface by Ni atom at incident angle in a range of 0°— 80° (with respect to the direction normal to the surface) is studied by molecular dynamics simulations. The atomic interaction potential obtained with embedded atom method is used in the simulation. The dependence of sputtering yield, energy and angular distribution of sputtered particles as well as sticking probability of Ni atom on incident angle are discussed. The dependence of sputtering yield on incident angle θ can be divided into three different regions in θ, i.e., θ ≤ 20°, 20° ≤ θ ≤ 60°, and θ ≥ 60°. Based on sticking probability and movement of incident atom, physical mechanism of low-energy sputtering at oblique particle bombardment is suggested. When the incident angle θ is smaller than 20°, the reflection of incident atom by target atom dominates the sputtering process of surface atom, which is similar to the sputtering mechanism for the case of θ = 0°. While for 20° ≤ θ ≤ 60°, the reflection of incident atom is no longer important for the low-energy sputtering. For the case of θ ≥ 60°, there occurs no sputtering.
A quasi transverse electromagnetic mode waveguide developed by using metal-patch electromagnetic bandgap structure
Ren Li-Hong, Luo Ji-Run, Zhang Chi
Acta Physica Sinica. 2011, 60 (8): 088401 doi: 10.7498/aps.60.088401
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To solve the narrow operation-band of the electromagnetic bandgap (EBG) waveguide, a quasi transverse electromagnetic (TEM) mode waveguide using the metal-patch EBG structure as sidewall is proposed in this paper. Theoretical analysis and numerical calculation show that broader bandwidth, better transportation property, and more uniform electric field distribution of the quasi-TEM mode can be reached. Simulation results using Ansoft HFSS indicate that the metal-patch EBG structure can convert TE10 mode into quasi-TEM mode in the central frequency of 14 GHz with a bandwidth of 1.7 GHz and the uniformity of electric field distribution reaches 84.7% in 83.9% cross section area.
An X band synthesizer for a few hundred megawatt level power microwaves
Fang Jin-Yong, Huang Hui-Jun, Zhang Zhi-Qiang, Zhang Xiao-Wei, Zhang Li-Jun, Zhang Qing-Yuan, Hao Wen-Xi, Huang Wen-Hua, Jiang Wei-Hua
Acta Physica Sinica. 2011, 60 (8): 088402 doi: 10.7498/aps.60.088402
Full Text: [PDF 2161 KB] Download:(1060)
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A synthesis method for a few hundred megawatt level power microwave is presented in this paper. Based on the coupling wave theory and the polarized wave orthogonal theory, the pulse series of one gigawatt level power microwave and one hundred megawatt level power microwave can be put in two separate ports and put out from one common port. The synthesizer is unitized by two cylindrical waveguides which are back to back combined; the cylindrical waveguide which is joined with the output port is named main channel, and the other cylindrical waveguide is called associate channel. The main channel transmits horizontally polarized TE011 mode microwave, and the operation frequency band is only limited to the barrier frequency λc. The associate channel transmits vertical polarized TE011 mode microwave, and the operation frequency band can reach up to several hundred mega hertz. High power experiment indicates that the transmission energy efficiency of the main channel is nearly 100% and the coupling energy efficiency of the associate channel is above 87%, the power capacity of the main channel is more than 1GW and that of the associate channel is about 300 MW.
Radiation-resistant bipolar n-p-n transistor
Zhai Ya-Hong, Li Ping, Zhang Guo-Jun, Luo Yu-Xiang, Fan Xue, Hu Bin, Li Jun-Hong, Zhang Jian, Su Ping
Acta Physica Sinica. 2011, 60 (8): 088501 doi: 10.7498/aps.60.088501
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Bipolar n-p-n transistor geometrical parameters are optimized based on the principle of minimizing the perimeter-to-area ratio (P/A). Three types of radiation-resistant n-p-n transistors are developed and fabricated in the 20 V bipolar process. The first is emitter-base junction hardened n-p-n transistor. The second has heavily boron doped base ring. And the last uses both radiation-resistant measurements. The experimental results indicate that after irradiated by the radiation of total dose of 1 kGy, in current gain, the common n-p-n(unhardened) transistor reduces about 60%—65%, while the first two hardened n-p-n transistors increases 10%—15%: the last hardened n-p-n transistors are 15%—20% greater than the common n-p-n transistors in current gain.
Efficient white polymeric light-emitting diodes by doping ionic iridium complex
Zhao Bao-Feng, Tang Huai-Jun, Yu Lei, Wang Bao-Zheng, Wen Shang-Sheng
Acta Physica Sinica. 2011, 60 (8): 088502 doi: 10.7498/aps.60.088502
Full Text: [PDF 1266 KB] Download:(629)
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A series of white polymer light-emitting diodes (WPLEDs) each with a single emitting layer using ionic iridium complex is fabricated. The white light is obtained via two complementary colors of orange light emitter ionic iridium complexe PF6 (Hnpy: 2-(naphthalen-1-yl)pyridine, c-phen: 1-ethyl-2-(9-(2-ethylhexyl)-9H-carbazol-3-yl)-1H-imidazo phenanthroline) and sky-blue light emitter Firpic (iridium bis(2-(4,6-difluorophenyl)-pyridinato-N,C(2)) picolinate. The emtting layer consists of poly(N-vinylcarbzole) (PVK) as host polymer, 1,3-bis -phenylene (OXD-7) as electron-transporting materials, Firpic and PF6. The structure of the WPLED is indium-tin-oxide/poly(3,4-ethylenedioxythiophene) doped with poly(styrenesulfonate)(40 nm)/emtting layer (80 nm)/CsF(1.5 nm)/Al(120 nm). When the mass ratio of PVK, OXD-7, Firpic, PF6 is 67 ∶23 ∶10 ∶0.25, the most efficient white light is obtained with a color coordinate of (0.31,0.40), a maximal luminance efficiency of 13.3 cd/A and a maximal luminance of 6032 cd/m2. Meanwhile, the color coordinate is unchanged with current density. The mechanism of the WPLED is discussed.
Dynamic properties of bilayer membrane
Peng Yong-Gang, Zheng Yu-Jun
Acta Physica Sinica. 2011, 60 (8): 088701 doi: 10.7498/aps.60.088701
Full Text: [PDF 1299 KB] Download:(553)
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We study the dynamics of bilayer membrane using the Fourier space Brownian dynamical equation. The surface of the bilayer membrane is demonstrated directly by using three-dimensional figure. Our results demonstrate that the slipping between the up-monolayer and bottom-monolayer is a very important dynamical process in the membrane dynamics, which strongly affects the height-height correlation function.
Oxygen and carbon behaviors in multi-crystalline silicon and their effect on solar cell conversion efficiency
Fang Xin, Shen Wen-Zhong
Acta Physica Sinica. 2011, 60 (8): 088801 doi: 10.7498/aps.60.088801
Full Text: [PDF 2673 KB] Download:(1538)
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Understanding and controlling the impurity behavior are important for low-cost and high-efficiency of multi-crystalline silicon solar cells. We employ the infrared spectroscopy to study the change of oxygen and carbon concentrations after thermal treatment in different parts of multi-crystalline silicon ingots grown by directional solidification technology. In correlation with the solar cell performances such as the minority carrier lifetime, photoelectric conversion efficiency and internal quantum efficiency, we investigate the physical mechanism of the effects of various concentrations of oxygen and carbon on cell performance. We propose an oxygen precipitation growth model considering the influence of carbon to simulate the size distribution and concentration of oxygen precipitation after the thermal treatment. It is found that carbon not only deteriorates the efficiency of the cells made from the silicon from the top part of the ingot, but also plays an important role in the effect of oxygen precipitation: enhancing the size and the quantity of oxygen precipitation in the silicon from the middle part of the ingot, which induces the defect and increases the recombination; while resulting in the small size and low quantity of oxygen precipitation in the silicon from the bottom part due to the low carbon content, thereby improving the cell efficiency through gettering impurities. We further demonstrate the complex behaviors of oxygen and carbon by a two-step thermal treatment technique, from which we point out that the two-step thermal treatment is applicable only to the improvement of the efficiency of solar cells from the bottom part of multi-crystalline silicon ingots.
Design of a highly efficient light-trapping structure for amorphous silicon solar cell
Zhou Jun, Sun Yong-Tang, Sun Tie-Tun, Liu Xiao, Song Wei-Jie
Acta Physica Sinica. 2011, 60 (8): 088802 doi: 10.7498/aps.60.088802
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A highly efficient light-trapping structure consisting of a diffractive grating, a MgF2 film, a ZnS film and Ag reflector, is used for a-Si:H solar cell. Using the rigorous coupled wave theory, the weighted absorptance photon number (ξAM1.5) of a 1 μm thick a-Si:H solar cell is calculated in a wavelength range from 400 to 1000 nm for the AM1.5 solar spectrum at 25 ℃. It is used to design the optimal parameters of the light-trapping structure. Results indicate that ξAM1.5 of the solar cell can reach 74.3%, if the period, the depth and the duty cycle of the diffractive grating, and the height of the MgF2 film and the ZnS film are 800 nm, 160 nm, 0.6125, 90 nm and 55 nm, respectively. If a ZnS/Ag film is fabricated on the rear surface of the solar cell, a larger ξAM1.5 (76.95%) can be obtained. It is demonstrated that the trapping structure is useful for elevating the efficiency of the solar cell.
Effect of 3,4,9,10-perylenetetracarboxylic dianhydride on the performance of ZnO nanorods/polymer hybrid solar cell
Yan Yue, Zhao Su-Ling, Xu Zheng, Gong Wei, Wang Da-Wei
Acta Physica Sinica. 2011, 60 (8): 088803 doi: 10.7498/aps.60.088803
Full Text: [PDF 3037 KB] Download:(699)
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ZnO nanoroods/poly (MEH-PPV) hybrid solar cells are fabricated and their properties are discussed. In order to improve the absorption of sunlight, a layer of 3,4,9,10-perylenetetracarboxylic dianhydride(PTCDA) is inserted between ZnO nanorods and MEH-PPV, and cells with the structure ITO/ZnO nanorods/PTCDA/MEH-PPV/Au are prepared with different thickness values of PTCDA. After introducing PTCDA, the devices show a strong and broad absorption in visible region, which increases the number of photo-induced excitons and thus results in an enlarged photocurrent. When the thickness of PTCDA is 40 nm, we can observe the smooth morphology of thin layer surface, and achieve the best performance of the device.
Some new integrable nonlinear dispersive equations and their solitary wave solutions
Yin Jiu-Li, Fan Yu-Qin, Zhang Juan, Tian Li-Xin
Acta Physica Sinica. 2011, 60 (8): 080201 doi: 10.7498/aps.60.080201
Full Text: [PDF 1060 KB] Download:(690)
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Under different system parameters, some new integrable models of the generalized modified Dullin-Gottwald-Holm are obtained by Painlevé analysis. Using the auto-Backlund transformation, solitary wave solutions of these integrable models are obtained.
Asymptotic solution to the delay sea-sir oscillator for El Niño/La Niña-southern oscillation mechnism
Mo Jia-Qi, Lin Wan-Tao, Lin Yi-Hua
Acta Physica Sinica. 2011, 60 (8): 080202 doi: 10.7498/aps.60.080202
Full Text: [PDF 1136 KB] Download:(656)
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A class of coupled system of the El Niño/La Niña-southern oscillation mechanism is studied. Using the asymptotic analytic perturbation method and the simple and valid technique, the asymptotic expansions of solution to the El Niño/La Niña-southern oscillation model are obtained and the asymptotic behavior of solution to the corresponding problem is considered.
Fidelity of the photon subtracted (or added) squeezed vacuum state and squeezed cat state
Lü Jing-Fen, Ma Shan-Jun
Acta Physica Sinica. 2011, 60 (8): 080301 doi: 10.7498/aps.60.080301
Full Text: [PDF 1972 KB] Download:(904)
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In this paper, the fidelity of photon subtracted (or added) squeezed vacuum state with arbitrary number of photons and squeezed cat state is derived analytically. The result shows that whether the photon is added or subtracted, the maximum fidelity increases with the increase of the change of photon number, and the amplitude of the superposition state corresponding to the maximum fidelity also increases. In addition, for the same number of subtracted or added photons, the amplitude of the superposition state corresponding to the maximum fidelity in the case of added photon is larger than in the case of subtracted photon, but the maximum fidelity in the case is smaller. Although it is more difficult to make photon added than photon subtracted the photon added can be used as an important method to obtain cat state of large amplitude.
Arbitrated quantum signature scheme based on entanglement swapping
Li Wei, Fan Ming-Yu, Wang Guang-Wei
Acta Physica Sinica. 2011, 60 (8): 080302 doi: 10.7498/aps.60.080302
Full Text: [PDF 1146 KB] Download:(949)
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An arbitrated quantum signature scheme based on entanglement swapping is proposed in this paper. On the foundation of Bell states, the message to be signed is coded with a unitary sequence, and consequently the unitary sequence is used to calibrate the Bell states between the signer and the arbitrator, finally the signature is generated through quantum cryptography. Using the correlation states generated through entanglement swapping on the arbitrator’s side, the receiver can verify the signature through Bell measurement on his own side. In this scheme, anyone except the authentic signer cannot forge a legal signature and the true receiver cannot deny his recipient because the security of underling quantum cryptography and the participants’privacy are effectively protected in this scheme.
Error correction and decoding for quantum stabilizer codes
Xiao Fang-Ying, Chen Han-Wu
Acta Physica Sinica. 2011, 60 (8): 080303 doi: 10.7498/aps.60.080303
Full Text: [PDF 1170 KB] Download:(828)
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Mapping the error syndromes to error operators is the core of quantum decoding network and the key step to realize quantum error correction. The definitions of the bit flip error syndrome matrix and the phase flip error syndrome matrix are presented, and then the error syndromes of Pauli errors are expressed in terms of the columns of the bit flip error syndrome matrix and the phase flip error syndrome matrix. It is also shown that the error syndrome matrix of a stabilizer code is determined by its check matrix, which is similar to the relationship between the classical error and the parity check matrix of classical codes. So, the techniques of error detection and error correction for classical linear codes can be applied to quantum stabilizer codes after some modifications. The error correction circuits are constructed based on the relationship between the error operator and error syndrom. The decoding circuit is constructed by reversing the encoding circuit because the encoding operators are unitary.
Generalized variational principles for Boussinesq equation systems
Cao Xiao-Qun, Song Jun-Qiang, Zhang Wei-Min, Zhu Xiao-Qian, Zhao Jun
Acta Physica Sinica. 2011, 60 (8): 080401 doi: 10.7498/aps.60.080401
Full Text: [PDF 1061 KB] Download:(800)
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The semi-inverse method is proposed by He to establish the generalized variational principles for physical problems, which can eliminate variational crisis brought by the Lagrange multiplier method. Via the He ’s semi-inverse method, a family of variational principles is constructed for the Boussinesq equation systems and variant Boussinesq equation systems of fluid dynamics. The obtained variational principles have also proved to be correct.
The brick-wall model unapplicable to the calculating of black hole entropy
Yang Xue-Jun, Zhao Zheng
Acta Physica Sinica. 2011, 60 (8): 080402 doi: 10.7498/aps.60.080402
Full Text: [PDF 1066 KB] Download:(696)
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The brick-wall model is widely used to calculate the entropies of static or stationary black holes. An ultraviolet cutoff factor needs to be introduced to remove the divergence of the result in brick-wall model. The cutoff factor has not been explained reasonably up to now. A study indicated that when the brick-wall model or thin film model was used to calculate the black hole entropy, the ultraviolet cutoff factor could be discarded if the generalized uncertainty relation was adopted. In this paper, it is proved that since the first term of Schwarzschild black hole entropy formula in the brick-wall model is not only the Bekenstein-Hawking term but also the term containing the ultraviolet cutoff factor, when the cutoff factor is removed, the Bekenstein-Hawking term is lost and the black hole entropy cannot be obtained by using the generalized uncertainty relation in brick-wall model.
A new global embedding approach to study Hawking and Unruh effects for higher-dimensional rotation black holes
Zhang Li-Chun, Li Huai-Fan, Zhao Ren
Acta Physica Sinica. 2011, 60 (8): 080403 doi: 10.7498/aps.60.080403
Full Text: [PDF 1074 KB] Download:(696)
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First, we effectively reduce the higher-dimensional rotation metric to a 2-dimensional metric near the event horizon which contains only the (t-r) sector. Then, we study the Unruh/Hawking temperature for (1+1)-dimensional space-time with the new global embedding method. It is shown that the viewpoint of Banerjee and Majhi is correct. We also extend the study to the case of higher-dimensional rotation black hole.
Thermodynamic properties of a weakly interacting Fermi gas in a strong magnetic field
Men Fu-Dian, Wang Bing-Fu, He Xiao-Gang, Wei Qun-Mei
Acta Physica Sinica. 2011, 60 (8): 080501 doi: 10.7498/aps.60.080501
Full Text: [PDF 1066 KB] Download:(729)
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Based on the "pseudopotential" method and the local-density approximation, the thermodynamic properties of a weakly interacting Fermi gas in a strong magnetic filed are studied, the integrated analytical expressions of thermodynamic quantities of the system are derived, and the effects of magnetic field as well as interparticle interactions on the thermodynamic properties of the system are analyzed. It is shown that at both high and low temperatures, magnetic field may adjust the effects of interacting. At low temperatures, magnetic field can lower the chemical potential, total energy and heat capacity of the system compared with the situation of Fermi gas in the absence of the magnetic field. The repulsive interactions may increase the chemical potential, but reduce the total energy and heat capacity of the system compared with the situation of non-interacting Fermi gas. At high temperatures, magnetic field as well as repulsive interactions can reduce the total energy and increase heat capacity of the system, moreover, strong magnetic field may change the effects of interaction on the total energy and the heat capacity of the system.
Merging crisis of chaotic saddle in a Duffing unilateral vibro-impact system
Feng Jin-Qian, Xu Wei
Acta Physica Sinica. 2011, 60 (8): 080502 doi: 10.7498/aps.60.080502
Full Text: [PDF 1698 KB] Download:(892)
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A computational investigation of chaotic saddles in a Duffing vibro-impact system is presented. Chaoctic saddle crisis is investigated in a duffing vibro-impact system considered. This crisis is due to the tangency of the stable and unstable manifolds of period saddle connecting two chaotic saddles. The threshold of tangency induces the merging crisis of chaotic saddle, that is, as the system parameter crosses the critical value, a larger boundary chaotic saddle appears due to the merging of two chaotic saddles located on the basin boundary and in the internal basin respectively. In fact, this chaotic saddle crisis is responsible for the merging crisis of chaotic attractors eventually.
Control of spiral waves in FitzHugh-Nagumo systems
Gao Jia-Zhen, Xie Ling-Ling, Xie Wei-Miao, Gao Ji-Hua
Acta Physica Sinica. 2011, 60 (8): 080503 doi: 10.7498/aps.60.080503
Full Text: [PDF 2717 KB] Download:(936)
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Control of spiral wave in two-dimensional FitzHugh-Nagumo equation is studied. The phase space compression approach is used to confine the system trajectory into a finite area and to annihilate spiral wave in the numerical simulation. Three stages are found in the control process. The spiral is driven to a homogenous stationary state when the compress limit is small; the spiral is stable with a fixed frequency when the compression limit is large; in the intermediate controlling parameter regime, the spatiotemporal turbulent state is observed. The controlling process is investigated by considering system pattern, variable evolution, phase space trajectory, etc, and the characteristics of amplitude function and oscillatory frequency are summarized as well.
Selective forgetting extreme learning machine and its application to time series prediction
Zhang Xian, Wang Hong-Li
Acta Physica Sinica. 2011, 60 (8): 080504 doi: 10.7498/aps.60.080504
Full Text: [PDF 1214 KB] Download:(1062)
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To solve the problem of extreme learning machine (ELM) on-line training with sequential training samples, a new algorithm called selective forgetting extreme learning machine (SF-ELM) is proposed and applied to chaotic time series prediction. The SF-ELM adopts the latest training sample and weights the old training samples iteratively to insure that the influence of the old training samples is weakened. The output weight of the SF-ELM is determined recursively during on-line training procedure according to its generalization performance. Numerical experiments on chaotic time series on-line prediction indicate that the SF-ELM is an effective on-line training version of ELM. In comparison with on-line sequential extreme learning machine, the SF-ELM has better performance in the sense of computational cost and prediction accuracy.
Cellular automaton simulation with directed small-world networks for the dynamical behaviors of spiral waves
Tian Chang-Hai, Deng Min-Yi, Kong Ling-Jiang, Liu Mu-Ren
Acta Physica Sinica. 2011, 60 (8): 080505 doi: 10.7498/aps.60.080505
Full Text: [PDF 1965 KB] Download:(1036)
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In this paper, we study the influence of the rewiring probability p of the directed small-world network on dynamical behavior of spiral wave using the Greenberg-Hastings cellular automaton model. The computer simulation results show that when p is small enough, the stable spiral wave under the regular networks keeps its stability unchanged, and when p is increased, the phenomenon such as meandering, breakup and disappearance of the spiral waves appears. The relation between the excitability and p leads to the conclusion that the occurrence of the above phenomenon originates from the reducing of the excitability. On the other hand, the period of cell is also related to p.
Spread spectrum communication based on hardware encryption
Xiao Bao-Jin, Tong Hai-Li, Zhang Jian-Zhong, Zhang Chao-Xia, Wang Yun-Cai
Acta Physica Sinica. 2011, 60 (8): 080506 doi: 10.7498/aps.60.080506
Full Text: [PDF 1288 KB] Download:(863)
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The scheme of 1 Gbit/s random sequence generated by chaotic laser as spreading code is proposed. Theoretical analysis indicates that the pseudo random sequence exhibited periodic behavior is eliminated by the random sequence and the capability of spreading code is enlarged. Meanwhile, the communication security can be improved by variable spreading code. The corresponding spread spectrum system is numerically simulated by Simulink software and the results demonstrate that the greater the used spreading gain, the lower the obtained error rate will be when the information speed is constant, which is consistent with the theoretical results. The anti-jamming ability of the spread spectrum system is strengthened and the security is enhanced compared with those of the traditional spreading system.
Modified function projective synchronization of a class of chaotic systems
Li Jian-Fen, Li Nong
Acta Physica Sinica. 2011, 60 (8): 080507 doi: 10.7498/aps.60.080507
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A general method of modifying function projective synchronization of a class of chaotic systems is proposed in this paper by designing a suitable response system. The two schemes of obtaining the response system from chaotic system are established based on unidirectional coupled synchronization. Since chaos synchronization can be achieved by transmitting only a single variable from driving system to response system, this method is more practical. The stability analysis in the paper is proved using Lyapunov stability theory. Numerical simulations of a hyperchaotic system verify the effectiveness of the proposed method.
Considering two-velocity difference effect for coupled map car-following model
Ge Hong-Xia, Cheng Rong-Jun, Li Zhi-Peng
Acta Physica Sinica. 2011, 60 (8): 080508 doi: 10.7498/aps.60.080508
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Based on the pioneer work of Konishi et al, a new coupled map car-following model is presented by considering the headway distance of two successive vehicles in front. The feedback control method presented by Zhao-Gao is utilized to suppress the traffic congestion in the coupled map car-following model. According to the control theory, the condition under which the traffic jam can be suppressed is analyzed. The results are compared with that presented by Konishi et al. The simulation results show that the new model with the feedback control method presented by Zhao-Gao could suppress the traffic jam effectively.
The stochastic energetics resonance of bistable systems and efficiency of doing work
Lin Min, Zhang Mei-Li, Huang Yong-Mei
Acta Physica Sinica. 2011, 60 (8): 080509 doi: 10.7498/aps.60.080509
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The interaction of work and heat between Brownian particles in a bistable system, the external periodic force and the thermal stochastic force are analyzed. The stochastic energy balance equation based on Langevin equation is established. For the Langevin equation subjected to periodic force, stochastic force and damping force, the method of combining dynamics and non-equilibrium thermodynamics is used. From force as the foothold changed into energy as the research core, the exchange of energy between system and environment and the efficiency of doing work are deeply analyzed with this method when the Brownian particle motion is along single trajectories, which reveals that the bistable system exhibits stochastic energetic resonance phenomenon.
Epidemic spreading in complex networks with spreading delay based on cellular automata
Wang Ya-Qi, Jiang Guo-Ping
Acta Physica Sinica. 2011, 60 (8): 080510 doi: 10.7498/aps.60.080510
Full Text: [PDF 1509 KB] Download:(923)
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In this paper, based on the cellular automata, we propose a new susceptible-infected-susceptible (SIS) model to study epidemic spreading in the networks with spreading delay. Theoretical analysis and simulation results show that the existence of spreading delay can significantly reduce the epidemic threshold and enhance the risk of outbreak of epidemics. It is also found that both the epidemic prevalence and the propagation velocity increase obviously with the spreading delay increasing. Moreover, the SIS model proposed in this paper describes not only the average propagation tendency of epidemics, but also the dynamic evolution process over the time of epidemics and the probability events such as outbreak and extinction of epidemics, and thus can overcome the limitations of the differential equation model based on mean-field method that describes only the average transmitting tendency of epidemics. Meanwhile, some suggestions of how to effectively control the propagation of epidemics are presented.
Ti:sapphire femtosecond comb with two spectral broadening parts
Cao Shi-Ying, Fang Zhan-Jun, Meng Fei, Wang Qiang, Li Tian-Chu
Acta Physica Sinica. 2011, 60 (8): 080601 doi: 10.7498/aps.60.080601
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The first generation of femtosecond optical frequency comb (FOFC) based on the Ti:sapphire femtosecond laser in National Institute of Metrology of China, is improved and optimized in this paper. By changing the repetition rate of the femtosecond laser, the spectral broadening parts and the beat frequency signal detection, the complexity of spectral broadening is reduced and the stability of FOFC and the convenience for the frequency measurement are improved. With such an FOFC, the absolute frequencies of the I2-stabilized Nd:YAG 532 nm laser and I2-stabilized He-Ne 633 nm laser are measured. The experimenal results are in accordance with the recommended values by the International Committee for Weights and Measures.
A novel large-orbit electron gun with gradually-changing reversal magnetic field
Wu Xin-Hui, Li Jia-Yin, Zhao Xiao-Yun, Li Tian-Ming, Hu Biao
Acta Physica Sinica. 2011, 60 (8): 080701 doi: 10.7498/aps.60.080701
Full Text: [PDF 1387 KB] Download:(742)
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A novel approach to achieve a large-orbit electron beam is demonstrated using a gradually-changing reversal magnetic field. On the basis of analyzing the general regularities of electron movement and various factors which lead to eccentricity and velocity spread in the gradually-changing reversal magnetic field, we design a large-orbit electron gun. Different from the traditional three-step method, our design does not pursuit the formation of thin tubular electron beam and the utilization of mutation reversal magnetic field, which reduces the difficulties in structure complexity and tube-making process. In addition, the cathode emission band can be placed in the axial magnetic field before the magnetic reversal point where its magnitude decreases gradually, by controlling the angular momentum difference between every trajectory starting points and using the offset effect of various unfavorable factors to reduce eccentricity and velocity spread. The simulation results are consistent with the theoretical analyses, which shows that the beam quality can be improved remarkably by fine-tuning electromagnetic fields, confirms that the efficiency and the applicability of the adjusting method we proposed, and provides a new technical way to obtain a high-quality large-orbit electron beam for high-efficiency large-orbit millimeter-wave devices.
Outgassing mass spectrum analysis with intense pulsed emission of carbon nanotube cathode
Shen Yi, Zhang Huang, Liu Xing-Guang, Xia Lian-Sheng, Yang An-Min
Acta Physica Sinica. 2011, 60 (8): 080702 doi: 10.7498/aps.60.080702
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The outgassing mass spectrum property with intense pulsed emission of the carbon nanotube(CNT) cathode is investigated on a 2 MeV linear induction accelerator injector by using the quadrupole mass spectrometer. The results show that the cathode has a capability of desorbing gases from the CNT cathode under pulsed high voltage. There are significant CO2, N2(CO) and H2 desorbing gases which play an important role in the formation of the cathode plasma. The mechanism of electron emission of CNT cathode is plasma-induced field emission, rather than explosive field emission, which is proved by analyzing the desorbed gases component.
X-ray Zernike apodized photon sieves for phase-contrast microscopy
Cheng Guan-Xiao, Hu Chao
Acta Physica Sinica. 2011, 60 (8): 080703 doi: 10.7498/aps.60.080703
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We present a kind of diffractive lens Zernike apodized photon sieves (ZAPS) whose structure is based on the combination of two concepts: apodized photon sieves and Zernike phase-contrast. Combined with the synchrotron light source, the ZAPS can be used as an objective for high-resolution phase-contrast X-ray microscopy in physical and life sciences. The ZAPS is a single optic unit that integrates the appropriate ±π/2 radians phase shift through selective zone placement shifts in an apodized photon sieve. The focusing properties of the ZAPS can be easily controlled by apodizing its pupil function. An apodized photon sieve with Gaussian pupil is fabricated by lithographic technique and shows that the side-lobes are significantly suppressed at the expense of slightly widening the width of the main lobe.
Research of Kramers-Krönig relationship for reconstruction of ultrashort electron longitudinal bunch profile by means of frequency domain measurements
Wu Dai, Liu Wen-Xin, Tang Chuan-Xiang, Li Ming
Acta Physica Sinica. 2011, 60 (8): 082901 doi: 10.7498/aps.60.082901
Full Text: [PDF 1229 KB] Download:(763)
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Kramers-Krönig (K-K) relationship is widely used in diagnostics of longitudinal bunch distribution with frequency domain measurement, and it has been used as a powerful tool in analysis of bunch profile and length. The study results show that the bunching beam parameters are severely affected by the choosing of the baseline of autocorrelation curve, power loss at low frequency and cutoff at high frequency, as well as extrapolation point, etc. As an example, the longitudinal bunching length is measured by means of coherent transition radiation in accelerator laboratory of Tsinghua University. We analyze the influence of the choice of parameters on measured experiment results and discuss the method of choosing the key parameters for the K-K transform.
High order harmonic generation from a two-dimensional model H atom in arbitrary polarized laser
Zhang Chun-Li, Feng Zhi-Bo, Qi Yue-Ying, Che Ji-Xin
Acta Physica Sinica. 2011, 60 (8): 083201 doi: 10.7498/aps.60.083201
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The two-dimensional time-dependent Schrödinger equation of arbitrary polarized laser pulse interacting with a model H atom is solved by using the two-dimensional asymptotic boundary condition (ABC) and symplectic algorithm. In order to investigate the influence of ellipticity on high order harmonic generation (HHG) of atom in arbitrary polarized laser field, we consider different ellipticities, and then compute the HHG for two-dimensional model H atom. Finally, we analyze the characteristics of HHG under different polarized laser fields. So it is reasonable and effective to extend the one-dimensional ABC and symplectic algorithm to the problem of laser interacting with a two-dimensional model atom.
Cross sections of Cq+(q=1—4)electron loss in collision with He, Ne and Ar investigating
Lu Yan-Xia, Xie An-Ping, Li Xiao-Hua, Xiang Dong, Lu Xing-Qiang, Li Xin-Xia, Huang Qian-Hong
Acta Physica Sinica. 2011, 60 (8): 083401 doi: 10.7498/aps.60.083401
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Electron-loss cross sections in Cq+(q=1—4)collisions with He,Ne,Ar are measured in the intermediate-velocity regime, and the ratio of the two-electron-loss cross section to the one-electrom-loss cross section, R21, are calculated. It is shown that a single-channel analysis is not sufficient to explain the results, and that projectile electron loss, electron capture by the projectile, and target ionization must be considered together to interpreter the data. The screening and antiscreening theory can annalyse the threshold velocity, but cannot totally explain the data of R21 with velocity increasing. The effective charge of target increases with velocity increasing. Ne and Ar have the same effective charges in the velocity regime, but He has a smaller one at the same velocity. The correlation between loss electrons is also analyzed.
Theoretical study of electronic structure and optical properties of OsnN0,±(n=1 — 6) clusters
Zhang Xiu-Rong, Wu Li-Qing, Rao Qian
Acta Physica Sinica. 2011, 60 (8): 083601 doi: 10.7498/aps.60.083601
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The possible geometrical and electronic structures of (OsnN)0,±(n=1—6) clusters are optimized by using the density functional theory (B3LYP) at the LANL2DZ level. For the ground state structures of (OsnN)0,±(n=1—6) clusters, The magnetic properties, the natural bond orbit (NBO), the spectrum and the aromatic characteristics are analyzed. The calculated results show that the magnetic moment of OsnN- cluster is quenched at n=1 and 5. Reversed ferromagnetic coupling between Os atom and N atom takes place in Os2N and Os4N0,± clusters. The NBO charge distribution of clusters depends on the relative position of the atom, for example, the charge transfer happening to N atoms in the endpoint is more obvious than that happening to the N atoms in the middle. There are obvious vibration peaks in IR and Raman spectra of (OsnN)0,±(n=1—6) clusters. The aromaticity of Os5N- cluster is the strongest.
Computational study on thermal stability of an AuCu249 alloy cluster on the atomic scale
Shao Chen-Wei, Wang Zhen-Hua, Li Yan-Nan, Zhao Qian, Zhang Lin
Acta Physica Sinica. 2011, 60 (8): 083602 doi: 10.7498/aps.60.083602
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Structural change of an AuCu intermetallic alloy cluster including 249 atoms during heating is studied by molecular dynamics simulation within the framework of embedded atom method. The analyses of pair-distribution function, atomic density function, and pair analysis technique show that the structural change of this cluster involves different stages from the outer part into the inner part owing to continuously interchanging positions among atoms at elevated temperature. During the change of the atom packing structure, gold atoms move from the inner part to the outer part of this cluster, whereas copper atoms move from the outer part into the inner part.
Numerical simulation of multipactor on dielectric surface in high direct current field
Cai Li-Bing, Wang Jian-Guo, Zhu Xiang-Qin
Acta Physica Sinica. 2011, 60 (8): 085101 doi: 10.7498/aps.60.085101
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The numerical simulation of multipactor in the dielectric surface breakdown in high direct current field is realized by using the particle-in-cell method. And the influence of the strength of the high direct current field, smoothness of the dielectric surface and secondary electron yield coefficient on the multipactor are researched through the simulation of multipactor. Finally, the influence of the tilting of high direct current field and external magnetic field on the multipactor are also investigated. The results show that selecting of the dielectric with low secondary electron yield coefficient and tilting of high direct current field can reduce the degree of multipactor, and for the external magnetic field the degree of multipactor decreases effectively only when the external magnetic field exceeds a certain value.
Analysis of formation mechanism of Li-like satellites in aluminum plasma and experimental application
Yu Xin-Ming, Cheng Shu-Bo, Yi You-Gen, Zhang Ji-Yan, Pu Yu-Dong, Zhao Yang, Hu Feng, Yang Jia-Min, Zheng Zhi-Jian
Acta Physica Sinica. 2011, 60 (8): 085201 doi: 10.7498/aps.60.085201
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Base on the code of steady K-shell model for collision radiative balance, we analyze the main mechanism of populating the Li-like satellite of aluminum plasma in detail. The evolutions of the Li-like satellite intensity ratios of 1s2p2 2P—1s22p2P, 1s2s2p(1S)2P—1s22s2S and 1s2p2 2D—1s22p2P with electron temperature and density are depicted from which we find that the intensity ratio is density sensitive and temperature unsensitive, which can be used to diagnose the electron density in hot dense plasma. At the same time, the evolution of the intensity ratio of He-like to He-like resonant with electron density is also given. The value of the ratio is large in the high electron density region due to the influence of the undistinguished Li-like statellite around the He-like intercombination. Finally, we simulate the spectra emitting from the hot dense aluminum plasmas generated by the "Shengguang-Ⅱ" equipment and give the electron temperature and density.
Magnetic shear effect on zonal flow generation in ion-temperature-gradient mode turbulence
Lu He-Lin, Chen Zhong-Yong, Li Yue-Xun, Yang Kai
Acta Physica Sinica. 2011, 60 (8): 085202 doi: 10.7498/aps.60.085202
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By decoupling the nonlinear fluid equations of ion-temperature-gradient (ITG) mode, the zonal flow-drift wave nonlinear dynamical equation including magnetic shear is derived. The role of magnetic shear for zonal flow generation by ITG mode turbulence is studied using a four-wave interaction model of modulational instability. Finally we can draw the conclusion that within a smaller range of k//, as |k//| increases, the growth rate of zonal flow is also increased.
Shock timing experiment in polystyrene target based on imaging velocity interferometer system for any reflector
Wang Feng, Peng Xiao-Shi, Liu Shen-Ye, Jiang Xiao-Hua, Ding Yong-Kun
Acta Physica Sinica. 2011, 60 (8): 085203 doi: 10.7498/aps.60.085203
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The shock timing experiment in ablator material wave, where the shock wave transferring and catching up are investigated, can be simulated with an inertial confinement fusion planar target. The photo ionization effect caused by the X-ray from Hohlraum target is explained. With the simulation data, the block effects of Au and Cu for shock wave are analyzed. The shock velocity and the transfer time of two-shock wave in CH material are compared by using two radiation sources. The shock signals of acceleration and deceleration are obtained after adding Au block layer with thickness 5 μm to the Al layer. The shock signals of acceleration, deceleration and reloading for two shock wave experiment are achieved after adding Au layer with thickness 2 μm and Cu layer with thickness 3 μm. The experimental data show that the two conditions should be satisfied in order to obtain the two shock signals. The first condition is that the difference in radiation between two steps source should be large, and the second condition is that the shield layer should be the combination of high impedance Au and low impedance Cu. These experimental results show the shock timing ability of "Shenguang-Ⅲ", and give the reference design for planar target in shock timing technique.
Simulation of protective effect of the buffer on the first mirror in HL-2A tokamak
Zheng Ling, Zhao Qing, Zhou Yan
Acta Physica Sinica. 2011, 60 (8): 085204 doi: 10.7498/aps.60.085204
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How to improve the protective effect of the first mirror is urgent for tokamaks at present. Buffer is one of the efficient methods to reduce the impurity redeposition on the first mirror. In order to study the protective effect of the buffer, the physical model of the particle redeposition on the first mirror is established with the Monte Carlo method on the basis of the parameters and experiments of the HL-2A tokamak. Particle redepositions on the first mirror are simulated under different conditions, including the cases without buffer and with cone buffer. The simulation results match well with the experimental results. The accuracy of the model is verified, and the simulation is efficient for providing theoretical basis for the protection of the first mirror.
Experimental study on emission spectra of air plasma induced by femtosecond laser pulses
Zhu Zhu-Qing, Wang Xiao-Lei
Acta Physica Sinica. 2011, 60 (8): 085205 doi: 10.7498/aps.60.085205
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The emission spectra produced in ambient air with focused strong femtosecond laser pulses are studied experimentally. The results show the emission spectrum presents the short-wavelength continuum spectrum with stronger intensity(cutoff wavelength 340 nm) and long-wavelength line spectrum with less weak intensity(near 800 nm). A similar spectrum shape can be acquired with fixed pulse width 50 fs and adjustable laser pulse energy. When the pulse energy is 1 mJ and the pulse width increases from 50 fs to 1ps, the peak value of continuum spectrum located in 500 nm wavelength becomes strong and gradually presents the line spectrum characteristics.
Numerical simulation of plasma immersion ion implantation for cubic target with finite length using three-dimensional particle-in-cell model
Wang Peng, Tian Xiu-Bo, Wang Zhi-Jian, Gong Chun-Zhi, Yang Shi-Qin
Acta Physica Sinica. 2011, 60 (8): 085206 doi: 10.7498/aps.60.085206
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Plasma immersion ion implantation (PIII) of the square target with finite length is simulated using a three-dimensional particle-in-cell (PIC) plasma simulation in this paper. The incident dose, the impact angle and the implanted energy on the target surface are investigated. The results show that the sheath around the square target with finite length becomes spherical rapidly during PIII. And the three-dimensional sheath width is small apparently compared with the one simulated by two-dimensional PIC. And it is found that the three-dimensional ion dose is not evenly distributed on the target surface during simulation time (50ω-1pi) in this work. The dose is smallest in the center of the target, and it is largest near the corner. This is due to spherical sheath where ions are focused and accelerated into near the corner. In the central zone, the ion incidence is nearly normal to the surface, and the impact average energy exceeds 90% of the maximum. But the impact angle near the corner is always nearly 45°, and the implanted energy is only about 50% of the maximum.
Development of percentile estimation formula for skewed distribution
Zhou Yun, Hou Wei, Qian Zhong-Hua, He Wen-Ping
Acta Physica Sinica. 2011, 60 (8): 089201 doi: 10.7498/aps.60.089201
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Order statistics establishes a relation between the position of the ranked data and corresponding cumulative probability, so it can be used to estimate the cumulative probability. Owing to the fact that different climatological data have different skewness degrees, in this paper, according to the cumulative probability function under the skewed distribution conditions, we perform theoretical analysis and numerical simulation to establish the position parameters of the regression model which are related to skewness index, then give an amperic percentile formula under the skewed distribution. By using the data about the summer temperature in global from 1980 to 2009, we compare the positions of ranked data corresponding to the 90th percentile, which are obtained by this formula and Jenkinson’s formula.
Influence of time delay on global temperature correlation
Zhi Rong, Gong Zhi-Qiang, Wang Qi-Guang, Xiong Kai-Guo
Acta Physica Sinica. 2011, 60 (8): 089202 doi: 10.7498/aps.60.089202
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With time delay under consideration, temperature correlation matrixes are constructed based on the reanalysis of temperature data provided by National Centers for Environmental Prediction/National Center for Atmospheric Research and European Centre for Medium-Range Weather Forecasts. Results indicate that the correlation of global temperature decreases with lag time, and the rate is dependent on time lag. We divide the lag time (1—30 d) into three segments, i.e., 1—7 d, 8—20 d and 21—30 d according to the decrease rate of global average correlation coefficient Cglb. When the lag time is in a specific interval (8—20 d), Cglb is unstable, which may explain the difficulty in long range weather forecast of 10—30 d. The spatial distribution of the global temperature correlation keeps stable for different lag times, while the numerical change shows zonal distribution on the whole, and that the most of Asia and the equatorial central and eastern Pacific show countertrend to other parts of similar latitudes.
Modeling of ultraviolet characteristics of deep space target
Yuan Yan, Sun Cheng-Ming, Huang Feng-Zhen, Zhao Hui-Jie, Wang Qian
Acta Physica Sinica. 2011, 60 (8): 089501 doi: 10.7498/aps.60.089501
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Ultraviolet detection has advantages of high sensitivity and low false alarm rate. Analysis of ultraviolet characteristics has great significance for space target detection. An accurate modeling method is proposed for the ultraviolet characteristics of space target. Based on the background environment and material properties of space target, region and grid division are generated, and the mathematical model of ultraviolet characteristics of space target is established by introducing bidirectional reflectance distribution function. The position relations of target, detector and background radiation source are determined by coordinate transformation algorithm. The calculation flow for ultraviolet characteristics of space target is derived. Finally, the observed ranges and irradiation distributions of Ziyuan-1 and Fengyun-3 satellite are calculated by the given parameters. The simulation results demonstrate the validity of the modeling method.
Characteristic periodicity analysis of the X-ray flux variability of quasar PKS 1510-089
Li Xiao-Pan, Zhang Hao-Jing, Zhang Xiong
Acta Physica Sinica. 2011, 60 (8): 089801 doi: 10.7498/aps.60.089801
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We present the X-ray light curve (1.5—12 keV) for quasar PKS 1510-089 with the observation of the all-sky monitor on the Rossi X-ray Timing Explorer from January 1996 to December 2009. Using the discrete correlation function method, the wavelet analysis method and the power spectrum method to analyze the data, we find that the light curve variability period of quasar PKS 1510-089 is (0.94±0.08)a. The super massive binary black hole model in used to analyze this source, and the results show that the orbit period of the super massive binary black hole system is (1.85±0.10)a and the super massive binary black hole model is the better model for describing the light curve periodic behaviour of quasar PKS 1510-089 at present.
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3f6753fc29def204 | tisdag 15 november 2016
realQM vs Hartree-Fock and DFT
I have put up an updated version of realQM (real Quantum Mechanics) to be compared with stdQM (standard QM).
stdQM is based on a linear Schrödinger equation in a $3N$ dimensional wave function with global support for an atom with $N$ electrons, which is made computable in Hartree-Fock and Density Functional Theory DFT approximations reducing the dimensionality to basically 3d.
realQM is based on a system of non-linear Schrödinger equations in $N$ 3d electron wave functions with local disjoint supports, which is computable without approximation. Evidence that realQM describes real physics is given.
onsdag 9 november 2016
Trump: End of Global Warming Alarmism
The new president of US Donald Trump expressed a clear standpoint against global warming alarmism during the presidential race:
• Any and all weather events are used by the GLOBAL WARMING HOAXSTERS to justify higher taxes to save our planet! They don't believe it is $\$\$\$\$$!
• This very expensive GLOBAL WARMING bullshit has got to stop. Our planet is freezing, record low temps,and our GW scientists are stuck in ice.
• It’s snowing & freezing in NYC. What the hell ever happened to global warming?
• Ice storm rolls from Texas to Tennessee - I'm in Los Angeles and it's freezing. Global warming is a total, and very expensive, hoax!
Trump says that he will end all federal clean energy development, all research on solar, wind, efficiency, batteries, clean cars, and climate science:
• I will also cancel all wasteful climate change spending from Obama-Clinton, including all global warming payments to the United Nations. These steps will save $100 billion over 8 years, and this money will be used to help rebuild the vital infrastructure, including water systems, in America’s inner cities.
This is hopeful to the world and to science. It says that you cannot fool all the people all the time, in a democracy with free debate and science.
This is the beginning of the end of global warming alarmism including its most aggressive form led by Sweden and Germany. The weather is now celebrating Trump's victory by heavy snow fall over Stockholm...
PS Trump picks top climate skeptic to lead EPA transition:
• Choosing Myron Ebell means Trump plans to drastically reshape climate policies.
• Ebell’s views appear to square with Trump’s when it comes to EPA’s agenda. Trump has called global warming “bullshit” and he has said he would “cancel” the Paris global warming accord and roll back President Obama’s executive actions on climate change (ClimateWire, May 27).
Finally, reason is taking over...
söndag 6 november 2016
Why are Scientists Openly Supporting Hillary?
Physicists and mathematicians such as Peter Woit, Leonard Susskind and Terence Tao have come out as strong supporters of Hillary in the presidential race, and then of course as strong opponents to Trump. This is unusual because scientists seldom (openly) take on political missions.
Why is that? Isn't science beyond politics? No, not in our time, and then not in particular climate science, which has become 100% politics. Climate scientists don't like Trump, because he says that climate science is 100% politics and not science.
Is it the same thing with physics and math? Is a pure mathematician like Tao and a string theorist like Susskind fearing that a questioning non-opportunist Trump would be more difficult to deal with than an opportunist Hillary representing (scientific) establishment? What if Trump would question the value of string theory, as he did with climate science?
lördag 5 november 2016
Weinberg: Why Quantum Mechanics Needs an Overhaul!
My new book Real Quantum Mechanics seems to fill a need: Nobel Laureate in Physics Steven Weinberg believes that quantum mechanics needs an overhaul because current debates suggest need for new approach to comprehend reality:
• I’m not as happy about quantum mechanics as I used to be, and not as dismissive of its critics.
• It’s a bad sign in particular that those physicists who are happy about quantum mechanics, and see nothing wrong with it, don’t agree with each other about what it means.
I hope this can motivate you to check out the new approach to quantum reality presented in the book, which addresses many of the issues raised by Weinberg.
Weinberg takes the first step to progress by admitting that quantum mechanics in its present form cannot be the answer to the physics of atoms and molecules.
Of course the witness by Weinberg is not well received by ardent believers in a quantum mechanics once and for all cut in stone by Heisenberg and Born, such as Lubos Motl.
But it may be that questioning a theory, in particular a theory supposedly being embraced by all educated, shows more brains and knowledge than simply swallowing it without any question.
PS1 I put up a comment on Lubos Reference frame, but the discussion was quickly cut by Lubos, us usual...any questioning of the dogma of Heisenberg-Bohr-Born is impossible to Lubos, but that is not in the spirit of real science and physics...
PS2 Here is my closing comment which will be censored by Lubos: It is natural to draw a parallel between Lubos defence of the establishment of QM and the defence of the Clinton establishment by Woit, Tao, Susskind et cet, (rightly questioned by Lubos) in both cases a defence with objective to close the discussion and pretend that everything is perfectly normal. Right Lobos?
PS3 Here is a link to Weinberg's talk. |
be1b26cc22ea28e1 | Saturday, July 26, 2014
What Born's rule can't be derived from
Sean Carroll continues to abuse his blog to promote his pseudoscientific would-be research:
Why Probability in Quantum Mechanics is Given by the Wave Function Squared
The article advertises his May 2014 preprint written along with a philosophy student, Charles Sebens. I have already discussed a text by these two authors in Measure for Measure... in May 2014. It turns out that they have written two very similar preprints. Yes, Sebens wrote another earlier paper – the title "Quantum Mechanics As Classical Physics" shows that this guy is hopeless, indeed.
First, sociologically, I think it is very unfortunate if the blogosphere is used for this self-promotion. The scientific community and the scientific public should evaluate the papers and ideas according to their quality and not according to the number of times when they are promoted in distorted blogs on the Internet. The Carroll-Sebens preprints are pure trash which is why, in an ideal world, they would immediately drop into the cesspool and no one would try to extract them again. We don't live in the ideal world. We live in a world where people are massively fed the objects from the cesspools by feeders such as Sean Carroll.
The claim is that they may derive Born's rule (that the probability is the squared absolute value of the inner product) from something deeper, namely from the many worlds fairy tale.
It doesn't make any sense whatsoever. I have discussed the numerous dimensions of why these claims are preposterous many times.
You can't really derive the probability rules from "anything deeper" because they're the most elementary part of a theory that makes fundamentally probabilistic predictions.
The actual reason behind Born's rule – the gem – was explained at the end of my May 2014 blog post and it has surely nothing to do with many worlds.
The point is that any physical theory, classical, quantum, or otherwise, has to define how its mathematical formalism expresses the fact that two states are mutually exclusive. In classical physics, two different points in the phase space are always mutually exclusive. In quantum mechanics, the mutual exclusiveness of two states is simply the orthogonality. For example, in the basis of eigenstates of an operator such as \(L_z\), the different coordinates (probability amplitudes) directly represent the mutually exclusive values \(L_z=-\ell\), \(L_z=-\ell+1\), and so on, up to \(L_z=+\ell\).
I don't have to explain to you that \((1,0,0,\dots)\) is orthogonal to \((0,1,0,\dots)\), do I?
Now, we need to know that the probability is a function of the length of the vector in the Hilbert space. This fact has to be assumed in one way or another. It may be argued to be necessary for any interesting theory. Unitary transformations form an interesting class of transformations and they are defined by keeping the length of a complex vector constant. One may eliminate other "alternative theories" that wouldn't represent transformations by unitary transformations, and so on. In all these cases, one would have to assume some extra things because one can't really cover all theories of unknown types. But surely among all the theories that have been proposed, the unitarity of the linear transformations may be shown to be necessary.
Again, we need to know that mutually exclusive states are orthogonal and the probability has something to do with the length of a state vector (or its projection to a subspace).
That's everything we need to assume if we want to prove Born's rule. The rest of Born's rule – I mean the choice of the second power – follows from the Pythagorean theorem. If you want me to be really specific and use my example (of course that it may be said completely generally), the probability that \(L_z\) is either \(m\) or \(m-1\) is equal to some function of the complex amplitudes \(c_m,c_{m-1}\).
On one hand, the probability of the "or" proposition merging two mutually exclusive possibilities has to be the sum of the probabilities of each and these individual probabilities are written as functions of the lengths of the projected vectors\[
P_{\rm or} = f[|c_m|]+f[|c_{m-1}|].
\] On the other hand, we should be able to calculate the probability of the "or" proposition directly from the length of the whole vector,\[
P_{\rm or} = f(\sqrt{|c_m|^2+|c_{m-1}|^2}).
\] You may see that the two formulae for \(P_{\rm or}\) are only equal if\[
f(c) = \alpha\cdot |c|^2
\] because the Pythagorean theorem implies that only the second powers behave "additively". The extra arbitrary parameter \(\alpha\) plays no role and one may set it to \(\alpha=1\).
That's the real reason why Born's rule works. The probabilities and mutual exclusiveness has to be expressed as a mathematical function or property of state vectors and the totally general rules for probabilities (like the additive behavior of probabilities under "or") heavily constrain what the map between the "human language" (probability, mutual exclusiveness) and the "mathematical properties" can be. The solution to these constraints is basically unique. The probabilities have to be given by the second powers of the moduli of the complex probability amplitudes. It's because only such "quadratic" formulae for the probabilities obey the general additive rules, thanks to the Pythagorean theorem.
(The derivation may be reverted, of course. If we know Born's rule – that the probabilities are given by the second power of the length – we may prove that mutually exclusive states are orthogonal because only orthogonal vectors and their hypotenuse are those that obey the Pythagorean theorem; much like the mutually exclusive options obey the additivity of probabilities under "or".)
Once we know the simple (a priori counterintuitive, maybe, but extremely important and universal) rule, it becomes meaningless to talk about its "origin" again. Shut up and calculate. There can't be anything that would be "deeper" yet "clearly independent" from the Born's axiom. The axiom, schematically \(P=|\psi|^2\), really has six characters and is based on some simplest concepts in linear algebra. You may hardly imagine a more concise, simpler, or more fundamental starting point! Such an even simpler starting point would have to be something like "OM". ;-) People who have a trouble with the fact that something like Born's rule is fundamental and true must clearly have different reasons than the "lack of simplicity" to invent non-existing problems.
Many of Carroll's readers manage to see through the cheap tricks. Carroll and Sebens aren't really deriving anything. One can't derive Born's rule from anything much deeper. You see that the proof above only used modest assumptions – no "second power" was directly included in any assumption – but it had to assume something. It is totally OK in science to assume something. Science is about formulating ("guessing", as Feynman would put it) competing hypotheses and deciding which of them is right by looking at the empirical evidence and thinking about it carefully enough. Quantum mechanics with its postulates was "guessed" and it has won the battle of science (against its proposed or just dreamed-about competitors) more clearly than any other general theory in the history of science.
An example of Carroll-Sebens circular reasonining is that they assume that small off-diagonal entries of a density matrix may be neglected – they assume it before they derive or admit that the small entries correspond to probabilities. That's, of course, illegitimate. If you want to replace a small quantity by zero, and to be able to see whether the replacement is really justified, you have to know what the quantity actually is. Moreover, these things are only negligible if classical physics becomes OK, so whatever you do with this approximation is clearly saying nothing whatsoever about the intrinsic, truly quantum, properties of quantum mechanics in the quantum regime!
Moshe Rozali and others re-emphasize that the "film with cats" illustration of the "splitting worlds" only works for a binary spectrum but many other observables have many eigenvalues and in many cases, the spectrum is actually continuous. Carroll never says how the worlds are split to a continuum of worlds and how e.g. the mutual exclusiveness is counted over there. He can't because there can't be any sensible answer. He doesn't ever answer whether the number of worlds today is higher than the number of worlds yesterday. He can't because there can't be any sensible answer. He never answers questions about the possibility for the "split branches" to reinterfere again in the future. He can't answer because there is no sensible answer: they clearly can reinterfere in the future in principle, quantum mechanics implies, while the very point of the "many worlds paradigm" is to make a major mistake and argue that the "splitting" is absolutely irreversible. It's never absolutely irreversible.
Incidentally, Moshe Rozali also says that the many worlds paradigm doesn't define – and, well, cannot really define – how often the splitting occurs. Rozali uses the example of particle collisions.
When two protons collide, they may be described as protons. But they may also be described with a higher resolution, as bound states of many quarks and gluons. The collision may produce a Higgs boson for a little while which decays to two tau leptons (yes, the Kaggle contest causes a professional deformation). Those later decay.
Now, we may ask whether the worlds split already when the taus are produced, or only when the taus decay to the final products, and so on. In all these cases, the answer of a correct quantum mechanical calculation is unequivocal: the most accurate quantum calculation doesn't allow any splitting whatsoever, at least not before the measurement is made. The Higgs boson and taus are strictly speaking virtual and the histories with these virtual particles interfere with other histories without Higgses or without taus (I am just saying that to calculate cross sections, you have to sum over Feynman diagrams with different, all allowed intermediate particles: every particle physicist who is not completely hopelessly incompetent knows that). It's just wrong to imagine that in a particular collision, the existence of a Higgs or the taus is a "strictly well-defined" piece of classical information. It's not. Saying that we're on a branch that either had this Higgs or didn't have the Higgs is a major mistake – which may only be harmless because the virtual particles are nearly on-shell (so that a particular Feynman diagram with a virtual particle is much greater than some other diagrams) and because the classical approximation is tolerable. But the virtual particles are never exactly on-shell and classical physics is never the exact description of the reality, so the "many worlds" description is always at least partially wrong.
Similar questions apply to the question whether the "splitting of the worlds" applies to the initial protons or initial gluons etc. In all cases, a "splitting" is just something that makes the calculation conceptually wrong, something that adds errors which may be small if classical physics is an OK approximation but which are very large and of order 100% in the strict quantum mechanical regime.
The mushy and sloppy reasoning doesn't appear just in some places of the Carroll-Sebens paper. Virtually every "idea" or sentence is flawed, illogical, or vacuous. For example, a principle is called "ESP":
ESP: The credence one should assign to being any one of several observers having identical experiences is independent of features of the environment that aren’t affecting the observers.
"ESP" isn't "extrasensorial perception", or at least Carroll and Sebens don't want to admit that it is. Instead, it is the "Epistemic Separability Principle". Pompous phrases is something that pompous fools enjoy.
There are problems with the "principle" at every level. First, the probability is interpreted as the "credence" which is a deliberately vague version of the "Bayesian probability". The problem is that at least in the world with many repetitions of the same experiment, the probability has to manifest itself in the frequentist way, too (the ratio of repetitions/worlds that have some property and those that don't). But in their picture, the frequentist picture never emerges. So they are actually assuming a "Bayesian" interpretation of the probability when they are claiming to derive that they can live without it, and so on.
The other obvious problem with the ESP quote above is that it says what the "credence" is independent of. But a usable theory should actually say what it does depend upon. Ideally, one should have a formula. If one has a formula, one immediately sees what it depends upon and what it doesn't depend upon. A person who actually has a theory would never try to make these unnecessarily weak statements that something does not depend on something else. Isn't it far more sensible and satisfactory to say what the quantity does depend upon – and what it's really equal to? Quantum mechanics answers all these questions very explicitly, Carroll and Sebens don't.
The statement is not only weak to the extent that it is useless. It is really intrinsically ill-defined. If we say that \(S\) is independent of \(T\), then we must say what other variables are kept fixed while we are testing the (in)dependence of \(S\) on \(T\). Do we keep \(p\) fixed or do we keep \(V\) fixed? I deliberately chose these letters because you should know this exact problem from thermodynamics. The entropy \(S(V,T)\) written in terms of the volume and the temperature may be independent of the temperature, but the entropy \(S(p,T)\) written in terms of the pressure and the temperature may depend on the temperature!
So without saying what are the other observables that \(S\) (or, in the quantum wars case, the probability) may depend upon, saying what they don't depend upon is absolutely vacuous and ill-defined.
If you want to strip the ESP proposition of all the nonsense, ill-defined words, and everything else, it really says:
ESP (partially fixed): If you measure some quantity \(X\), the result is independent of some completely different quantities \(Y\) that you don't measure.
Nice but it's a completely worthless tautology. Yes, if you're looking at a cat, you're not looking at a dog. The pompous language may prevent one from seeing that the original ESP sentence is the very same crap but if you think about it at least for a minute, you must be able to see that it is the same crap.
Moreover, even the partiually fixed version of ESP is wrong in quantum mechanics, in certain important respects. What's important is that if you also measure \(Y\), you do this measurement first, and if \(X,Y\) don't commute with each other, then the result for \(Y\) will influence the outcome for \(X\). More precisely, the best prediction for the \(X\) measurement must take the result of the \(Y\) measurement into account even if they are different quantities. They may really be thought of as "truly independent" if they commute with each other. (This "independent observables have to be mutually commuting" is some sort of an operator counterpart of the statement for states that "truly mutually exclusive, different states must be orthogonal to one another".)
The Carroll-Sebens papers are meaningless zero-citation crackpot tirades, if we exclude self-citations (something that certain crackpots love to collect). Every genuine physicist knows that but Carroll abuses the traffic on his blog and misleads the thousands of people who visit it into thinking that he is something else than a crank. I think that it is immoral.
1. You are doing such important work exposing this. Of course Carroll can never win the war with his twisted MW interpretation since nature trumps will always ultimately trump trash. But over time he and his buddies can cause a shitload of confusion that will take years to clean up. He should have stuck to GR.
2. Hello. Your proof of the Born rule with Pythagoras is a little bit similar in spirit to Gleason's theorem. Do you think Gleason's theorem alone is enough to justify the Born rule?
3. Us simple civil/sanitary engineers from a previous generation thought that physics was the ultimate science, the high church of (empirical) reason, with actual saints like Bohr and Einstein. It makes us afraid, lost on a darkling plain, to realize there are lunatics even that cloister.
4. Dear Andrew, it is a little bit similar, perhaps, but I find such an assertion to be too vague if you want to elaborate upon it. It's similar but it's also different. Gleason's theorem is about the formula for probabilities of P using density matrices, Prob=Tr(P.rho). There is no additional "squaring" in this formula.
But the general approach to prove that the quantum formulae are the only consistent ones is the same. The assumptions are also additive properties of the probability, and so on.
I wasn't constructing my argument based on Gleason's theorem because I have never considered such results to be important. Gleason's theorem is another case of too much ado about nothing. The content of it was surely clear to the founders of quantum mechanics. In the same way, i don't really claim any inevitable originality, anyway. I am sure that everyone who understands QM decently enough could write down such an argument. It's just that unfortunately, almost no one is doing such things so the public discourse is drowning in nonsense which is emitted by very many people, indeed.
5. In the local right-wing newspaper the headlines are comedy gold.
The Friday edition features ITS GONE PEAR SHAPED, AND NOW PUTIN HAS HIS BACK TO THE WALL (in which Putin is claimed to fear not the West but his own people) followed by (in the same edition) PUTIN'S VORACIOUS APPETITE IS NOT SATED (in which it is claimed that the Russian people will circle the wagons around their embattled hero and cry foul at foreign attempts to denounce him)
The PM Abbott, in the style of the true venal pollie, has jumped on the bandwagon of international self-aggrandisement with unaccustomed alacrity in order to distract attention away from his troubles at home. Like a true Aussie hero he's leading the pack for truth and justice against no less an enemy than Satan himself.
Its all gone off rather well, the Saturday edition features an in-depth article from the paper's in-house nutter G. Sheridan entitled ABBOTT ACCRUES DIPLOMATIC CAPITAL FOR CRISIS LEADERSHIP, followed by PURER VIEW OF CHARACTER ON DISPLAY AS POLITICS LAID ASIDE (in which the various aspects of the PM's greatness of personality is discussed).
The whole effect of this relentless and absurd beat-up is like watching a cartoon. It's clear that the printed media is now nothing more than an outlet for the favoured political views of whoever controls them. As sources of rational information they are completely worthless. I gave up reading the left-wing newspaper decades ago. It looks like its time to give up reading the right-wing newspaper as well.
6. Lubos, you’re unending battles with these QM interpreters is becoming reminiscent of those WWI guys going over the top to charge entrenched defenders - a very nobel effort indeed. It’s a mystery to me how a CalTech professor of physics can get away with babbling obvious BS about multiple universes, but it sure looks like he can and in doing so even acquire more prestige, at least among the uninitiated.
Have you ever seen a formulation of QM based on the quaternions? I’ve long wondered about that and after reading the Dirac interview you posted the other day I was happy to see Dirac did also. Like he said, the quaternions provide the simplest example of a non-communative field and so seem natural for QM given the centrality of commutators.
7. I wish Sheldon would go around to his office and explain how QM actually works ;-)
8. Great points. You touched on this in the post but to me the issue is really even more fundamental than the details of the argument. The entire strategy doesn't constitute a "derivation" from the beginning. Basically, what all of these arguments do is find some reasonable assumptions and then show that the Born rule is the unique choice for probabilities consistent with these assumptions. Then, I'm supposed to believe that this means that unitary evolution implies the Born rule.
Of course, there is another option, which is that the entire exercise is pointless because there is no sensible probabilistic interpretation of many worlds to begin with. Thus, it doesn't matter that there is a unique rule that this nonexistent probabilistic interpretation hypothetically must obey.
To me, it's like arguing that the basic postulates of gauge theory imply that there can be no anomalies. You "derive" this factoid by demonstrating that if there was an anomaly with a non-zero coefficient it would destroy unitarity and render the theory inconsistent. Therefore, the only consistent possibility for the coefficient of any anomaly is 0, so we've proven that gauge theories can never be anomalous! Of course, we've neglected the option that maybe some gauge theories are just not well-behaved physical theories at the quantum level. The "derivations" of the Born rule are equally absurd in my mind.
9. To Caroll issues like "how and when branching happens" are "relatively tractable technical challenges"
Once Wolfgang Pauli drew a blank rectangle and said that this proved he could paint like Titian "Only technical details are missing".
Pauli was joking. Carroll isn't. That is unfortunate. Lubos is as usual excellent in pointing out that the "technical details" are that the theory is DOA ... the usual meaning.
10. We are the thin strong thread, for our scientific method. The rest is drama, art, and poetry.
11. It used to be eminent physicists would get letters and documents by mail from cranks, scribblings and rantings using diagrams and pictures and grade school math. Now they sit next to you on the faculty. Now they publish alongside you in peer reviewed journals. Next you will be in the audience politely applauding as they accept an award your work deserves. Finally, Physics itself will be rewritten in crankery.
12. "But Obama was elected, Lubos. That’s democracy!"
Yes, so were Bliar, and Cameron, and any other number of complete shits. Just because all the alternatives to democracy are far worse doesn't mean it doesn't suck. Not that you claimed it didn't though.
The Zeroth Axiom of Governance
All governments, however constituted, are run by irredeemably traitorous scum who should at best be regarded as extremely dangerous recidivists let out on parole by mistake.
Much follows from this. In particular, institutions of governance which in their construction do not fully reflect all the implications of this Axiom for the wellbeing of the people so governed should be regarded as a clear and present danger and eliminated accordingly, i.e. with extreme prejudice, and pronto.
Now, since there are NO institutions of governance anywhere in the world which in their construction fully reflect all the implications of this Axiom for the wellbeing of their respective peoples, they should all therefore be immediately eliminated with extreme prejudice and replaced forthwith.
OK, that's the plan in a nutshell. I leave it as an exercise for others to flesh out the details and put it into execution.
Talking of executions, I hope to see Bliar, Brown and Cameron hanged one day for sapping and impurifying all of our precious bodily fluids, but mostly of course for filling the country with all those hideous turd-world aliens.
13. I am confused. AFAIK there is no standard theory(but many conjectures, see references below) the derives Schrodinger equation, so how is it that people try to "explain" the Born rule and yet SE itself has not been explained.
from introduction, see also the references(4,5,6,7)
"Most students and professors will
tell you that the Schr¨odinger equation cannot be derived. Beyond the standard approaches
in modern textbooks there have been several noteworthy attempts to derive the Schr¨odinger
equation from different principles 4,5,6,7, including a very compelling stochastic method8
, as well as useful historical expositions9."
14. Such fundamental equations obviously can't be derived "out of nothing". At most, one may show that they are not independent,. that some subset of these fundamental rules or equations is enough to derive others. But if some others follow from the first ones, anyway, it's not terribly important - it is a pure human convention - to decide which of the claims and equations are fundamental axioms and which of them are derived. It's only the whole structure that has a physical meaning.
Aside from deriving from nothing which is impossible and deriving from some other, comparably fundamental propositions and equations, one may also do what Schrodinger did and "induce" the equation from its desired consequences - the empirical observations and logical necessities. In this sense, Schrodinger "derived" it by working to combine de Broglie's wave with some equations that make packets move similarly as particles move in the external potential. That was his specific non-relativistic equation at the end. But it's not really a derivation of the general Schrodinger equation for any Hilbert space, not necessarily one non-relativistic particle; it is not really a derivation in the mathematical sense. It's just a way to make the right guess look like less shocking a victory in a lottery and more unavoidable. None of these "derivations" may ever be rock-solid because right theories in general can't be "uniquely" extracted from the observations. If this were possible, Newton would have derived quantum mechanics himself.
15. LOL, right. Now people - often without such affiliations - are bombarded by letters and blog posts by cranks who often sit on faculties.
16. Yes, Obama was elected, Gene, and so was Putin and most others.
Accidentally, I just got this video
where Obama says that individuals are too small insects who have to surrender all their rights in the name of the New World Order. Is that video genuine? I can't believe it. I've been listening it 10 times, trying to see some discontinuity proving it's fake, but so far I failed!
17. Who is brainwashed? Go to Donbass, open eyes. I have many family there. My father was teacher of math physics for all his life then go retired and had a shop in home village. Now the shop is robbed by separatists and he is called a jew, even if he is not a jew. All jew properties will be nationalised - they say. Poroshenko is the jew, they write it on the internet, on posters, they say it on gatherings. American jews, european jews, banker jews, ukrainian jews want to destroy ukraine. nazi-jews - sepraratist live in paranoia. Jews everywhere but there are rally almost no jews in ukraine (really very few) -only in mind of separatists. Jews pay for nazis - they scare people with Azov Batalion, but if i were jew i would be scare more of separatist that want take all from businessman, they fight capitalism, not the Azov Batalion nobody never not saw. They live in fantasy world, all enemies are controled by USA jews, Poland has military bases in Odessa, Sweden has special soldiers sent, CIA control ukrainian army, ukraine shot areaoplane on Malaysia to kill Putin, capitalist fight agaisnt Russia last bastion of freedom from jewish bankers, they pin hammer and sickles to saint George ribbons -this all is insane. All is fighting against them, Americans, Poles, Lithuanians, Latvians, Estonians, Swedes Canadians. But for them its ok that they have Serbians, Bulgarians or even Chechens. They supported by nazi skinheads from Russia, Poland, Hungary. Nazis proud to fight nazis.. Deep, deep, deep paranoia. Come do Donetsk. I was there already. See with your own eyes and then respond how you can support all of this mindless amok with no sense. Go, buy ticket, I want to see you there. I want to know if after talking with these people your mind and intelligence wont feel ofended by their bullshit, and propaganda. Did you saw russian national tv. Are you not offended by this? They treat people like morons and you clap them? It so easy to write all those thing if you were not there, so buy a ticket and please go! Or come to Russia if you afraide of coming to Donetsk. Come to Moscow. See what people say like that Poland, Hungaria and Romania are anexing west ukraine - so many people believe this sick bullshit. Come see this, and then say if you are not offended.
18. As a Canadian I'm a bit sensitive to U.S. extraterritoriality. I would have thought the European's would have had bigger ballz though. With a little foresight I think the French could have owned a U.S. bank or two. Simply hedge against the Euro big time and then destroy it's value (i.e. trade war, piss the Germans off). From this position of strength they could have sued the U.S. for economic peace. Perhaps have the dollar cleared in a DMZ. Too clever?
19. You dont have to be the Russian President, because he's clearly following your line of thinking - he's bringing Russia down with the idea of preserving its "greatness" - a task worthy of a true ayatollah. If he was truly peaceful, reasonable and even smart, there would be no rockets in the separatist's hands - only trade agreements. But things are not simple, right? One have to consider the mass stupidity and ignorance of the population a.k.a national interests and use it to some advantage. There are also clever people dreaming of war glory, invasions and empires (which all fell due to economic reasons - the last colonial country in Europe - Portugal - is one of the poorest states now). The good leader will try to find the balance, while the bad one will run for the popularity. Russia would have invaded Ukraine if only it was capable of doing so, and obviously it's not (otherwise it would be fact), contrary to the sport-level of euphoria in some people.
Well, anyone is free to choose a side to defend and I guess someone has to take care of the Third World's interests. As the proud son of Mother Russia Sergey Brin has puted it: Russia is Nigeria with snow. Of course, he was not correct. In Index of Economic Freedom (very informative indicator) Nigeria is higher than Russia, which shares place with Burundi. So if we hear from a source, that the world have arrived at a time when a backward state is loaded with the mission of saving the world from the leading, developed nations, maybe we have to check the credibility of the source. Putin makes all efforts to keep Russia at the level of its well-deserved fame. He is making exactly the opposite of what he actually tries to achieve - he is bringing NATO to its borders. That's the only result from his actions, besides the increasing financial isolation - another sife effect of his cop-turned-thug level of thinking.
Nobody would benefit from severed relationships with Russia, it must be noted. But this statement alone is serving the Russian madness. Russia is the one which will lose the most from any form of isolation. Lets look at some numbers from the period 2011-2013. The Russian export for EU is twice the size of the EU export - about 230 billions. Fifty-five percents of all Russian export is to the EU. The european investments in Russia constitutes about 80% of all investments (almost 190 bil), while Russian investments are 76 bil and represents a negligible part of all investments in the EU. Europe imports 45% of its gas and 33% of its oil from Russia, while Russia exports 88% oil and 70% gas to Europe and so on... Lets have the data in mind when we praise the wild lands at East, lest they enlight our non-brainwashed minds with their darkness.
20. Luboš M. I'm fully with you. But what I learned from the commentators is that all logic is not enough if somebody is growing up intolerant.
21. Not sure if I am on the right track here - but is it not possible to derive Einstein's field equations from a small set of fundamental assumptions (like equivalence principle, invariance of the speed of light in vacuum)?. Obviously not its source term (the energy momentum tensor) which needs input from the physical setup of the system, but pretty much all the rest of the equations are derived from something more fundamental. Einstein set up his equations, then he extracted predictions like gravity waves, light deflection which were subsequently found in corresponding measurements.
That would be different from the Schrödinger equation, which had been designed to yield numbers in agreement with existing spectra. In this sense: Is not the Schrödinger equation a rather empirical product, while Einstein's equations are from first principles?
22. A historical aside: Some time ago I had a look at Born's original paper "Quantenmechanik der Stoßvorgänge" which is in German. I realized, that interestingly he first got "his" rule wrong, he considered Φ instead of |Φ|^2. But then he corrected it by adding a footnote:
"Anmerkung bei der Korrektur: Genauere Überlegung zeigt, daß die Wahrscheinlichkeit dem Quadrat der Größe Φ proportional ist."
23. Accepting that America supported Maidan, and accepting that shooting down MH17 was a mistake, there is still only one warmonger here. Russians are taking over key positions in the "separatist" movement, as Putin moves to take by force that which he could not win any other way. No matter how much justification he may feel, there is only one way to see what is happening - Russia is taking territory from neighboring countries by use of force. The fig leaf of a native separatist movement has been pushed aside in recent days. Maybe Europe needs a good bloodletting, a new war to focus their attention, but I fear that any attempt to soft pedal what Mr. Putin is doing will be seen as apology for a coming mass murder, and, no matter how lofty or justified his goals, he is the bad guy, now.
24. You totally miss the real reason for Putin’s actions in accepting the dogma of his megalomania. I urge you to try and look at things from his point of view on the assumption that he is a reasonable and, perhaps, even a kind man. Even if you do not believe this, give it a try.
Putin’s world is vastly different from yours in that he has to worry about internal stability in mother Russia. It has been true for centuries that Russia has needed secure, stable and friendly neighbors in order to preserve its own, internal, national integrity. He is not “the bad guy” but just a leader doing the best in the position he occupies. I do not envy him.
Europe certainly does not need a good bloodletting; they have enough of that and western interference can only serve to increase that likelihood.
25. The "sphere of influence” serves the vital need to increase Russia’s internal cohesiveness/stability. Russia does not enjoy the huge geographical advantages of my own country, the US.
26. I’m sure you know that Winston Churchill said that the best argument against democracy is a five minute conversation with the average voter but that all other forms of governance are worse.
It is not an ideal world.
27. kashyap vasavadaJul 27, 2014, 6:01:00 PM
This is little bit off topic for this blog post. But mention of quaternion, by Dirac and you, reminded me of a paper by my colleague. He (Horia Petrache) is professionally an experimental biophysicist but he is very good mathematician. He likes to do such things in his spare time after he is done with biophysics and physics teaching! The paper is pedagogical. It discusses how hyper complex numbers can be arrived at from simple group theory. He thinks that this may be known to mathematicians but may be possibly new and interesting to physicists. He would like to get comments from anyone who is interested in such stuff. His e-mail
address is given in the paper.
28. The US, from about 1991 until about 2008, was unarguably the world's dominant power. After the banker-created financial crash of 2008 - and the $16 trillion bank bailout - the economic pieces on BigZ's grand chessboard of power began to crumble and fall, and by 2013 China had become the dominant world economy. So the US had its generation of unchallenged supremacy, but the US neoliberalcon kleptocrats drove the US ship of state aground with ill considered, expensive wars; incessant bank bailouts which continue to the tune of some $50 billion per month; trade policies which exported US jobs to India and China; and economic policies that funneled 95% of income gains into the pockets of the 1%. What is happening now in Ukraine (and Africa as well) is a last ditch attempt to finish looting the world before the whole crumbling US edifice collapses. Putin is demonized because he has failed to cooperate with that grand plan.
29. Dear Holger, I am convinced it's right to say that the postulates of quantum mechanics deserve the label "fundamental principles" much more than Einstein's equations - and in this sense, they are analogous to the "equivalence principle" from which the equations of GR are deduced.
But the particular simple form of Einstein's equations, just with the Einstein tensor, and a stress-energy tensor, isn't fundamental or exact in any way. General equations obeying all the good principles also contain arbitrary higher-derivative terms (higher order in the Riemann tensor and its derivatives, with proper contractions), and may be coupled to many forms of matter including extended objects and things not described by fields at all.
So the simplicity of Einstein's equations - the fact that only the Einstein tensor appears on the lef hand side - is nothing fundamental at all. It's really a consequence of approximations. At long enough distances, all the more complicated terms that are *exactly* equally justified, symmetric, and beautiful become negligible.
On the other hand, the form of Schrodinger's equations or other universal laws of quantum mechanics is *exact* and *undeformable*, so it's much more fundamental.
Schrodinger's equation itself is just one among numerous ways - not exactly the deepest one - to describe dynamics in quantum mechanics - the equation behind Schrodinger's picture (there's also the Heisenberg picture and the Feynman approach to QM, not to mention the Dirac interaction picture and other pictures).
The wisdom inside Schrodinger's equation may perhaps be divided to several more "elementary" principles and insights. The wave function, when its evolution carries the dynamical information, is evolving unitarily with time. And the generator of the unitary transformations is the Hamiltonian. These two pieces combine to Schrodinger's equation.
The unitarity of all transformations as represented in QM is a very general principle that could again be called a universal postulate, or it's derivable from other closely related principles that are the postulates. It holds for all transformations, including rotations etc., not just for the time translations generated by the Hamiltonian.
The map between the evolution in time and the Hamiltonian is really due to Emmy Noether, so the Hamiltonian's appearance in this equation in QM is due to the quantum mechanical reincarnation of Noether's theorem. The theorem is very deep by itself, even in classical physics.
Again, I am not saying that the principles behind GR aren't deep. But Einstein's equations *are not* these principles. They're just a random product obeying some principles and its simplicity is only due to people's laziness, not because this simplified form would be fundamentally exact. It's not. The postulates of quantum mechanics however *are* and have to be exact. I feel that you get these things upside down.
30. Gary EhlenbergerJul 27, 2014, 7:06:00 PM
Thanks for that paper.
31. I agree completely even if I am only a dinosaur physicist. It has been fifty years since I learned any new math but it seems to me that there are only two fundamental things in physics, quantum mechanics and statistical mechanics. Both have to be inviolate else reason itself has no meaning.
I’m sue these two must be connected at some very deep level as well.
32. Sort of similar to Dirac's "derivation" of his equation. He derived it by playing around with equations and matrices.
33. ESP--- sounds like apartheid to me :)
Also it sounds like a good sound bite for Alan Sokal to use in his next paper for Social Context mag.
34. Thanks very much for this articulate rebuttal. At the end of Carroll's latest post, I notice he sounds like some fervent devotee -- a regular True Believer. He's all filled with parallel universes evangelical fervor. Sigh.
35. Hi Gene, I am still not seeing why relativity would not be regarded fundamental. Historically, QM was initially non-relativistic, and it worked OK to explain the spectrum of hydrogen on the accuracy level available at that time. Once the measurements turned more accurate, physicists had to sit down again and modify QM to include relativistic corrections. This procedure was then repeated a few times until they could finally reproduce effects like the Lamb shift. On the other hand, Einstein did not derive relativity from QM as a limit case. It was the opposite, relativity was there first, and the founders of QM were forced to incorporate its mechanisms into their equations.
Imagine, spectroscopy had not been invented by the 1920s - would the Schrödinger equation have been set up nonetheless? Or QED, without the availability of precision measurements like Lamb shift or the gyromagnetic ratio of the electron? I don't think so, because the formalisms first had to be tweaked for quite some time to bring theory in close agreement with those measurements (see also the book of Schweber about "QED and the men who made it").
Einstein, however, derived his equations essentially through thought experiments, based on very basic principles. They were available prior to those precision experiments, which subsequently verified his ideas to an incredible level of accuracy (an obvious exception was the invariance of the speed of light, which was known at that time and taken by Einstein as a building block of his theory). Nobody had ever been thinking about gravity waves before Einstein presented his equations. Now, their indirect measurement through the Hulse-Taylor binary pulsar represents one of the most precise agreements between experiment and theory ever observed in physics - without any prior fine tuning of the theory. Einstein himself could have done the corresponding calculations using his original formalism.
That is why I would regard relativity truly fundamental.
36. Well there are many schools of thought. I subscribe to the one that believes that nervous countries on the perimeter of Russia, "the country that does not know where it ends" in the words of Vaclav Havel, actively seek any kind of protection available. Without the comfort of geographical separation that we Czechs now enjoy, I can imagine why they run under the NATO umbrella. And given the NATO history and doctrine, peaceful and non-expansionist Russia has nothing to worry about.
37. NATO is an attractive club for the nervous countries on Russian perimeter. And there would be nothing wrong if every former Eastern bloc countries were in it. I do not recall any "faux outrage", if anything the butchering of separatist Chechens together with their civilians would have deserved even more attention than it got at that time.
38. I have to admit being out of my depth here. It’s just that I cannot imagine a non-quantum world; it is absurd to think there could be such a thing and statistical mechanics/thermodynamics amounts only to careful counting.
I can envision violations of GR but QM and SM seem as inviolate as logic itself.
39. Hi Lubos, have you seen this recent preprint (, which aims to derive the rules of quantum mechanics from string interactions (instead of the other way around as you have explained in past blog posts)?
Note that Itzhak Bars is the persistent advocate of physical theories with two time coordinates. He is also a conservative to some extent.
41. :)
Yes, I'd heard that one. I don't know what the setting was but presumably clearly it reflects his rather dim view of the average voter.
I say he was absolutely right of course. Indeed, I take the effective dimness of voters en masse as a given — the evidence is stunningly overwhelming. Moreover it is highly unlikely that things will ever change in that respect. Incidentally, how much of this dimness one should attribute to apathy, ignorance, gullibility, pre-occupation with more immediate concerns etc, or just plain intellectual incapacity is a separate issue. Dunno.
But there's a flip side to all this, one that maybe Churchill would naturally prefer to downplay, namely treacherous governors — and it's those whom I prefer to concentrate on. By way of contrast, I believe a lot can be done about them. Or preferably to them. At least I know what I'd like to do to them, some of them anyway. :)
That was really my point.
BTW, WC, though a great man and just what we needed at the time, was certainly not without a giant streak of shit running down his back. He was quite happy for instance to have Londoners locked out of the tube stations during the Blitz. Yes, the little people could just f##king well cower in those useless Anderson shelters—if they could build them, that is—or under the fcuking stairs. Well, the mob thought otherwise and smashed their way through, plod or no plod. So that was the end of that load of nonsense.
You don't hear much about it though. It doesn't fit the 'disciplined' (i.e. fquit) plucky-Londoner characterisation of the officially endorsed 'narrative' (a word I have come to despise).
Still, the lesson remains: king or no king, don't push your luck, because we'll cut your f##king head off if you do, so be nice.
42. GR is a classical theory that works for large distances. If you go to very small distances it is clear that you will be in the QM realm and it is obvious that gravity cannot vanish into the thin air at small distances, so that makes it clear that gravity is quantum mechanical in origin. Then QM must be the more fundamental. At least that is my simple minded conclusion.
43. Maybe I said that wrong. Europe does not so much need a bloodletting as much as they deserve it. All this crap about making warfare illegal, followed by substantial disarmament, relying on America for a defense-on-the-cheap. Well like Spock famously said, that was never gonna work as long as humans were involved. Now America has been driven into some neo-isolationist cocoon and western Europe stands alone. Whatever happens, they did it to themselves. Putin is following a perfectly good rule book. So are all the militias, warlords, proxies, and whatever else. Gonna be fun to watch - from far away.
44. Both, GR and QM, break down at distances of the Planck length. There has to be a theory replacing them, and at lower energies that theory has to cover properties of both. It will also cover the origin of gravity, but it is impossible to tell off hand how exactly that would look like. Sure such a gravity would share properties with quantum mechanics, but it has to be more than that. Otherwise one could simply quantize gravity and carry everything over to distances below the Planck length, but it doesn't work out, because the so called "classical " gravity, with its rampaging curvature, is spoiling the party at these length scales.
45. No, Holger, you are totally wrong. All evidence makes it overwhelmingly clear that QM never breaks down, not even at the Planck scale. We can do perfectly consistent calculations of quantum gravity physical phenomena at the Planck scale and the postulates of QM are 100% valid.
46. Putin played it shrewdly. Had he quickly invaded the rest of Ukraine, little NATO could have done, but would have bankrupted Russia (Steve Piechenik analysis) -- As it is, NATO is stuck with financial support. Front door closed, back door trading with the BRICS is increasing. To protect the Rotschild dollar, war is needed.
47. 9 veteran US intelligence officers now confirm, that Kerry is not telling the truth (again).
"We are hearing indirectly from some of our former colleagues
that what Secretary Kerry is peddling does not square with the real intelligence."
This corresponds with a previous posting on that website from investigative journalist Robert Parry (probably relying on the same source), that US satellite photos show the fatal missile fired from a launcher which appeared to be controlled by Ukrainian soldiers in Ukrainian uniforms.
48. Hmmm, Vladimir should appoint her Foreign Affairs Secretary---
given her pic, Kerry would be so flummoxed, Vlad could do what he liked :)
49. not me, another Gordon.
50. I am the real Gordon!
51. Russia's character comes from 1000 years of bloodletting of it's own people. |
1eafaa6eb3e11016 | Forget what you know | Jacob Barnett | TEDxTeen
Translator: Jaime Ochoa
Reviewer: Capa Girl Hey! I’m Jacob Barnett,
are you guys excited? (Cheers) Alright! I am here to tell you why you guys
should forget everything you know, right now! So, first thing you guys need to know: suppose you guys are
all doing your homework. OK, you know,
it’s something you have to do; and, you’re doing great
on your homework, you are getting great grades,
fabulous prizes, such as you know, Benjamins and all this great stuff. I’m here to tell you
that you’re doing it all wrong! That’s right, I did just say that,
you’re doing it all wrong! In order to succeed you have to look at everything with your
own unique perspective. OK, what does that mean? That means that, when you think, you must think in your own creative way, not accepting everything
that’s already out there. By the way, the people I’m showing you in the background
are my little brothers Ethan and Wesley, one of them is a chemist and the
other one is a meteorologist. So, your perspective might be
the only way you can see art or history
or music, or whatever. So, let me show you one of the
ways in which I can see math. that’s 32 and the rotations represent: addition, subtraction,
division, multiplication, etc. My main reason of coming out here is to do some quantum mechanics, OK? So, today, what we’re gonna do is, we’re gonna do the Schrödinger equation, split it into time independent components, and we’re gonna solve it for the boundary conditions of a
lattice and a particle in the box. So, let’s get to work! So, I have some lecture notes,
which I’d like you guys to pass out. I’m gonna split them into two rows. So, if I can have some people
come up and get these? No, wait.
Before you come up here I need to let you know about
something very quickly. OK, just stay there.
I’m kidding! (Laughter) I didn’t —
(Applause) I did not come here to frighten you all
with quantum mechanics — not yet. So, let’s think about something simpler. How many of you here
have heard about circles? OK, good. So, why are circles important? They are the shape of cookies. They are the shape of
skateboard wheels, and most importantly, they’re the shape of the thing
that turns on your X-box 360. (Laughter) So, what do we know
from school about circles? We know Pi r2,
we know they’re round. Do we know anything else? Not really. (Laughter) So, let me tell you something cool
you can do with circles. It’s called Johnson’s Theorem. It’s not really a theorem,
it’s just, you know, a way mathematicians
can think of stuff. So, what Johnson said was, “You take three circles, you overlap them in a way
so that there’s six blue lines” — where I call each
of the circles blue; so there’s six lines
coming in one point. The other three points
are in a circle of the same size; Interesting. So, this isn’t just Pi r2,
This is something new. So because Johnson didn’t just think: “Oh, it’s gotta be Pi r2 and round,
that’s it,” he created math. And he did it in his own
unique perspective way. So, now I know not all of you are
necessarily mathematically gifted, so — (Laughter) so, let’s move on to some
more interesting stuff. By now you might have heard about
Isaac Newton in your High School career. You might have heard
about him from prisms or whatever he might have done. So, in 1665, Isaac Newton
was at the University of Cambridge. Now, for those of you who
really know your history, at that time Cambridge
had closed due to the plague. So, Isaac Newton,
he didn’t have a way to learn. He had to stop learning,
and he was probably, hiding in a dormitory with
his cat running from the plague. Now, while he was doing this
he decided he had to stop learning, but he didn’t want
to stop thinking. OK? So, because of that he was thinking
about this problem in astrophysics. And specifically I think
he wanted to calculate the motion of the Moon
around the Earth, so I sort of revamped that problem
into the case of Mercury around the Sun. So, OK. What he did was, in order to solve
this problem he created calculus, Newton’s three laws,
the universal law of gravitation, the reflecting telescope
to check his work, and optics, and all this crazy stuff in that two years
that he had stopped learning. So, I guess that was really good for us, because at that time
Newton had to stop learning; but when he stopped learning he started
thinking and he created science. And, OK, that’s just great,
we now have a theory of physics! So, OK. He could have probably
been some top scholar, he could have had a 4.0 GPA, he could have been
on the dean’s list, he could have had
his professors proud; but he wouldn’t have created anything if he didn’t stop learning. Newton needed to start thinking, and think of things out of his
own unique perspective, in order to create his theory. So, now let me formally introduce myself because I did not do that
at the beginning of the talk. So, about 11 years ago I was diagnosed
with this thing called autism. that it seemed I wasn’t thinking at all. Basically I’d be like,”Oh, look here’s
this reflection of that light, so there’s light up here,
but, oh, there’s my shadow, so there’s a light back there”
and I looked over and it’s over there. (Laughter) OK. So, because of that, you know, people thought I would never learn because it just looked
I was just staring into the opening; it looked like
I wasn’t doing anything at all. So, people told me I would
never learn, I’d never think, I’d never talk, I’d never
tie my shoes, which — OK, they might have had
a point, you know, So —
(Laughter) You know, but however,
at that age, I went to the Barnes and Noble,
and I got a textbook, and from the data that was in that textbook
I derived Kepler’s laws. When I wasn’t supposed to be
learning or thinking at all. So, basically from the other
people’s point of view it wasn’t really looking too good, I wasn’t
fingerpainting, or doing story time, or any of the other stuff
the 2-3-4 year olds would do; but, you know,
what they did was, because I — they took me to special Ed., which is extremely special in the fact that it didn’t educate me. (Laughter) So, during that time
I had to stop learning because I didn’t have a
way to learn, you know, I was just in special Ed. So what they would do is — So, I wasn’t able to learn
anything at all. However, at that age I started
thinking about things and sort of the way
of all of these shadows, and I think that’s why I like astrophysics,
and physics, and math today; because I had to stop learning, I believe that’s why
I do what I do today. OK, so let me continue
about gravity. It’s a very exciting topic for those
of us who are in physics. So let me continue. Now, what happened was, about a couple of centuries later, the physicists had enough experimental
technology to test Newton’s orbit. Now, Newton predicted that
the orbit of Mercury was an oval, or as scientists like to say
“an ellipse.” However, when we pointed our
telescopes out, we saw that thing. For those of you who are scientists you know
that’s extremely exaggerated, but — This was not looking good,
Newton had failed. One of the greatest physicists,
of all minds, had failed, he failed! (Laughter) So, we needed someone else,
just like Newton had done, to forget everything they knew! And you know, recreate this. That man’s name was Albert Einstein. Albert Einstein, what he would
— he was also — he was stopped in his tracks,
he was not doing very well. He was Jewish
and it was pre-Nazi Germany, so, he was not able to get
a position at the local university. He had to work at a patent office; which, OK, that’s not theoretical physics,
and we’re talking about Einstein here. So, yeah, what happened was, Einstein,
he had all this time to think all of a sudden. He had to stop learning,
but he had all this time to think; he liked to have these
thought experiments and liked to think about
all these different things. So what Einstein thought was, OK, he pictured himself on
a trampoline with a couple of friends, which — they are actually —
a failure of my sentence there, and the fact that physicists that’s usually a couple
more than they have. (Laughter) Albert Einstein was probably on
a trampoline with one of his friends, and you know, they were
probably playing some, I don’t know, tennis or something. So, however, you know,
they are physicists, they don’t have very good
hand to eye coordination, so they probably didn’t, you know, catch the tennis ball, and
it went rolling around them; and Einstein looked at this and said, “Without friction, this is gravity!” He realized, “This is just gravity.” So, afterwards,
he predicted the motion which is gonna end up
like that crazy thing; but that crazy thing is exactly
that other crazy thing. So, Einstein had solved the problem just by thinking about it in his
own unique perspective, in his own unique way. He stopped learning,
and he started thinking, and he started creating. So now let me get back
on the story, you know, I wasn’t really looking too good, so I just kind of brush it over there. So, about three years ago, I — OK, there was a calculus class
I wanted to sit in the back of, so, I decided,
in order to sit in the back of this, I am going to learn:
algebra, trigonometry, all this other middle school stuff, all the high school math, and first year undergrad
calculus in two weeks, so I could sit in the
back of this class. I was ten. (Laughter) Okay — So, also at that time, proving this, I got accepted into the University; and yet again I was still ten. So, OK, then I had to go to
an entrance interview, you know, that’s what you gotta do,
its a university. So, I had to go
to this entrance interview, and because of parking,
I had all these coins, and, you know, I dropped them all
over the guy’s office; making him think I had
no common sense and he pretty much held
me back for a semester. So, I also had to stop
learning at that time. OK, what did I do? Did I stop learning and just, you know,
start playing video games and stuff? No! I started thinking about shapes! (Laughter) And I was thinking about this
specific problem in astrophysics that I was really
interested in at that time, which I still kind of am. Now, what I did was, over the next two weeks I started
thinking about these shapes, I started thinking about this problem, and after a while I had solved it. So, I have solved this
problem in astrophysics, which basically is similar to, you know, what’s happening with
Einstein and Newton right now. I am not going to tell
you the exact problem due to the fact that I have
not published it, yet. When my paper gets published,
you may figure out about it; (Laughter) for those who read scientific papers. (Laughter) I thought about all these problems
and you know, I only has a 500 cheap
thing of paper from Officemax; and since I was thinking about
these multidimensional things, it filled them up really quickly. So, then I moved on to white boards
because I was out of paper. But the white board,
it also filled up pretty quickly, so then I moved on
to my parents’ windows. After that I got chased down by
all this Windex and stuff and, you know, my equations would get erased
by these horrible Windex creators but, so, because of that, after about a month or so, my parents realized I was
not going out to the park, I was just drawing these
weird shapes on the windows. And basically I was
trying to disprove myself, you know, I didn’t want to end up
like Newton; I did not want to, you know, be proven a hundred years
down the road, disproved. So, what I did was,
I was going on the windows, I was trying to disprove
myself, but to no avail. After that, my parents, you know, they figured
I should be on the park, so they called some
guy up at Princeton, and they told him to disprove
what I was doing. Unfortunately that wasn’t the case,
and he said I was on the right track; (Laughter)
(Applause) Then because I had to stop learning, I started thinking
and I solved the problem. After that I decided to create
a calculus video for other people who wanted to
still do calculus; the three others out there, and, so that way they could also learn. OK, so, I made this calculus video, people noticed that I was 12
and I was doing a calculus video. After that, the first people that
noticed was the Indianapolis Star, and they put me on the front
page of some newspaper and as you can see
from this picture, I was eating a sandwich,
it was really yummy. So, OK. After that, my calculus
video, it went viral. At the time of this photo
it had some two million views. So, first of all a calculus video going viral,
who would have ever thought? (Laughter) So, after that it got translated
into whatever this language is. Is there anybody who can tell
me what language this is? I can’t read it.
(Audience) Chinese OK, it’s Chinese,
OK, good to know. (Laughter) So, then after that, I had some
guy from Fox TV call me up, and I was able to draw on his
windows, and he was Glen Beck. (Laughter) The thing special about that experience
was that the windows were huge, 23 floors above the ground,
and overlooked the Chrysler Building, so that was a fun experience. (Laughter) Then after that I started having some really
strange visitors show up to my house. (Laughter) I had Morley Safer show up,
and he’s from CBS Sixty Minutes. Now, for those of you who can
really see this picture very well, you may notice that I am
wearing the same sandals. (Laughter) Now, let’s sort of recap
what we’ve done. Have Einstein, and Johnson, and Newton, and everyone
I talked about, are they really geniuses? Is that really what has
made them so special? Is that really why they
did all their work? Absolutely not! They, no! That’s not why!
(Laughter) OK, so, what happened was,
all they did was, they made the transition from learning, to thinking; to creating, which by now the media has
translated into, you know, genius. Now, I’m pretty sure they
had relatively high IQs; but, as some of you may know, there are lots of people out there with high
IQs who don’t create this sort of thing, they usually just end up memorizing a
couple hundred thousand digits of Pi. So, first of all, my question to them is:
why not memorize a different number? Like, I mean,
I am wearing Phi right now. So, in conclusion, I am not supposed to be here at all, you know, I was told that I wouldn’t talk. There’s probably some therapist watching
this who’s freaking out right now. (Laughter) (Cheers)
(Applause) OK, I am not supposed to be talking,
I am not supposed to be learning; but because I made that transition
from learning to thinking, to creating, I am here today; and I am talking to some four hundred
to eight hundred people in New York. OK. Now, what would I want you
guys to get out of this speech? What I want you guys to do is,
for the next 24 hours, I know you guys may have school
or what not, even though it’s a Saturday; for the next 24 hours
don’t learn anything! You are not allowed to learn
anything for the next 24 hours. (Audience) Yes! (Laughter) However, what I’d like you to do is, I’d like you to go into some field, I mean, you all have some
passion, I don’t know about it, I’ve been talking to
you for 11 minutes. I have no idea what you
guys are interested in. But, you guys have some
passion and all out there and you all know what it is. So, I want you to think about that field instead of learning in that field; and instead of being a student of that field, be the field! Whether it’s music or architecture, or science or whatever; and I want you to think about that field and, who knows,
maybe you can create something. Thank you very much.
I’m Jacob Barnett. (Cheers) (Applause)
100 thoughts on “Forget what you know | Jacob Barnett | TEDxTeen
1. Don't know when or which field would it be, but he's gonna win the nobel prize
2. While reading his mom's book The Spark about him, came here to see if this is even possible! May God bless him, his family and anyone with Autism.
3. What makes autistic people geniuses is they do what they love. and they never get bored thinking about it.
4. Now he's 21 and despises his mum for wasted childhood. "You promised me that comedy routine would get me right up there with Adam Sandler"
5. I am a sixty year old grandma and you are just magnificent young man.I am raising my grandson and there is nothing in life is rather do.Thank you could listen to you all night.
6. wow! Bold and beautiful! So much to learn! So much to appreciate!
Undoubtedly speechless!
7. I've got short term memory loss, so forgetting what I know always happens.
8. " Learn , create , think , be the field "
My interpretation in Jakes overall message is that it takes more than intelligence alone to make real change .
He mentions people who can recite Pi yet never make an advancement , create new ideas / make a difference .
Maybe savant like intelligence can be unlocked in anyone to a degree and that a key lies in finding motivation to be driven mentally enough to acquire an obsessive immersion and tap into the subconscious . Point the mind at a target and never give up .
Kim Peek was classified as a megasavant . He travelled the world impressing people with his infinite recitation of facts .
Yet to my knowledge never created like Newton or Einstein .
9. But somehow this so much intelligence like 170 iq can be heavy to carry. I can see he has thunders inside and how difficult it is. He is so lovely by the way.
10. This kid is so not offensive and, the 5500 people who down voted this must belong to a part of society that appreciates nothing.
11. obivously a genious but he has to learn to restrain himself abit haha xD
12. Actually, I find it more difficult to stop learning than just memorizing the others' ideas. At the same time, it almost makes me feel like I'm reborn in a world of my own.
13. Hey Jacob – think about how our genius-level acceleration toward Progress (as Ape-men) is inextricably tumbling us toward extinction. Hmm… Is there a "dark side" to learning beyond our most recent genetic mutation 200,000 years ago, or am I perhaps just evolving epigenetically toward a more sustainable mode for our planet and the answers/solutions to our manifestly diseased place within it? P.S. And don't forget Leibnitz!
14. Math, Physics, Chemistry (AKA God) are the true firmament. Biology, Anthropology, Archaeology, History, Film and Women's Studies (AKA All life on Earth) are terra firma. And your genius will always be suspect if you can't compassionately merge "Heaven" and "Earth." [Einstein did]
15. People with HFA have problem learning, therefore, they have a lot of opportunity to think and be smart
16. This amazing boy's huge potential is the result of loving parents and compassionate teachers. Kudo to his dad+mom and his teachers throughout school and college. You embrace the darkness and seek the light of it so this boy can be a ray of hope and light for many many kids with difficult conditions in the future.
17. This boy is amazing! Early challenges must have been difficult on his parents and him. Being told your son won’t speak and here he is on a Ted Talk. And that is surpassing all expectations the experts put on him. He’s not perfect but no one is regardless. 👏🏽👏🏽👨🏻🎓 I’m proud of him and I don’t even know his family. Congrats!
18. Everytime the audience laugh:😂🤣🤣🤣🤣
Me:😂 I don't get it 🤔
19. Can't watch this kid without smiling. It hardly even matters how smart he is…what's most impressive is that joie de vivre.
20. He’s thinking like Isaac Newton, Albert Einstein or other important scientists. And i think he’ll change the world.
21. Amazing child…such charisma too…Awesome is so mild a word…❤💯
22. The coolest guy ever! I want a friend who get this crazy with physics!
23. He is the most amazing and sweat person I've ever seen. He's a genius.
24. 考えるんだ！脳みそをシワくちゃにしろ！
25. Such little guy has such many things to tell. As person with Aspereger all I can say about him is "Wow".
26. This kid is insanely smart and it’s astonishing. He’s kind of cringy tho.
27. I've been looking at videos on autism due to my child's physician hinting at autism in my 2 year old, I've noticed that each child's parent says in their own world or something similar. Then I came across a video of several people in their 20s that have autism that you would never guess had it and each one said they learned in a very different way, a lot of these videos talk about spectrums and it's intesting to me because it's not a disease it's just a different way of learning like everybody else just because you have freckles does not mean that the other person is going to have freckles or red hair or green eyes. It just means that you're learning differently than everybody else I have three other kids and each heart individually unique to one another my youngest is the one being diagnosed with autism yet from what I have observed he is highly intelligent he does things that his other brothers would have never done at that age so brings me to believe that each parent should take their individual time to try and find ways to teach their kids so they can take in as much as I can and run with what they know and keep growing instead of just labeling their child as a disabled child for handicapped when in reality they're perfectly normal it's just their spectrum of looking at something is totally different from how you see it
28. Who knows, he might end up re-deriving the CTMU, and perhaps parallel it, just in other words and/or syntax, without having known at all. Interesting stuff, there.
29. なんて明るい少年！
30. This kid look so frightened and uncomfortable on stage it makes me sad. Like he’s definitely on the spectrum hard. And is FORCING himself to smile.
31. I'm just looking at the comment section and thinking;
people who use the word 'cringe' are probably at the lowest level on the intelligence quotient… meaning they won't watch the video b/c they're focusing on shallow stuff instead of the actual message which is how far their focus goes.
Inspiring kid, especially with his condition. Thinking out of the mold is just as important as working within a paradigm.
32. "La transición desde el aprendizaje hasta el pensamiento y la creación".
"Sean la disciplina".
33. I think first he needs to control his motions .
Sorry ,no offense intended
34. This kid is genius there is no doubt in it but unfortunately he lacks in confidence that's why he looks too nervous though he is correct in everything.
35. unbelievable I had watched this talk years ago and I have understood his concept now his mind is approaching stars MASHALLAH
36. この子めっちゃ盛り上がってていききれてしかも自分でわらっっちゃっててかわいいい🤭❤️❤️
37. この子めっちゃ可愛い
38. To see any child this excited about what he can know and excited about telling others about it is spectacular.
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Josef Loschmidt
Ernst Mach
Donald MacKay
Henry Margenau
James Clerk Maxwell
Ernst Mayr
John McCarthy
Warren McCulloch
George Miller
Stanley Miller
Ulrich Mohrhoff
Jacques Monod
Emmy Noether
Alexander Oparin
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
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
The Measurement Problem
The "Problem of Measurement" in quantum mechanics has been defined in various ways, originally by scientists, and more recently by philosophers of science who question the "foundations of quantum mechanics."
Measurements are described with diverse concepts in quantum physics such as:
• wave functions (probability amplitudes) evolving unitarily and deterministically (preserving information) according to the linear Schrödinger equation, (von Neumann's Process 2)
• superposition of states, i.e., linear combinations of wave functions with complex coefficients that carry phase information and produce interference effects (Dirac's principle of superposition),
• quantum jumps between states accompanied by the "collapse" of the wave function that can destroy or create information (Dirac's projection postulate, von Neumann's Process 1),
• probabilities of collapses and jumps given by the square of the absolute value of the wave function for a given state,
• values for possible measurements given by the eigenvalues associated with the eigenstates of the combined measuring apparatus and measured system (Dirac's axiom of measurement),
• Heisenberg indeterminacy principle.
The original problem, said to be a consequence of Niels Bohr's "Copenhagen interpretation" of quantum mechanics, was to explain how our measuring instruments, which are usually macroscopic objects and treatable with classical physics, can give us information about the microscopic world of atoms and subatomic particles like electrons and photons.
Bohr's idea of "complementarity" insisted that a specific experiment could reveal only partial information - for example, a particle's position. "Exhaustive" information requires complementary experiments, for example to also determine a particle's momentum (within the limits of Werner Heisenberg's indeterminacy principle).
Some define the problem of measurement simply as the logical contradiction between two laws describing the motion of quantum systems; the unitary, continuous, and deterministic time evolution of the Schrödinger equation versus the non-unitary, discontinuous, and indeterministic collapse of the wave function. John von Neumann saw a problem with two distinct (indeed, opposing) processes.
The mathematical formalism of quantum mechanics provides no way to predict when the wave function stops evolving in a unitary fashion and collapses. Experimentally and practically, however, we can say that this occurs when the microscopic system interacts with a measuring apparatus.
Others define the measurement problem as the failure to observe macroscopic superpositions.
Decoherence theorists (e.g., H. Dieter Zeh and Wojciech Zurek, who use various non-standard interpretations of quantum mechanics that deny the projection postulate - quantum jumps - and even the existence of particles), define the measurement problem as the failure to observe macroscopic superpositions such as Schrödinger's Cat. Unitary time evolution of the wave function according to the Schrödinger wave equation should produce such macroscopic superpositions, they claim.
Information physics treats a measuring apparatus quantum mechanically by describing parts of it as in a metastable state like the excited states of an atom, the critically poised electrical potential energy in the discharge tube of a Geiger counter, or the supersaturated water and alcohol molecules of a Wilson cloud chamber. (The pi-bond orbital rotation from cis- to trans- in the light-sensitive retinal molecule is an example of a critically poised apparatus).
Excited (metastable) states are poised to collapse when an electron (or photon) collides with the sensitive detector elements in the apparatus. This collapse is macroscopic and irreversible, generally a cascade of quantum events that release large amounts of energy, increasing the (Boltzmann) entropy. But in a "measurement" there is also a local decrease in the entropy (negative entropy or information). The global entropy increase is normally orders of magnitude more than the small local decrease in entropy (an increase in stable information or Shannon entropy) that constitutes the "measured" experimental data available to human observers.
The creation of new information in a measurement thus follows the same two core processes of all information creation - quantum cooperative phenomena and thermodynamics. These two are involved in the formation of microscopic objects like atoms and molecules, as well as macroscopic objects like galaxies, stars, and planets.
According to the correspondence principle, all the laws of quantum physics asymptotically approach the laws of classical physics in the limit of large quantum numbers and large numbers of particles. Quantum mechanics can be used to describe large macroscopic systems.
Does this mean that the positions and momenta of macroscopic objects are uncertain? Yes, it does, although the uncertainty becomes vanishingly small for large objects, it is not zero. Niels Bohr used the uncertainty of macroscopic objects to defeat Albert Einstein's several objections to quantum mechanics at the 1927 Solvay conference.
But Bohr and Heisenberg also insisted that a measuring apparatus must be a regarded as a purely classical system. They can't have it both ways. Can the macroscopic apparatus also be treated by quantum physics or not? Can it be described by the Schrödinger equation? Can it be regarded as in a superposition of states?
The most famous examples of macroscopic superposition are perhaps Schrödinger's Cat, which is claimed to be in a superposition of live and dead cats, and the Einstein-Podolsky-Rosen experiment, in which entangled electrons or photons are in a superposition of two-particle states that collapse over macroscopic distances to exhibit properties "nonlocally" at speeds faster than the speed of light.
These treatments of macroscopic systems with quantum mechanics were intended to expose inconsistencies and incompleteness in quantum theory. The critics hoped to restore determinism and "local reality" to physics. They resulted in some strange and extremely popular "mysteries" about "quantum reality," such as the "many-worlds" interpretation, "hidden variables," and signaling faster than the speed of light.
We develop a quantum-mechanical treatment of macroscopic systems, especially a measuring apparatus, to show how it can create new information. If the apparatus were describable only by classical deterministic laws, no new information could come into existence. The apparatus need only be adequately determined, that is to say, "classical" to a sufficient degree of accuracy.
How Classical Is A Macroscopic Measuring Apparatus?
As Landau and Lifshitz described it in their 1958 textbook Quantum Mechanics"
The possibility of a quantitative description of the motion of an electron requires the presence also of physical objects which obey classical mechanics to a sufficient degree of accuracy. If an electron interacts with such a "classical object", the state of the latter is, generally speaking, altered. The nature and magnitude of this change depend on the state of the electron, and therefore may serve to characterise it quantitatively...
1. A non-causal process 1, in which the measured electron winds up randomly in one of the possible physical states (eigenstates) of the measuring apparatus plus electron.
This process came to be called the collapse of the wave function or the reduction of the wave packet.
The probability for finding the electron in a specific eigenstate is given by the square of the coefficients cn of the expansion of the original system state (wave function ψ) in an infinite set of wave functions φ that represent the eigenfunctions of the measuring apparatus plus electron.
Information physics says that the particle "shows up" only when a new stable information structure is created, information that subsequently can be observed.
2. A causal process 2, in which the electron wave function ψ evolves deterministically according to Schrödinger's equation of motion for the wavelike aspect.
(ih/2π) ∂ψ/∂t = .
The particle path can not be observed.
Von Neumann claimed there is another major difference between these two processes. Process 1 is thermodynamically irreversible. Process 2 is reversible. This confirms the fundamental connection between quantum mechanics and thermodynamics that information physics finds at the heart of all information creation.
Information physics can show quantum mechanically how process 1 creates information. Indeed, something like process 1 is always involved when any information is created, whether or not the new information is ever "observed" by a human being.
Process 2 is deterministic and information preserving.
Just as the new information recorded in the measurement apparatus cannot subsist unless a compensating amount of entropy is transferred away from the new information, something similar to Process 1b must happen in the mind of an observer if the new information is to constitute an "observation."
It is only in cases where information persists long enough for a human being to observe it that we can properly describe the observation as a "measurement" and the human being as an "observer." So, following von Neumann's "process" terminology, we can complete his theory of the measuring process by adding an anthropomorphic
Process 3 - a conscious observer recording new information in a mind. This is only possible if there are two local reductions in the entropy (the first in the measurement apparatus, the second in the mind), both balanced by even greater increases in positive entropy that must be transported away from the apparatus and the mind, so the overall increase in entropy can satisfy the second law of thermodynamics.
For some physicists, it is the wave-function collapse that gives rise to the problem of measurement because its randomness prevents us from including it in the mathematical formalism of the deterministic Schrödinger equation in process 2.
The randomness that is irreducibly involved in all information creation lies at the heart of human freedom. It is the "free" in "free will." The "will" part is as adequately and statistically determined as any macroscopic object.
Designing a Quantum Measurement Apparatus
The first step is to build an apparatus that allows different components of the wave function to evolve along distinguishable paths into different regions of space, where the different regions correspond to (are correlated with) the physical properties we want to measure. We then can locate a detector in these different regions of space to catch particles travelling a particular path.
We do not say that the system is on a particular path in 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 von Neumann process 2.
Let's consider the separation of a beam of photons into horizontally and vertically polarized photons by a birefringent crystal.
We need a beam of photons (and the ability to reduce the intensity to a single photon at a time). Vertically polarized photons pass straight through the crystal. They are called the ordinary ray, shown in red. Horizontally polarized photons, however, are deflected at an angle up through the crystal, then exit the crystal back at the original angle. They are called the extraordinary ray, shown in blue.
We have not actually measured yet, so a single photon passing through our measurement apparatus is described as in a linear combination (a superposition) of horizontal and vertical polarization states,
| ψ > = ( 1/√2) | h > + ( 1/√2) | v > (1)
See the Dirac Three Polarizers experiment for more details on polarized photons.
An Information-Preserving, Reversible Example of Process 2
To show that process 2 is reversible, we can add a second birefringent crystal upside down from the first, but inline with the superposition of 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 crystal, they emerge in a state identical to that before entering the first crystal (black arrow).
Note that the two crystals can be treated classically, according to standard optics.
An Information-Creating, Irreversible Example of Process 1
But now suppose we insert something between the two crystals that is capable of a measurement to produce observable information. We need a detector that locates the photon in one of the two rays.
We can now create an information-creating, irreversible example of process 1. Suppose we insert something between the two crystals that is capable of a measurement to produce observable information. We need detectors, for example two charge-coupled devices that locate the photon in one of the two rays.
We can write a quantum description of the CCDs, one measuring horizontal photons, | Ah > (shown as the blue spot), and the other measuring vertical photons, | Av > (shown as the red spot).
We treat the detection systems quantum mechanically, and say that each detector has two eigenstates, e.g., | Ah0 >, corresponding to its initial state and correlated with no photons, and the final state | Ah1 >, in which it has detected a horizontal photon.
When we actually detect the photon, say in a horizontal polarization 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 photon in equation (1) into the horizontally polarized state | h >.
The second is the quantum jump of the horizontal detector from | Ah0 > to | Ah1 >.
These two happen together, as the quantum states have become correlated with the states of the sensitive detectors in the classical apparatus.
One can say that the photon has become entangled with the sensitive horizontal detector area, so that the wave function describing their interaction is a superposition of photon and apparatus states that cannot be observed independently.
| ψ > + | Ah0 > => | ψ, Ah0 > => | h, Ah1 >
These jumps destroy (unobservable) phase information, raise the (Boltzmann) entropy of the apparatus, and increase visible information (Shannon entropy) in the form of the visible spot. The entropy increase takes the form of a large chemical energy release when the photographic spot is developed (or a cascade of electrons in a CCD).
Note that the birefringent crystal and the parts of the macroscopic apparatus other than the sensitive detectors are treated classically.
We can animate these irreversible and reversible processes,
We see that our example agrees with Von Neumann. A measurement which finds the photon in a specific state n is thermodynamically irreversible, whereas the deterministic evolution described by Schrödinger's equation is reversible.
The Boundary between the Classical and Quantum Worlds
Some scientists (John von Neumann and Eugene Wigner, for example) have argued that in the absence of a conscious observer, or some "cut" between the microscopic and macroscopic world, the evolution of the quantum system and the macroscopic measuring apparatus would be described deterministically by Schrödinger's equation of motion for the wave function | ψ + A > with the Hamiltonian H energy operator,
(ih/2π) ∂/∂t | ψ + A > = H | ψ + A >.
Our quantum mechanical analysis of the measurement apparatus in the above case allows us to locate the "cut" or "Schnitt" between the microscopic and macroscopic world at those components of the "adequately classical and deterministic" apparatus that put the apparatus in an irreversible stable state providing new information to the observer.
John Bell drew a diagram to show the various possible locations for what he called the "shifty split." Information physics shows us that the correct location for the boundary is the first of Bell's possibilities.
The Role of a Conscious Observer
In 1941, Carl von Weizsäcker described the measurement problem as an interaction between a Subject and an Object, a view shared by the philosopher of science Ernst Cassirer.
Fritz London and Edmond Bauer made the strongest case for the critical role of a conscious observer in 1939:
So far we have only coupled one apparatus with one object. But a coupling, even with a measuring device, is not yet a measurement. A measurement is achieved only when the position of the pointer has been observed. It is precisely this increase of knowledge, acquired by observation, that gives the observer the right to choose among the different components of the mixture predicted by theory, to reject those which are not observed, and to attribute thenceforth to the object a new wave function, that of the pure case which he has found.
We note the essential role played by the consciousness of the observer in this transition from the mixture to the pure case. Without his effective intervention, one would never obtain a new function.
In 1961, Eugene Wigner made quantum physics even more subjective, claiming that a quantum measurement requires a conscious observer, without which nothing ever happens in the universe.
When the province of physical theory was extended to encompass microscopic phenomena, through the creation of quantum mechanics, the concept of consciousness came to the fore again: it was not possible to formulate the laws of quantum mechanics in a fully consistent way without reference to the consciousness All that quantum mechanics purports to provide are probability connections between subsequent impressions (also called "apperceptions") of the consciousness, and even though the dividing line between the observer, whose consciousness is being affected, and the observed physical object can be shifted towards the one or the other to a considerable degree [cf., von Neumann] it cannot be eliminated. It may be premature to believe that the present philosophy of quantum mechanics will remain a permanent feature of future physical theories; it will remain remarkable, in whatever way our future concepts may develop, that the very study of the external world led to the conclusion that the content of the consciousness is an ultimate reality.
Other physicists were more circumspect. Niels Bohr contrasted Paul Dirac's view with that of Heisenberg:
Landau and Lifshitz said clearly that quantum physics was independent of any observer:
In this connection the "classical object" is usually called apparatus, and its interaction with the electron is spoken of as measurement. However, it must be most decidedly emphasised that we are here not discussing a process of measurement in which the physicist-observer takes part. By measurement, in quantum mechanics, we understand any process of interaction between classical and quantum objects, occurring apart from and independently of any observer.
David Bohm agreed that what is observed is distinct from the observer:
If it were necessary to give all parts of the world a completely quantum-mechanical description, a person trying to apply quantum theory to the process of observation would be faced with an insoluble paradox. This would be so because he would then have to regard himself as something connected inseparably with the rest of the world. On the other hand,the very idea of making an observation implies that what is observed is totally distinct from the person observing it.
And John Bell said:
It would seem that the [quantum] theory is exclusively concerned about 'results of measurement', and has nothing to say about anything else. What exactly qualifies some physical systems to play the role of 'measurer'? Was the wavefunction of the world waiting to jump for thousands of millions of years until a single-celled living creature appeared? Or did it have to wait a little longer, for some better qualified system...with a Ph.D.? If the theory is to apply to anything but highly idealised laboratory operations, are we not obliged to admit that more or less 'measurement-like' processes are going on more or less all the time, more or less everywhere? Do we not have jumping then all the time?
Three Essential Steps in a "Measurement" and "Observation"
We can distinguish three required elements in a measurement that can clarify the ongoing debate about the role of a conscious observer.
1. In standard quantum theory, the first required element is the collapse of the wave-function. This is the Dirac projection postulate and von Neumann Process 1.
As we showed above for photons, the detector in the upper half of a Stern-Gerlach apparatus will fire, indicating detection of an electron with spin up. As with photons, if the probability amplitude | ↑ > in the upper half does not collapse as the electron is detected, it can still be recombined with the probability amplitude | ↓ > in the lower half to reconstruct the unseparated beam.
When the apparatus detects a particle, the second required element is that it produce a determinate record of the event. But 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 teh information created.
3. Finally, the third required element is an indelible determinate record that can be looked at by an observer (presumably conscious, although the consciousness itself has nothing to do with the measurement).
For Teachers
For Scholars
John Bell on Measurement
Does [the 'collapse of the wavefunction'] happen sometimes outside laboratories? Or only in some authorized 'measuring apparatus'? And whereabouts in that apparatus? In the Einstein—Podolsky-Rosen—Bohm experiment, does 'measurement' occur already in the polarizers, or only in the counters? Or does it occur still later, in the computer collecting the data, or only in the eye, or even perhaps only in the brain, or at the brain—mind interface of the experimenter?
David Bohm on Measurement
John von Neumann on Measurement
cn = < φn | ψ >
(ih/2π) ∂ψ/∂t =
Process 2 is deterministic and information preserving.
It gave rise to the so-called problem of measurement because its randomness prevents it from being a part of the deterministic mathematics of process 2.
Information physics has solved the problem of measurement by identifying the moment and place of the collapse of the wave function with the creation of an observable information structure.
The presence of a conscious observer is not necessary. It is enough that the new information created is observable, should a human observer try to look at it in the future. Information physics is thus subtly involved in the question of what humans can know (epistemology).
The Schnitt
von Neumann described the collapse of the wave function as requiring a "cut" (Schnitt in German) between the microscopic quantum system and the observer. He said it did not matter where this cut was placed, because the mathematics would produce the same experimental results.
There has been a lot of controversy and confusion about this cut. Some have placed it outside a room which includes the measuring apparatus and an observer A, and just before observer B makes a measurement of the physical state of the room, which is imagined to evolve deterministically according to process 2 and the Schrödinger equation.
von Neumann contributed a lot to this confusion in his discussion of subjective perceptions and "psycho-physical parallelism, which was encouraged by Neils Bohr. Bohr interpreted his "complementarity principle" as explaining the difference between subjectivity and objectivity (as well as several other dualisms). von Neumann wrote:
the Schnitt
Quantum Mechanics, by Albert Messiah, on Measurement
Messiah says a detailed study of the mechanism of measurement will not be made in his book, but he does say this.
The dynamical state of such a system is represented at a given instant of time by its wave function at that instant. The causal relationship between the wave function γ(to) at an initial time to, and the wave function γ(t) at any later time, is expressed through the Schrödinger equation. However, as soon as it is subjected to observation, the system experiences some reaction from the observing instrument. Moreover, the above reaction is to some extent unpredictable and uncontrollable since there is no sharp separation between the observed system and the observing instrument. They must be treated as an indivisible quantum system whose wave function Ψ(t) depends upon the coordinates of the measuring device as well as upon those of the observed system. During the process of observation, the measured system can no longer be considered separately and the very notion of a dynamical state defined by the simpler wave function γ(t) loses its meaning. Thus the intervention of the observing instrument destroys all causal connection between the state of the system before and after the measurement; this explains why one cannot in general predict with certainty in what state the system will be found after the measurement; one can only make predictions of a statistical nature1
1) The statistical predictions concerning the results of measurement are derived very naturally from the study of the mechanism of the measuring operation itself, a study in which the measuring instrument is treated as a quantized object and the complex (system + measuring instrument) evolves in causal fashion in accordance with the Schrödinger equation. A very concise and simple presentation of the measuring process in Quantum Mechanics is given. in F. London and E. Bauer, La Théorie de l'Observation en Mécanique Quantique (Paris, Hermann, 1939). More detailed discussions of this problem may be found in J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton, Princeton University Press, 1955), and in D. Bohm, (Quantum Theory New York, Prentice-Hall, 1951).
Quantum Mechanics, vol. I, p. 157
Decoherence Theorists on Measurement
In general, decoherence theorists see the problem of measurement as why do we not see macroscopic superpositions of states. Why do measurements always show a system and its measuring apparatus to be in a particular state - a "pointer state," and not in a superposition?
Our answer is that we never see microscopic systems in a superposition of states either. Dirac's principle of superposition says only that the probability (amplitudes) of finding a system in different states has non-zero values for different states. Measurements always reveal a system to be in one state. Which state is found is a matter of chance. [Decoherence theorists do not like this indeterminism.] The statistics from large numbers of measurements of identically prepared systems verify the predicted probabilities for the different states. The accuracy of these quantum mechanical predictions (1 part in 1015) shows quantum mechanics to be the most accurate theory ever known.
Guido Bacciagaluppi summarized the view of decoherence theorists in an article for the Stanford Encyclopedia of Philosophy. He defines the measurement problem as the lack of macroscopic superpositions:
Maximilian Schlosshauer situates the problem of measurement in the context of the so-called "quantum-to-classical transition," namely the question of exactly how deterministic classical behavior emerges from the indeterministic microscopic quantum world.
In this section, we shall describe the (in)famous measurement problem of quantum mechanics that we have already referred to in several places in the text. The choice of the term "measurement problem" has purely historical reasons: Certain foundational issues associated with the measurement problem were first illustrated in the context of a quantum-mechanical description of a measuring apparatus interacting with a system.
However, one may regard the term "measurement problem" as implying too narrow a scope, chiefly for the following two reasons. First, as we shall see below, the measurement problem is composed of three distinct issues, so it would make sense to rather speak of measurement problems. Second, quantum measurement and the arising foundational problems are but a special case of the more general problem of the quantum-to-classical transition, i.e., the question of how effectively classical systems and properties around us emerge from the underlying quantum domain.
On the one hand, then, the problem of the quantum-to-classical transition has a much broader scope than the issue of quantum measurement in the literal sense. On the other hand, however, many interactions between physical systems can be viewed as measurement-like interactions. For example, light scattering off an object carries away information about the position of the object, and it is in this sense that we thus may view these incident photons as a "measuring device." Such ubiquitous measurement-like interactions lie at the heart of the explanation of the quantum-to-classical transition by means of decoherence. Measurement, in the more general sense, thus retains its paramount importance also in the broader context of the quantum-to-classical transition, which in turn motivates us not to abandon the term "measurement problem" altogether in favor of the more general "problem of the quantum-to-classical transition."
As indicated above, the measurement problem (and the problem of the quantum-to-classical transition) is composed of three parts, all of which we shall describe in more detail in the following:
1. The problem of the preferred basis (Sect. 2.5.2). What singles out the preferred physical quantities in nature—e.g., why are physical systems usually observed to be in definite positions rather than in superpositions of positions?
2. The problem of the nonobservability of interference (Sect. 2.5.3). Why is it so difficult to observe quantum interference effects, especially on macroscopic scales?
3. The problem of outcomes (Sect. 2.5.4). Why do measurements have outcomes at all, and what selects a particular outcome among the different possibilities described by the quantum probability distribution? [The answer (since Einstein, 1916) is chance.]
Familiarity with these problems will turn out to be important for a proper understanding of the scope, achievements, and implications of decoherence. To anticipate, it is fair to conclude that decoherence has essentially resolved the first two problems. Since these problems and their resolution can be formulated in purely operational terms within the standard formalism of quantum mechanics, the role played by decoherence in addressing these two issues is rather undisputed.
By contrast, the success of decoherence in tackling the third issue — the problem of outcomes — remains a matter of debate, in particular, because this issue is almost inextricably linked to the choice of a specific interpretation of quantum mechanics (which mostly boils down to a matter of personal preference). In fact, most of the overly optimistic or pessimistic statements about the ability of decoherence to solve "the" measurement problem can be traced back to a misunderstanding of the scope that a standard quantum effect such as decoherence may have in resolving the more interpretive problem of outcomes.
The main concern of the decoherence theorists then is to recover a deterministic picture of quantum mechanics that would allow them to predict the outcome of a particular experiment. They have what William James called an "antipathy to chance."
Max Tegmark and John Wheeler made this clear in a 2001 article in Scientific American:
The discovery of decoherence, combined with the ever more elaborate experimental demonstrations of quantum weirdness, has caused a noticeable shift in the views of physicists. The main motivation for introducing the notion of wave-function collapse had been to explain why experiments produced specific outcomes and not strange superpositions of outcomes. Now much of that motivation is gone. Moreover, it is embarrassing that nobody has provided a testable deterministic equation specifying precisely when the mysterious collapse is supposed to occur.
H. Dieter Zeh, the founder of the "decoherence program," defines the measurement problem as a macroscopic entangled superposition of all possible measurement outcomes:
Because of the dynamical superposition principle, an initial superposition Σ cn | n > does not lead to definite pointer positions (with their empirically observed frequencies). If decoherence is neglected, one obtains their entangled superposition Σ cn | n > | Ψ n >, that is, a state that is different from all potential measurement outcomes | n > | Ψ n >. This dilemma represents the "quantum measurement problem" to be discussed in Sect. 2.3. Von Neumann's interaction is nonetheless regarded as the first step of a measurement (a "pre-measurement"). Yet, a collapse seems still to be required - now in the measurement device rather than in the microscopic system. Because of the entanglement between system and apparatus, it would then affect the total system.
Zeh continues:
2.3 The Measurement Problem
The superposition of different measurement outcomes, resulting according to a Schrodinger equation when applied to the total system (as discussed above), demonstrates that a "naive ensemble interpretation" of quantum mechanics in terms of incomplete knowledge is ruled out.
It's not clear why the standard ensemble interpretation is "ruled out," but Zeh offers a solution, which is to deny the projection postulate of standard quantum mechanics and use an unconventional interpretation that makes wave-function collapses only "apparent":
A way out of this dilemma within quantum mechanical concepts requires one of two possibilities: a modification of the Schrodinger equation that explicitly describes a collapse (also called "spontaneous localization" - see Chap. 8), or an Everett type interpretation, in which all measurement outcomes are assumed to exist in one formal superposition, but to be perceived separately as a consequence of their dynamical autonomy resulting from decoherence.
It was John Bell who called Everett's Many-Worlds Interpretation "extravagant."
While this latter suggestion has been called "extravagant" (as it requires myriads of co-existing quasi-classical "worlds"), it is similar in principle to the conventional (though nontrivial) assumption, made tacitly in all classical descriptions of observation, that consciousness is localized in certain semi-stable and sufficiently complex subsystems (such as human brains or parts thereof) of a much larger external world.
Everett rejected the intuitively simple collapse of multiple possibilities to one actual situation. Instead he proposed the instantaneous creation of multiple universes, each with all the matter and energy of the observable universe. Surely his Many Worlds was the most absurd anti-Occam proposal ever made!
Occam's razor, often applied to the "other worlds", is a dangerous instrument: philosophers of the past used it to deny the existence of the interior of stars or of the back side of the moon, for example. So it appears worth mentioning at this point that environmental decoherence, derived by tracing out unobserved variables from a universal wave function, readily describes precisely the apparently observed "quantum jumps" or "collapse events" (as will be discussed in great detail in various parts of this book).
Jeffrey Bub worked with David Bohm to develop Bohm's theory of "hidden variables." They hoped their theory might provide a deterministic basis for quantum theory and support Albert Einstein's view of a physical world independent of observations of the world. The standard theory of quantum mechanics is irreducibly statistical and indeterministic, a consequence of the collapse of the wave function when many possibilities for physical outcomes of an experiment reduce to a single actual outcome.
This is a book about the interpretation of quantum mechanics, and about the measurement problem. The conceptual entanglements of the measurement problem have their source in the orthodox interpretation of 'entangled' states that arise in quantum mechanical measurement processes...
All standard treatments of quantum mechanics take an observable as having a determinate value if the quantum state is an eigenstate of that observable. If the state is not an eigenstate of the observable, no determinate value is attributed to the observable. This principle - sometimes called the 'eigenvalue-eigenstate link' - is explicitly endorsed by Dirac (1958, pp. 46-7) and von Neumann (1955, p. 253), and clearly identified as the 'usual' view by Einstein, Podolsky, and Rosen (1935) in their classic argument for the incompleteness of quantum mechanics (see chapter 2). Since the dynamics of quantum mechanics described by Schrödinger's time-dependent equation of motion is linear, it follows immediately from this orthodox interpretation principle that, after an interaction between two quantum mechanical systems that can be interpreted as a measurement by one system on the other, the state of the composite system is not an eigenstate of the observable measured in the interaction, and not an eigenstate of the indicator observable functioning as a 'pointer.' So, on the orthodox interpretation, neither the measured observable nor the pointer reading have determinate values, after a suitable interaction that correlates pointer readings with values of the measured observable. This is the measurement problem of quantum mechanics.
Bacciagaluppi, Guido, The Role of Decoherence in Quantum Mechanics, first published Mon Nov 3, 2003; substantive revision Mon Apr 16, 2012
Jeffrey Bub, Interpreting the Quantum World. Cambridge University, 1997, p.2.
Adriana Daneri, A. Loinger, and G. M. Prosperi, Nuclear Physics, 33 (1962) pp.297-319. (W&Z, p.657)
Erich Joos, H. Dieter Zeh, et al., Decoherence and the Appearance of a Classical World in Quantum Theory. Springer, 2010,
Gunter Ludwig, Zeitschrift für Physik, 135 (1953) p.483
Maximilian Schlosshauer, Decoherence and the Quantum-to-Classical Transition. Springer, 2007, pp.49-50
Leo Szilard, Behavioral Science, 9 (1964) pp.301-10. (W&Z, p.539)
Max Tegmark and John Wheeler, Scientific American, February (2001) pp.68-75.
John von Neumann, The Mathematical Foundations of Quantum Mechanics, (Princeton, NJ, Princeton U. Press, 1955), pp.347-445. (W&Z, p.549)
John Wheeler and Wojciech Zurek, Quantum Theory and Measurement (Princeton, NJ, Princeton U. Press, 1983) (= W&Z)
Eugene Wigner, "The Problem of Measurement," Symmetries and Reflections (Bloomington, IN, Indiana U. Press, 1967) pp.153-70. (W&Z, p.324)
Chapter 5.4 - Immortality Chapter 5.6 - Mind-Body
Part Four - Knowledge Part Six - Solutions
Normal | Teacher | Scholar |
7384c0fad7e3e9e6 | A Single-atom optical excitation
Entanglement of neutral-atom chains by spin-exchange Rydberg interaction
Conditions to achieve an unusually strong Rydberg spin-exchange interaction are investigated and proposed as a means to generate pairwise entanglement and realize a SWAP-like quantum gate for neutral atoms. Ground-state entanglement is created by mapping entangled Rydberg states to ground states using optical techniques. A protocol involving SWAP gate and pairwise entanglement operations is predicted to create global entanglement of a chain of atoms in a time that is independent of .
03.67.Bg, 03.65.Ud, 32.80.Ee, 32.80.Xx
I introduction
Entanglement is a property of multiparticle quantum states that is essential for implementing quantum information or computation protocols Einstein et al. (1935); Nielsen and Chuang (2000). As a result, schemes for the fast and efficient generation of entanglement among many quantum systems are the subject of intense theoretical and experimental efforts. Atoms offer an ideal arena for the demonstration of quantum protocols given the stability of their ground states and the powerful optical and trapping techniques that have been developed to control their internal and external degrees of freedom Adams and Riis (1997). The excitation of Alkali atoms to energy levels of large principal quantum number, generically named Rydberg states, provides a way to enhance by several orders of magnitude otherwise weak neutral atom interactions. The resulting dipole-dipole or van der Waals interaction between two highly excited Rydberg atoms allows the creation of stable entangled states of atoms in their ground electronic level Møller et al. (2008); Saffman and Mølmer (2009); Zhao et al. (2010); Saffman et al. (2010); Isenhower et al. (2010); Urban et al. (2009); Zhang et al. (2012); Carr and Saffman (2013); Bhaktavatsala Rao and Mø lmer (2013). The methods employed to date Møller et al. (2008); Saffman and Mølmer (2009); Zhao et al. (2010); Saffman et al. (2010); Isenhower et al. (2010); Urban et al. (2009) rely on the Rydberg blockade effect, in which two-atom energy levels with two Rydberg excitations experience a large interaction induced shift, while energy levels with a single Rydberg excitation are unperturbed. As a consequence, under conditions in which a single Rydberg excitation is resonantly excited, the pair excitation probability is strongly suppressed Tong et al. (2004); Heidemann et al. (2007); Vogt et al. (2007); Schwarzkopf et al. (2011); Dudin and Kuzmich (2012); Dudin et al. (2012); Viteau et al. (2012); Barredo et al. (2014).
The excitation of two atoms into Rydberg -orbitals with different principal quantum numbers, and , and opposite electron spin orientation, produces not only the Rydberg blockade shift van Ditzhuijzen et al. (2008); Han and Gallagher (2009); Bariani et al. (2012); Günter et al. (2013); Gorniaczyk et al. (2014); Tiarks et al. (2014); Li et al. (2014); Paredes-Barato and Adams (2014), but also a coupling that exchanges the electron spin states. While the blockade shift is usually the dominant effect, the spin-exchange coupling can be made almost equally strong with the right choice of and . In this regime, when nearby atoms are optically driven, the probability to create double Rydberg excitations can be as large as one, in sharp contrast to the case when . Furthermore, the two-atom Rydberg state created by this mechanism is one of two entangled Bell states, the triplet denoted , in the subspace of the two-atom Rydberg excitations. The orthogonal Rydberg Bell state, the singlet experiences a strong level shift and is effectively decoupled from the excitation. The entangled Rydberg state created in this way is metastable, but the entanglement can be mapped optically to the ground state in a time short compared to the metastable state lifetime. In the following, we explain how to produce ground state entanglement of a pair of atoms by a sequence of three pulses, and then discuss a protocol for global entanglement of a chain of atoms involving a sequence of twelve pulses which minimizes the blockade shifts due to multiple Rydberg excitations along the chain.
Figure 1: (Color online) Schematic illustration for generating entanglement between two Rb atoms. (a) Geometry considered with atoms (black dots) in tightly focused optical traps (shaded regions) along the quantization axis . (b) Laser excitation of atom by two-photon transitions. The two-photon transition via the 5P (6P) state of atom has an effective Rabi frequency (). (c) Relevant levels in the excitation of Rydberg Bell state .
The main body of the paper gives an account of the novel interaction mechanism and its applications to quantum computation. Section II introduces the spin-exchange interaction and its potential for two-atom entanglement. Section III discusses an example to achieve pairwise entanglement via optical pumping. In Sec. IV, we investigate the pairwise entanglement efficiency by numerical simulation. In Sec. V, we study a quantum gate that is similar to the SWAP gate, and introduce a protocol of global entanglement of atoms in a chain. Section VI gives a summary. Additional details of the theory are given in the appendices.
Ii Two-atom entanglement
Consider two Rb atoms, denoted and , respectively, which are loaded into two far-off-resonant traps created by tightly focused laser beams. The interatomic axis is defined by the vector . Each of the atoms can be independently driven by laser light, as shown in Fig. 1. We suppose atom is prepared in a hyperfine level of the ground state manifold identified as “spin-up” while atom is prepared in the state “spin-down”: (. These states may be optically coupled via two-photon transitions to the Rydberg states ( of atoms and , respectively. Here we use the hyperfine notation to label ground states and the fine structure notation for Rydberg levels, according to the spectroscopic resolution usually achieved in experiments (see Appendix A).
Consider two Rydberg atoms prepared in the states and , respectively, and separated by a distance large enough that the states are coupled by the van der Waals interaction Saffman et al. (2010); Walker and Saffman (2008). In the case , this coupling is dominant and induces a shift of the doubly excited Rydberg state commonly referred to as the blockade shift. When , and under special conditions, the coupling , which exchanges the electronic spin states, may become equally large. In the two-atom product basis , the total van der Waals interaction is then,
By using the measured results for the relevant quantum defects Li et al. (2003) and a semiclassical expression for the radial matrix elements Kaulakys (1995), we numerically evaluate and for and . We find four pairs of states where the interaction coefficients differ by less than ; see Table 1.
The occurrence of a strong spin-exchange interaction in these cases arises through an interference effect involving a small number of dominant intermediate and states (see Appendix B). We find that the transition matrix elements for spin exchange constructively interfere whereas a partially destructive interference limits the blockade shift. This results in almost equal magnitude of and .
59 61 -196 194
73 75 4080 -4025
97 100 -59780 58800
121 124 1104000 -1124000
Table 1: Strength of blockade and spin-exchange interactions between two atoms of principle quantum numbers and . are in unit of .
The eigenstates and eigenvalues of Eq. (1) are , and . If the interaction coefficients have equal magnitudes, one of the two energy eigenvalues is unshifted from the non-interacting value. For the four cases shown in Table 1, and have opposite signs, and by choosing an appropriate atomic separation , () can be made much larger (smaller) than the excitation Rabi frequency. As a result, atoms and will be excited to the entangled two-atom Rydberg state . The lifetime of the entanglement created can be enhanced by coupling it to the two-atom ground state, by driving Rabi oscillations and simultaneously on both atoms [Fig. 1(b)], so that the Rydberg triplet state is mapped to the ground level Bell state
Figure 2: (Color online) Schematic of the entanglement protocol via pulses 1, 2 and 3. During pulse 1 (2), only the two-photon transition via 5P (6P) state of atom () is excited with Rabi frequency (). During pulse 3, the four two-photon transitions via 5P and 6P states of atoms and are turned on and the magnitudes of these two-photon Rabi frequencies are set equal to . On the left column we show the bare two-atom basis, while on the right we highlight the superposition states involved.
Iii Pairwise entanglement protocol
We now describe the complete protocol creating ground state entanglement via a three--pulse sequence.
Pulse 1 on atom : We take the initial state to be . The first pulse acts on atom and excites to via the state, as shown in Fig. 2. Since atom is in its ground state, there is no Rydberg interaction. Thus, by applying a pulse of duration , we generate the product state .
Pulse 2 on atom B: Following pulse 1, apply a two-photon laser pulse to atom with Rabi frequency , as shown in Fig. 2. This pulse excites to via the state as in Fig. 1(b). The evolution of the two-atom wave function is governed by the Hamiltonian
where the basis vectors are , and . In the case of , we can adiabatically eliminate the state . The population on the Rydberg Bell state reaches its maximum
for a pulse of duration . The prefactor indicates that this is a two-atom -pulse, while the correction results from the small shift of .
Pulse 3 on both atoms: The final step is the mapping of the entangled state of Rydberg atoms onto the ground states. In contrast to Pulses 1 and 2, switch on all four Rabi channels simultaneously exciting the two atoms, as shown in Fig. 1(b).
When the Rabi frequencies satisfy only the intermediate states and are coupled to and where and , see Fig. 2. Ordering the states and the Hamiltonian matrix during Pulse 3 reads,
At the end of Pulse 3, of duration , the entangled Rydberg state would be completely mapped onto the entangled ground state with fidelity limited by the residual shift , and the radiative decay rate of the Rydberg level.
Figure 3: (Color online) Evolution of the two-atom state during pulses 2 and 3 of the entanglement protocol, starting from . We plot the populations of the relevant states from the basis in Eq. (7): the red (gray) solid line is , the blue (dark gray) dashed line is , the green (light gray) dotted line is .
Iv Numerical simulation and fidelity
The proposed scheme relies on creation of the Rydberg entangled state, which requires
In order to show that the ground Bell state can be prepared with high fidelity for finite , we numerically study the time evolution of the atomic state following the procedure discussed above. We choose the Rydberg state pair with principal quantum numbers , atomic separation m, and set the single-atom Rabi frequency kHz. We expand the two-atom wavefunction as
and we numerically solve the Schrödinger equation for Pulses 2 and 3 (see Appendices C and D), since pulse 1 is trivial. The achieved ground state fidelity is . To improve the fidelity of the prepared ground Bell state, we numerically optimize by varying Rabi frequencies and pulse durations. Figure 3 shows the state evolution with kHz and kHz: a fidelity of for the ground Bell state is achieved in less than ten microseconds.
The main practical difficulty is to have all four Rabi frequencies for the excitation channels equal to each other. In order to show that the the protocol is robust against dispersion in the Rabi frequencies, we numerically integrate the Schrödinger equation varying and in the interval for Pulse 3, with , , and . By performing such simulations, we find that almost all fidelities are larger than () for (see Appendix D).
V SWAP gate and entanglement of atomic chains
The spin-exchange Rydberg interaction may be used together with Rydberg blockade to implement a quantum logic gate based on a simple combination of three single-atom laser interaction processes: two -pulses applied to atom , are separated by an intermediate -pulse applied to atom A. If both atoms are initially prepared in the same ground state, we obtain the state transformations,
During stages 1 and 3, a single Rydberg excitation is created and removed, while in stage 2, Rydberg blockade of the states and , respectively, prevents any double excitation. Alternatively, when the atoms are initially prepared in opposite spin ground states, there is a crucial difference. As a consequence of the strong spin-exchange interaction, the 2 pulse resonantly couples to the triplet state and flips the spin of both the Rydberg excitation and the atomic ground state; see Fig. 4. In this case the combination of van der Waals interaction and single atom 2 pulse creates a resonant “lambda” transition between two-atom states:
The three stage protocol completes the quantum SWAP gate transformation, with a phase shift of the swapped states. By choosing m and Rabi frequency kHz for the pulse on atom A, we can demonstrate a gate fidelity for an operation time s.
Figure 4: (Color online) Schematic of the SWAP-like gate protocol. During the pulse, , and are hardly excited due to Rydberg blockade, while will be excited, forming a system with and .
The combination of quantum SWAP gate with pairwise atom entanglement operations can be used as the basis of a protocol to entangle a chain of an arbitrary number of atoms. The key observation is that the atoms should be entangled sequentially in pairs, allowing gaps between the pairs in order to minimize spurious level shifts due to Rydberg blockade. After the pair-entanglement, a series of SWAP gate operations link all the pairs in a fully entangled state. Such a procedure may be easily sketched for the case of atoms, where the atoms are labeled sequentially: . We first use the two-atom entanglement protocol described above to entangle atoms and , where . This is followed by a similar protocol that entangles atoms and . Because all atoms are now entangled with one of their two neighboring atoms, the two-qubit SWAP gate is used to entangle atoms and , followed by another SWAP operation for entanglement of and . In this way we may entangle all atoms by a twelve-pulse protocol as shown in Fig. 5. In the case of 4 atoms, this protocol generates the entangled state after the application of 9 pulses (see Appendix E).
During the pairwise entanglement and SWAP gate operations, the metastable Rydberg states are populated for a time , thus the fidelity of the entanglement of the -atom chain may be estimated as , where is the lifetime of the Rydberg level. When keeping the leading order term after expanding the exponential of under the condition of , the total error of the prepared many-atom entangled state scales linearly with . Numerical simulations show that pairwise entanglement and SWAP gate operations can each be carried out in around 10s. Thus a 4-atom chain may be entangled in a time s with m. For (a) , ms for and , while for (b) , ms for and Saffman et al. (2010). Decreasing to makes four times larger, and reduces to . Then For , a 4-atom chain can be entangled with , while an 8-atom chain can be entangled with . These fidelities are comparable to the values for 4 and 8 atom entanglement by asymmetric blockade in Ref. Saffman and Mølmer (2009) and dissipation in Ref. Carr and Saffman (2013). The dissipative protocol in Carr and Saffman (2013) does not suffer from the spontaneous emission issue, but in comparison the present coherent process is almost three orders of magnitude faster. Furthermore, the linear scaling of the error with the total number of entangled atoms is similar to the the blockade-based situation of Ref. Saffman and Mølmer (2009): in that case the error scales cubically at low and then saturates to a linear behavior. By contrast to the multi-atom entanglement based on Ryberg interactions and adiabatic passage proposed in Ref. Møller et al. (2008), the duration and the Rabi frequency of our scheme are independent of . The requirements on the Rabi frequencies and principal quantum numbers are well within experimental reach and the individual atomic addressing allows to tailor the distance between the atoms as well as the desired target state. All these considerations show that the proposed spin-exchange mechanism represents a valid candidate to realize fast quantum operations with Rydberg atoms.
Figure 5: (Color online) Schematic illustration of a protocol to generating entanglement between 12 Rb atoms loaded in a one-dimensional lattice. The pairwise entanglement protocol creates entanglement between atoms and ( and ) during step 1 (2), while the SWAP-like gate protocol entangles atoms and ( and ) during step 3 (4).
Vi Conclusion
Throughout this work we have considered the case of individually trapped atoms addressed by external lasers. An interesting alternative realization of these protocols relies on the creation of a small super-atom in an elongated ensemble. This option has the advantage of solving the probabilistic loading of the individual traps as well as to provide a boost to the single-atom procedure by a factor where is the number of individual atoms in the super-atom. Quantum gates between super-atoms have been recently proposed Paredes-Barato and Adams (2014) and the multiplexing of different qubits in an elongated ensemble has already been realized experimentally Lan et al. (2009).
In conclusion, we have proposed a fast and robust mechanism to entangle neutral atoms. It is based on a variation of the van der Waals interaction between atoms excited to Rydberg states: for different principal quantum numbers, the spin-exchange interaction may be comparable to the Rydberg blockade shift thus induce a resonance between ground state levels and an entangled Rydberg state. This metastable state may then be mapped to a stable, entangled ground state. Furthermore, the entanglement efficiency may be improved by using small ensembles as well as by manipulating the structure of the Rydberg manifolds via external fields. Based on the spin-exchange interaction, pairwise entanglement along with a SWAP-like gate form the basis of a protocol for the generation of ground-state entanglement of many atoms in a chain configuration. These protocols may be implemented in present experiments leading to quantum manipulation of many-body systems.
XFS and TABK acknowledge support from AFOSR and the Quantum Memories MURI of the Air Force Office of Scientific Research. FB acknowledges support from the DARPA QuASAR program and the US NSF.
Appendix A Single-atom optical excitation
This section shows how to realize the transitions in Fig 1(b) of the main text. We consider an atom optically excited via two linearly polarized light fields, one -polarized, the other -polarized, traveling along and direction, respectively. To have such a pair of light fields, a beam splitter is placed along the axis as shown in Fig. 6. For each incoming light field, the following optical devices are used, (), beam splitter, () and (), mirror, and (), a quarter-wave plate, or a wave plate whose thickness makes it effectively a combination of a half wave-plate and a quarter-wave plate, depending on the specific low-lying intermediate P state. By tuning the positions of the two mirrors so that they are symmetric to each other about the plane of the beam splitter, we have
where is the distance between and , where . Here denotes the position of the atom. Assuming that the light field impinging on the beam splitter is , one can show that the electric field on the atom is:
In order to excite the the Rydberg states and defined in the main text for either atom or atom , one shall use waveplates so that and , respectively. As a result, the transition from the ground level to the Rydberg level via an intermediate level () can be realized by a two-photon Rabi process via a pair of effective left (right)-hand polarized light fields.
The non-interacting Hamiltonian for atom A (the Hamiltonian for atom B is similar) in the dipole approximation and rotating-wave approximation for the atom-field coupling is
where indicates right (left)-hand polarization, with is the energy of a specific atomic manifold, and are the central frequencies of the lasers for the lower and upper transitions, and they satisfy
The Rabi frequencies read,
where or 6, is the electric field of the laser for the lower (upper) transition in Eq. (9), and are the reduced matrix elements of the electric dipole operator obtained via the Wigner-Eckart theorem Rose (1957); Jenkins et al. (2012), with the elementary charge, and a Clebsch-Gordan coefficient Rose (1957).
Here, the column and row indices for are for quantum numbers (from left to right) and (from top to bottom), and the column and row indices for are for quantum numbers , (from left to right) and (from top to bottom).
Figure 6: (Color online) Schematic illustration for generating an effective right-hand or left-hand polarized light field from two linearly polarized light fields related by a or phase difference.
Below, we derive the effective two-photon Rabi frequency when the two-photon detuning is zero James (2000); Brion et al. (2007); Han et al. (2013). Using the method of Ref. James (2000), we can derive a Hamiltonian for far off resonant optical driving. The method of Ref. James (2000) is essentially an adiabatic approximation. The Hamiltonian in a rotating frame and rotating-wave approximation for a three level system with basis is (the subscripts d and u denote down and up, respectively)
where is the length defined in Eq. (8) for the transition from to (from to ). Here we assume that is real, and the two-photon transition is resonant. We write the wave function as , and starting from state , one can find and
which indicates that Max and Min achieve maximum difference only when . This can guide setting up the condition in experiments. In fact, when the difference of Max and Min reaches its maximum while adjusting one of , the condition is met. When , the Hamiltonian in a rotating frame is
When the lower and upper Rabi transitions have the same Rabi frequency, the effective Rabi frequency between and is . By assuming , we can write the effective Rabi frequency as .
For the system studied in the main text, we shall identify as for the two-photon transition via 5P state of atom , while as for the two-photon transition via 6P state of atom . But since both of these two-photon transitions are resonant, and the ground states or the Rydberg states are degenerate, we immediately find if one sets a common for exciting both of the two Rydberg state and , with or .
Appendix B van der Waals interaction
Since we consider the uncommon situation of the interaction between two Rydberg levels of different principal quantum numbers, we will briefly outline a perturbation calculation here. Consider two Rb Rydberg atoms, one prepared in state , and the other in state , where . We consider the following four channels for the dipole-dipole interaction, each characterized by its energy defect Walker and Saffman (2008),
Here denote the principal quantum numbers of the pair state produced by the scattering process. These four couplings are known to be the dominant ones in our case. In the van der Waals interaction the atoms then go back to the initial levels and the magnetic quantum number of either atom can change up to unit Walker and Saffman (2008). We can separate the angular dependence of the interaction from the principal quantum numbers. Its matrix representation in the basis of is given by
where diag means a diagonal matrix with as diagonal matrix elements. In the basis of ,, , and , Eq. (13) becomes,
The van der Waals interaction strength for each channel is
We can identify two types of coefficients:
with or . Here is the radial part of the atomic wave function, and the integration about can be approximated as in Ref. Kaulakys (1995).
Value of
1 71841.8
2 71922.5
3 71928.5
6 71929.9
10 71930
15 71930
20 71930
Table 2: Convergence of the van der Walls interaction strength depending on the channels considered (up to ) for a pair of atoms in the state . Unit: .
When , we have . Also, the two channels with energy defect and are the same. We report in Table 2 the values of or for different for two atoms at |
5af4769e1ddceeb9 | Causal fermion system - Wikiwand
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Causal fermion system
From Wikipedia, the free encyclopedia
The theory of causal fermion systems is an approach to describe fundamental physics. Its proponents claim it gives quantum mechanics, general relativity and quantum field theory as limiting cases[1][2][3][4] and is therefore a candidate for a unified physical theory.
Causal fermion systems were introduced by Felix Finster and collaborators.
Motivation and physical concept
The physical starting point is the fact that the Dirac equation in Minkowski space has solutions of negative energy which are usually associated to the Dirac sea. Taking the concept seriously that the states of the Dirac sea form an integral part of the physical system, one finds that many structures (like the causal and metric structures as well as the bosonic fields) can be recovered from the wave functions of the sea states. This leads to the idea that the wave functions of all occupied states (including the sea states) should be regarded as the basic physical objects, and that all structures in space-time arise as a result of the collective interaction of the sea states with each other and with the additional particles and "holes" in the sea. Implementing this picture mathematically leads to the framework of causal fermion systems.
More precisely, the correspondence between the above physical situation and the mathematical framework is obtained as follows. All occupied states span a Hilbert space of wave functions in Minkowski space . The observable information on the distribution of the wave functions in space-time is encoded in the local correlation operators which in an orthonormal basis have the matrix representation
(where is the adjoint spinor). In order to make the wave functions into the basic physical objects, one considers the set as a set of linear operators on an abstract Hilbert space. The structures of Minkowski space are all disregarded, except for the volume measure , which is transformed to a corresponding measure on the linear operators (the "universal measure"). The resulting structures, namely a Hilbert space together with a measure on the linear operators thereon, are the basic ingredients of a causal fermion system.
The above construction can also be carried out in more general space-times. Moreover, taking the abstract definition as the starting point, causal fermion systems allow for the description of generalized "quantum space-times." The physical picture is that one causal fermion system describes a space-time together with all structures and objects therein (like the causal and the metric structures, wave functions and quantum fields). In order to single out the physically admissible causal fermion systems, one must formulate physical equations. In analogy to the Lagrangian formulation of classical field theory, the physical equations for causal fermion systems are formulated via a variational principle, the so-called causal action principle. Since one works with different basic objects, the causal action principle has a novel mathematical structure where one minimizes a positive action under variations of the universal measure. The connection to conventional physical equations is obtained in a certain limiting case (the continuum limit) in which the interaction can be described effectively by gauge fields coupled to particles and antiparticles, whereas the Dirac sea is no longer apparent.
General mathematical setting
In this section the mathematical framework of causal fermion systems is introduced.
Definition of a causal fermion system
A causal fermion system of spin dimension is a triple where
• is a complex Hilbert space.
• is the set of all self-adjoint linear operators of finite rank on which (counting multiplicities) have at most positive and at most negative eigenvalues.
• is a measure on .
The measure is referred to as the universal measure.
As will be outlined below, this definition is rich enough to encode analogs of the mathematical structures needed to formulate physical theories. In particular, a causal fermion system gives rise to a space-time together with additional structures that generalize objects like spinors, the metric and curvature. Moreover, it comprises quantum objects like wave functions and a fermionic Fock state.[7]
The causal action principle
Inspired by the Langrangian formulation of classical field theory, the dynamics on a causal fermion system is described by a variational principle defined as follows.
Given a Hilbert space and the spin dimension , the set is defined as above. Then for any , the product is an operator of rank at most . It is not necessarily self-adjoint because in general . We denote the non-trivial eigenvalues of the operator (counting algebraic multiplicities) by
Moreover, the spectral weight is defined by
The Lagrangian is introduced by
The causal action is defined by
The causal action principle is to minimize under variations of within the class of (positive) Borel measures under the following constraints:
• Boundedness constraint: for some positive constant .
• Trace constraint: is kept fixed.
• The total volume is preserved.
Here on one considers the topology induced by the -norm on the bounded linear operators on .
The constraints prevent trivial minimizers and ensure existence, provided that is finite-dimensional.[8] This variational principle also makes sense in the case that the total volume is infinite if one considers variations of bounded variation with .
Inherent structures
In contemporary physical theories, the word space-time refers to a Lorentzian manifold . This means that space-time is a set of points enriched by topological and geometric structures. In the context of causal fermion systems, space-time does not need to have a manifold structure. Instead, space-time is a set of operators on a Hilbert space (a subset of ). This implies additional inherent structures that correspond to and generalize usual objects on a space-time manifold.
For a causal fermion system , we define space-time as the support of the universal measure,
With the topology induced by , space-time is a topological space.
Causal structure
For , we denote the non-trivial eigenvalues of the operator (counting algebraic multiplicities) by . The points and are defined to be spacelike separated if all the have the same absolute value. They are timelike separated if the do not all have the same absolute value and are all real. In all other cases, the points and are lightlike separated.
This notion of causality fits together with the "causality" of the above causal action in the sense that if two space-time points are space-like separated, then the Lagrangian vanishes. This corresponds to the physical notion of causality that spatially separated space-time points do not interact. This causal structure is the reason for the notion "causal" in causal fermion system and causal action.
Let denote the orthogonal projection on the subspace . Then the sign of the functional
distinguishes the future from the past. In contrast to the structure of a partially ordered set, the relation "lies in the future of" is in general not transitive. But it is transitive on the macroscopic scale in typical examples.[5][6]
Spinors and wave functions
For every the spin space is defined by ; it is a subspace of of dimension at most . The spin scalar product defined by
is an indefinite inner product on of signature with .
A wave function is a mapping
On wave functions for which the norm defined by
is finite (where is the absolute value of the symmetric operator ), one can define the inner product
Together with the topology induced by the norm , one obtains a Krein space .
To any vector we can associate the wave function
(where is again the orthogonal projection to the spin space). This gives rise to a distinguished family of wave functions, referred to as the wave functions of the occupied states.
The fermionic projector
The kernel of the fermionic projector is defined by
(where is again the orthogonal projection on the spin space, and denotes the restriction to ). The fermionic projector is the operator
which has the dense domain of definition given by all vectors satisfying the conditions
As a consequence of the causal action principle, the kernel of the fermionic projector has additional normalization properties[9] which justify the name projector.
Connection and curvature
Being an operator from one spin space to another, the kernel of the fermionic projector gives relations between different space-time points. This fact can be used to introduce a spin connection
The basic idea is to take a polar decomposition of . The construction becomes more involved by the fact that the spin connection should induce a corresponding metric connection
where the tangent space is a specific subspace of the linear operators on endowed with a Lorentzian metric. The spin curvature is defined as the holonomy of the spin connection,
Similarly, the metric connection gives rise to metric curvature. These geometric structures give rise to a proposal for a quantum geometry.[5]
A fermionic Fock state
If has finite dimension , choosing an orthonormal basis of and taking the wedge product of the corresponding wave functions
gives a state of an -particle fermionic Fock space. Due to the total anti-symmetrization, this state depends on the choice of the basis of only by a phase factor.[10] This correspondence explains why the vectors in the particle space are to be interpreted as fermions. It also motivates the name causal fermion system.
Underlying physical principles
Causal fermion systems incorporate several physical principles in a specific way:
• A local gauge principle: In order to represent the wave functions in components, one chooses bases of the spin spaces. Denoting the signature of the spin scalar product at by , a pseudo-orthonormal basis of is given by
Then a wave function can be represented with component functions,
The freedom of choosing the bases independently at every space-time point corresponds to local unitary transformations of the wave functions,
These transformations have the interpretation as local gauge transformations. The gauge group is determined to be the isometry group of the spin scalar product. The causal action is gauge invariant in the sense that it does not depend on the choice of spinor bases.
• The equivalence principle: For an explicit description of space-time one must work with local coordinates. The freedom in choosing such coordinates generalizes the freedom in choosing general reference frames in a space-time manifold. Therefore, the equivalence principle of general relativity is respected. The causal action is generally covariant in the sense that it does not depend on the choice of coordinates.
• The Pauli exclusion principle: The fermionic Fock state associated to the causal fermion system makes it possible to describe the many-particle state by a totally antisymmetric wave function. This gives agreement with the Pauli exclusion principle.
• The principle of causality is incorporated by the form of the causal action in the sense that space-time points with spacelike separation do not interact.
Limiting cases
Causal fermion systems have mathematically sound limiting cases that give a connection to conventional physical structures.
Lorentzian spin geometry of globally hyperbolic space-times
Starting on any globally hyperbolic Lorentzian spin manifold with spinor bundle , one gets into the framework of causal fermion systems by choosing as a subspace of the solution space of the Dirac equation. Defining the so-called local correlation operator for by
(where is the inner product on the fibre ) and introducing the universal measure as the push-forward of the volume measure on ,
one obtains a causal fermion system. For the local correlation operators to be well-defined, must consist of continuous sections, typically making it necessary to introduce a regularization on the microscopic scale . In the limit , all the intrinsic structures on the causal fermion system (like the causal structure, connection and curvature) go over to the corresponding structures on the Lorentzian spin manifold.[5] Thus the geometry of space-time is encoded completely in the corresponding causal fermion systems.
Quantum mechanics and classical field equations
The Euler-Lagrange equations corresponding to the causal action principle have a well-defined limit if the space-times of the causal fermion systems go over to Minkowski space. More specifically, one considers a sequence of causal fermion systems (for example with finite-dimensional in order to ensure the existence of the fermionick Fock state as well as of minimizers of the causal action), such that the corresponding wave functions go over to a configuration of interacting Dirac seas involving additional particle states or "holes" in the seas. This procedure, referred to as the continuum limit, gives effective equations having the structure of the Dirac equation coupled to classical field equations. For example, for a simplified model involving three elementary fermionic particles in spin dimension two, one obtains an interaction via a classical axial gauge field [2] described by the coupled Dirac- and Yang-Mills equations
Taking the non-relativistic limit of the Dirac equation, one obtains the Pauli equation or the Schrödinger equation, giving the correspondence to quantum mechanics. Here and depend on the regularization and determine the coupling constant as well as the rest mass.
Likewise, for a system involving neutrinos in spin dimension 4, one gets effectively a massive gauge field coupled to the left-handed component of the Dirac spinors.[2] The fermion configuration of the standard model can be described in spin dimension 16.[1]
The Einstein field equations
For the just-mentioned system involving neutrinos,[2] the continuum limit also yields the Einstein field equations coupled to the Dirac spinors,
up to corrections of higher order in the curvature tensor. Here the cosmological constant is undetermined, and denotes the energy-momentum tensor of the spinors and the gauge field. The gravitation constant depends on the regularization length.
Quantum field theory in Minkowski space
Starting from the coupled system of equations obtained in the continuum limit and expanding in powers of the coupling constant, one obtains integrals which correspond to Feynman diagrams on the tree level. Fermionic loop diagrams arise due to the interaction with the sea states, whereas bosonic loop diagrams appear when taking averages over the microscopic (in generally non-smooth) space-time structure of a causal fermion system (method of microscopic mixing).[4] The detailed analysis and comparison with standard quantum field theory is work in progress.
1. ^ a b Finster, Felix (2006). The principle of the fermionic projector. Providence, R.I: American Mathematical Society. ISBN 978-0-8218-3974-4. OCLC 61211466.Chapters 1-4Chapters 5-8Appendices
2. ^ a b c d Finster, Felix (2016). "The Continuum Limit of Causal Fermion Systems". Fundamental Theories of Physics. 186. Cham: Springer International Publishing. arXiv:1605.04742. doi:10.1007/978-3-319-42067-7. ISBN 978-3-319-42066-0. ISSN 0168-1222.
3. ^ Finster, Felix (19 February 2011). "A Formulation of Quantum Field Theory Realizing a Sea of Interacting Dirac Particles". Letters in Mathematical Physics. Springer Science and Business Media LLC. 97 (2): 165–183. arXiv:0911.2102. doi:10.1007/s11005-011-0473-1. ISSN 0377-9017.
4. ^ a b Finster, Felix (2014). "Perturbative quantum field theory in the framework of the fermionic projector". Journal of Mathematical Physics. AIP Publishing. 55 (4): 042301. arXiv:1310.4121. doi:10.1063/1.4871549. ISSN 0022-2488.
5. ^ a b c d Finster, Felix; Grotz, Andreas (2012). "A Lorentzian quantum geometry". Advances in Theoretical and Mathematical Physics. International Press of Boston. 16 (4): 1197–1290. arXiv:1107.2026. doi:10.4310/atmp.2012.v16.n4.a3. ISSN 1095-0761.
6. ^ a b Finster, Felix; Kamran, Niky (2019). "Spinors on Singular Spaces and the Topology of Causal Fermion Systems". Memoirs of the American Mathematical Society. American Mathematical Society (AMS). 259 (1251). arXiv:1403.7885. doi:10.1090/memo/1251. ISSN 0065-9266.
7. ^ Finster, Felix; Grotz, Andreas; Schiefeneder, Daniela (2012). "Causal Fermion Systems: A Quantum Space-Time Emerging From an Action Principle". Quantum Field Theory and Gravity. Basel: Springer Basel. pp. 157–182. arXiv:1102.2585. doi:10.1007/978-3-0348-0043-3_9. ISBN 978-3-0348-0042-6.
8. ^ Finster, Felix (2010). "Causal variational principles on measure spaces". Journal für die reine und angewandte Mathematik. Walter de Gruyter GmbH. 2010 (646): 141–194. arXiv:0811.2666. doi:10.1515/crelle.2010.069. ISSN 0075-4102.
9. ^ Finster, Felix; Kleiner, Johannes (17 March 2016). "Noether-like theorems for causal variational principles". Calculus of Variations and Partial Differential Equations. Springer Science and Business Media LLC. 55 (2): 35. arXiv:1506.09076. doi:10.1007/s00526-016-0966-y. ISSN 0944-2669.
10. ^ Finster, Felix (26 August 2010). "Entanglement and second quantization in the framework of the fermionic projector". Journal of Physics A: Mathematical and Theoretical. IOP Publishing. 43 (39): 395302. arXiv:0911.0076. doi:10.1088/1751-8113/43/39/395302. ISSN 1751-8113.
Further reading
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Causal fermion system
Listen to this article |
9a27059341c754b2 | Thursday, March 21, 2019
actual infinite falling (all-)together &/or chaosmos COLLAPSE
Musée d'Orsay/January 2018 (for more A/Z photography see portfolio here);
Clara Colosimo in Fellini's Prova d'orchestra;
Arquipélago dos Pombos Correios (o soverdouro);
The great abyss inframince (by A/Z, for more see here);
"... the term quantum mechanics is very much a misnomer. It should, perhaps, be called quantum nonmechanics..."
David Bohm
"But he wants The Cold like he wants His Junk—NOT OUTSIDE where it does him no good but INSIDE so he can sit around with a spine like a frozen hydraulic jack..."
William S. Burroughs
"Ihr verehrt mich: aber wie, wenn eure Verehrung eines Tages umfällt?"
"Ich komme aus Höhen, die kein Vogel je erflog, ich kenne Abgründe, in die noch kein Fuß sich verirrt hat..."
"Vor mir giebt es diese Umsetzung des dionysischen in ein philosophisches Pathos nicht."
"... ein vollkommnes Ausser-sich-sein mit dem distinktesten Bewusstsein einer Unzahl feiner Schauder und Überrieselungen bis in die Fusszehen; eine Glückstiefe, in der das Schmerzlichste und Düsterste nicht als Gegensatz wirkt, sondern als bedingt, als herausgefordert, sondern als eine nothwendige Farbe innerhalb eines solchen Lichtüberflusses; ein Instinkt rhythmischer Verhältnisse, der weite Räume von Formen überspannt..."
"... la majorité est travaillé par une minorité proliférante et non dénombrable qui risque de détruire la majorité dans son concept même, c'est-à-dire en tant qu'axiome... le étrange concept de non-blanc ne constitue pas un ensemble dénombrable... Le propre de la minorité, c'est de faire valoir la puissance du non-dénombrable, même quand elle est composée d'un seul membre. C'est la formule des multiplicités. Non-blanc, nous avons tous à le devenir, que nous soyons blancs, jaunes ou noirs."
Deleuze & Guattari
"Eighteenth-century masters achieved most pleasing effects with forgrounds of warm brown and fading distances of cool, silvery blues... Constable wanted to try out the effect of respecting the local color of grass somewhat more, in his Wivenhoe Park he is seen pushing the range more in the direction of bright greens. Only in the direction of, it is a transposition, not a copy."
Ernst H. Gombrich (Art and Illusion)
"Oboukhoff acumule alors en un étroit espace les demi-tons; la sensation est déchirante, horripilante... Avec le glissando, c'est tout l'infini du monde sonore qui fait irruption dans la musique tempérée..."
Boris de Schloezer (1925)
"Note the parallels between ordinary awareness, classical physics, and the natural and counting integers..."
Dean Radin (Real Magic)
This is AGAINST Carlo Rovelli's dictum or pseudo-problem: "visto que tudo se atrai, a única maneira de um Universo finito não desmoronar sobre si mesmo é que se expanda" [since all things attract one another, the only way a finite Universe can avoid collapse is to expand] (A realidade não é o que parece, p. 105)—but why should one use the term "finite" (or even "infinite") to describe a universe with no definite borders (like a 3-sphere, or something even more complex)? The infinite is not equivalent to the huge. The infinite is simply (according to Dedekind) what can be matched up to its own parts (the only reason to deny this is hysteria, paradox-freakishness). The universe (the chaosmos) both expands & collapse! As a whole and at the length of its space-time infinitesimals (or epsilon-delta limits, whatever), the macro/micro contractions, the revolving ruminations (what Rovelli confusedly calls "granulations," as if they were incompatible with any notion of continuity) of an autophagic real-virtual Einsteinian mollusk. If you have three fundamental constants (as Rovelli suggests, A realidade não é o que parece, p. 229), velocity [of light], information and Planck's length (c, ħ, ℓp), what matters is the relation among them (which might be revealed in established, finite proportions) not each one of their supposedly fixed (absolute) values (and even the relation might vary, fluctuate). Otherwise you behave like a very stupid painter arguing over the positive value of (what we call) green or brown in the transposition of tonal gradations to canvas.
Main Hall:
Time out of joints or the excessive solution (academically and sophistically called 'the measurement problem'):
"If quantum state evolution proceeds via the Schrödinger equation or some other linear equation, then, as we have seen in the previous section, typical experiments will lead to quantum states that are superpositions of terms corresponding to distinct experimental outcomes. It is sometimes said that this conflicts with our experience, according to which experimental outcome variables, such as pointer readings, always have definite values. This is a misleading way of putting the issue, as it is not immediately clear how to interpret states of this sort as physical states of a system that includes experimental apparatus, and, if we can’t say what it would be like to observe the apparatus to be in such a state, it makes no sense to say that we never observe it to be in a state like that," Wayne Myrvold's "Philosophical Issues in Quantum Mechanics," Stanford Encyclopedia of Philosophy
"... von Neumann makes the logical structure of quantum theory very clear by identifying two very different processes, which he calls process 1 and process 2... Process 2 is the analogue in quantum theory of the process in classic physics that takes the state of a system at one time to its state at a later time. This process 2, like its classic analogue, is local and deterministic. However, process 2 by itself is not the whole story: it generates a host of ‘physical worlds’, most of which do not agree with our human experience. For example, if process 2 were, from the time of the big bang, the only process in nature, then the quantum state (centre point) of the moon would represent a structure smeared out over a large part of the sky, and each human body–brain would likewise be represented by a structure smeared out continuously over a huge region. Process 2 generates a cloud of possible worlds, instead of the one world we actually experience...," Jeffrey M. Schwartz's, Henry P. Stapp's and Mario Beauregard's "Quantum physics in neuroscience and psychology: a neurophysical model of mind–brain interaction," Philosophical Transactions of the Royal Society (2005).
"... a seminal discovery by Heisenberg... in order to get a satisfactory quantum generalization of a classic theory one must replace various numbers in the classic theory by actions (operators). A key difference between numbers and actions is that if A and B are two actions then AB represents the action obtained by performing the action A upon the action B. If A and B are two different actions then generally AB is different from BA: the order in which actions are performed matters. But for numbers the order does not matter: AB=BA. The difference between quantum physics and its classic approximation resides in the fact that in the quantum case certain differences AB–BA are proportional to a number measured by Max Planck in 1900, and called Planck’s constant. Setting those differences to zero gives the classic approximation," Jeffrey M. Schwartz's, Henry P. Stapp's and Mario Beauregard's "Quantum physics in neuroscience and psychology: a neurophysical model of mind–brain interaction," Philosophical Transactions of the Royal Society (2005).
"At their narrowest points, calcium ion channels are less than a nanometre in diameter... The narrowness of the channel restricts the lateral spatial dimension. Consequently, the lateral velocity is forced by the quantum uncertainty principle to become large. This causes the quantum cloud of possibilities associated with the calcium ion to fan out over an increasing area as it moves away from the tiny channel to the target region... This spreading of this ion wave packet means that the ion may or may not be absorbed on the small triggering site. Accordingly, the contents of the vesicle may or may not be released... the quantum state of the brain splits into a vast host of classically conceived possibilities, one for each possible combination of the release-or-no-release options at each of the nerve terminals... a huge smear of classically conceived possibilities," Jeffrey M. Schwartz's, Henry P. Stapp's and Mario Beauregard's "Quantum physics in neuroscience and psychology: a neurophysical model of mind–brain interaction," Philosophical Transactions of the Royal Society (2005).
"... waves make diffraction patterns precisely because multiple waves can be at the same place at the same time, and a given wave can be at multiple places at the same time... by definition particles are localized entities that take up space, they can be here or there, but not in two places at once. However it turns out that particles can produce diffraction patterns under specific circumstances... a given particle can be in a state of superposition... to be in a state of superposition between two positions, for exemple, is not to be here or there or even here and there, but rather it is to be indeterminately here-there. That is, it is not simply that the position is unknown, but rather there is no fact of the matter to whether it is here or there... it is a matter of ontological indeterminacy and not merely epistemological uncertainty... patterns of difference... are arguably at the core or what matter is and are at the heart of how quantum physics understands the world... the quantum probabilities are calculated by taken account of all the possible paths connecting the points. In other words, a given particle that starts out here and winds up there is understood as is understood to be in a superposition of all possible paths between two points. Or in its four dimensional quantum field theory elaboration, all possible space-time histories... the very meaning of superposition is that all possible histories are happening together, they all coexist and mutually contribute to this overall pattern or else there wouldn't be a diffraction pattern..." Karen Barad's "Troubling Time's & Ecologies of Nothingness," European Graduate School Video Lectures (YouTube), my transcription.
"Quantum physics opens up another possibility beyond the relatively familiar phenomena of spatial diffraction, namely, temporal diffraction. The existence of temporal diffraction is due to a less well-known indeterminacy principle than the usual position/momentum indeterminacy principle... something call the energy/time indeterminacy principle. This indeterminacy principle plays a key role in quantum field theory... temporalities are not merely multiple, but rather temporalities are specifically entangled and threaded through one another such that there is no determinate answer to the question what time is it? Karen Barad's "Troubling Time's & Ecologies of Nothingness," European Graduate School Video Lectures (YouTube), my transcription.
"During the waning decades of the 20th century, the most murdering century by some accounts in history, the notion that the past might be open to revision through a quantum erasure came to the fore. The quantum erasure experiment is a variation of the two slit diffraction experiment, an experiment which Feynman said contains all the mysteries of quantum physics. Against this fantastic claim of the possibility of erasure, I will claim that in paying close attention to the material labours entailed the claim of erasure possibility fades, at least full erasure, while at the same time bringing to the forth a relational ontology sensibility to questions of time, memory and history... the nature of time and being, or rather time-being itself is in question and can't be assumed. What this experiment tells us is not simply that a given particle would have done something different in the past, but that the very nature of its being, its ontology, in the past remains open to future reworkings... In particular I argue that this experiment offers empirical evidence for a relational ontology or perhaps more accurately a hauntology as against a metaphysics of presence... Remarkably this experiment makes evident that entanglement survives the measurement process and further more that material traces of attempts at erasure can be found in tracing the entanglements... While the past is never finished, and the future is not what would unfold, the world holds or rather is the memories of its iterated reconfigurings" Karen Barad's "Troubling Time's & Ecologies of Nothingness," European Graduate School Video Lectures (YouTube), my transcription.
"If classical physics insists that the void has no matter and no energy, the quantum principle of ontological indeterminacy, and particularly the indeterminacy relation between energy and time, pose into question the existence of such a zero energy, zero matter state... the indeterminacy principle allows for fluctuations of the vacuum... the vacuum is far from empty, it is fill with all possible indeterminate yearnings of space-time mattering... we can understand vacuum fluctuation in terms of virtual particles. Virtual particles are the quanta of the vacuum fluctuations... the void is a spectral ground, not even nothing can be free of ghosts... there is infinite number of possibilities, but not everything is possible. The vacuum isn't empty but neither is anything in it... particles together with their antiparticles and pairs can be created out of the vacuum by putting the right amount of energy into the vacuum... So, similarly, particles together with their antiparticles and pairs can go back into the vacuum, emitting the excess energy" Karen Barad's "Troubling Time's & Ecologies of Nothingness," European Graduate School Video Lectures (YouTube), my transcription.
Labyrinthine corridors, rooms:
"This was on Friday afternoon. Saturday morning I awoke early and read the two papers. Bohm, in simple clear language, declared that indeed there were conceptual problems in both macro- and microphysics, and that they were not to be swept under the carpet... And, further, Bohm suggested that the root of those problems was the fact that conceptualizations in physics had for centuries been based on the use of lenses which objectify (indeed the lenses of telescopes and microscopes are called objectives). Lenses make objects, particles," Karl Pribram's "The Implicate Brain";
"An equally important step in understanding came at a meeting at the University of California in Berkley, in which Henry Stapp and Geoffrey Chew of the Department of Physics pointed out that most of quantum physics, including their bootstrap formulations based on Heisenberg's scatter matrices, were described in a domain which is the Fourier transform of the spacetime domain. This was of great interest to me because Russell and Karen DeValois of the same university had shown that the spatial frequency encoding displayed by cells of the visual cortex was best described as a Fourier transform of the input pattern. ***The Fourier theorem states that any pattern, no matter how complex, can be analyzed into regular waveform components of different frequencies, amplitudes, and (phase) relations among frequencies. Further, given such components, the original pattern can be reconstructed. This theorem was the basis for Gabor's invention of holography," Karl Pribram's "The Implicate Brain";
[***when different wave patterns meet, they add up to form new patterns; you can analyse complex wave patterns as if they were a superposition of more simple waves, which have, for instance a definite, uniforme wavelength; the illustration at left is taken from the site of professor John D. Norton (Pittsburg University): "Einstein for Everyone"; it is important to note that real wave patterns studied in physics are much more complex than this two dimensional representation, and that they are ultimately formed by something that is neither strictly speaking a wave nor a particle as these are classically understood; I shall also say that not all John D. Norton's explanations given in the referred site seem very enlightening to me]
(see picture above)
From Maxwell's equations, we should expect an infinite number of frequencies of electromagnetic waves (or radiation, which includes visible light, and waves whose frequencies are bellow the one which produces the red colour, such as radio waves, and also waves whose frequencies are above the one which produces the violet colour, such gama rays). All these electromagnetic waves travel at what is called the speed of light (the frequencies can vary because the wavelength also varies proportionally) and constitute the electromagnetic spectrum. High frequency means also high photon energy. The photon energy is related to how single atoms of different material objects can absorb and emit electromagnetic waves, which happens always in quantum discrete amounts. As concrete musical instruments, atoms can produce oscillations only in certain restricted ways, and they do so very energetically. The physical production of what we perceive as forms and colours has to do, however, more directly with the way electromagnetic waves travel much more freely and continuously in space, through, for instance, air or water, interfering (constructively or destructively) with one another, interacting with molecules—and we are talking about electromagnetic waves of lower energy and frequencies, which are visible. What we see, although, isn't everything.
Does the continuum (infinitely divisible) preclude plurality? Does the discrete precludes unity? Of course no! Except for the lack of imagination of the purist & prudish. But thanks gosh, in philosophy we also have Leibniz's "Natura non facit saltus," and Peirce's synechism (everything is connected), the immemorial and unending, irreducible battles between the one and the many. Why should people be so afraid of a conundrum of straight lines, curves, and points (which besides going for these one- and two-dimensions, can be extrapolated to n-)? Infinitesimals, differentials, and limits, what is the real difference? Epsilon-delta definition (Cauchy, Bolzano, Weierstrass) and nonstandard analysis (Abraham Robinson) are all in the end perfectly compatible. Add to that synthetic differential geometry or the smooth infinitesimal (F. W. Lawvere), whatever! The actual infinite—everything else starts from it! Just don't be afraid of lingo—the science wars are an affair of securing university bonus in times of economic havoc. And don't forget that continuity doesn't have to be only local, that is, the chaosmos is full of nonlocal connections, the innermost separations! What matters is attitude, not content or specific formulations.
["Whenever a point x is within δ units of c, f(x) is within ε units of L," graphic and definition from the Wikipedia's epsilon-delta entry ]
["Infinitesimals (ε) and infinites (ω) on the hyperreal number line (1/ε = ω/1)," graphic and definition from Wikipedia's hyperreal number entry]
(see picture above)
"Cusanus... took the circle to be an infinilateral regular polygon, that is, a regular polygon with an infinite number of (infinitesimally short) sides... The idea of considering a curve as an infinilateral polygon was employed by a number of later thinkers, for instance, Kepler, Galileo and Leibniz... Traditionally, geometry is the branch of mathematics concerned with the continuous and arithmetic (or algebra) with the discrete. The infinitesimal calculus that took form in the 16th and 17th centuries, which had as its primary subject matter continuous variation, may be seen as a kind of synthesis of the continuous and the discrete, with infinitesimals bridging the gap between the two. The widespread use of indivisibles and infinitesimals in the analysis of continuous variation by the mathematicians of the time testifies to the affirmation of a kind of mathematical atomism which, while logically questionable, made possible the spectacular mathematical advances with which the calculus is associated. It was thus to be the infinitesimal, rather than the infinite, that served as the mathematical stepping stone between the continuous and the discrete," John L. Bell's "Continuity and Infinitesimals" (Stanford Encyclopedia of Philosophy) [I like this passage very much, and this is a very useful article, but I'm not subscribing in detail to all ideas Bell developed there];
"... science needs calculus; calculus needs the continuum; the continuum needs a very careful definition; and the best definition requires there to be actual infinities (not merely potential infinities) in the micro-structure and the overall macro-structure of the continuum... Informally expressed [for Dedekind], any infinite set can be matched up to a part of itself; so the whole is equivalent to a part. This is a surprising definition because, before this definition was adopted, the idea that actually infinite wholes are equinumerous with some of their parts was taken as clear evidence that the concept of actual infinity is inherently paradoxical... [Cantor's] new idea [similar to Dedekind's] is that the potentially infinite set presupposes an actually infinite one. If this is correct, then Aristotle’s two notions of the potential infinite and actual infinite have been redefined and clarified," Bradley Dowden's "The Infinite" (Internet Encyclopedia of Philosophy) [I like this passage very much, and this is a very useful article, but I'm not subscribing in detail to all ideas Dowden developed there];
"... in Quantum Electrodynamics... processes of much greater complexity [than a simple electron-electron scattering] could intervene in the scattering process. For example, the exchanged photon could convert to an electron-positron pair which would subsequently recombine... or one of the incoming electrons might emit a photon and reabsorb it on the way out... in general, the exchange of arbitrarily large numbers of photons, electrons and positrons can contribute to electromagnetic interactions... very complicated multiparticle exchanges have to be taken into account in the analysis of physical systems. Indeed, no exact solutions to the Quantum Electrodynamics are known, nor have such solutions ever been shown rigorously to exist [but precise approximations are possible]," Andrew Pickering's Constructing Quarks (p. 63);
"... in quantum field theory all forces are mediated by particle exchange... It is equally important to stress that the exchanged particles... are not observable... To explain why this is so, it is necessary to make a distinction between 'real' and 'virtual' particles... particles with unphysical values of energy and momentum are said to be 'virtual' or 'off mass-shell' particles. In classical physics they could not exist at all... In quantum physics, in consequence of the Uncertainty Principle, virtual particles can exist, but only for an infinitesimal and experimentally undetectable length of time. In fact, the lifetime of a virtual particle is inversely dependent upon how far its mass diverges from its physical value," Andrew Pickering's Constructing Quarks (p. 64-65);
"In quantum mechanics the particles themselves can be represented as fields. An electron, for example, can be considered a packet of waves with some finite extension in space. Conversely, it is often convenient to represent a quantum mechanical field as if it were a particle. The interaction of two particles through their interpenetrating fields can then be summed up by saying the two particles exchange a third particle, which is called the quantum of the field. For example, when two electrons, each surrounded by an electromagnetic field, approach each other and bounce apart, they are said to exchange a photon, the quantum of the electromagnetic field. The exchanged quantum has only an ephemeral existence... The larger their energy, the briefer their existence. The range of an interaction is related to the mass of the exchanged quantum. If the field quantum has a large mass, more energy must be borrowed in order to support its existence, and the debt must be repaid sooner lest the discrepancy be discovered. The distance the particle can travel before it must be reabsorbed is thereby reduced and so the corresponding force has a short range. In the special case where the exchanged quantum is massless [such as a photon] the range is infinite," Gerard 't Hooft's "Gauge Theories of the Forces between Elementary Particles" (Scientific American, vol. 242, n. 6, 1980, pp. 104-141);
"It was not immediately apparent that quantum electrodynamics could qualify as a physically acceptable theory. One problem arose repeatedly in any attempt to calculate the result of even the simplest electromagnetic interactions, such as the interaction between two electrons. The likeliest sequence of events in such an encounter is that one electron emits a single virtual photon and the other electron absorbs it. Many more complicated exchanges are also possible, however; indeed, their number is infinite. For example, the electrons could interact by exchanging two photons, or three, and so on. The total probability of the interaction is determined by the sum of the contributions of all the possible events... Perhaps the best defense of the theory is simply that it works very well. It has yielded results that are in agreement with experiments to a n accuracy of about one part in a billion, which makes quantum electrodynamics the most accurate physical theory ever de vised," Gerard 't Hooft's "Gauge Theories of the Forces between Elementary Particles" (Scientific American, vol. 242, n. 6, 1980, pp. 104-141);
"If an electron enters a medium composed of molecules that have positively and negatively charged ends, for example, it will polarize the molecules. The electron will repel their negative ends and attract their positive ends, in effect screening itself in positive charge. The result of the polarization is to reduce the electron's effective charge by an amount that in creases with distance... The uncertainty principle of Werner Heisenberg suggests... that the vacuum is not empty. According to the principle, uncertainty about the energy of a system increases as it is examined on progressively shorter time scales. Particles may violate the law of the conservation of energy for unobservably brief instants; in effect, they may materialize from nothingness. In QED [Quantum Electrodynamics] the vacuum is seen as a complicated and seething medium in which pairs of charged "virtual" particles, particularly electrons and positrons, have a fleeting existence. These ephemeral vacuum fluctuations are polarizable just as are the molecules of a gas or a liquid. Accordingly QED predicts that in a vacuum too electric charge will be screened and effectively reduced at large distances," Chris Quigg's Elementary Particles and Forces (Scientific American, vol. 252, n. 4, 1985, pp. 84-95);
"A nuvem de probabilidades que acompanha os elétrons entre uma interação e outra é um pouco parecida com um campo. Mas os campos de Faraday e Maxwell, por sua vez, são feitos de grãos: os fótons. Não apenas as partículas estão em certo sentido difusas no espaço como campos, mas também os campos interagem como partículas. As noções de campo e de partícula, separadas por Faraday e Maxwell, acabam convergindo na mecânica quântica. A forma como isso acontece na teoria é elegante: as equações de Dirac determinam quais valores cada variável pode assumir. Aplicadas à energia das linhas de Faraday, dizem-nos que essa energia pode assumir apenas certos valores e não outros... As ondas eletromagnéticas são de fato vibrações das linhas de Faraday, mas também, em pequena escala, enxames de fótons... Por outro lado, também os elétrons e todas as partículas de que é feito o mundo são 'quanta' de um campo... semelhante ao de Faraday e Maxwell," Carlo Rovelli's A realidade não é o que parece (Objetiva, 2014, p. 125);
"A 'nuvem' que representa os pontos do espaço onde é provável encontrar o elétron é descrita por um objeto matemático chamado 'função de onda.'O físico austríaco Erwin Schrödinger escreveu uma equação que mostra como essa função de onda evolui no tempo. Schrödinger esperava que a 'onda' explicasse as estranhezas da mecânica quântica... Ainda hoje alguns tentam entender a mecânica quântica pensando que a realidade é a onda de Schrödinger. Mas Heisenberg e Dirac logo compreenderam que esse caminho é equivocado. A função [de onda] não está no espaço físico, está em um espaço abstrato formado por todas as possíveis [virtuais!] configurações do sistema... A realidade do elétron não é uma onda [?]: é esse aparecer intermitente nas colisões," Carlo Rovelli's A realidade não é o que parece (Objetiva, 2014, p. 271);
"When we say that we wish to make sense of something we meant o put it into spacetime terms, the terms of Euclidean geometry, clock time, etc. The Fourier transform domain is potential to this sensory domain. The waveforms which compose the order present in the electromagnetic sea which fills the universe make up an interpenetrating organization similar to that which characterizes the waveforms "broadly cast" by our radio and television stations. Capturing a momentary cut across these airwaves would constitute their hologram. The broadcasts are distributed and at any location they are enfolded among one another. In order to make sense of this cacophany of sights and sounds, one must tune in on one and tune out the others. Radios and television sets provide such tuners. Sense organs provide the mechanisms by which organisms tune into the cacophany which constitutes the quantum potential organization of the elecromagnetic energy which fills the universe," Karl Pribram's "The Implicate Brain";
"... the cloud chamber photograph does not reveal a “solid” particle leaving a track. Rather it reveals the continual unfolding of process with droplets forming at the points where the process manifests itself. Since in this view the particle is no longer a point-like entity, the reason for quantum particle interference becomes easier to understand. When a particle encounters a pair of slits, the motion of the particle is conditioned by the slits even though they are separated by a distance that is greater than any size that could be given to the particle. The slits act as an obstruction to the unfolding process, thus generating a set of motions that gives rise to the interference pattern," Basil J. Hiley's "Mind and matter: aspects of the implicate order described through algebra" (in K. H. Pribram's and J. King's Learning as Self-Organisation, New Jersey, Lawrence Erlbaum Associates, 1996, pp. 569-86);
"Let us... ask what the algebraic structure tells you about the underlying phase space. Because the algebra is non-commutative there is no single underlying manifold. That is a mathematical result. Thus if we take the algebra as primary then there is no underlying manifold we can call the phase space. But we already know this. At present we say this arises because of the 'uncertainty principle,' but nothing is 'uncertain,'" Basil Hiley's "From the Heisenberg Picture to Bohm: a New Perspective on Active Information and its relation to Shannon Information" (in A. Khrennikov, Proc. Conf. Quantum Theory: reconsideration of foundations, Sweden, Växjö University Press, pp. 141-162, 2002).
"What Gelfand showed was that you could either start with an a priori given manifold and construct a commutative algebra of functions upon it or one could start with a given commutative algebra and deduce the properties of a unique underlying manifold. If the algebra is non-commutative it is no longer possible to find a unique underlying manifold. The physicist’s equivalent of this is the uncertainty principle when the eigenvalues of operators are regarded as the only relevant physical variables. What the mathematics of non-commutative geometry tells us is that in the case of a non-commutative algebra all we can do is to find a collection of shadow manifolds... The appearance of shadow manifolds is a necessary consequence of the non-commutative structure of the quantum formalism," Basil Hiley's "Phase Space Descriptions of Quantum Phenomena" (in A. Khrennikov, Quantum theory: Reconsiderations of Foundations, Vaxjo University Press, 2003).
the odd transformation of Der Herr Warum (Gödel with Resnais);
the only three types of ingenuity;
why self-help books are not to be dismissed;
the most auspicious tetrahedron;
what is REAL space? what is REAL number?
Timothy Leary in the 1990s;
5G?! Get real...
list of charming scientists/engineers;
pick a soul (ass you wish);
- en profane: Orsay & Centre Pompidou;
view from Berthe Trépat's apartment;
list des déclencheurs musicaux;
Dark Consciousness;
The Doors of Perception;
Structuralism, Poststructuralism;
List des figures du chaos primordial (Deleuze);
Brazilian Perspectivism;
Piano Playing (Kochevitsky);
- L'Affirmation de l'âne (review of Smolin/Unger's The Singular Universe);
And also:
1. Anonymous3:52 PM
2. Anonymous12:00 AM
concerning my study and knowledge.
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f97920433cdcc43d | Beauty-full exotic bound states at the LHC
Article: Beauty-full Tetraquarks
Authors: Yang Bai, Sida Lu, and James Osborn
Good Day Nibblers,
As you probably already know, a single quark in isolation has never been observed in Nature. The Quantum Chromo Dynamics (QCD) strong force prevents this from happening by what is called ‘confinement. This refers to the fact that when quarks are produced in a collision for example, instead of flying off alone each to be detected separately, the strong force very quickly forces them to bind into composite states of two or more quarks called hadrons. These multi-quark bound states were first proposed in 1964 by Murray Gell-Mann as a way to explain observations at the time.
The quarks are bound together by QCD via the exchange of gluons (e.g. see Figure 1) and there is an energy associated with how strongly they are bound together. This binding energy between the quarks contributes to the ‘effective mass’ for the composite states and in fact it is what is largely responsible for the mass of ordinary matter (Footnote 1). Most of the theoretical and experimental progress has been in two or three quark bound states, referred to as mesons and baryons respectively. The most familiar examples of quark bound states are the neutron and proton, both of which are baryons composed of three quarks bound together and form the basis for atomic nuclei.
Figure 1: Bound state of four bottom quarks (blue) held together by the QCD strong force which is transmitted via the exchange of gluons (pink).
Of course four and even more quark bound states are possible and some have been observed, but things get much trickier theoretically in these cases. For four quark bound states (called tetra-quarks) the theoretical progress had been largely limited to the case where at least one of the quarks was a light quark, like an up or a down quark.
The paper highlighted here takes a step towards understanding four quark bound states in the case where all four quarks are heavy. These heavy four body systems are extra tricky because they cannot be decomposed into pairs of two body systems which we could solve much more easily. Instead, one must solve the Schrödinger equation for the full four body system for which approximation methods are needed. The example the current authors focus on is the four bottom quark bound state or 4b state for short (see Figure 1). In this paper they use sophisticated numerical methods to solve the non-relativistic Schrödinger equation for a four-body system bound together by QCD. Specifically they solve for the energy of the ground state, or lowest energy state, of the 4b system. This lowest energy state can effectively be interpreted as the mass of the 4b composite state.
In the ground state the four bottom quarks arrange themselves in such a way that the composite system appears as spin-0 particle. So in effect the authors have computed the mass of a composite spin-0 particle which, as opposed to being an elementary scalar like the Standard Model Higgs boson, is made up of four bottom quarks bound together. They find the ground state energy, and thus the mass of the 4b state, to be about 18.7 GeV. This is a bit below the sum of the masses of the four (elementary) bottom quarks which means the binding energy between the quarks actually lowers the effective mass of the composite system.
The interesting thing about this study is that so far no tetra-quark states composed only of heavy quarks (like the bottom and top quarks) have been discovered at colliders. The prediction of the mass of the 4b resonance is exciting because it means we know where we should look at the LHC and can optimize a search strategy accordingly. This of course increases the prospects of observing a new state of matter when the 4b state decays, which it can potentially do in a number of ways.
For instance it can decay as a spin-0 particle (depicted as \varphi in Figure 2) into two bound states of pairs of b quarks, which themselves are referred to as \Upsilon mesons. These in turn can be observed in their decays to light Standard Model particles giving many possible signatures at the LHC. As the authors point out, one such signature is the four lepton final state which, as I’ve discussed before, is a very precisely measured channel with small backgrounds. The light mass of the 4b state also allows for it to potentially be produced in large rates at the LHC via the strong force. This sets up the exciting possibility that a new composite state could be discovered at the LHC before long simply by looking at events with four leptons with total energy around 18 – 19 GeV.
Figure 2: Production of a four bottom quark bound state (\varphi) which then decays to two bound states of bottom quark pairs called \Upsilon mesons.
Of course, one could argue this is less exciting than discovering a new elementary particle since if the 4b state is observed it won’t be the discovery of a new particle but instead of yet another manifestation of the QCD strong force. At the end of the day though, it is still an exotic state of nature which has never been observed. Furthermore, these exotic states could be interesting testing grounds for beyond the Standard Model theories which include new forces that communicate with the bottom quark.
We’ll have to wait and see if the QCD strong force can indeed manifest itself as a four bottom quark bound state and if the prediction of its mass made by the authors indeed turns out to be correct. In the meantime, it gives plenty of motivation to experimentalists at the LHC to search for these and other exotic bound states and gives us perhaps some hope for finding physics beyond the Standard Model at the LHC.
Footnote 1: I know what you are thinking, but I thought the Higgs gave mass to matter!? Well yes, but…The Higgs gives mass to the elementary particles of the Standard Model. But most of the matter (that is not dark!) in the universe is not elementary, but instead made up of protons and neutrons which are composed of three quarks bound together. The mass of protons and neutrons is dominated by the binding and kinetic energy of the three quarks systems and therefore it is this that is largely responsible for the mass of normal matter we see in the universe and not the Higgs mechanism.
Other recent studies on heavy quark bound states:
Further reading and video:
1) TASI 2014 has some great introductory lectures and notes on QCD:
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Roberto Vega-Morales is currently a post-doctoral researcher in high energy theory at the University of Granada in Spain. Previously he was at the Laboratoire de Physique Thèorique in Paris France. He conducted his Ph.D studies at Northwestern University as well as Fermilab and was awarded the 2014 J.J. and Noriko Sakurai Dissertation Award in Theoretical Particle Physics. His research focuses on the phenomenology of the Higgs boson at the LHC as well as models of Supersymmetry and extended Higgs sectors. He struggled mightily with French and is happy to be speaking Spanish nowadays.
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33036bb8f25ff0ee | What does quantum mechanics tell us about reality? Part II
This blog post follows on from a previous post, “What does quantum mechanics tell us about reality”, in which I tried to give a balanced and non-technical overview of some of the interpretations of quantum mechanics. In this post I will take a different approach: this will be a slightly biased, critical, and more technical follow-up. I recommend reading the earlier post first, but those already familiar with the interpretations of quantum mechanics should be able to dive straight into this.
In the Schrödinger’s cat thought experiment, a cat is placed in a box with a device that contains a radioactive atom and a vial of poison. If the atom decays, then the device is designed to release the poison, thus killing the cat. It is now well known that such an atom can be put into a state in which it has decayed, and not decayed simultaneously – this is known as a superposition state. Now, if this system is studied using the central equation in quantum mechanics, the Schrödinger equation, then the following result will be found: if the atom is in a superposition state, then this will lead to the cat being in a superposition state. The cat will be dead and alive simultaneously! Now suppose you open the box – what will you find? The Schrödinger equation again predicts that, if the cat was in a superposition of being dead and alive, then when you open the box you will also enter into a superposition. You will be in a superposition of either seeing the dead cat, whilst simultaneously seeing the alive cat.
Artwork by Joe Hollis
This clearly does not fit with our experience of the real world. We never see objects in superpositions, and indeed we never seem to experience superpositions ourselves. And while the above experiment is far too challenging to perform using a real cat, conceptually similar experiments have been performed in which an object is put into superposition, and then observed. The result of these experiments fits with our experience and intuition: we never see a superposition state. So what has gone wrong here? Have we misapplied the Schrödinger equation? Is the Schrödinger equation incorrect? The standard resolution, which can be found in most quantum mechanics textbooks, is to introduce the “collapse postulate”: On observation a superposition state collapses, meaning that only one outcome of an observation, or a measurement, is ever observed. I.e. we only ever see the cat as being dead or alive. But the collapse postulate raises as many problems as it solves. What exactly constitutes an observation or measurement? If macroscopic objects are made of quantum particles, what is so special about a measuring device or a conscious human observer to cause collapse? (This questions are together often termed the measurement problem.)
Is the Schrödinger equation sufficient to solve the problem?
Despite how quantum mechanics is often discussed, there is now a widely accepted and carefully studied solution to these problems that utilises the Schrödinger equation alone, and does not have to introduce the troublesome collapse postulate. The solution lies in the theory of decoherence. Elsewhere I give a more thorough introduction to decoherence, in particular in relation to Schrödinger’s cat. But in this post I will try to give a simple and minimal introduction that still captures the main ideas.
Imagine you have a single atom and some cutting-edge experimental equipment capable of putting this atom into a superposition of two locations, A and B. The crucial question here, which is at the heart of decoherence, is: how do you know it is in a superposition? If you directly measure the atom then you will either see it at position A, or position B, but not both. To confirm the superposition a more advanced step needs to be taken: we must do an interference experiment. This involves the idea of constructive and destructive interference of waves, which can be seen by throwing two stones into a pond close to one another. The waves coming from one stone interfere with the waves coming from the other stone. If two peaks meet they reinforce one another creating a larger peak, whereas if a peak and trough meet they cancel each other out. Quantum mechanical objects, such as the atom we are trying to interfere, are described by equations known as wave functions. As the name suggests, these particles act like waves, and just like the stones in the pond they can demonstrate interference. I’m unsure myself how the exact experiment would work to interfere the two parts of an atom that has been put into a superposition, but by measuring the interference between the two wavefunctions the superposition can indeed be confirmed, and this is now an extremely well measured phenomenon in experiments.
Now suppose you are given two atoms, and you prepare the atoms in the following superposition state: both atoms are in position A, in superposition with both atoms in position B. Again, how can we confirm the superposition? If we directly measure the position of the atoms, then we either find both of them in position A, or both in position B (this is known as an entangled state – the position of the first atom is “entangled” with the position of the second, because we always find them together). Again we do not see, and cannot confirm, the superposition in this way, and we must perform an interference experiment. Now comes the crucial point: the interference experiment must be done on both atoms simultaneously, otherwise we will never see an interference pattern. If we just take the first atom, and try and interfere it with itself, then this will not work. (I explain this in more detail here.)
We can now return to Schrödinger’s cat. The cat is in a superposition state of being dead and alive, but how can we confirm the superposition? First imagine that the only thing in the box is the cat –it is in a complete vacuum with no air particles or photons or anything. In this case, it is in principle possible to perform an interference experiment with the cat. The dead part of the superposition interferes with the alive part of the superposition, and an interference pattern would be observed, confirming the superposition. This is not practically possible because we would have to interfere every single particle in the cat, and this involves precisely controlling and manipulating every single particle. But according to the laws of physics this is at least in principle possible.
But it is not realistic that the cat could be in a complete vacuum, and no matter how hard we tried there would always be at least a few particles in the box with the cat. These unwanted particles (and photons etc) are often termed the environment, and we assume that we do not have control nor access to them. Now again put the cat into a superposition. The cat will inevitably interact with the unwanted particles in the box, and as soon as they interact the cat and unwanted particles will become entangled with one another. Then, if we want to do an interference experiment, we would have to not only interfere all the particles in the cat, but also all the extra particles and photons in the box. We would have to precisely control and manipulate all of these particles, but as stated above we are assuming that we cannot control them and cannot access them. Therefore, in this case it is not even in principle possible to do an interference experiment. We cannot ever confirm that the cat was in a superposition.
Now what happens when we open the box? As soon as we look at the cat we become entangled with it, and enter into the superposition. The cat is dead and we see a dead cat, in superposition with the cat being alive whilst we see an alive cat. But again there will be unwanted particles and photons, and very quickly the cat and ourselves will become entangled with these particles and photons. Again, if we want to confirm that we are in a superposition, we would have to be able to manipulate and control all of these particles and photons, which is clearly not possible. Therefore, again, we cannot ever confirm that we are in a superposition. Furthermore, it is likely that some of the photons that have interacted with you will escape from the room through window, flying off to space at the speed of light! In this case, seeing that we can’t travel at the speed of light to collect these photons, it is not even in principle possible to confirm the superposition.
We have now solved the main problems in the Schrödinger’s cat thought experiment. Is the Schrödinger equation wrong? No – we can explain our observations, i.e. that we never see the cat in a superposition, just using the Schrödinger equation. Why do we never see the cat in a superposition? You must do an interference experiment to confirm the superposition, but this is not possible when we factor in the other particles in the box with the cat. The question of “what constitutes a measurement?” has not really been answered yet, but I will address this in a future post in which I defend the many worlds interpretation.
I have not yet fully addressed what happens when you open the box – whether you are really in a superposition, and if so, why you don’t “experience” this superposition. The answer to this really depends on how you interpret quantum mechanics, and this is what I will turn to next.
Many worlds interpretation
The introduction above to Schrödinger’s cat and decoherence has, in a sense, been written in the language of the many worlds interpretation. In the many worlds interpretation we firstly assume that the Schrödinger equation is sufficient in itself to explain paradoxes such as Schrödinger’s cat, and secondly we assume that quantum mechanics is a theory that tells us about real objects in the real world. The first of these points is justified in the above introduction to decoherence, and nowadays this explanation is widely accepted. The second point is the usual way we interpret science – normally we assume that our equations and theorems are telling us something about a real world that exists independent of ourselves.
These two assumptions might seem quite straightforward, but they lead to quite a radical picture of the world in which we live. For example, in the many worlds interpretation we say that the cat is indeed in a superposition of being dead and alive. There is technically just one cat, but it is dead and alive simultaneously. However, we have seen that the two parts of the superposition cannot ever interfere with each other. Interference is the only way of confirming that an object in is in a superposition, so the dead and alive cats cannot ever know of each other’s existence. Furthermore, the equations of quantum mechanics are such that the future life of the cat (at least the alive one) does not depend on whether the cat is in a superposition or not. Therefore, for all intents and purposes we can think of this as two cats, one dead and one alive. This is where the idea of “many worlds” comes from. For all intents and purposes there are two worlds, one containing an alive cat and one containing a dead cat.
The same idea holds when you open the box. You split into a superposition of seeing a dead cat and seeing an alive cat. But again the two parts of the superposition cannot ever know of the other’s existence, because they would have to interfere with one another to confirm this, and this isn’t possible. Therefore we can again treat this as being two separate worlds, one in which the cat is dead and you are presumably emotionally and morally scarred by the experience, and another in which the cat is alive and you will be relieved.
This picture of the universe is clearly unintuitive, and often people reject many worlds outright and come up with all kinds of criticisms of this interpretation. In my opinion most of the standard criticisms are either ill-founded or result from a lack of understanding of the basic theory, and in a future post I will try to flesh out many worlds theory and provide straightforward responses to many of the criticisms.
QBism – does the wavefunction represent reality?
Before continuing, an important comment is needed. Just before uploading this post I was in contact with Chris Fuchs – one of the founders and main promoters of QBism. To cut a long story short, he said (politely but firmly) that (referring to my previous post) “you capture none of the flavor of QBism at all in what you write. You present QBism as a kind of lifeless prediction machine (a positivism or instrumentalism), rather than as an attempt to make a deep statement about the character of the world”. He recommended reading https://arxiv.org/abs/1601.04360 and https://arxiv.org/abs/1207.2141. I have decided to keep my description of QBism in this current post unedited, but bear in mind Chris’s comment when you read this! And please comment on this post if you have an opinion about whether/how I misrepresent QBism…
As introduced above, we can represent quantum mechanical objects using an equation known as a wavefunction. The wavefunction tells us everything we know about this object. For example, we could write down the wavefunction for a single particle in an (equal) superposition of two locations. This wavefunction can then be used to predict what we will see if we perform certain measurements. For example, using the wavefunction we can calculate that, assuming the superposition is equal, if we measure the position of the particle then it will be in position A with 50% probability, or position B with 50% probability. Furthermore, we can use the wavefunction to predict what will happen if we perform an interference experiment. In particular, it will tell us the properties of certain outcomes: it will say that if we perform interference experiment X, then outcome Y will happen with probability Z.
Numerous experiments over the years have confirmed that quantum mechanics is extremely good at correctly predicting outcomes to experiment. But, in a sense, QBism says that this is all that quantum mechanics is good for. It says that we should not interpret the wavefunction as describing a real object, and therefore it is meaningless to ask if the cat is really dead and alive simultaneously. We simply cannot know – all we know is the probability of what will happen if we open the box. More specifically, the wavefunction represents our state of knowledge. It tells us what we know, not what exists. This is similar to Bayesian probability theory, in which probabilities this represent our knowledge of the world, not the world itself. For this reason QBism can also be called quantum Bayesianism.
I certainly have some sympathy with QBism. It takes quantum mechanics seriously, and in particular the Schrödinger equation, and does not try to modify the formulae. And it certainly has a strong point: how do we ever really know what exists? The answer is that we observe it, and we perform measurements on it, and we devise clever experiments to perform measurements on the extremes of scale and energy. But until we measure anything, we cannot truly know what it is, and whether it exists. So in this sense QBism is right that quantum mechanics is just a toolbox for predicting experiments.
But is this all quantum mechanics is? Throughout most of human history the goal of science has been to learn more about the world. We do astronomy and astrophysics to learn about stars and galaxies; we smash particles into one another in colliders to learn about what matter is made of; and we do quantum experiments to learn about the weird and wonderful properties of the quantum world. QBism therefore is a radical departure from how we normally treat the scientific endeavour. It is not necessarily the wrong way to interpret quantum mechanics, but Qbists should at least acknowledge that it is an extreme philosophical position.
To take this further, imagine the Schrödinger’s cat thought experiment, but with your friend opening the box rather than yourself. QBism is perfectly good at predicting what your friend will see when they open the box. But, presumably, you believe that your friend exists, and you might be interested in what happens to them when they open the box. QBism cannot tell us this – you can write down the wavefunction for your friend, but this is only a tool for calculating what you will see when you interact with your friend. Many worlds, on the other hand, is perfectly well-equipped to ask questions about your friend. The answer may be disturbing – that they in effect split into two versions – but at least it is a consistent and coherent answer. And this idea can be extended: many worlds theory predicts that almost continuously the world – and therefore your friend – splits into almost infinite parts of a vast superposition, which we can think of as parallel universes.
Would my assumption that my friend exists be incorrect? Perhaps. Maybe in the “real” world it is meaningless to ask about the state of things before we interact with them. But my friend certainly does exist in my head – I can imagine them walking towards the box, opening it, and looking inside. We can then call this world the “imaginary” world. Even though it might not exist outside my mind, I am still interested in what my imaginary friend is doing in this imaginary world. Removing yourself from the picture, now imagine a scene familiar to yourself, such as your house, or your pet, or your favourite sports team. I wonder what they are doing right now? Are there near infinite numbers of them, in near infinite parallel universes? Or is it meaningless to ask what they are doing right now, and only meaningful to think about what happens when you interact with them in some way?
My favourite thing about QBism is this: the wavefunction is normally written using the Greek letter psi, which is often pronounced “sigh”. Ontology is the study of the existence of things, whereas epistemology is concerned with knowledge rather than existence. Therefore, a Qbist is a psi-epistemist. Whereas someone like me who believes in many worlds and therefore that the wavefunction is real, can be termed a psi-ontologist. It deeply troubles me that I am a psi-ontologist (say this sentence out loud to yourself if you don’t get the joke!).
Collapse theories
Until reasonably recently it was not fully appreciated that the Schrödinger equation alone can lead to the appearance of collapse. Therefore, to explain why we either see the cat as dead or alive a “collapse postulate” was introduced into quantum mechanics. Initially it was just a postulate, and no explanation was given of how collapse takes place, or what causes it. But this introduces many difficult questions: What causes the collapse? It is usually assumed that a measurement causes collapse: but what is a measurement? Often it is said that a “measuring device”, or even a conscious observer, is what causes the collapse. But if macroscopic objects are made of quantum particles, what is so special about a measuring device or a conscious human observer to cause collapse?
Over the years various theories have been introduced to explain collapse with the hope of answering the above questions. Various mechanisms have been proposed: complexity causes collapse – the more complex a system, the more likely it is to collapse; or consciousness itself causes collapse; or gravity causes collapse – the larger the mass, the more likely collapse will occur. These models therefore can explain why Schrödinger’s cat is never seen, or measured, as being in a superposition state.
But now, with the theory of decoherence that I introduced above, we can explain the appearance of collapse without having to add extra postulates into the theory. Collapse theories are therefore unnecessary to explain our observations. So why do they still exist? I have never met anyone who both understands decoherence, and thinks that it is wrong, so collapse would presumably happen in addition to decoherence. And if you are uncomfortable with the conclusion that the cat is in a superposition (many worlds), or that it is meaningless to ask about the state of the cat (QBism), then you can modify quantum mechanics – specifically, modify the Schrödinger equation – so that the state collapses. But for me this seems like a case of changing the science in order to fit our wishes.
This might not be a problem if quantum mechanics was a young and underdeveloped theory. But this is certainly not the case, and the Schrödinger equation itself is responsible for quantum mechanics often being termed “our most successful theory ever”. Do we really want to modify such an equation? Quantum mechanics also works relativistically (i.e. combining it with Einstein’s special relativity), and it has been extended to quantum field theory, which has successfully predicted the Higgs boson. But collapse theories are far from achieving such extensions.
To be fair to gravity-induced-collapse, at some point quantum mechanics, as with any other theory, will be surpassed by some other theory. Quantum mechanics will still be an excellent approximation in many regimes, but in the extremes it will surely break down. But what are these extremes? Potentially the fact that general relativity and quantum mechanics cannot yet fit together gives a clue to this. In this case, might gravity in fact collapse the wavefunction? In my understanding this is at the heart of Roger Penrose’s suggestions to both explain collapse and unify general relativity and quantum mechanics.
For me the main positive to collapse theories is that they are testable. This is especially true for gravity-induced-collapse. If we put bigger and bigger systems into a superposition, while sufficiently isolating them from the environment so that decoherence doesn’t cause the appearance of collapse, then eventually at a certain mass threshold these systems should spontaneously collapse. These experiments should be possible in the relatively near future, and will serve to either confirm this theory, or give extra weight to non-collapse theories such as many worlds.
Consciousness-induced-collapse is in principle testable, but this is far beyond current experiments. To confirm this we would have to put a conscious entity into a superposition. We would have to isolated sufficiently it from the environment so that there is no decoherence, and we would have to be able to control and manipulate every particle in the conscious entity so that we can do an interference experiment. If the consciousness spontaneously collapses, thereby preventing interference, this will be strong evidence that consciousness does induced collapse. The best route to this could be using quantum computers. If we can simulate consciousness on a computer, then we could upload this program to a quantum computer, and subsequently put the consciousness into a superposition. But we don’t even know what consciousness is and such a test is infeasible for now. In addition, I argue elsewhere that if consciousness did cause collapse then the reality this would lead to would be far more bizarre and absurd than even many worlds theory predicts!
Pilot wave theory
Einstein famously stated that “God does not play dice”. He simply couldn’t believe that a fundamental theory of nature such as quantum mechanics could really be probabilistic. For example, generally in quantum mechanics we would say that on opening the box containing Schrödinger’s cat it would be random whether the cat is observed as dead or alive (with a certain probability of each). In many worlds theory both outcomes may exist, but it is random whether you end up in the part of the superposition with the dead cat or with the alive cat, so in this sense it is still random. In contrast, theories such as general relativity and Newtonian mechanics are deterministic. For example, if you know all of the positions and velocities of the planets in the solar system, then you can predict with certainty where the planets will be at any given time in the future.
To prevent the randomness of quantum mechanics a “deterministic hidden variable theory” was devised (named Bohmian/De Broglie/pilot wave theory). Taking again the example of the cat, in this theory there are additional variables beyond those in the Schrödinger equation. If we knew the values of all these variables, then we would know with certainty whether the cat will be dead or alive when we open the box. However, these variables are “hidden”, meaning they are fundamentally beyond our measurements and observations. We cannot, and will not, ever be able to determine these values, and therefore quantum mechanics will always appear to be random.
For me this is an even worse case than collapse theories of changing the science so that it more closely fits with our intuition. For protagonists of this theory it is so important that nature must not be random that they are willing to invent an underlying deterministic world that we cannot ever even in principle see. But why should nature be deterministic? In addition, the Schrödinger equation itself is deterministic, so in fact many worlds theory is a deterministic theory. We know with certainty that the cat will be dead and alive. The randomness just comes in when you ask “which universe will I end up in?”. But it is still, from the outside, deterministic.
There are some further complications/criticisms to this theory. John Bell famously showed that, if these hidden variables exist, then they must communicate with one another faster than the speed of light. Furthermore, in a recent paper Renato Renner showed that hidden variable models cannot be self-consistent (although this might not necessarily mean that they are wrong?!).
There are many other interpretations of quantum mechanics, and many more seem to be invented year-on-year. My personal view is that quantum physicists need to stop inventing new interpretations, and consolidate the old ones. Indeed both many worlds and QBism have some features that are unsatisfactory to some and unintuitive to all. But in my understanding there is nothing fundamentally wrong with either of these. Sure there are small problems that need to be ironed out, but this is the same for any theory. My personal prediction is that in 100 years from now, if we survive existential risks such as nuclear war or artificial intelligence taking over the world, pretty much every quantum physicist will either be a Qbist, or believe that we live in a fantastic quantum multiverse!
About P A Knott
I currently hold a Research Fellowship from the Royal Commission for the Exhibition of 1851. My research project will tackle a key challenge in the quantum technology revolution by designing computer algorithms that automate the engineering of useful quantum states. These algorithms will enable the design of novel experiments to bring forward the development of new technologies such as quantum computing, communications and metrology. In my previous post I worked at the University of Nottingham on a project entitled "Sentient observers in the quantum regime and the emergence of objective reality", with Gerardo Adesso, Marco Piani, and Tommaso Tufarelli. This project involved using quantum information theory to investigate foundational questions concerning the role of the observer in physical theories. More generally, my research interests include quantum metrology, quantum state engineering, quantum sensing networks, and optical interferometry.
This entry was posted in Entanglement, Philosophy, Quantum foundations, Superposition. Bookmark the permalink.
9 Responses to What does quantum mechanics tell us about reality? Part II
1. juan m. jones says:
Qbism says a lot about the world but requires that you can change your mind about what is knowledge, science and reality.
2. P A Knott says:
It’s certainly true that QBism says a lot about the world – I was a bit harsh in my representation of it. Though I think it is accurate to say that QBism doesn’t say anything about the state of Schrödinger’s cat, before you open the box?
• juan m. jones says:
It says that the Hilbert space associated with the cat is something about “reality”, but the cat, or any system, is pure potential, not actuality. Actuality arises, as experience, when systems interact. Hilbert space is objetive.
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3. Pingback: Why the many worlds interpretation of quantum mechanics is fantastic | quanta rei
4. Hello Paul,
I have just discovered your very interesting blog via your last paper about decoherence and quantum Darwinism (http://arxiv.org/pdf/1811.09062.pdf), the one you mention here in this post.
Please excuse my layman question (being at the same time a friendly provocation), but introducing your paper you write:
> Together these theories explain how our classical reality emerges from an underlying quantum mechanical description
and in the last sentence of the abstract you announce that finally you are going to
> demonstrate how decoherence and quantum Darwinism can shed significant light on the measurement problem
(and indeed you write about the role of the observer and about the measurement problem just near the end, just before comparisions with Everett Interpretation and Conclusion)
How can you state the problem of emergence of classical reality solved when you can only (!) ,,shed significant light” on the measurement problem?
(what is more quirky you seem to admit that even having read almost the whole paper ,,it may not be immediately obvious” that decoherence and quantum Darwinism can shed that light)
Is not the measurement problem (with ubiquitous questions about the role of the observer) the core, the essence of the problem of why and how our classical reality emerges?
I know your answer:
> in decoherence and quantum Darwinism this is not the case
but is it a truly honest answer? or rather, is it the answer of someone from the Decoherence Church? I doubt it as in another paper (https://knottquantum.weebly.com/uploads/9/0/9/4/90944896/does_consciousness_collapse_the_quantum_state.pdf) you write:
> the so-called measurement problem of quantum mechanics […] lies at the very heart of the theory
and BTW, I do agree that ,,we cannot ever confirm that we are in a superposition”, but from completely different reasons, I just feel that trying to conduct such measurement would be like trying to bootstrap oneself up, what is more, using bootstrap in a superposition 😉
Best regards,
5. P A Knott says:
Hi Wojciech,
Thanks for your comment and interest!
In my opinion decoherence does completely solve the measurement problem, but I didn’t fully explain this in my essay, hence the wording I used. But apologies if what I wrote in the abstract was misleading.
Generally I do agree with this! It depends on exactly how you define the measurement problem, and what aspect of classical reality you are trying to explain, but I agree that generally the measurement problem is a key part.
So, why do you think that decoherence does not solve the measurement problem? Which specific part of the measurement problem is not solved? If you tell me this, I will try to answer as best I can.
6. Hi Paul,
Thank you so much for your reply.
Why do I think that decoherence does not solve the measurement problem?
Well, thre is so much fuss about decoherence and now quantum Darwinism, so many serious people working in this field, that I can not simply ignore them.
I have started studying their contributions in earnest with strong feeling that there must be something important about the world that decoherence can reveal, yet the more I read the more I am puzzled and worried.
I am not very advanced in my study of the formalism and its consequences, but it is not something wrong with the formalism that keeps me awake at nights 🙂
I have rather serious difficulties to grasp the meaning, the foundations.
To be honest, I am not a 100% layman, but now I am more interested in philosophy of physics (and were my first name be different, I would have lost my initial enthusiasm for the works of Wojciech H. Zurek long time ago 😉 )
Reading more about the philosophy of quantum theory I have got infected with doubts that seems fatal and incurable for me.
The main infector for me was Chris Fields, take a look on his argumentation here: http://arxiv.org/abs/1402.6629v6
(about decoherence in the last paragraph of chapter 4, I do not understand what he writes in chap 5 and I suspect him of being wrong there but it does not matter)
or here: https://chrisfieldsresearch.com/quant-freedom.pdf
As you can see, it is not some specific part of the measurement problem that decoherence have not elucidated yet, the problem is deeper (I feel it must be so, as the observer is the keystone)
But it is so good to have such problem to think about, what a sad world would it be, were such problems could be solved so easily with decoherence or something like that!
Best regards,
• P A Knott says:
It would be great to discuss your problems with decoherence, but you haven’t yet told me what they are! Unfortunately I don’t have time to read this paper you mentioned — I read the paragraph you suggested but it didn’t make sense to me without having read the rest of his argument.
Perhaps you could start by explaining what is wrong with the examples I gave of decoherence in my paper in the section titled Decoherence, starting on page 2. (Here’s the paper: https://arxiv.org/pdf/1811.09062.pdf)
7. Pingback: Discussion and review of Shadows of the Mind by Roger Penrose | quanta rei
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81e07ce2679f2375 | fredag 28 augusti 2015
Finite Element Quantum Mechanics 4: Spherically Symmetric Model
I have tested the new atomic model described in a previous post in setting of spherical symmetry with electrons filling a sequence of non-overlapping spherical shells around a kernel. The electrons in each shell are homogenized to spherical symmetry which reduces the model to a 1d free boundary problem with the free boundary represented by the inter-shell spherical surfaces adjusted so that the combined wave function is continuous along with derivates across the boundary. The repulsion energy is computed so as to take into account that electrons are not subject to self-repulsion, by a corresponding reduction of the repulsion within a shell.
The remarkable feature of this atomic model, in the form of a 1d free boundary problem with continuity as free boundary condition and readily computable on a lap-top, is that computed ground state energies show to be surprisingly accurate (within 1%) for all atoms including ions (I have so far tested up to atomic number 54 and am now testing excited states).
Recall that the wave function $\psi (x,t)$ solving the free boundary problem, has the form
• $\psi (x,t) =\psi_1(x,t)+\psi_2(x,t)+...+\psi_S(x,t)$ (1)
with $(x,t)$ a common space-time coordinate, where $S$ is the number of shells and $\psi_j(x,t)$ with support in shell $j$ is the wave function for the homogenized wave function for the electrons in shell $j$ with $\int\vert\psi_j(x,t)\vert^2\, dx$ equal to the number of electrons in shell $j$.
Note that the free boundary condition expresses continuity of charge distribution across inter-shell boundaries, which appears natural.
Note that the model can be used in time dependent form and then allows direct computation of vibrational frequencies, which is what can be observed.
Altogether, the model in spherical symmetric form indicates that the model captures essential features of the dynamics of an atom, and thus can useful in particular for studies of atoms subject to exterior forcing.
I have also tested the model without spherical homogenisation for atoms with up to 10 electrons, with similar results. In this case the the free boundary separates diffferent electrons (and not just shells of electrons) with again continuous charge distribution across the corresponding free boundary.
In this model electronic wave functions share a common space variable and have disjoint supports and can be given a classical direct physical interpretation as charge distribution. There is no need of any Pauli exclusion principle: Electrons simply occupy different regions of space and do not overlap, just as in a classical multi-species continuum model.
This is to be compared with standard quantum mechanics based on multidimensional wave functions $\psi (x_1,x_2,...,x_N,t)$ typically appearing as linear combinations of products of electronic wave functions
• $\psi (x_1,x_2,...,x_N,t)=\psi_1(x_1,t)\times \psi_2(x_2,t)....\times\psi_N(x_N,t)$ (2)
for an atom with $N$ electrons, each electronic wave function $\psi_j(x_j,t)$ being globally defined with its own independent space coordinate $x_j$. Such multidimensional wave functions can only be given statistical interpretation, which lacks direct physical meaning. In addition, Pauli's exclusion principle must be invoked and it should be remembered that Pauli himself did not like his principle since it was introduced ad hoc without any physical motivation, to save quantum mechanics from collapse from the very start...
More precisely, while (1) is perfectly reasonable from a classical continuum physics point of view, and as such is computable and useful, linear combination of (2) represent a monstrosity which is both uncomputable and unphysical and thus dangerous, but nevertheless is supposed to represent the greatest achievement of human intellect all times in the form of the so called modern physics of quantum mechanics.
How long will it take for reason and rationality to return to physics after the dark age of modern physics initiated in 1900 when Planck's "in a moment of despair" resorted to an ad hoc hypothesis of a smallest quantum of energy in order to avoid the "ultra-violet catastrophe" of radiation viewed to be impossible to avoid in classical continuum physics. But with physics as finite precision computation, which I am exploring, there is no catastrophe of any sort and Planck's sacrifice of rationality serves no purpose.
PS Here are the details of the spherical symmetric model starting from the following new formulation of a Schrödinger equation for an atom with $N$ electrons organised in spherical symmetric form into $S$ shells: Find a wave function
as a sum of $N$ electronic complex-valued wave functions $\psi_j(x,t)$, depending on a common 3d space coordinate $x\in R^3$ and time coordinate $t$ with non-overlapping spatial supports $\Omega_1(t)$,...,$\Omega_N(t)$, filling 3d space, satisfying
• $i\dot\psi (x,t) + H\psi (x,t) = 0$ for all $(x,t)$, (1)
where the (normalised) Hamiltonian $H$ is given by
• $V_k(x)=\int\frac{\vert\psi_k(y,t)\vert^2}{2\vert x-y\vert}dy$, for $x\in R^3$,
• $\int_{\Omega_j}\vert\psi_j(x,t)\vert^2 =1$ for all $t$ for $j=1,..,N$.
Assume the electrons fill a sequence of shells $S_k$ for $k=1,...,S$ centered at the atom kernel with $N_k$ electrons on shell $S_k$ and
• $\int_{S_k}\vert\psi (x,t)\vert^2 =N_k$ for all $t$ for $k=1,..,S$,
• $\sum_k^S N_k = N$.
The total wave function $\psi (x,t)$ is thus assumed to be continuously differentiable and the electronic potential of the Hamiltonian acting in $\Omega_j(t)$ is given as the attractive kernel potential together with the repulsive kernel potential resulting from the combined electronic charge distributions $\vert\psi_k\vert^2$ for $k\neq j$, with total electronic repulsion energy
• $\sum_{k\neq j}\int\frac{\vert\psi_k(x,t)\vert^2\vert\psi_k(y,t)\vert^2}{2\vert x-y\vert}dxdy=\sum_{k\neq j}V_k(x)\vert\psi_k(x)\vert^2\, dx$.
Assume now that the electronic repulsion energy is approximately determined by homogenising the $N_k$ electronic wave function $\psi_j$ in each shell $S_k$ into a spherically symmetric "electron cloud" $\Psi_k(x)$ with corresponding potential $W_k(y)$ given by
• $W_k(y)=\int_{\vert x\vert <\vert y\vert}R_k\frac{\vert\Psi_k(x)\vert ^2}{\vert y\vert}\, dx+\int_{\vert x\vert >\vert y\vert}R_k\frac{\vert\Psi_k(x)\vert ^2}{\vert x\vert}\, dx$,
and $R_k(x)=\frac{N_k-1}{N_k}$ for $x\in S_k$ is a reduction factor reflecting non self-repulsion of each electron (and $R_k=1$ else): Of the $N_k$ electrons in shell $S_k$, thus only $N_k-1$ electrons contribute to the value of potential in shell $S_k$ from the electrons in shell $S_k$. We here use the fact that the potential $W(x)$ of a uniform charge distribution on a spherical surface $\{y:\vert y\vert =r\}$ of radius $r$ of total charge $Q$, is equal to $Q/\vert x\vert$ for $\vert x\vert >r$ and $Q/r$ for $\vert x\vert <r$.
Our model then has spherical symmetry and is a 1d free boundary problem in the radius $r=\vert x\vert$ with the free boundary represented by the radii of the shells and the corresponding Hamiltonian is defined by the electronic potentials computed by spherical homogenisation in each shell. The free boundary is determined so that the combined wave function $\psi (x,t)$ is continuously differentiable across the free boundary.
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Chemistry LibreTexts
Wave Function of Multi-electron Atoms
• Page ID
• Unlike hydrogenic atoms, the wavefunctions satisfying Schrödinger's equation for multi-electron atoms cannot be solved analytically. Instead, various techniques are used for giving approximate solutions to the wave functions.
First Approximation.
The wavefunctions of multi-electron atoms can be considered, as a first approximation, to be built up of components, where the combined wavefunction for an atom with k electrons is of the form:
\(\Psi = \Psi(1) \; \Psi(2) \;...\; \Psi(k)\)
Here, \(\Psi(1)\), \(\Psi(2)\), up to \(\Psi(k)\) represent wave functions for the first, second, up to kth electrons. Each \(\Psi(i)\) is considered to be in the form of a wave function for the single electron of the hydrogenic atom subject to the Pauli Exclusion Principle and after making adjustments to account for shielding and penetration.
The Pauli Exclusion Principle allows at most two electrons in any one orbital. This is explained by postulating an additional quantum number for electron spin, ms , which can have values of +1/2 or -1/2. The Dirac theory of quantum mechanics, applied to electron orbitals, more naturally explains this spin magnetic quantum number because the theory goes beyond the assumptions of the Schrödinger equation by also accounting for the relativistic behavior of orbiting electrons.
Shielding occurs because other electrons that are closer to the nucleus shield an electron from the attractive force of the nucleus. Adjusting the hydrogenic orbitals can be done by reducing the value of Z to Zeff. The amount of the reduction depends on which inner orbitals are occupied and how much the orbital being calculated is able to penetrate the shielding. As the amount of shielding increases, Zeff becomes smaller, and the energy levels of the orbital increase. Conversely, as the amount of penetration increases, the shielding is reduced, Zeff becomes bigger, and the energy level of the orbital decreases.
For example, when the 1s orbital is fully occupied by two electrons, the third electron could occupy any of 2s or 2p orbitals, which would have the same energy level in a hydrogenic atom. However, as illustrated in the diagrams below, the 2s orbital concentrates more of its probability near the nucleus than the 2p orbital does. As a result, the 2s orbital penetrates the 1s orbital shielding more than the 2p orbital does. Hence, a third electron will occupy the 2s orbital where it has a slightly lower energy than in the 2p orbital.
The values of Zeff can be calculated by various techniques. Slater's rule is a relatively simple ad hoc method of estimating Zeff for values of n up to 4, that is for electrons in the s, p, d, or f orbitals.
Other approaches
Other approaches for calculating wave functions for multi-electron atoms use numerical methods to make successive approximations to solutions for Schrödinger's equation, using calculation-intensive computer programs. These methods often use some other approximation as a starting point. Two such approaches are the variation methods and perturbation methods.
Even without precise numerical solutions for the wave functions, these concepts can still provide qualitative and conceptual guidance for understanding chemical bonding and other phenomena that are based on electron orbitals.
• Atkins, P., & de Paula, J. (2006). Physical Chemistry for the Life Sciences. New York: W. H. Freeman and Company.
• Ladd, M. (1998). Introduction to Physical Chemistry (3rd ed). Cambridge, UK: Cambridge University Press.
• McMahon, D. (2005). Quantum Mechanics Demystified. NewYork: McGraw-Hill Professional.
• McQuarrie, D. A., & Simon, J. D. (1997). Physical Chemistry: A molecular approach. Sausalito, CA: University Science Books.
• Thanh Hua (UCD) |
3d2c9be5437a169c | Planck constant
(Redirected from Planck's constant)
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File:MaxPlanckWirkungsquantums20050815 CopyrightKaihsuTai.jpg
The Planck constant (denoted ) is a physical constant that is used to describe the sizes of quanta. It plays a central part in the theory of quantum mechanics, and is named after Max Planck, one of the founders of quantum theory. A closely related quantity is the reduced Planck constant (also known as Dirac's constant and denoted , pronounced "h-bar"). The Planck constant is also used in measuring energy emitted as photons, such as in the equation E=hf, where E is energy, h is Planck's constant, and f is frequency.
The Planck constant and the reduced Planck constant are used to describe quantization, a phenomenon occurring in subatomic particles such as electrons and photons in which certain physical properties occur in fixed amounts rather than assuming a continuous range of possible values.
Significance of the size of Planck's constant
Expressed in the SI units of joule seconds (J·s), the Planck constant is one of the smallest constants used in physics. The significance of this is that it reflects the extremely small scales at which quantum mechanical effects are observed, and hence why we are not familiar with quantum physics in our everyday lives in the way that we are with classical physics. Indeed, classical physics can essentially be defined as the limit of quantum mechanics as the Planck constant tends to zero.
In natural units, the Dirac constant is taken as 1 (i.e., the Planck constant is 2·π), as is convenient for describing physics at the atomic scale dominated by quantum effects.
Units, value and symbols
The Planck constant has dimensions of energy multiplied by time, which are also the dimensions of action. In SI units, the Planck constant is expressed in joule seconds (J·s). The dimensions may also be written as momentum times distance (N·m·s), which are also the dimensions of angular momentum. The value of the Planck constant is:
The value of the Dirac constant is:
The figures cited here are the 2006 CODATA-recommended values for the constants and their uncertainties. The 2006 CODATA results were made available in March 2007 and represent the best-known, internationally-accepted values for these constants, based on all data available as of 31 December 2006. New CODATA figures are scheduled to be published approximately every four years.
Unicode reserves codepoints U+210E () for the Planck constant, and U+210F () for the Dirac constant.
More recent values
In October 2005, the National Physical Laboratory reported initial measurements of the Planck constant using a newly improved watt balance. They report a value of:
Origins of Planck's constant
The Planck constant, , was proposed in reference to the problem of black-body radiation. The underlying assumption to Planck's law of black body radiation was that the electromagnetic radiation emitted by a black body could be modeled as a set of harmonic oscillators with quantized energy of the form:
is the quantized energy of the photons of radiation having frequency (Hz) of (nu) or angular frequency (rad/s) of (omega).
This model proved extremely accurate, but it provided an intellectual stumbling block for theoreticians who did not understand where the quantization of energy arose — Planck himself only considered it "a purely formal assumption"Template:Fix/category[citation needed]. This line of questioning helped lead to the formation of quantum mechanics.
In addition to some assumptions underlying the interpretation of certain values in the quantum mechanical formulation, one of the fundamental corner-stones to the entire theory lies in the commutator relationship between the position operator and the momentum operator :
where is the Kronecker delta. For more information, see the mathematical formulation of quantum mechanics.
The Planck constant is used to describe quantization. For instance, the energy (E) carried by a beam of light with constant frequency () can only take on the values
It is sometimes more convenient to use the angular frequency , which gives
Many such "quantization conditions" exist. A particularly interesting condition governs the quantization of angular momentum. Let J be the total angular momentum of a system with rotational invariance, and Jz the angular momentum measured along any given direction. These quantities can only take on the values
Thus, may be said to be the "quantum of angular momentum".
The Planck constant also occurs in statements of Heisenberg's uncertainty principle. Given a large number of particles prepared in the same state, the uncertainty in their position, , and the uncertainty in their momentum (in the same direction), , obey
Dirac constant
The Dirac constant or the "reduced Planck constant", , differs only from the Planck constant by a factor of . The Planck constant is stated in SI units of measurement, joules per hertz, or joules per (cycle per second), while the Dirac constant is the same value stated in joules per (radian per second).
In essence, the Dirac constant is a conversion factor between phase (in radians) and action (in joule-seconds) as seen in the Schrödinger equation. The Planck constant is similarly a conversion factor between phase (in cycles) and action. All other uses of Planck's constant and Dirac's constant follow from that relationship.
See also
1. "Planck constant". 2006 CODATA recommended values. NIST. Retrieved 2007-08-08.
2. "Planck constant in eV s". 2006 CODATA recommended values. NIST. Retrieved 2007-08-08.
• Barrow, John D. (2002). The Constants of Nature; From Alpha to Omega - The Numbers that Encode the Deepest Secrets of the Universe. Pantheon Books. ISBN 0-375-42221-8.
• Robinson, I. A. (2007). "An initial measurement of Planck’s constant using the NPL Mark II watt balance". Metrologia. 44: 427–440. doi:10.1088/0026-1394/44/6/001.
External links
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cfe091e540033ec8 | Photosynthesis – storage of solar energy by plants
We have seen how the body takes in energy and how it uses it. But before we can consume food to obtain the energy stored in it, that energy must have been stored there. This is the result of photosynthesis, which leads us to consider the chloroplast.
Chloroplast structure
Chloroplasts occur inside the cells of plants, like the nucleus or the mitochondria. Like mitochondria, they contain their own simple form of DNA, because, like mitochondria, they originated as bacteria which moved into another cell, felt at home and stayed. Within a double cell membrane, they have a number of closed membranes called thylakoids arranged in stacks, each of which is called a granum. There is fluid inside all these spaces; that inside the membrane and in which the thylakoids are arranged is called the stroma.
Chloroplast structure, from Wikimedia Commons
Chloroplast structure, from Wikimedia Commons
Photosynthesis takes place in two steps: light reactions in the thylakoid membranes and the Calvin cycle in the stroma.
1. The light reactions use energy from sunlight in two ways: to store energy as ATP; and to transfer electrons to form NADPH. Both are passed to the Calvin cycle.
2. The Calvin cycle uses the electrons and ATP plus CO2 from the air to make glucose.
The light reactions thus furnish the energy and the fuel used by the Calvin cycle.
The two steps of photosynthesis, from Openstax College
The two steps of photosynthesis, from Openstax College
Light reactions
The light-reaction phase of photosynthesis is also called the Z-scheme, but since the Z usually is shown lying on its side, it looks much more like an N-scheme. Light reactions take place in three steps.
Photophosphorylation (Z-scheme), by author, after Kratz
Photophosphorylation (Z-scheme), by author, after Kratz (2009)
In steps (1) and (3), called Photosystem II (PII) and Photosystem I (PI),1For historical reasons, photosynthesis II comes before photosynthesis I. energy from light excites an electron in chlorophyll to a higher energy level. Since the most important form of chlorophyll, chlorophyll a, absorbs red and blue light but reflects green, leaves are most often green. Other pigments may absorb light of other frequencies and so give different colors. These other pigments (called the antenna complex) transfer any energy they absorb to the chlorophyll a in what is called the reaction center, which can thus collect energy from light of different wavelengths, extending the sensitivity range of the process. Only in the reaction center are excited electrons passed to the next phase.
Although Photosystem II and Photosystem I are similar in operation, they differ in a number of ways. For one thing, their reaction centers contain different pigments: P680 in PII and P700 in PI. (The P numbers refer to the wavelength in nano-meters of maximum light sensitivity of each pigment.)
In photosynthesis II, light energy serves two purposes.
1. It forces a reaction-center electron to be released to the electron transport chain of the next step, an electron transport chain, like those in mitochondria.
2. It also powers water photolysis, the separation of water molecules into O2, protons and electrons.
All this takes place inside the thylakoid membrane.
Each of the products of step 2 has its own destination. A small part of the oxygen is used by the plant’s mitochondria for energy, the rest is released into the atmosphere where, for instance, we breathe it. The protons serve in the next step. And the electrons replace the electrons lost by chlorophyll in step 1. This process is historically and evolutionarily quite old, having already taken place over 3 Gya in cyanobacteria, where the plentiful source of electrons was water.
Photoloysis, the breakup of water to yield electrons occurs as follows
2 H2O → 4 H+ + 4 e + O2
I.e, four electrons at a time. But P680+ can only receive one electron. A process called the oxygen-evolving process exists which allows this to take place, but unfortunately, it is well beyond the scope of this document. Also, alas, it is not completely understood. If it were, it might enable us to extract hydrogen from water in an energy-efficient way, which could put an end to our energy problems.2It could also completely shake up the world economic and political situation, but that is way beyond the scope of this document.
The electron released by PII then goes through photophosphorylation, an electron transport chain similar to that in mitochondria, but now taking place in the thylakoid membrane of the chloroplast. At each step, some of the electron energy is used to pump protons across the thylakoid membrane. At the end of the chain, the electrochemical gradient of the protons across the membrane serves to turn ATP synthase which converts ADP into ATP by the process of chemiosmosis. So at the end of step 2, we have ATP and a free but weak electron.
PI again uses solar energy to kick an electron up to higher energy where it is released. This time, it can be replaced by the electron leaving the ETC. The electron released by PI has enough energy to go through a process which stores its energy on the electron carrier NADPH, a close relative of our old pal NADH. The solar energy is now stored in the NADPH and the ATP from the ETC and both move to the next step, the Calvin cycle.
In the light reactions, electrons and energy have different fates. Electrons from water wind up in NADPH; solar energy is transferred to ATP. So the overall effect of light reactions is to store solar energy in ATP for use by the plant or in the Calvin cycle, and to energize NADPH for the Calvin cycle. The complete chemical formula for the light reactions is the following.
Calvin cycle
The second step of photosynthesis, the Calvin cycle, takes place in the stroma of the chloroplast. It takes in CO2 and uses the chemical energy produced by the light reactions to make sugar molecules, usually glucose.
The Calvin cycles takes place in three stages, which are indicated in the figure.
The Calvin cycle, from Openstax College
The Calvin cycle, from Openstax College
In stage 1, carbon fixation, the enzyme whose “much-needed nickname” is RuBisCO3Kratz (2009), 197., catalyzes the reaction of CO2 and 5-carbon RuBP into a 6-carbon compound which immediately splits into two 3-carbon compounds called 3-PGA. Then, in the reduction step, ATP and NADPH from the light reaction photosystem I reduce 3-PGA to G3P. On each tour of the cycle, one G3P separates from the cycle and these molecules eventually (at the end of six tours of the cycle) form a carbohydrate molecule, usually glucose (C6H12O6). The other G3P molecule and ATP regenerate RuBP, so the cycle can begin again. So it takes six tours of the Calvin cycle to convert CO2 into glucose. The complete formula is therefore the following.
6 CO2 + 12 (NADPH + H+) → C6H12O6 + 12 NADP+ + 6 H2O
ignoring the energy from ATP going to ADP and Pi.
It is impossible to stress overly much the importance of these reactions. They are essential for life on Earth. Not only is our oxygen-rich atmosphere originally due to photosynthesis by cyanobacteria and stromatolites, the current maintenance of oxygen levels depends on it. And the very energy we run on, as we have seen in this chapter, comes from the glucose made in the Calvin cycle.
This is worth repeating.
• The Calvin cycles takes in CO2 from the air and uses the energy-rich products of the light reactions to form glucose and prepare for the next tour of the cycle. This cycle depends on the enzyme RuBisCO, which therefore is essential to life on planet Earth.
• We and other animals eat the plants – and other animals which have eaten plants. After breakdown of food by digestion, the glucose originating in photosynthesis is used by cellular respiration to provide energy in the form of ATP which powers our muscles, our neurons and other metabolic functions. The waste from this conversion is CO2, which goes back into the Calvin cycle.
• Light reactions use the energy from sunlight to take in water and break it down into O2, protons and electrons. The electrons are energized by light to go through chemiosmosis and form energy-rich products which are passed to the next step, the Calvin cycle.
Notice that CO2 is produced as waste in cellular respiration, then taken in by the Calvin cycle to be reconverted into glucose and O2. This process must remain in equilibrium
Now, on to more physiology subjects, this time about communication.
Notes [ + ]
1. For historical reasons, photosynthesis II comes before photosynthesis I.
2. It could also completely shake up the world economic and political situation, but that is way beyond the scope of this document.
3. Kratz (2009), 197.
DNA expression and regulation– protein synthesis
DNA, RNA and ribosomes, in that order, are essential components in the synthesis of proteins. DNA contains the information necessary not only for reproduction, but also for daily cell growth and maintenance. Messenger RNA carries the information to the ribosomes. With the help of yet another kind of RNA, the ribosomes assemble the proteins. All this depends on gene regulation.
The use of DNA to initiate protein synthesis is called DNA expression.
This sequence of events is summed up in the so-called central dogma of molecular biology, often paraphrased as “DNA makes RNA and RNA makes protein.” More precisely, DNA is transcribed inside the nucleus to make mRNA, which is expelled from the nucleus to the cytoplasm, where it is translated to protein by ribosomes.
DNA –> transcription (nucleus)→ mRNA→ translation (ribosome)→ protein.
The recipe is expressed in “bytes” of three nucleobases; one three-base byte is referred to as a code-word. When transcribed to its complementary form in mRNA, it is called a codon. Since each base can have one of four values (C, G, A or T, in DNA), the codon can take on 64 values.
RNA transcription
The enzyme which does the work of “reading” a gene on the DNA and building a corresponding gene of RNA is called RNA polymerase1In fact, there are several forms of RNA polymerase, but that complexity is well beyond the scope of this document.. There are at least four types of RNA and transcription makes them all. For protein synthesis, the RNA constructed is called mRNA, or messenger RNA. The DNA recipe begins with a sequence called the promoter. RNA polymerase contains a complementary sequence which binds to the promoter and launches transcription. As will soon be seen, transcription is started only if it is allowed by gene regulation. RNA polymerase unwinds a part of the DNA chain and reads code-words, starting with the promoter. As it reads the DNA, it constructs a complementary chain, called pre-mRNA, from nucleotides. It is complementary in the sense that if the DNA contains a C (or A or G or T) then the pre-mRNA contains a G (or U or C or A – remembering that RNA replaces T by U).
The raw materials RNA polymerase uses to construct RNA are nucleoside triphosphates (NTPs). An NTP molecule has two of its phosphate groups which contain a significant amount of energy from ATP. This energy is used to bond the nucleotides together to form RNA.
The RNA polymerase moves along the DNA, unwinding sections as it goes, reads the code words and assembles the appropriate pre-mRNA codons from NTPs. The separated DNA strands recombine in its wake. Eventually, it reaches a transcription-terminator sequence in the DNA and ends transcription. It now has gone through three steps, known as initiation, elongation (of the produced pre-mRNA) and termination. The pre-mRNA then is released into the nucleoplasm.
Splicing mRNA, from Openstax College
Splicing mRNA, from Openstax College
Before leaving the nucleus, the pre-mRNA must be cleaned up. This is needed because DNA contains non-coding, or junk, sequences. The codons which should be kept are called exons (like “expressed”) and those which should be deleted are called introns (like “interrupted”).2I would have preferred for exon to mean “exclude” and intron to mean “include”, but some contrary biologist decided otherwise. He could at least have taken a vote! Small particles called “snurps” (for snRNPs, or small nuclear ribonucleoproteins), made up of RNA and proteins, bind together to form spliceosomes, which remove introns and splice the exons back together again, resulting in a cleaned-up form of mRNA.3Are you wondering how the snurps can recognize the introns an exons? So am I. All I can say is that it is quite complicated and has something to do with methylation of the DNA strands. It is currently not completely understood why there are introns at all, but there are indications that they may be of importance.
The mRNA is then moved out of the nucleus for the next step.
Protein synthesis – translation
After the mRNA leaves the nucleus, it is used to provide the input data for the synthesis of proteins. This takes place on ribosomes.
There are two sorts of ribosomes in eukaryotic cells, depending on their location.
• Free ribosomes float in the cytoplasm and make proteins which will function there.
• Membrane-bound ribosomes are attached to the rough endoplasmic reticulum; they are what makes it look “rough”. Proteins produced there will either form parts of membranes or be released from the cell.
In most cells, most proteins are released into the cytoplasm.
Ribosomes are made of ribosomal RNA, or rRNA (one more kind of RNA), and proteins. They are constructed within the nucleolus as two subunits, which are released through the nuclear pores into the cytoplasm.
In addition to the mRNA and the ribosome subunits, a method Is needed for supplying the appropriate amino acids to be linked by peptide bonds to make up the protein or enzyme being constructed. Enter still one more kind of RNA, transfer RNA, or tRNA.
A molecule of tRNA is a molecule of RNA folded into a double strand with loops which give it a precise 3-dimensional shape. The loop on one end has an anticodon, the function of which is to match its complement codon on mRNA. The other end has a binding site (adenylic acid) for a specific amino acid. So the tRNA is the “dictionary” which converts codons into amino acids4This of course poses the question, where do the tRNA molecules come from? Good question.. A tRNA molecule is “charged” with an amino acid molecule by a tRNA-activating enzyme which uses energy from ATP to covalently bond the appropriate amino acid from molecules in the cytoplasm. Such tRNA molecules, carrying an amino acid, are called aminoacyl tRNA.
The ribosome itself contains three assembly areas or spaces called, in order of occupation, the A-site, the P-site and the E-site. Initially, the ribosome subunits are floating independently in the cytoplasm or attached to the RER. The initiation of translation begins when the small subunit binds at its P-site to the START codon of the mRNA strand. Then the corresponding tRNA (methionine) binds to the START codon and the large ribosome subunit is attached, completing assembly of the ribosome. Now the first tRNA is in the P-site and next mRNA codon in the A-site. The methinone constitutes the beginning of the peptide chain which will become the protein. Then a cycle takes place in which the ribosome reads in the mRNA strand, like computers of my youth read in paper tape, each new codon arriving in A.
Gene translation in the ribosome, from Openstax College
Gene translation in the ribosome, from Openstax College
The process then pursues the elongation stage of translation. The aminoacyl tRNA for the codon in the A-site is carried in, so the first two amino acids are now in the P and A sites. The ribosome then catalyzes the formation of a peptide bond between these two amino acids. The ribosome then moves the mRNA so the P-site amino acid enters the E-site, the A-site one enters the P-site, and a new one enters the A-site. It continues like that until a STOP codon enters the A-site and brings about termination of translation and release of the completed peptide chain.
All these steps of transcription and translation require energy, so protein synthesis is one of the most energetically costly of cell processes. Much of this energy is used to make enzymes essential to the functioning of the cell. Most enzymes are proteins.
Once part of a strand of mRNA has left one ribosome, it can enter another. One strand may actually be in 3 to 10 ribosomes at once, in a different step of translation in each one. Such clusters of ribosomes translating the same mRNA strand are called polyribosomes.
Regulation of gene expression
Every cell in an organism has the same complete genome in its nucleus and so has access to all the same protein “recipes”. But, for example, heart cells should not produce proteins used only by the liver and no cell should produce proteins in quantities beyond what it can use. Cells change over time too: Think of the adaptation to pregnancy or disease.
Cells are specialized and so express different genes: Differential gene expression leads to cell differentiation. Controlling which proteins to express and when is called regulation. Note that this is one more instance of communication in the body, telling genetic machinery what to express when.
Regulation of prokaryotic cells
Regulation in prokaryotic cells is relatively simple, as there is no nucleus. so transcription and translation take place almost in the same place and at the same time. Regulation in prokaryotic cells, though, almost always concerns transcription.
An example from a prokaryotic cell will show how this works – and introduce some new terminology.
The bacterium E. Coli normally uses glucose for energy. But if glucose is absent and lactose is present, it can use the lactose. Bacteria arrange groups of genes to be controlled together into a structure called an operon. The set of proteins necessary for the use of lactose are part of the lac operon. The operon begins with a promoter, which indicates the beginning of the operon and is the site where RNA polymerase binds to begin the transcription. In between the promoter and the set of genes, of which there may be any number, is a sequence called the operator, which is where DNA-binding genes bind to regulate transcription5Look out for the terminology: Sean B. Carroll refers to the operator as a genetic switch, a term we will meet with in the disussion of regulation in eukaryotes.
When no lactose is present, a protein called the lac repressor is bound to the operator and the state of the lac operon is “off”. (Figure.) This is because the repressor blocks access to the rest of the operon.
The gene for the lac repressor is a constitutive gene: It is always expressed because it is the recipe for an essential protein. On the other hand, a regulated gene, is expressed selectively.
Regulation of the lac operon, from Openstax College
Regulation of the lac operon, from Openstax College
In addition to the binding site for the operator, the lac repressor has a second, allosteric, binding site. When lactose is present, an isomer of lactose binds to the allosteric site of the repressor, which causes it to change its form and unbind from the operator. The lac operon is now in the “on” state. This form of regulation is called induction: Lactose is said to be the inducer of the lac operon and acts through the allosteric site of the lac repressor. Transcription now occurs, but slowly. Because some glucose still may be present, it is not certain that the proteins mapped by the lactose-digesting genes are needed. This depends on how much glucose is lacking.
The second part of this process depends on the presence of glucose and is regulated by a second DNA-binding protein, CAP (catabolite activator protein). CAP is also an allosteric protein with one DNA-binding site and one allosteric site which binds to cyclic AMP (cAMP). CAP is only active when it is bound to cAMP. You guessed it, cAMP levels are high when glucose levels are low.6If we go one step back, we see that glucose binds to an allosteric site on the enzyme adenylate cyclase, which makes cAMP from ATP, and disables it. So lack of glucose stimulates production of cAMP, which binds to CAP, which binds to the promoter to enhance synthesis. In that case, cAMP-CAP binds to the promoter and enhances transcription of the genes. So lactose can be considered the “on-off” switch for transcription of genes for lactose-digestion and cAMP-CAP, the “volume control”.
Regulation of eukaryotic cells
In eukaryotic cells, transcription occurs inside the nucleus and translation outside, so mRNA is shuttled across the nuclear membrane in between the two processes. Regulation in eukaryotic cells therefore may take place inside or outside the nucleus or at any step in the expression pathway, including control of access to the gene in the DNA, control of transcription, pre-mRNA processing, mRNA lifetime and translation, and modification of the final proteins. Even the activity levels of enzymes which facilitate expression can be controlled.
Pre-transcription regulation
Inside the nucleus, histones, around which chromatin is wound to make nucleosomes, can wind or unwind to change spacing of the nucleosomes and thereby allow or deny access to genes. This process is a form of epigenetic regulation.7Look out, epigenetic regulation is used for somewhat different notions, too, and they are not all necessarily so. Since histones are positively charged and DNA, negatively, modifying the charge by adding chemical “tags” to either modifies the configuration of the DNA.
Transcription regulation
The most frequent regulation of expression in eukaryotes is during transcription. Control of transcription by the prokaryotic lac operon is a relatively simple process: In order to begin transcription of a gene, RNA polymerase must bind with the gene’s promoter but cannot do so if a lac repressor is bound to the operator region which follows the promoter on the gene.
Eukaryotic gene transcription is regulated similarly, but with more of everything. Instead of a repressor which binds to an operon, eukaryotes have a slew of transcription factors which bind to multiple regulatory sequences. Regulatory sequences, sometimes referred to as switches, may be almost anywhere on the DNA strand, even far from the gene. The existence of multiple switches for each gene allows the gene to be present In different types of cells, but activated selectively in each by different switches.
There are two types of transcription factors.
• general transcription factors affect any gene in all cells and are part of the transcription-initiation complex;
• regulatory transcription factors affect genes specific to the type of cell.
The two types of transcription factors work together with three types of regulatory sequences (transcription-factor binding sites).
• promoter proximal elements are, of course, near the promoter and turn transcription on;
• enhancers are far from the regulated genes or in more than one place and also turn the transcription on;
• silencers are also far away from the regulated genes but turn transcription off.
Activator transcription factors are those which bind to enhancers to promote expression, repressor transcription factors, to silencers to decrease expression.
The promoter in eukaryotic cells is more complex too, The basal promoter begins with the TATA box, recognized by its beginning which contains the seven-nucleotide sequence TATAAAA, followed by a set of transcription-factor binding sites.
The whole set of transcription factors is summed combinatorially to determine whether or how much the gene will be expressed. Selective promotion or inhibition at combinations of these sites can therefore bring about tissue-specific gene expression. Each tissue type may have its own specific enhancer or silencer sequence for the same gene. For instance, the neuron-restrictive silencing element (NRSE) is a repressor which prevents genes from being expressed in any cells which are not neurons. In addition, environmental changes may bring about different gene expression according to current, perhaps temporary needs.
Coactivator proteins bind with general and regulatory transcription factors to form the transcription-initiation complex. RNA polymerase only binds to the transcription-factor complex.
Transcription factors in eukaryotic cells
Transcription factors in eukaryotic cells8Author’s own work
The above figure shows the case of an enhancer bound by activator transcription factors.9The bobby-pin curl is an idealization; DNA shapes are far more complex than that. The enhancer, on the left, originally is quite far from the promoter, until DNA bending causes it to change its shape, allowing the enhancer to come in contact with the promoter and the rest of the transcription-initiation complex.
Since transcription factors are proteins, they too are coded by genes and these genes are regulated in turn by other transcription factors. Eventually, it is the original cell (such as a fertilized egg) plus the environment which start the chain going. Of course, only genes which are present can be influenced; it’s nature and nurture.
Transcription factors and signaling elements coded by some of these genes make up the genetic toolkit, as we will see in a moment.
Splicing regulation
In between transcription and translation, proteins may interfere with spliceosomes to modify splicing of pre-mRNA. Different intron selections can allow different mRNAs to be produced from the same pre-mRNA, a phenomenon known as alternative splicing.
Pre-translation and translation regulation
RNA does not hang around forever, nor should it. Eventually, it is degraded and is no longer functional. So controlling its lifetime is another way of regulating its activity.
Yet another type of RNA, very short-stranded microRNA or miRNA, can bind with complementary mRNA before it is translated and signal that it should be destroyed by the cell. For this purpose, miRNA also associates with RISC (RNA-induced silencing complex).
Other proteins, RNA-binding proteins (RBPs), can bind with the 5′ cap or the 3′ tail of the mRNA and either increase or decrease its stability.
Phosphorylation or attachment of other chemicals to the mRNA protein initiator complex also inhibit translation.
Similar bindings may take place on the protein after translation and modify its stability, lifetime or function.
The development genetic toolkit – what evo devo tells us
Development, meaning embyronic development, is the process in which a genotype, a set of genes, becomes a phenotype, a particular living organism. Mutation works on genes, but natural selection works on phenotypes, the results of development. So evolution and development work hand in gene, so to speak, and the branch of biology which studies them together is called “evo-devo“.
The homeobox is a genetic sequence of DNA about 180 bases long. It is a sequence of DNA, of genetic material. It codes for about 60 amino-acid residues and these proteins are the homeobox domain, or the homeodomain. The reason the homeobox is really special is that it is “conserved” across most species of eukaryotes, meaning they all have similar homeobox sequences. Some such sequences are the same in frogs and mice by up to 59 of 60 base pairs. It is thought that there are about two dozen types of homeodomains, therefore of homeoboxes. The ubiquity of the sequence is fairly astounding in itself. It means that the sequence could not have evolved independently all those many times – think of the number of species concerned. So the homeobox must date back to be on the order of 500 millions years old.
But there’s more. The homeobox is not a gene by itself, but exists within many different, much larger genes – indeed, hundreds of times larger. Since they all contain the homeobox, they are called homeobox genes.
Many animals have a disposition of body parts along an axis, such as the antennae, wings and legs along the body axis of a fruit fly, or the existence or not of ribs along the vertebrae of a vertebrate animal. It turns out that the choice of body part at each segment along the axis is regulated by a transcription factor coded by a single gene – the Hox gene. Hox genes are an example of homeobox genes; they contain the homeobox sequence. These Hox “master” genes control the developmental differentiation of, for instance, a fruit fly’s serially homologous body parts;10The front legs of a cat and our arms are considered homologous body parts. Structures along a body axis, similar but different, are called serially homologous with respect to each other. in simpler terms, its body pattern. They are “master” genes because they determine whether a given part will form or not, leaving the details to genes farther down the chain. But such “detail” genes will not function at all without the “master” gene, which therefore regulates quite a large number of genes. Hox genes are sufficiently similar that introduction of mouse Hox genes into a fly can cause the growth of the indicated organ — in fly format. They also control the very different serial structure of snakes.
Homeotic genes occur in clusters. One more amazing fact is that the genes of a cluster are in the same order as that of the body segments they control. It is sufficient to replace the gene in a given cluster, say at the antenna position on a fruit fly, with another, say a leg gene, and a leg develops at the antenna position on the fly. Since the transcription factors coded by such genes can change the cells they regulate into something else, they are called homeotic11Homeosis is the transformation of one organ into another. transcription factors and their genes are homeotic genes. The protein domain they express is therefore a homeotic domain, Hox, for short.
Other homeobox families also exist, as we will see a few in a moment. Hox genes are just one family of them.
It is remarkable that quite similar homeodomains have been found in almost all animals. Such conservation of homeobox genes across species shows that embryonic development of most animals, fungi and plants is controlled at some level by approximately the same genes. They must have been around since animals diverged from each other over 500 Mya. The original Hox gene was duplicated and then each copy took on slightly different functions. Subsequent duplications and modifications have led to the diversity of animals today. Comparison of the genes can contribute to building at least a partial tree of life.
Because the homeobox genes code for transcription factors, each of which is used in so many organisms in similar ways, the proteins coded by homeobox genes are constituents of what some biologists call the genetic toolkit. It’s like using a common screwdriver to drive screws in different contexts. Just as one screwdriver serves in many contexts, so does each type of homeobox gene. A homeobox gene is therefore in some way a “master” gene. The toolkit is common to almost all animals, with only little variation from one to another. It contains genes not only for transcription factors, but also for various molecules which are signaling elements. They play important roles in embryonic development, or embryogenesis.
In other animals also, the genes exist in clusters, with the gene order in the cluster corresponding to that of the organism’s parts. Different Hox genes, being similar but slightly different, bind to different regulatory sequences on DNA and therefore regulate different genes. One homeobox protein may regulate many genes and a number of homeobox proteins may work together to refine selection. Because of this possibility of multiple binding, a small change in activation of toolkit genes can bring about a large change in the phenotype. So the genetic toolkit may explain development more simply than if all genes had to be specific to each different part, location and development time of an organism.
Toolkit genes themselves have multiple switches. Switches are the means by which a relatively limited set of toolkit genes may be used differently in different regions, or even different animals, or at different times in embryonic development – which furnishes material for evolution.
The existence of different layers of transcription factors also explains how a small genetic change (in a transcription factor) can bring about a relatively important change in the phenotype of the organism.
A specific bodily environment (liver, heart, blood, …) contains some set of organic molecules specific to that environment. These molecules or a sub-set thereof will serve as transcription factors to activate a particular sub-set of the toolkit. In other words, the environment chooses which tools to use.1 The proteins expressed by toolkit genes will activate or suppress expression of body-part proteins at that place and time.
Environmental molecules ==> toolkit proteins ==> body parts
Each arrow indicates that the object to the left switches on expression of the object to the right.
Some terminology helps to understand the evo-devo literature.
• Transcription factors are proteins and so are not on the DNA string, therefore not on the same molecule as the DNA which is regulated. They therefore are called trans-acting regulatory elements (TRE).12In Latin, “cis” means “this side of” and “trans” means “the other side of”. Think of cis-Alpine (this side of the Alps) and trans-Alpine.
• Switches are on the same string of DNA as the regulated gene and are called cis-acting regulatory elements (CRE).
So one can say that TREs bind to CREs to regulate gene expression. Got that?
The following table lists just a few of the homeobox families and the organism components they regulate. As is clear, they regulate quite different structures.
Protein name Penotype regulated
Hox body regions (e.g., head, thorax or abdomen)
Pax-6 eyes
Distal-less (Dll) limbs
Sonic organogenesis (tissue patterns)
Ulrabithorax (Ubx) represses insect wing formation
In all animals, there exist similar gene sequences corresponding to protein domains which are transcription factors for that animal’s version of some phenotype. A Pax-6 gene from a mouse makes an eye form in a fruit fly – a fruit-fly eye, not a mouse eye.
Cell differentiation
Stem cells are those which may split and form any kind of cell.13Usually to make an identical stem cell and another cell, which may or may not be a stem cell. But once a specific type of cell is made, it can only do certain things. This is because it no longer has access to the entire recipe book (genome), but only those recipes which it needs. The cell is then said to be differentiated and the process for making it is differential gene expression. Such gene regulation or differentiation depends on the cell’s environment. We have seen an example where the presence of lactose induces the expression of the lac operon.
Gene regulation can fill a book. And cell differentiation can fill another.
Continue with cell division and the cell cycle.
Notes [ + ]
1. In fact, there are several forms of RNA polymerase, but that complexity is well beyond the scope of this document.
4. This of course poses the question, where do the tRNA molecules come from? Good question.
7. Look out, epigenetic regulation is used for somewhat different notions, too, and they are not all necessarily so.
8. Author’s own work
9. The bobby-pin curl is an idealization; DNA shapes are far more complex than that.
11. Homeosis is the transformation of one organ into another.
13. Usually to make an identical stem cell and another cell, which may or may not be a stem cell.
Some basic biochemistry
Understanding physiology and neuroscience requires knowing a certain amount of biochemistry. Most of the building blocks of our bodies are macromolecules composed of proteins (long chains of amino acids), polysaccharides (carbohydrates), lipids (fats) and nucleic acids (which make up DNA and RNA).
Amino acids and proteins
Amino acids are the basic building blocks for proteins. In a way, they are quite simple, all being variations on the same basic formula.
Common formula for amino acids, by "GyassineMrabetTalk" via Wikimedia Commons
Common formula for amino acids, by “GyassineMrabetTalk” via Wikimedia Commons
Each amino acid consists of a central carbon atom, an amino group (NH3+), a carboxyl group (COO) and a variable group, designated by the letter “R”, for residue. In the figure, the third H in NH3 has been transferred to one O on the COO to make COOH and balance out the total charge.
The complete set of amino acids comprises only 21 acids and is shown in the following figure. It is often stated that there are only 20 amino acids, in which case selenocystein, which occurs rarely, has been omitted.
The amino acids, by Dan Cojocari via Wikimedia Commons
Amino acids are the basic building blocks of proteins. A protein is a polypeptide, that is a polymer (a chain of linked subunits) formed by condensation (ejection of water molecules) so as to link the amino acids by peptide bonds. Schematically, it looks like the example in the following figure, which shows the OH on the left combining with an H+ on the right to make a water molecule and leave the two amino acids connected by a peptide bond. Actually, the process is not that direct, but goes through several enzyme-assisted steps in order to achieve the peptide bond. (We’ll get back to enzymes shortly.)
Peptide formation by condensation of two amino acids, by "GyassineMrabet" via Wikimedia Commons
Peptide formation by condensation of two amino acids, by “GyassineMrabet” via Wikimedia Commons
The bonding properties of proteins depend largely on their shape. The shape of the protein depends first on the sequence of amino acids, which constitutes the protein’s primary structure. The polypeptide chain may then coil up into a secondary structure called the alpha helix. The R groups may interact among themselves, bringing about a change in the 3-dimensional shape, or conformation, the tertiary structure. Different polypeptide chains then may bind to form a quaternary structure. In this way, proteins can take on very intricate shapes.
Four hierarchical structures of hemoglobin, by OpenStax College via Wikimedia Commons
Symmetry does not seem to be well respected in biology. A helical protein with a right-handed twist can not generally be substituted for one with a left-handed twist: It just will not work the same way. This differing of the two versions is called chirality.
3D structure of myoglobin protein. Alpha helices are turquoise. By AzaToth via Wikimedia Commons
Proteins may be enormously long polypeptide chains.
Enzymes, which are usually proteins1RNA can also function as an enzyme and it is not a protein., serve as organic catalysts, meaning that they help to bring about reactions that otherwise would not happen or would happen far too slowly. They only bring about reactions which are energetically possible but which nevertheless need a “push” to get started. Enzymes provide the push by lowering the activation energy of the reaction. Complete equations for different reactions would include the enzymes on both sides, but they are usually omitted. Every physiological process in the body depends on enzymes. Enzymes themselves only work under rather strict conditions of temperature and acidity. If the pH or temperature is not just right, the enzymes will not work, the reactions will not take place and the organism will suffer. The names of enzymes generally end in -ase, for example, lactase.
An enzyme can do its work because of its shape. It folds itself so as to form a pocket called the active site. A molecule which fits into the active site is called a substrate. The enzyme can then usher the substrate through the reaction. This “lock and key” model of enzyme-subtrate interaction is refined further in the induced-fit model, wherein dynamic modifications in the enzyme’s structure enable It to exactly fit the substrate, like a glove stretching to fit a hand.
The body can regulate the rate of such reactions by regulating the efficiency of the enzymes which catalyze it. One way to do this is to have a molecule similar in shape to the substrate and use it to block the active site. Or a molecule can bind to what is called an allosteric site on the enzyme, meaning a site which is not the active site. Binding to such a site changes the shape of the molecule and thereby renders it ineffective for binding with its usual substrate. If the enzyme catalyzes a reaction too much, so that there is an excedent of end products, the end products themselves may attach to an allosteric site and block further reactions, resulting in a feedback mechanism which reduces the rate of the reaction.
Reactions catalyzed by enzymes generally take place in a number of small steps rather than all at once. This has a double advantage:
• At each step, the enzyme can bring the reactants together, reducing the activation energy, the amount of energy needed for the reaction to begin.
• The energy output from each small step will not be so much as to harm the cell.
The sum of all the small steps is referred to as a metabolic pathway.
Carbohydrates are molecules composed of carbon, hydrogen and oxygen, usually with the latter two elements in the same relative amounts as in water. So a generic “carb” could be represented by the formula
Carbohydrates are saccharides, or sugars, and referred to as monosaccharides or polysaccharides, depending on the length of the molecule.
The most important monosaccharides in the body are: two, ribose and deoxyribose, based on rings of five carbon atoms (pentoses) and three, glucose, fructose and galactose, based on rings of six carbon atoms (hexoses).2In fact, glucose, galactose and fructose all have the same formula, C6H12O6, but differ in their conformations. Similarly, ribose and deoxyribose share the same formula, C5H10O5, but different conformations.
The five common monosaccharides, from Openstax College
The five common monosaccharides, from Openstax College
Saccharides formed from two monosaccharides are called disaccharides. Important ones for the human body are sucrose (table sugar), lactose (milk sugar) and maltose (malt sugar).
Polysaccharides may contain thousands of monosaccharides. Common ones are starches (polymers of glucose found in plant foods), glycogen (a polymer of glucose used for storage in the body) and cellulose (“fiber”, found in the cell walls of plants).
We will be considering the importance of carbohydrates in the body’s production of energy from food.
Lipids are mostly hydrocarbons with very little oxygen and so forming only non-polar C-C or C-H bonds, making them hydrophobic. They consist of triglycerides, phospholipids, cholesterol and small quantities of other substances. Lipids are necessary for the formation of cell membranes and for other functions within cells.
The commonest form of lipid (“fat”) in the body is triglyceride, consisting of a glycerol nucleus covalently bonded to the ends of three fatty-acid chains, long hydrocarbon chains terminated at one end by a carboxyl group (COO-) and at the other by a methyl group (CH3). The C=O link to the glycerol is an ester linkage.
Triglyceride structure, with three fatty acids (orange background) attached to glycerol (pink), adapted from Openstax College
Fatty acids may be saturated or unsaturated, meaning saturated in bonds with hydrogen. A saturated fatty acid has only single bonds between carbon molecules, leaving two bonds free to connect with hydrogen. An unsaturated acid may have a double bond between carbons, meaning each one can only bond with one hydrogen. The double bonds between carbons may change the shape of the fatty acid.
Saturated and unsaturated fatty acids, from Openstax college
Saturated and unsaturated fatty acids, from Openstax college
Saturated fatty acids pack tighter and so exist generally as semi-solid substances called fats. Unsaturated fatty acids pack more loosely (because of the kinks) and are the constituents of more liquid oils
It is currently thought3Or, at least, recently. It’s hard to keep up with what nutritionists tell us. that saturated fats lead to increased risk of heart disease, relative to unsaturated fats. The worst, though, is thought to be so-called trans fats.4The word trans comes from biochemistry and indicates functional groups on opposite sides of the carbon chain. In order to ensure longer shelf life, food producers sometimes convert unsaturated fats into saturated ones by hydrogenation, the addition of hydrogen atoms5The first such hydrogenated shortening was marketed under the brand name Crisco. It was partially hydrogenated cottonseed oil.. Trans fats are those which have only been partially hydrogenated6In more detail, a cis double bond is converted to a trans double bond, hence the trans.. On the other hand, there is evidence that omega-3 unsaturated fats are effective in reducing the risk of heart disease and perhaps beneficial in other ways. They are called omega-3 because the word “omega” is used in biochemistry to refer to the methyl end of the fatty acid chain and the double carbon bond is the third from that end.
Phospholipids are similar to triglycerides, but the glycerol is attached to only two fatty acids, the third being replaced by a “head group” containing phosphate.
Phospholipid structure, from Openstax College, via Wikimedia Commons
Phospholipid structure, from Openstax College, via Wikimedia Commons
The phosphate “head” is negatively charged and therefore hydrophilic but the fatty acid tails are hydrophobic, so the molecule is ampiphatic (as discussed in the chemistry chapter) and forms micelles or membranes in an aqueous environment. Of major importance for life, phospholipids are the principal component of cell membranes.
Just as proteins are polymers formed from chains of amino acids, nucleic acids – DNA and RNA – are polymers made up of chains of linked nucleotides. A nucleotide is composed of a pentose (five-carbon) sugar molecule like deoxyribose (which gives the “D” in DNA) or ribose (in RNA), a nitrogenous base (or nucleobase) and one phosphate group.7Common usage employs the term nucleotide for those with more than one phosphate group. Different nucleotides contain different bases.
Nucleotides, from Openstax College
Nucleotides, from Openstax College
There are five possible nucleobases in two groups:
• pyrimidines – cytosine, thymine and uracil, with a single-ring structure; and
• purines – adenine and guanine, with two rings and therefore two nitrogen atoms.
Another, very special nucleotide is adenosine monophosphate, or AMP. When a second phosphate group is added to AMP, it makes ADP (adenosine diphosphate); addition of a third phosphate group makes adenosine triphosphate, or ATP, the “energy currency” or energy carrier in cells of all living organisms. Like all nucleotides, AMP consists of a nitrogenous base attached to a pentose sugar attached to a phosphate group; in this case, the nitrogenous base is adenine and the pentose sugar is ribose. It takes energy to add a Pi (phosphate) to make ADP or a Pi to ADP to make ATP. This energy is stored in the ATP molecule as chemical potential energy and can be recovered later to do useful biological work, such as to flex muscles (including heart muscles), make blood flow, power peristaltic movement of the intestines or permit action potentials in neurons. We will talk much more of this in the next chapter.
A nucleotide without the phosphate group is called a nucleoside, so ATP may also be referred to as a nucleoside triphosphate. Nucleoside triphosphates are the raw materials for building RNA molecules.
ATP and ADP, from Openstax College
ATP and ADP, from Openstax College
Nucleic acids – DNA and RNA
The nucleic acids, DNA and RNA, are assembled from nucleotides. They differ in three ways:
• DNA, deoxyribonucleic acid, contains deoxyribose as its sugar; RNA, ribonucleic acid, contains ribose.
• The “allowed” nucleobases for DNA are cytosine (referred to in this context as C), guanine (G), adenine (A), and thymine (T); in RNA, T is replaced by uracil (U).
• DNA molecules form a double strand; RNA, a single one.
The IUPAC (International Union of Pure and Applied Chemistry) has a rather hairy set of rules for numbering carbon atoms in organic compounds. In the case of the sugar in a nucleotide, the 1′ carbon (one-prime, prime to denote sugar) is the one attached to the nitrogenous base. The count moves around the ring away from the oxygen apex.
Nucleic acids are formed by dehydration (or condensation, removal of a water molecule) between a pentose sugar of one molecule (the 3′ carbon) with the phosphate (on the 5′ carbon of the pentose) of another. The result is called a phosphodiester bond. The chain is thus held together by a sugar-phosphate backbone, independently of attached nucleobases, which protrude out from the chain.
DNA chains form double strands due to hydrogen bonds between nucleobases on each chain, with C bonding only to G and A only to T. So a purine (A or G) is always bonded to a pyrimidine (C, T or U). The result forms a double helix, like a twisted ladder. Note from the preceding figure that there are three hydrogen bonds between guanine and cytosine, but only two between adenine and thymine.
The combination of two DNA strands into a double helix offers the advantage that the nucleobases are not sticking out into the cytoplasm where they may be more easily mutated. Rather, the bases of the two strands are “holding hands” (through hydrogen bonds) to protect each other from mutation. This increased security may explain why DNA, which stores genetic information, forms a double helix, but RNA does not.
Some detail: The nucleic acid strand is polar, i.e., the ends are not the same. One end has a phosphate group attached to the 5′ carbon of the sugar; this is called the 5′ end. The other end has a hydroxyl group (OH) attached to the 3′ carbon of the sugar, so this is called the 3′ end. When combining into a double helix, the ends are reversed, i.e., the 3′ end of one is opposite the 5′ end of the other.
Since the total length of all the DNA strands in a human nucleus would equal 2-3 m, it must be compacted in order to fit into the nucleus. The helical strand is wrapped around histone proteins to form nucleosomes. The string of nucleosomes is then twisted and re-twisted, like a piece of cord, until it forms a compact string called chromatin. The chromatin will be used to form chromosomes (only) when needed for reproduction.
DNA compaction, from Openstax College
DNA compaction, from Openstax College
Oxidation-reduction and electron carriers
The concept of oxidation and reduction is essential to biochemistry, so let’s beat on it a while. Actually, it also is important to other domains of chemistry. Oxidation and reduction occur together in oxidation-reduction, or redox, reactions.
An entity which loses electrons is said to be oxidized; if it gains electrons, it is reduced. Think of its charge, which becomes more negative as it gains an electron. Oxygen likes to gain electrons, so when it pinches one from another substance, that substance is oxidized. A simple example is Na and Cl going together:
Na + Cl → Na+ + Cl
The Na loses an electron and becomes positive; it is an electron donor and is oxidized. The Cl gains an electron, becoming negative, and is reduced.
A substance which is oxidized, i.e., gives up electrons, is an electron donor or reducing agent or reductant. One which is reduced, i.e., gains electrons, is an electron receptor or oxidizing agent or oxidant. Schematically, we can write
donor (reductant) <—> e- + electron receptor (oxidant)
where the reductant and oxidant together are said to constitute a conjugate redox pair.
The term oxidation is understood perhaps most clearly in a reaction like the rusting (oxidation) of copper:
2 Cu + O2 → 2 CuO
One can see that:
• At the same time as oxygen receives electrons from Cu and so is reduced,
• copper, in releasing electrons to oxygen, adds on oxygen to become copper oxide (rust).
So we can see that adding oxygen is also oxidation and releasing it is reduction.
H2 + F2 → 2 HF
which is perhaps not easy to recognize as an oxidation of hydrogen. But consider the two half-reactions, the obvious oxidation part
H2 → 2 H+ + 2 e
and the reduction part
F2 + 2 e → 2 F
Put them together to get
H2 + F2 → 2 H+ + 2 F → 2 HF
But now, look at combustion (oxidation) of propane. It can be complete combustion
C3H8 + 5 O2 → 3 CO2 + 4 H2O
or partial combusion
C3H8 + 2 O2 → 3 C + 4 H2O.
Carbon is obviously oxidized in complete combustion, since it adds oxygen. Partial combustion, even though it does not add oxygen, is still considered oxidation. So we can add:
• Releasing hydrogen is also a sign of oxidation.
It’s inverse therefore must be reduction.
A less obvious example is oxidation of copper oxide:
2 Cu2O + O2 → 4CuO.
The thing to notice here is that the first oxide of copper is cuprous oxide, where copper has a valency state of +1. On the right, though, it has valency state of +2 and so this is cupric oxide. Copper has changed valency states and In so doing has given up an electron, which indicates oxidation.
Chemists also talk about oxidation-reduction in term of oxidation states, but those are complicated too, so we will ignore them. Finally oxidation-reduction criteria can bet represented in the following table.
redox process electron oxygen hydrogen
oxidation release add release
reduction add release add
Getting back to biochemistry, more interesting examples, which are important in cellular respiration, are those of the coenzymes8A coenzyme is a non-protein compound that is necessary for the functioning of an enzyme. Enzymes are macromolecular catalysts, most of which are proteins. nicotinamide adenine dinucleotide and flavin adenine dinucleotide, better and more simply known as NAD and FAD. These two molecules are electron carriers and they pick up and leave off their electrons through redox reactions. If a reaction such as
C6H12O6 + 6O2 → 6CO2 + 6H2O
is allowed to take place all at once, it releases a useless and dangerous amount of energy. So the reaction is broken up into intermediate steps with these cofactors as intermediate oxidizing and reducing substances. Their oxidized forms are NAD+ and FAD and they are reduced to NADH and FADH2.
NAD is a dinucleotide, meaning it is composed of two nucleotides, which are joined by a phosphate group. One nucleotide has an adenine base, the other, nicotinamide.
NAD molecule by "NEUROtiker" via Wikimedia Commons
NAD molecule by “NEUROtiker” via Wikimedia Commons
During cellular respiration (explained later), a molecule referred to as the substrate gives up two H atoms, and so is oxidized, bringing about the reduction of NAD+ in the following way, where R means “residue” and indicates a substrate:
RH2 + NAD+ → NADH + H+ + R
Ignoring R on both sides
NAD+ + 2H → NADH + H+
or, in more detail,
NAD+ + 2H → NAD+ + H + H+ → NADH + H+
which shows that one of the H atoms is in the form of hydride, H, with two electrons.The NAD+ absorbs the hydride, equivalent to two electrons electrons and one proton, thereby gaining electrons and so being reduced to NADH. (Remember, NAD+ and NADH are abbreviations, not chemical forumulas.) In a later step, both the H atoms will be used for energy transfer and the NADH will give up two electrons and so be re-oxidized to NAD+. In this way, NAD transports electrons from one reaction to another.
NAD oxidation-reduction by Fvasconcelllos via Wikimedia Commons
The equivalent formula for the reduction of FAD goes in two steps:
FAD + e + H+ → FADH
FADH + e + H+ → FADH2
to make
FAD + 2H → FAD + H + H+ → FADH2
which shows that FAD is reduced to FADH2 since it adds both electrons and hydrogen.
Now we are ready to look at cells, the basic units of life.
Notes [ + ]
1. RNA can also function as an enzyme and it is not a protein.
3. Or, at least, recently. It’s hard to keep up with what nutritionists tell us.
4. The word trans comes from biochemistry and indicates functional groups on opposite sides of the carbon chain.
5. The first such hydrogenated shortening was marketed under the brand name Crisco. It was partially hydrogenated cottonseed oil.
6. In more detail, a cis double bond is converted to a trans double bond, hence the trans.
7. Common usage employs the term nucleotide for those with more than one phosphate group.
What biochemistry and cellular biology tell us
We have seen how the universe grew from a tiny point to become the enormous – probably infinite – place we see about us. We have focused on a small part of this huge entity and have seen how our solar system has formed and then our planet; how the Earth evolved to reach its current – but temporary – state of support of life; when life was born and how it evolved from bacteria to plants and marine creatures, then land creatures like dinosaurs, then mammals and primates and – currently – us.
So now what? Well, there’s us. But a complete study of that subject is well beyond the domain of this document, so let’s concentrate on a limited subset of it. As a former physicist and informaticien, and so naturally interested in energy and communications, I will emphasize those two threads in studying the human body. That, at least, is the goal. This route should lead to the ultimate and most subjective-seeming domain, cognitive science – the study of the brain.
We must start small, though, with cells, as all else follows from them. And in order to understand them, we need to know
Osmosis and buffering
Diffusion and osmosis
Collective or intensive properties like pressure or boiling and melting points, which are independent of the amount of a substance, are called colligative properties. The concentration of a solute is such a property. A solution wants to have the same concentration everywhere, as this represents the state of highest entropy (randomness). So in case of non-equilibrium, a solute will migrate from any region of higher concentration to regions of lower concentration — just like heat energy flows from hotter to cooler, and for similar reasons. When both regions are mixed and at the same concentration, the result is less ordered and so of higher entropy. This is diffusion.
Another very important colligative property is osmotic pressure. This is only a bit tricky to understand.
Normally, one expects a solute to diffuse from a region of higher concentration to one of lower concentration in order to bring about equal concentrations of the solute. But if the two regions are separated by a membrane which the solute cannot cross but the water can, then the opposite happens. Water flows from the region of lower concentration, i.e., where there is less solute, to the region of higher concentration, which has the effect of diluting the latter and lowering its concentration. At the same time, the solute concentration on the other (source) side goes up. This process is osmosis. The force or pressure driving the water across the membrane is called osmotic pressure.
So, in diffusion, the solute migrates; in osmosis, the solute cannot cross the membrane, so water migrates.
The concentration of solute depends not on its mass but only on the number of atoms or ions.
If the membrane is a cell membrane, then water flows into or out of the cell, depending on the solute concentration inside and outside. Cells usually have a higher solute concentration of biomolecules inside than out, which drives water into the cells. If unchecked, the inflow of water could cause the cell to expand until it exploded, but nature has come up with mechanisms to prevent this catastrophe, including reinforcement of cell walls and pumps to remove water from the cell.
In plants, osmotic pressure stiffens cells with reinforced cell walls, giving the plant rigidity to support it standing up. The opposite thing happens when a salad leaf wilts.
Buffering — acids and bases
Water is naturally somewhat ionized.
Water ionization
Auto-ionization of water, by Cdang via Wikipedia
There are no free protons in water (even though we often will write them as such), hence the hydronium ion, H3O+, with an extra proton. An acid is defined as a proton donor (furnishes H+) and a base as a proton acceptor (consumes H+), so it can be seen that water is weakly both: H3O+ is a donor and OH is an acceptor. The degree of acidity is frequently indicated by the pH value, where
pH = log(1/[H+]) = -log([H+])
where [H+] is the concentration of H+ in moles per liter.1A mole is the mass of a substance contianing the same number of fundamental units (atoms, molecules, etc.) as there are atoms in 12.000 g of 12C. This number is 6.023×1023, which is called Avogadro’s number, designated by NA. Water at 25°C has a pH of 7; ph < 7 means more H+ and therefore more acidic; pH > 7 means basic. Like all such chemical transformations, there is an equilibrium point for the above reactions. This is also true for any other weak acid dissolved in water. Consider acetic acid,
which occurs in an equilibrium state of acetic acid itself (an acid, therefore a proton donor) and CH3OOO (a base, or proton acceptor). These two substances constitute a conjugate acid-base pair. When this weak acid is dissolved in water, two equilibria must be established at the same time, for water and for acetic acid, here represented simply as HAc.
H2O <-> H+ + OH
HAc <-> H+ + Ac
Now if we add a small quantity of a base, say NaOH, to this solution, the base will decompose into Na+ and OH-, the latter a proton acceptor or base. This will change the pH of the solution. But the equilibrium of HAc will adjust so as to decrease the pH and the basicity. The resulting overall increase in basicity will be, in the best of cases, less than expected just considering the addition of a small quantity of strong base. Seen from the point of view of radicals, the OH- from the strong base will combine with some of the protons from the water and acetic acid. But then the acetic acid will be out of equilibrium, so it will produce more free protons to re-establish its equilibrium, thereby attenuating the effects of the added NaOH. A similar but opposite mechanism acts to maintain pH if a small quantity of strong acid is added.
This ability to reduce induced acidity is called buffering. A buffer is an aqueous system which resists changes in acidity from a small amount of added base or acid. It is composed of a weak acid and its conjugate base. It is important as the mechanism by which living beings adjust the acidity of cells. If body acidity is not within rather strict limits, enzymes will not function and so neither will we. The body uses a buffer system based on the conjugate pair carbonic acid and bicarbonate:
H2CO3 <-> H+ + HCO3
If blood acidity starts to become too high, bicarbonate leaps in and absorbs protons. If it becomes too low, carbonic acid supplies them.2It’s really a tad more complex because of another equilibrium: H2C03 ↔ CO2 + H2O. See Lehninger, 63. This is one of many regulative mechanisms the body has for maintaining the proper equilibrium of certain solutions and processes needed by the body in order to stay alive. We will see more.
The global water cycle
Let us briefly leave the microscopic considerations of chemistry and look at water on the scale of the Earth. Water circulates through the ground, streams, oceans and lakes and the atmosphere in what is called the water cycle.
The water cycle, from USGS
This is just one of a number of transformational processes which assure the distribution of an essential component of life on Earth. The diagram is pretty much self-explanatory.
That’s it for the introductory material. Now let’s look at the history of it all. That starts in the past. Way back in the past, about 13.7 billion years ago (Gya).
Notes [ + ]
2. It’s really a tad more complex because of another equilibrium: H2C03 ↔ CO2 + H2O. See Lehninger, 63.
Cheat sheet
Some generally useful information you may want to look up occasionally.
Geological time scale, eons, eras, periods and epochs
Geological time scale and
Types of hominins
Timeline and grouping of principal fossil hominid species
Biological species classification
Classification of modern humans and house cats, after Wikipedia
Classification of modern humans and house cats, after Wikipedia
The periodic table of the elements
Periodic table of the elements
Particles of the standard model
Standard Model particle zoo
Hominoid clades
Hominoid families with dates.
Hominoid families with dates.
Phylogenetic tree
Phylogenetic tree By MPF [Public domain], via Wikimedia Commons
Phylogenetic tree By MPF [Public domain], via Wikimedia Commons
Now we are ready to understand how it is that carbon is such a versatile element. It is at the basis of all organic chemistry and, in particular, biochemistry. The functioning of all living things depends on water and on the versatility of the carbon atom.
We saw that the carbon atom’s electron-shell configuration was
12C: 1s22s22p2
so it has four electrons in its valence shell (n=2). That enables it to share its four electrons with four others from other atoms. The bonds tend to be equally spaced around the carbon atom in the form of a tetrahedron, like those little creamer packets you get in cheap restaurants. For instance, a carbon atom can bond with four hydrogens, sharing each of its four valence electrons with one hydrogen, so each hydrogen has two and the carbon has eight and everybody is happy. This is called methane and looks like this.
"Methane-2D-stereo" by SVG version by Patricia.fidi - Own work. Licensed under Public Domain via Wikimedia Commons.
Methane molecule, CH4 by Patricia.fidi via Wikimedia Commons.
You should see one of the lower-right-hand hydrogens as pointing up out of the page; the other, down into it. The angles between any two adjacent connecting lines (which of course are only imagined by us) are about 109.5°. Carbon’s versatility in binding is illustrated by the examples in this diagram.
Versatiliy of carbon bonding, after Lehninger.
The dots represent valence electrons and the right-hand column is another way of looking at the product in terms of bonds rather than electrons. Each line between atoms is a shared pair of electrons. Note the double and triple inter-carbon bonds in the last two examples. This large number of ways of bonding is the key to carbon’s versatility. In fact, compared to the huge number of such molecules possible, only a relatively small number of the same biomolecules occur in living organisms. This is the first example we see of nature using the same set of techniques or tools all over the biosphere.
Single bonds between carbons also exist, of course, and have the particular advantage that the carbons and whatever is bonded to them can rotate around the axis linking the two carbons. This is more important than one might think. It turns out that some proteins function differently in their left-handed and right-handed versions. Since rotation can change the shape of the molecule, this enables biomolecules with hundreds of atoms to take on specific shapes with definite mechanical or fluid properties. (We will see some of this in the biochemistry chapter.)
The importance of water is not just because we drink it. Let’s go look at that.
Water is tremendously important to us if only because around 70% of the surface of the globe is covered with it. Each of us is 55-75% water (by weight) and life most likely arose in water. Two properties of water are of fundamental importance for biochemistry and, therefore, for life.
• the attractive force between water molecules and
• the tendency of water to ionize slightly.
Polarization and hydrogen bonds
As we saw, the electron configuration of oxygen’s eight electrons is:
16O: 1s22s22p4
So it needs two more electrons in order to fill its valence shell. As everybody knows, it bonds with two atoms of hydrogen to make H2O. Each hydrogen atom shares its electron with the oxygen, making two covalent bonds. The oxygen atom now has the desired eight electrons in its valence shell. The resulting arrangement is triangular.
Oxygen is more electronegative1Electronegativity depends on the number of electrons and on the distance of the valence electrons from the nucleus. than hydrogen, meaning it has a stronger attraction for electrons, so the electrons spend more time in the vicinity of the oxygen, making that end of the molecule slightly more negative. The molecule is said to be polarized.
Since one end of the molecule is more negative than the other, the negative end of one molecule is electrostatically attracted to the positive end of another and this forms a weak bond called a hydrogen bond. In this image, one sees the proposed tetrahedral form of the molecule as well as the hyrdrogen bonds between molecules.
Model of hydrogen bonds between water molecules, from Wikimedia Commons
Hydrogen bonds are strongest when the electrostatic interaction of the participant atoms is maximized, as shown in the above figure. This directionality is responsible for the geometric structure of hydrogen-bonded molecules into crystals.
Hydrogen bonds do not only occur in water. They also form between an electronegative atom and a hydrogen atom covalently bonded to another electronegative atom, be it the same or different.
Base pair GC” by YikrazuulOwn work. Licensed under Public Domain via Wikimedia Commons.
Hydrogen bonds are much weaker than covalent bonds, typically on the order of a twentieth. But when there are many of them, their combined strength can be great indeed. A striking example is DNA, in which the opposing strands are held together by hydrogen bonds between the bases, as in the preceding figure. But more on that later.
If the molecules are rushing about (as in water), they are relatively independent and the substance is a liquid. Hydrogen bonds are constantly formed and broken, forming so-called “flickering clusters”. Heat them some more and they separate entirely and the water becomes a gas — water vapor. The hydrogen bonds between molecules hold them together pretty well, though, and this accounts for the rather high boiling temperature of water. Chill them down to a temperature where they do not move much any more and the hydrogen bonds assemble the molecules into a solid lattice or crystal — ice.
“Hex ice” by NIMSoffice (talk). via Wikipedia Commons
Ionization, hydrophobic and hydrophilic molecules
Because of its polarization, water can pull apart polar molecules, such as table salt, NaCl, where the positive sodium Na+ is attracted by the negative end of the water molecule and the negative chlorine Cl by the positive end. This is what makes water a good solvent. One can see the advantage of this from another angle. Remember entropy? Nature wants higher entropy, meaning more disorder. But NaCl forms a highly ordered crystal structure. When the molecules are pulled apart in water, a more disordered state is achieved and entropy increases. Voilà!
On the other hand, non-polar molecules are not soluble. They are called hydrophobic, because they do not “like” water. NaCl likes it and so is called hydrophilic. This has some amazing and important consequences.
The behavior of solvents in aqueous solutions is a very important subject in biochemistry — and a fairly vast one. Let us look at one interesting and essential type of compound: Ampiphatic compounds have some regions that are polar or charged, therefore hydrophilic, and others that are not polar or charged and so are hydrophobic. In the figure below, we consider molecules illustrated as having a green hydrophilic head and long, yellow hydrophobic tails. When they are dissolved in water, the hydrophobic parts flee the water and tend to group together (like people grouped together facing outwards in the midst of a pack of threatening wolves), leaving the hydrophilic parts on the outside turned towards the water. The result is a spherical blob called a micelle.
Micelle scheme-en” by SuperManuOwn work. Licensed under CC BY-SA 3.0 via Wikimedia Commons.
One can understand this from thermodynamics, too. The water molecules are highly ordered around the hydrophobic parts of the ampiphatic molecule. Hiding these on the interior of the micelle reduces the ordering and therefore represents a state of higher entropy.2Lehninger, 49.
And there is another possibility. Think of the micelle opened up, like an orange, and flattened out and another one put alongside it, so that the hydrophobic ends are against each other and isolated from the water by the hydrophilic ends on the outside, as in part 1 of this diagram.
Lipid bilayer and micelle by Stephen Gilbert via Wikipedia Commons
This ampiphatic substance could be a lipid (organic fat), in which case this is a lipid bilayer, which is what forms cell membranes. So we are ready to start looking at cells in the physiology chapter. And all that is due to electrostatics, QM and thermodynamics — it’s all simple physics.
There are a couple more, slightly more complex, attributes of water we should know about. They are the important ideas of osmosis and buffering.
Notes [ + ]
1. Electronegativity depends on the number of electrons and on the distance of the valence electrons from the nucleus.
2. Lehninger, 49.
Atomic energy levels and chemical bonding
Atomic structure is the basis of chemistry. It is explained by Quantum Mechanics, which is part of physics. We will see that physics explains chemistry, which explains physiology, which at least start to explain neurobiology. It’s one thing that leads to another.
In QM, the properties of a system, that is, a given object or set of objects, such as an atom, are given by the solution to the Schrödinger equation for the system. For atoms, there are a set of solutions, corresponding to different energy states of the atoms. What follows may smack of numerology.
Consider the hydrogen atom, composed of one negatively-charged electron in orbit around a nucleus containing one positively-charged proton. (This is an experimental result.) Look out, the orbit is not a well-defined path around the nucleus like those animations you see in TV ads, but rather a cloud of probability which indicates the likelihood that the electron will be found at any given point in the cloud. This is due to the probabilistic character of QM and the Uncertainty Principle. The different solutions to the Schrödinger equation express the possible energy values of the atom. Each one is specified by a set of integer numbers called quantum numbers. In the case of the hydrogen atom, they are the following:
1. The principal quantum number, designated by the symbol n, takes on integer values from 1 on up, but in practice only to 7. It indicates the shell, or level of the cloud, in which the electron is found. The values 1-7 are often indicated by the letters K, L, M…Q.
2. The orbital quantum number, l, indicates a level within the shell which is called the subshell, It can take on values from 0 up to n-1. The values 0-3 are often referred to as s, p, d and f.1The notations s, p, d and f come from spectroscopy and are abbreviated forms of sharp, principal, diffuse and fundamental.
3. The orbital magnetic quantum number, m, refers to the magnetic orientation of the electron. It can range from -l up through +l.
4. The electron spin, ms, can take on only two values, ½ or -½.
So the only allowed values for the quantum numbers are
n = 1, 2, 3, …
l = 0…n-1 (for a given value of n)
m = -l…+l (for a given value of l)
because those are the ones for which the Schrödinger equation has solutions. It is actually quite simple.
The QM exclusion principle forbids two electrons to occupy the same state. So each set of values (n, l, m, ms) can correspond to only one electron. The result is illustrated in the following table.
n (shell)
l (subshell)
m (orbital)
Max no. electrons
1 0 0 2
2 0
3 0
-1, 0, 1
-2. -1, 0, 1, 2
4 0
-1, 0, 1
-2, -1, 0, 1, 2
The fact that the quantum numbers do not vary continuously from, say, 0 to 0.001 and then 0.002 and on, but but jump from one integer value to another means that the energy of the electron in the electric field of the nucleus also takes on non-continuous values. These are called quantum states and are a feature, or if you prefer, a peculiarity, of QM.
The chemical properties of an atom depend only on the number of electrons. Unless the atom is chemically combined with another, the number of electrons is equal to the number of protons and is called the atomic number. All atoms except hydrogen have nuclei which also contain neutrons. As everyone knows, the number of neutrons can vary and atoms of an element with different numbers of neutrons are called isotopes. As we shall see, these are extraordinarily useful due to the fact that the properties of the element can vary from isotope to isotope.
In specifying which subshells are occupied by the electrons in an atom, one often uses the format
where l is specified as s, p, d or f and # is the number of electrons in the subshell. In its minimum energy state, called the ground state, the carbon atom (atomic number = 12, nucleus contains 6 protons and 6 neutrons) has the following electron configuration:
12C: 1s22s22p2
which indicates the maximum number of two electrons in shell 1, again in subshell s of shell 2 and the remaining two in subshell p of shell 2. Similarly, oxygen (atomic number = 16, 8 each of protons and neutrons) is
16O: 1s22s22p4
the meaning of which should now be clear.
What is interesting is that, for energetic reasons, each atom would like to have its outside subshell filled. If a few electrons are missing, it wants more; if most are missing, it might be willing to give up the rest in order to have an empty outside shell, referred to as the valence shell. (The number of electrons in this outer shell is called the valence.2Officially, the maximum number of univalent atoms (originally hydrogen or chlorine atoms) that may combine with an atom of the element under consideration, or with a fragment, or for which an atom of this element can be substituted.) For instance, hydrogen
1H: 1s1
wants two electrons or none in its 1s shell, so it could give up its electron or gain one. What happens is, two H atoms share their electrons to make a molecule of H2, so each has two electrons half the time. Better than nothing.3To continue the anthropomorphisms, this is a kind of solidarity in which humans are often lacking.
Since oxygen already has shell 2 half-filled, it would probably prefer to gain electrons to fill it. And carbon… but carbon is special and will be considered in a moment.
Look at sodium (Na, atomic number 11) and chlorine (Cl, atomic number 17):
Na: 1s22s22p63s1
Cl: 1s22s22p63s23p5
Sodium could happily give up that 3s electron and chlorine could use it to fill up its 3p valence shell. And this is what happens in table salt, NaCl. If you put salt in water, it separates (for reasons which will be discussed shortly) into charged ions, Na+ and Cl, because chlorine is greedy and keeps that negative 3s electron it took away from sodium. This attraction for electrons is called electronegativity. This is very important in biochemical reactions in cells, as we shall see.
In brief, it turns out that elements with two, ten or eighteen electrons are particularly stable.
Chemistry is the study of chemical systems (atoms, molecules) and chemical bonding between such objects. In the case of NaCl, the sodium and chlorine have opposite electrical charge and the attractive electric force is what holds the molecule together. This is called ionic bonding. Sometimes, when atoms cannot decide which has more right to an electron, the electron is shared between them, as in H2, making both atoms relatively happy. Bonding based on shared electrons is called covalent bonding; it is a sort of consensus situation, if we may go on with the anthropomorphism.
Elements with the same number of electrons in their outer shells have similar chemical properties. So they are arranged in columns in that wonderful physical/chemical tool, the periodic table of the elements.
Periodic table of the elements
It is easy to see that each element in the first column is like hydrogen in having one electron in its valence shell.
H: 1s1
Li: 1s22s1
Na: 1s22s22p63s1
K: 1s22s22p63s23p64s1
… and so on.
The extra elements in the middle are rule-breakers. Instead of filling one subshell before moving on to the next, they start one, add a small number (often only one) of electrons to the next, then go back to finish filling the next-to-last.
Columns in the table are called groups; rows, periods.
The subshell configurations we have been giving are for the lowest energy state of the atom, called the ground state, in which subshells are filled from the “bottom” up (with some exceptions, as just mentioned). But if that hydrogen electron is struck by a photon, enough energy may be transferred from the photon to the 1s electron to push it into a higher-energy subshell. The atom is then said to be in an excited state. The electron may then re-descend spontaneously to the lower subshell, emitting a photon of energy equivalent to the difference in energy levels of the subshells. In QM, photons behave like waves whose energy is a function of their frequency, so the frequency – equivalently, the color – of the light emitted is characteristic of the difference in energy of the two subshells. Any atom’s subshells will therefore correspond to a given set of photon frequencies emitted and these are seen as colors, although not all these colors will be visible to a human eye. The set of frequencies constitute the spectrum of the atom and may be used to analyze the identity of a light source. In this way, we can identify the chemical components of light-emitting objects like distant stars.
There are two other types of bonding. We will consider hydrogen bonds very shortly in the discussion of water. The fourth form is due to the shifting electron density distribution around an atom. At times, this may form a temporary dipole even in a neutral atom. This may in turn induce a dipole in nearby atom in such a way that the two dipoles attract each other very weakly. This is London, or van der Waal’s, bonding.
The functioning of all living things depends on water and on the versatility of the carbon atom. So let’s start with carbon.
Notes [ + ]
1. The notations s, p, d and f come from spectroscopy and are abbreviated forms of sharp, principal, diffuse and fundamental.
3. To continue the anthropomorphisms, this is a kind of solidarity in which humans are often lacking.
What atomic physics and chemistry tell us
I am, reluctantly, a self-confessed carbon chauvinist. Carbon is abundant in the Cosmos. It makes marvelously complex molecules, good for life. I am also a water chauvinist. Water makes an ideal solvent system for organic chemistry to work in and stays liquid over a wide range of temperatures. But sometimes I wonder. Could my fondness for materials have something to do with the fact that I am made chiefly of them?
– Carl Sagan, Cosmos
The early stages of the universe and the lives of stars are the matter of physics and astronomy and their offspring, astrophysics and cosmology. By the time the first living things showed up on Earth, processes were occurring which require our knowing about the phenomena described by the science of chemistry. QM is the basis of atomic physics and that is the basis of chemistry, so we are ready for it.
To do even begin a comprehensive survey of chemistry is well beyond the scope of this document. We will illustrate its usefulness and some of its fruits by considering two subjects of great importance not only to Carl Sagan but to all of us – carbon and water.
In order to do that, it is necessary to know about several sujects:
Then we move on to consider, first, the past, starting almost 14 Gya. |
61a871db7be81942 | Main Group Theory in Solid State Physics and Photonics: Problem Solving with Mathematica
Book cover Group Theory in Solid State Physics and Photonics: Problem Solving with Mathematica
While group theory and its application to solid state physics is well established, this textbook raises two completely new aspects. First, it provides a better understanding by focusing on problem solving and making extensive use of Mathematica tools to visualize the concepts. Second, it offers a new tool for the photonics community by transferring the concepts of group theory and its application to photonic crystals.
Clearly divided into three parts, the first provides the basics of group theory. Even at this stage, the authors go beyond the widely used standard examples to show the broad field of applications. Part II is devoted to applications in condensed matter physics, i.e. the electronic structure of materials. Combining the application of the computer algebra system Mathematica with pen and paper derivations leads to a better and faster understanding. The exhaustive discussion shows that the basics of group theory can also be applied to a totally different field, as seen in Part III. Here, photonic applications are discussed in parallel to the electronic case, with the focus on photonic crystals in two and three dimensions, as well as being partially expanded to other problems in the field of photonics.
The authors have developed Mathematica package GTPack which is available for download from the book's homepage. Analytic considerations, numerical calculations and visualization are carried out using the same software. While the use of the Mathematica tools are demonstrated on elementary examples, they can equally be applied to more complicated tasks resulting from the reader's own research.
Year: 2018
Language: english
ISBN 13: 9783527411337
ISBN: 352741133X
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Wolfram Hergert and
R. Matthias Geilhufe
Group Theory in Solid State
Physics and Photonics
Wolfram Hergert and R. Matthias Geilhufe
Group Theory in Solid State Physics
and Photonics
Problem Solving with Mathematica
Prof. Wolfram Hergert
Martin Luther University Halle-Wittenberg
Von-Seckendorff-Platz 1
06120 Halle
Dr. R. Matthias Geilhufe
Roslagstullsbacken 23
10691 Stockholm
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Printed on acid-free paper.
Introduction 1
Symmetries in Solid-State Physics and Photonics 4
A Basic Example: Symmetries of a Square 6
Part One Basics of Group Theory
Symmetry Operations and Transformations of Fields 11
Rotations and Translations 11
Rotation Matrices 13
Euler Angles 16
Euler–Rodrigues Parameters and Quaternions 18
Translations and General Transformations 23
Transformation of Fields 25
Transformation of Scalar Fields and Angular Momentum 26
Transformation of Vector Fields and Total Angular Momentum
Spinors 28
Basic Definitions 33
Isomorphism and Homomorphism 38
Structure of Groups 39
Classes 40
Cosets and Normal Divisors 42
Quotient Groups 46
Product Groups 48
Basics Abstract Group Theory
Point Groups 52
Notation of Symmetry Elements 52
Classification of Point Groups 56
Space Groups 59
Lattices, Translation Group 59
Symmorphic and Nonsymmorphic Space Groups 62
Site Symmetry, Wyckoff Positions, and Wigner–Seitz Cell 65
Color Groups and Magnetic Groups 69
Magnetic Point Groups 69
Magnetic Lattices 72
Magnetic Space Groups 73
Noncrystallographic Groups, Buckyballs, and Nanotubes 75
Structure and Group Theory of Nanotubes 75
Buckminsterfullerene C60 79
Discrete Symmetry Groups in Solid-State Physics and Photonics
Definition of Matrix Representations 84
Reducible and Irreducible Representations 88
The Orthogonality Theorem for Irreducible Representations 90
Characters and Character Tables 94
The Orthogonality Theorem for Characters 96
Character Tables 98
Notations of Irreducible Representations 98
Decomposition of Reducible Representations 102
Projection Operators and Basis Functions of Representations 105
Direct Product Representations 112
Wigner–Eckart Theorem 120
Induced Representations 123
Representation Theory
Symmetry and Representation Theory in k-Space 133
The Cyclic Born–von Kármán Boundary Condition
and the Bloch Wave 133
The Reciprocal Lattice 136
The Brillouin Zone and the Group of the Wave Vector k 137
Irreducible Representations of Symmorphic Space Groups 142
Irreducible Representations of Nonsymmorphic Space Groups 143
Part Two
Applications in Electronic Structure Theory
The Schrödinger Equation 151
The Group of the Schrödinger Equation 153
Degeneracy of Energy States 154
Time-Independent Perturbation Theory 157
General Formalism 159
Crystal Field Expansion 160
Crystal Field Operators 164
Transition Probabilities and Selection Rules 169
Generalization to Include the Spin
Solution of the Schrödinger Equation
The Pauli Equation 177
Homomorphism between SU(2) and SO(3) 178
Transformation of the Spin–Orbit Coupling Operator 180
The Group of the Pauli Equation and Double Groups 183
Irreducible Representations of Double Groups 186
Splitting of Degeneracies by Spin–Orbit Coupling 189
Time-Reversal Symmetry 193
The Reality of Representations 193
Spin-Independent Theory 194
Spin-Dependent Theory 196
Solution of the Schrödinger Equation for a Crystal 197
Symmetry Properties of Energy Bands 198
Degeneracy and Symmetry of Energy Bands 200
Compatibility Relations and Crossing of Bands 201
Symmetry-Adapted Functions 203
Symmetry-Adapted Plane Waves 203
Localized Orbitals 205
Construction of Tight-Binding Hamiltonians 210
Hamiltonians in Two-Center Form 212
Hamiltonians in Three-Center Form 216
Inclusion of Spin–Orbit Interaction 224
Tight-Binding Hamiltonians from ab initio Calculations 225
Hamiltonians Based on Plane Waves 227
Electronic Energy Bands and Irreducible Representations 230
Examples and Applications 236
Calculation of Fermi Surfaces 236
Electronic Structure of Carbon Nanotubes 238
Tight-binding Real-Space Calculations 240
Spin–Orbit Coupling in Semiconductors 245
Tight-Binding Models for Oxides 247
Electronic Structure Calculations
Part Three
Applications in Photonics
Maxwell’s Equations and the Master Equation for Photonic
Crystals 254
The Master Equation 254
One- and Two-Dimensional Problems 256
Group of the Master Equation 257
Master Equation as an Eigenvalue Problem 259
Models of the Permittivity 260
Reduced Structure Factors 264
Convergence of the Plane Wave Expansion 266
Solution of Maxwell’s Equations
Photonic Band Structure and Symmetrized Plane Waves 270
Empty Lattice Band Structure and Symmetrized Plane Waves 270
Photonic Band Structures: A First Example 273
Group Theoretical Classification of Photonic Band Structures 276
Supercells and Symmetry of Defect Modes 279
Uncoupled Bands 283
Three-Dimensional Photonic Crystals
Two-Dimensional Photonic Crystals
Empty Lattice Bands and Compatibility Relations 287
An example: Dielectric Spheres in Air 291
Symmetry-Adapted Vector Spherical Waves 293
Part Four
Other Applications
Vibrations of Molecules 301
Permutation, Displacement, and Vector Representation
Vibrational Modes of Molecules 305
Infrared and Raman Activity 307
Lattice Vibrations 310
Direct Calculation of the Dynamical Matrix 312
Dynamical Matrix from Tight-Binding Models 314
Analysis of Zone Center Modes 315
Group Theory of Vibrational Problems
Introduction to Landau’s Theory of Phase Transitions 320
Basics of the Group Theoretical Formulation 324
Examples with GTPack Commands 326
Invariant Polynomials 326
Landau and LifshitzCriterion 327
Landau Theory of Phase Transitions of the Second Kind
Complex Spherical Harmonics 332
Definition of Complex Spherical Harmonics 332
Cartesian Spherical Harmonics 332
Transformation Behavior of Complex Spherical Harmonics 333
Tesseral Harmonics 334
Definition of Tesseral Harmonics 334
Cartesian Tesseral Harmonics 335
Transformation Behavior of Tesseral Harmonics 336
Electronic Structure Databases 337
Tight-Binding Calculations 337
Pseudopotential Calculations 338
Radial Integrals for Crystal Field Parameters 339
Molecular Databases 339
Database of Structures 339
Appendix A Spherical Harmonics
Appendix B Remarks on Databases
Calculation of Band Structure and Density of States 341
Calculation of Eigenmodes 342
Comparison of Calculations with MPB and Mathematica 343
Appendix D Technical Remarks on GTPack
Structure of GTPack 345
Installation of GTPack 346
Appendix C Use of MPB together with GTPack
Symmetry principles are present in almost all branches of physics. In solid-state
physics, for example, we have to take into account the symmetry of crystals, clusters, or more recently detected structures like fullerenes, carbon nanotubes, or
quasicrystals. The development of high-energy physics and the standard model
of elementary particles would have been unimaginable without using symmetry
arguments. Group theory is the mathematical approach used to describe symmetry. Therefore, it has become an important tool for physicists in the past century.
In some cases, understanding the basic concepts of group theory can become
a bit tiring. One reason is that exercises connected to the definitions and special
structures of groups as well as applications are either trivial or become quickly
tedious, even if the concrete calculations are mostly elementary. This occurs, especially, when a textbook does not offer additional help and special tools to assist
the reader in becoming familiar with the content. Therefore, we chose a different
approach for the present book. Our intention was not to write another comprehensive text about group theory in solid-state physics, but a more applied one
based on the Mathematica package GTPack. Therefore, the book is more a handbook on a computational approach to group theory, explaining all basic concepts
and the solution of symmetry-related problems in solid-state physics by means of
GTPack commands. With the length of the manuscript in mind, we have, at some
points, omitted longer and rather technical proofs. However, the interested reader is referred to more rigorous textbooks in those cases and we provide specific
references. The examples and tasks in this book are supposed to encourage the
reader to work actively with GTPack.
GTPack itself provides more than 200 additional modules to the standard Mathematica language. The content ranges from basic group theory and representation theory to more applied methods like crystal field theory and tight-binding
and plane-wave approaches to symmetry-based studies in the fields of solid-state
physics and photonics. GTPack is freely available online via The package is designed to be easily accessible by providing a complete Mathematica style
documentation, an optional input validation, and an error strategy. Therefore, we
believe that also advanced users of group theory concepts will benefit from the
book and the Mathematica package. We provide a compact reference material
and a programming environment that will help to solve actual research problems
in an efficient way.
In general, computer algebra systems (CAS) allow for a symbolic manipulation
of algebraic expressions. Modern systems combine this basic property with numerical algorithms and visualization tools. Furthermore, they provide a programming language for the implementation of individual algorithms. In principle, one
has to distinguish between general purpose systems like, e.g., Mathematica and
Maple, and systems developed for special purposes. Although the second class of
systems usually has a limited range of applications, it aims for much better computational performance. The GAP system (Groups, Algorithms, and Programming)
is one of these specialized systems and has a focus on group theory. Extensions
like the system Cryst, which was built on top of GAP, are specialized in terms of
computations with crystallographic groups.
Nevertheless, for this book we decided to use Mathematica, as Mathematica
is well established and often included in the teaching of various Physics departments worldwide. At the Department of Physics of the Martin Luther University
Halle-Wittenberg, for example, specialized Mathematica seminars are provided
to accompany the theoretical physics lectures. In these courses, GTPack has been
used actively for several years.
During the development of GTPack, two paradigms were followed. First, in the
usual Mathematica style, the names of commands should be intuitive, i.e., from
the name itself it should become clear what the command is supposed to be applied for. This also implies that the nomenclature corresponds to the language
physicists usually use in solid-state physics. Second, the commands should be intuitive in their application. Unintentional misuse should not result in longer error
messages and endless loop calculations but in an abort with a precise description
of the error itself. To distinguish GTPack commands from the standard Mathematica language, all commands have a prefix GT and all options a prefix GO. Analogously to Mathematica itself, commands ending with Q result in logical values, i.e.,
either TRUE or FALSE. For example, the new command GTGroupQ[list] checks if
a list of elements forms a group.
The combination of group theory in physics and Mathematica is not new in its
own sense. For example, the books of El-Batanouny and Wooten [1] and McClain [2] also follow this concept. These books provide many code examples of
group theoretical algorithms and additional material as a CD or on the Internet.
However, in contrast to these books, we do not concentrate on the presentation
of algorithms within the text, but provide well-established algorithms within the
GTPack modules. This maintains the focus on the application and solution of real
physics problems. References for the implemented algorithms are provided whenever appropriate.
In addition to applications in solid-state physics we also discuss photonics, a
field that has undergone rapid development over the last 20 years. Here, instead of
discussing the symmetry properties of the Schrödinger, Pauli, or Dirac equations, Maxwell’s equations are in the focus of consideration. Analogously to the
periodic crystal lattice in solids, periodically structured dielectrics are discussed.
GTPack can be applied in a similar manner to both fields.
The book itself is structured as follows. After a short introduction, the basic
aspects of group theory are discussed in Part One. Part Two covers the application
of group theory to electronic structure theory, whereas Part Three is devoted to
its application to photonics. Finally, in Part Four two additional applications are
discussed to demonstrate that GTPack will be helpful also for problems other than
electronic structure and photonics.
GTPack has a long history in terms of its development. In this context, we would
like to thank Diemo Ködderitzsch, Markus Däne, Christian Matyssek, and Stefan
Thomas for their individual contributions to the package. We would especially
like to acknowledge the careful work of Sebastian Schenk, who contributed significantly to the implementation of the documentation system. Furthermore, we
would like to thank Kalevi Kokko, Turku University Finland, who provided a silent
work place for us on several occasions. At his department, we had the opportunity
to concentrate on both the book and the package and many parts were completed
in this context. This was a big help. We acknowledge general interest and support
from Martin Hoffmann and Arthur Ernst. Also we would like to thank WileyVCH, especially Waltraud Wüst, Martin Preuss and Stefanie Volk.
Lastly, we would like to thank our families for their patience and support during
this long-term project.
Stockholm and Halle (Saale),
October 2017
R. Matthias Geilhufe, Wolfram Hergert
When the original German version was first published in 1931, there was a
this reluctance has virtually vanished in the meantime and that, in fact, the
younger generation does not understand the causes and the bases of this reluctance.
E.P. Wigner (Group Theory, 1959)
Symmetry is a far-reaching concept present in mathematics, natural sciences
and beyond. Throughout the chapter the concept of symmetry and symmetry
groups is motivated by specific examples. Starting with symmetries present in
nature, architecture, fine arts and music a transition will be made to solid state
physics and photonics and the symmetries which are of relevance throughout
this book. Finally the square is taken as a first explicit example to explore all
transformations leaving this object invariant.
Symmetry and symmetry breaking are important concepts in nature and almost
every field of our daily life. In a first and general approach symmetry might be
defined as: Symmetry is present when one cannot determine any change in a system
after performing a structural or any other kind of transformation.
Nature, Architecture, Fine Arts, and Music
One of the most fascinating examples for symmetry in nature is the manifold and
beauty of the mineral skeletons of Radiolaria, which are tiny unicellular species.
Figure 1.1a shows a table from Haeckel’s “Art forms in Nature” [4] presenting a
special group of Radiolaria called Spumellaria.
The concept of symmetry can also be found in architecture. Our urban environment is characterized by a mixture of buildings of various centuries. However,
every epoch reflects at least some symmetry principles. For example, the Art déco
style buildings, like the Chrysler Building in New York City (cf. Figure 1.1b), use
symmetry as a design element in a particularly striking manner.
Group Theory in Solid State Physics and Photonics, First Edition. Wolfram Hergert and R. Matthias Geilhufe.
© 2018 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2018 by WILEY-VCH Verlag GmbH & Co.
1 Introduction
Figure 1.1 Symmetry in nature and architecture. (a) Table 91 from HAECKEL’s ‘Art forms in
Nature’ [4]; (b) Chrysler Building in New York City [5] (© JORGE ROYAN,,
CC BY-SA 3.0).
Within the fine arts, the works of M.C. Escher (1898–1972) gain their special
attraction from an intellectually deliberate confusion of symmetry and symmetry
In Escher’s woodcut Snakes [6], a threefold rotational symmetry can be easily
detected in the snake pattern. A rotation by 120◦ transforms the painting into itself. A considerable amount of his work is devoted to mathematical principles and
symmetry. The series “Circle Limits” deals with hyperbolic regular tessellations,
but they are also interesting from the symmetry point of view. The woodcut, entitled Circle Limit III [6], the most interesting under the four circle limit woodcuts,
shows a twofold rotational axis. If the figure is transformed into a black and white
version a fourfold rotational axis appears. Obviously, the color leads to a reduction of symmetry [7]. The change of symmetry by inclusion of additional degrees
of freedom like color in the present example or the spin, if we consider a quantum
mechanical system, leads to the concept of color or Shubnikov groups. A comprehensive overview on symmetry in art and sciences is given by Shubnikov [8].
Weyl [9] and Altmann [10] start their discussion of symmetry principles from
a similar point of view.
Also in music symmetry principles can be found. Tonal and temporal reflections, translations, and rotations play an important role. J.S. Bach’s crab canon
from The Musical Offering (BWV1079) is an example for reflection. The brilliant
effects in M. Ravel’s Boléro achieved by a translational invariant theme represent
an impressive example as well.
1 Introduction
The conservation laws in classical mechanics are closely related to symmetry. Table 1.1 gives an overview of the interplay between symmetry properties and the
resulting conservation laws.
A general formulation of this connection is given by the Noether theorem.
That symmetry principles are the primary features that constrain dynamical laws
was one of the great advances of Einstein in his annus mirabilis 1905 [11]. The
relevance of symmetry in all fields of theoretical physics can be seen as a major
achievement of twentieth century physics.
In parallel to the development of quantum theory, the direct connection between quantum theory and group theory was understood. Especially E. Wigner
revealed the role of symmetry in quantum mechanics and discussed the application of group theory in a series of papers between 1926 and 1928 [11] (see also
H. Weyl 1928 [12]). Symmetry accounts for the degeneracy of energy levels of a
quantum system. In a central field, for example, an energy level should have a degeneracy of 2l + 1 (l – angular momentum quantum number) because the angular
momentum is conserved due to the rotational symmetry of the potential. However, considering the hydrogen atom a higher ‘accidental’ symmetry can be found,
where levels have a degeneracy of n2 , the square of the principle quantum number. The reason was revealed by Pauli [13, 14] in 1926 using the conservation
of the quantum mechanical analogue of the Lenz–Runge vector and by Fock
in 1935 by the comparison of the Schrödinger equation in momentum space
with the integral equation of four-dimensional spherical harmonics [15]. Fock
showed that the electron effectively moves in an environment with the symmetry
of a hypersphere in four-dimensional space. The symmetry of the hydrogen atom
is mediated by transformations of the entire Hamiltonian and not of its parts,
the kinetic and the potential energy alone. Such dynamical symmetries cannot be
found by the analysis of forces and potentials alone. The basic equations of quantum theory and electromagnetism are time dependent, i.e., dynamic equations.
Therefore, the symmetry properties of the physical systems as well as the symmetry properties of the fundamental equations have to be taken into account.
Table 1.1 Conservation laws and symmetry in classical mechanics.
Symmetry property
Homogeneity of time (translations in time)
Homogeneity of space (translations in space)
Isotropy of space (rotations in space)
Invariance under Galilei transformations
Conserved quantity
Angular momentum
Center of gravity
1 Introduction
Symmetries in Solid-State Physics and Photonics
In Figure 1.2, two representative examples of solid-state systems are shown. The
scanning tunneling microscope (STM) image in Figure 1.2a depicts two monolayers of MgO on a Ag(001) surface in atomic resolution. The quadratic arrangement
of protrusions representing one sublattice is clearly revealed. One of the main
tasks of solid-state theory is the calculation of the electronic structure of systems
starting from the real-space structure.
However, the many-particle Schrödinger equation, containing the coordinates of all nuclei and electrons of a solid cannot be solved directly, neither analytically nor numerically. This problem can be approached by discussing effective
one-particle systems, for example, in the framework of density functional theory
(cf. [16]). Therefore, it will be sufficient to study Schrödinger-like equations in
the following to investigate implications of crystal symmetry.
In the first years of electronic structure theory of solids, principles of group theory were applied to optimize computations of complex systems as much as possible due to the limited computational resources available at that time. Although
this aspect becomes less important nowadays, the connection between symmetry
in the structure and the electronic properties is one of the main applications of
group theory.
Figure 1.2 Symmetry in solid-state physics
and photonics. (a) Atomically resolved
STM image of two monolayers of MgO on
Ag(001) (from [17], Figure 1) (With permission, Copyright © 2017 American Physical
Society.)(b) SEM image of a width-modulated
stripe (a) of macroporous silicon on a silicon
substrate. The increasing magnification in (b)–
(d) reveals a waveguide structure prepared by
a missing row of pores. (from [18]). (With permission, Copyright © 1999 Wiley-VCH GmbH.)
1.1 Symmetries in Solid-State Physics and Photonics
Next to the optimization of numerical calculations, group theory can be applied to classify promising systems for further investigations, like in the case of the
search for multiferroic materials [19, 20]. In general, four primary ferroic properties are known: ferroelectricity, ferromagnetism, ferrotoroidicity, and ferroelasticity. The magnetoelectric coupling, of special interest in applications, is a secondary ferroic effect. The occurrence of multiple ferroic properties in one phase
is connected to specific symmetry conditions a material has to accomplish.
Defects in solids and at solid surfaces play a continuously increasing role in basic research and applications (diluted magnetic semiconductors, p-magnetism in
oxides). For example, group theory allows to get useful information in a general
and efficient way (cf. [21, 22]) treating defect states in the framework of perturbation theory.
More recently, a close connection between high-energy physics and condensed
matter physics has been established, where effective elementary excitations within a crystal behave as particles that were formally described in elementary particle
physics. A promising class of materials are Dirac materials like graphene, where
the elementary electronic excitations behave as relativistic massless Dirac fermions [23, 24]. Degeneracies and crossings of energy bands within the electronic
band structure together with the dispersion relation in the neighborhood of the
crossing point are closely related to the crystalline symmetry [25, 26].
In Figure 1.2b, a scanning electron microscope (SEM) image of macroporous
silicon is shown. The special etching technique provides a periodically structured
dielectric material that is referred to as a photonic crystal. The propagation of
electromagnetic waves in such structures can be calculated starting from Maxwell’s equations [27, 28]. The resulting eigenmodes of the electromagnetic field
are closely connected to the symmetry of the structured dielectric. Group theory
can be applied in various cases within the field of photonics. Subsequently, a few
examples are mentioned. The photonic bands of two-dimensional photonic crystals can be classified with respect to the symmetry of the lattice. The symmetry
properties of the eigenmodes, found by means of group theory, decide whether
this mode can be excited by an external plane wave [29]. Metamaterials are composite materials that have peculiar electromagnetic properties that are different
from the properties of their constituents. Group theory can be used for design
and optimization of such materials [30]. Group theoretical arguments also help
to discuss the dispersion in photonic crystal waveguides in advance. Clearly, this
approach represents a more sophisticated strategy in comparison to relying on a
trial and error approach [31, 32]. If a magneto-optical material is used for a photonic crystal, time-reversal symmetry is broken due to the intrinsic magnetic field.
In this case, the theory of magnetic groups can be used to study the properties of
such systems [33].
The goal of this book is to discuss the variety of possible applications of computational group theory as a powerful tool for actual research in photonics and
electronic structure theory. Specific examples using the Mathematica package GTPack will be provided.
1 Introduction
A Basic Example: Symmetries of a Square
As a first example, the symmetry of a square is discussed (Figure 1.3). The square
is located in the x y-plane. In general, the whole x y-plane could be covered completely by squares leading to a periodic arrangement like that of the STM image
from the two MgO layers on Ag(001) in Figure 1.2a. Subsequently, operations that
leave the square invariant are identified. 1)
First, rotations of 0, π∕2, π, and 3π∕2 in the mathematical positive direction
around the z-axis represent such operations. A rotation by an angle of 0◦ induces
no change at all and is therefore named identity element E. Instead of the rotation
by 3π∕2 a rotation by −π∕2 can be considered. Furthermore, a rotation by an angle
of 𝜑 + n2π, n = 1, 2, … is equivalent to a rotation by 𝜑 and is not considered as
a new operation. In total, four inequivalent rotational operations are found.
Next to rotations leaving the square invariant, reflection lines can be identified.
Performing a reflection, the perpendicular coordinates with respect to the line
change their sign. In the present example, the x-axis is such a reflection line and
furthermore a symmetry operation. By a reflection along this line, the point 1
becomes 4, 2 becomes 3, and vice versa. If the symmetries are considered in three
dimensions, a reflection might be expressed by a rotation with angle π around
the normal direction of the reflection line (here it is the y-axis) followed by an
inversion (the inversion changes the signs of all coordinates). A rotation around
the y-axis interchanges the points 1 and 2 and 4 and 3 as well. After applying an
inversion the points 1 and 3 and 2 and 4 are interchanged. Additionally, the y-axis
and the two diagonals of the square are reflection lines.
In total there are eight inequivalent symmetry elements, four rotations and four
reflections. Those elements form the symmetry group of the square. The combination of two symmetry elements, i.e., the application one after another, leads to
another element of the group.
In Figure 1.4, a square is presented with different coloring schemes. It can be
verified that the use of color in Figure 1.4b–d reduces the symmetry. The symmetry groups of the colored squares are subgroups of the group of the square of
Figure 1.3 Square with coordinate system and reflection
lines. The vertices are numbered only to explain the effect of
symmetry operations.
1) Symmetry operations are restricted here to the x y-plane, i.e., are orthogonal coordinate
transformations in x and y represented by 2 × 2 matrices.
1.2 A Basic Example: Symmetries of a Square
Figure 1.4 Symmetry of a square: Square colored in different ways.
Figure 1.4a. As an example: In Figure 1.4c the diagonal reflection lines still exist,
but but the mirror symmetry along the x- and y-axis is broken. Furthermore, the
fourfold rotation axis is reduced to a twofold rotation axis. While the square itself represents a geometrical symmetry, the color scheme might be thought to be
connected with a physical property like the spin, in terms of spin-up (black) and
spin-down (white).
In the next sections, the basics of group theory are introduced. The symmetry
group of the square will be kept as an example. Referring to Figure 1.2b, a hexagonal arrangement of pores can be seen for the photonic crystal. The symmetry
group of a hexagon has 12 elements.
Task 1 (Symmetry of the square and the hexagon). The Notebook GTTask_1.nb
contains a discussion of the symmetry properties of the colored squares of Figure 1.4. Extend the discussion to a regular hexagon and its different colored versions to get familiar with Mathematica and GTPack.
Part One
Basics of Group Theory
Symmetry Operations and Transformations of Fields
Wer die Bewegung nicht kennt, kennt die Natur nicht.
Aristoteles (Phys. III; 1 200b 15–16)
The symmetry of a physical system is described by operations leaving the system invariant. Throughout the chapter such operations are introduced and
discussed. In general, symmetry operations can be distinguished in rotations
and translations. Furthermore, rotations can be subdivided into proper and
improper rotations depending on the sign of the determinant of the rotation
matrix. Besides rotation matrices, alternative representations of rotations can
be derived. In particular, Euler angles, Euler–Rodrigues parameters and
quaternions are discussed. Many physical theories like electrodynamics and
quantum mechanics are field theories. To provide a basic framework for later
chapters, the transformation properties of scalar, vector and spinor fields are
Rotations and Translations
Rotations, EULER angles, quaternions
Change between active and passive definition of transformations
Prints whether active or passive definition is used
Checks if the input is a list of Euler angles
Gives Euler angles for a given symmetry element
Checks if the input is a quaternion
Multiplication of quaternions
Gives the inverse of a quaternion
Gives the absolute value of a quaternion
Gives the conjugate of a quaternion
Gives the polar angle of a quaternion
2 Symmetry Operations and Transformations of Fields
e z e'z
Figure 2.1 Illustration of active and passive rotation. (a) Active rotation; (b) passive rotation.
A coordinate transformation in three-dimensional Euclidean space ℝ3 can be
written as a combination of a pure rotation or reflection and a translation. Such
transformations can be discussed as active or passive (Figure 2.1). Subsequently,
both terms will be explained for pure rotations. A Cartesian coordinate system,
fixed in space, is given in terms of the mutually orthogonal unit vectors (ex , e y , ez ).
A vector r1 points from the origin O to the point P1 . An active rotation moves
the point P1 to P2 and the vector r1 is transformed to r2 satisfying |r1 | = |r2 |
̂ (a) (δ) r1 .
r2 = R
̂ (a)
Here, R
z (δ) represents the corresponding rotational operator, discussed in detail
in Section 2.1.1, where the superscript indicates the active nature of the rotation
and the subscript the rotation axis ez . A rotation about an angle δ in the mathematical positive sense (counterclockwise) is performed. The inverse of this operation is an operation around the same axis but by an angle −δ, or, alternatively, by
an angle δ but about the opposite direction of the rotation axis
̂ (a) (−δ) = R
̂ (a) (δ) .
̂ (a)
(δ) = R
Hence, P2 can be transformed to P1 by the inverse operation
̂ (a)
(δ)r2 .
r1 = R
In comparison to active rotations, a passive rotation is defined by a rotation of the
coordinate frame itself. Hence, by fixing a point P1 with vector r1 , and rotating to
a new coordinate frame (ex′ , e y′ , ez ′ ), the rotation can be formulated as
̂ ( p) (δ) r1 .
r′1 = R
Even though the rotations (2.1) (active) and (2.4) (passive) are not equivalent, they
can be related to each other. There is a one-to-one correspondence, or an isomorphism (in the group theory language), between both formulations
̂ (a)
̂ ( p) (−δ) ,
̂ (a) (−δ) = R
̂ ( p) (δ) .
̂ (a) (δ) = R
(δ) = R
2.1 Rotations and Translations
A similar concept also applies for translations. In the active formulation, a point
P2 gets shifted to P3 by a vector t. In the passive formulation, the origin O of a
rotated coordinate frame is shifted to O ′ by a vector −t.
The calculation of physical, i.e., measurable quantities, is independent of choosing the active or passive formulation. Therefore, both conventions can be found
in different references. For example, the books of Cornwell [35, 36] and ElBatanouny and Wooten [1] use the passive definition, while the active formulation is used by Inui et al. [37]. See also Altman [38] and Morisson and
Parker [39] for a detailed discussion of the topic.
Within GTPack the passive definition is used as the standard setting. Therefore,
with the exception of this chapter, the passive formulation will be used throughout
this book. The superscripts (a) and ( p) are suppressed in the following as long as
no specific reason for their use is present.
Example 1 (Active and passive rotations in GTPack). As a standard, GTPack uses
the passive formulation of rotations. That means, asking for the rotation matrix
of a symmetry element (e.g., a threefold rotation about ez , denoted by C3z ) using
the command GTGetMatrix gives a matrix describing a clockwise rotation of the
coordinate system about the z-axis. However, GTPack offers the option to switch
between active and passive definition by means of the command GTReinstallAxes.
An example is shown in Figure 2.2. To check which formulation is currently used,
GTWhichAxes can be applied.
Rotation Matrices
Throughout this section, the active formulation of rotations will be used to discuss rotations in more detail. A rotation of a vector preserves the length of the
vector itself and leaves the angle between any two transformed vectors invariant.
Such a transformation (cf. Figure 2.3) is represented by an orthogonal matrix, i.e.,
̂ (a)
also the operator R
z (δ) is represented by an orthogonal matrix. A matrix is orthogonal if the columns (or rows) form an orthonormal set. The eigenvalues of
orthogonal matrices are of absolute value 1. They are either real or appear in conjugate complex pairs. If the first column of a 2-dimensional matrix is given by the
normalized vector (cos 𝜑, sin 𝜑), two choices for the second column are possible
satisfying the orthogonality of the matrix,
cos 𝜑 − sin 𝜑
cos 𝜑
sin 𝜑
, R2 =
R1 =
sin 𝜑 cos 𝜑
sin 𝜑 − cos 𝜑
The determinants of these two matrices are ±1,
det R1 = 1 ,
det R2 = −1 .
Rotations with a determinant +1 are called proper rotations. Accordingly, rotations with a determinant −1 are called improper rotations. The eigenvalues of R1
2 Symmetry Operations and Transformations of Fields
Figure 2.2 Change from the passive definition to the active definition of rotations by means of
Figure 2.3 Result of the rotation by means of
the matrices (2.6) on the vector r1 .
2.1 Rotations and Translations
and R2 , respectively, are given by
r11 = cos 𝜑 − i sin 𝜑 ,
r12 = cos 𝜑 + i sin 𝜑
r21 = −1 ,
r22 = +1 .
Now, a vector r1 = (r0 cos 𝜑0 , r0 sin 𝜑0 ) being transformed under R1 and R2 with
an angle 𝜑 = 2α is considered. The results of the transformations are
cos(α + (α + 𝜑0 ))
r2 = R 1 ⋅ r1 = r 0
sin(α + (α + 𝜑0 ))
cos(α + (α − 𝜑0 ))
r3 = R 2 ⋅ r1 = r 0
sin(α + (α − 𝜑0 ))
as can be seen in Figure 2.3. Matrix R1 rotates r1 counterclockwise by an angle
𝜑 = 2α. Hence, it is a proper rotation. In case of R2 , the vector r1 is getting reflected
along a line with angle α to the x-axis. This reflection represents an improper
The equations (2.6) can be extended to the three-dimensional space, via
⎛cos 𝜑
= ⎜ sin 𝜑
⎜ 0
− sin 𝜑
cos 𝜑
0⎟ ,
c ⎟⎠
⎛cos 𝜑
= ⎜ sin 𝜑
⎜ 0
sin 𝜑
− cos 𝜑
0⎟ .
c ⎟⎠
To obtain orthogonal matrices the parameter c has to be ±1. For c = 1, R1′ and R′2
represent the same transformations as before.
For c = −1, the transformation R1′ represents a so-called rotoreflection. That
means, with respect to x and y the transformation is a proper rotation, but the
z-component changes sign. Unlike R2 , the determinant of the matrix R2′ is +1 for
c = −1. The transformation R2′ represents a binary rotation meaning a rotation
by π around a certain axis. Here, the axis of rotation is not the z-axis as will be
discussed in Task 2.
On the basis of properties of orthogonal matrices the following cases are possible for 3 × 3 transformation matrices:
a) The matrix has one real and two conjugate complex eigenvectors (type A).
b) All eigenvectors of the matrix are real (type B).
The various kinds of rotations are summarized in Table 2.1. The set of all proper and improper rotations forms a group called O(3), the group of all threedimensional orthogonal matrices. The subset of all proper rotations, i.e., 3dimensional orthogonal matrices with determinant 1 forms the group SO(3).
The definition of a group is given in Chapter 3.
2 Symmetry Operations and Transformations of Fields
Table 2.1 The different cases of general rotations. The set of eigenvectors of type A contains
one real and two mutually complex conjugate eigenvectors, while set B contains real eigenvectors only (complex conjugation ist denoted by ⋆ ).
Type A
Type B
Proper rotation
Rotoreflection (improper rotation)
Binary rotation
EULER Angles
In general, every rotation can be expressed by a series of three rotations characterized by the Euler angles.
Definition 1 (Euler angles). A rotation R(α, β, γ) can be described in the active
convention by three successive rotations. A rotation of the system ex , e y , ez (or a
vector r) with respect to the coordinate system ex , e y , ez fixed in space is defined
∙ a rotation of ex , e y , ez by γ around ez ,
∙ a rotation of the transformed ex , e y , ez by β around e y ,
∙ finally a rotation of the transformed ex , e y , ez by α around ez .
The angles are restricted to:
−π < γ ≤ π ,
−π < α ≤ π .
Figure 2.4 With the help of GTEulerAnglesQ it can be checked if the input is a list of EULER
angles, {{α, β, γ }, det R}.
2.1 Rotations and Translations
Within GTPack Euler angles are implemented as lists of the form {{α, β, γ},
det R}. It is important to include the determinant of the rotation matrix, det R to
also cover the set of improper rotations (det R = −1). To check if an input is in the
appropriate form used by GTPack, GTEulerAnglesQ can be applied. An example is
shown in Figure 2.4.
According to the definition of the Euler angles, a rotation matrix can be written
as the product of three orthogonal transformation matrices, describing rotations
about the corresponding axes (cf. (2.12)),
R(α, β, γ)
⎛cos α
= ⎜ sin α
⎜ 0
− sin α
cos α
0⎞ ⎛ cos β
0⎟ ⎜ 0
1⎟⎠ ⎜⎝− sin β
sin β ⎞ ⎛cos γ
0 ⎟ ⎜ sin γ
cos β ⎟⎠ ⎜⎝ 0
− sin γ
cos γ
⎛cos α cos β cos γ − sin α sin γ − cos α cos β sin γ − sin α cos γ
= ⎜sin α cos β cos γ + cos α sin γ − sin α cos β sin γ + cos α cos γ
− sin β cos γ
sin β sin γ
cos α sin β⎞
sin α sin β ⎟ .
cos β ⎟⎠
However, the choice of Euler angles is not unique and the following relations can
be verified,
R(α, 0, γ) = R(α + γ, 0, 0) = R(0, 0, α + γ) ,
R(α, π, γ) = R(α + δ, π, γ + δ) .
Here, δ is an arbitrary angle. The inverse rotation R(α, β, γ)−1 is defined via,
R(α, β, γ)−1 ⋅ R(α, β, γ) = 1 ,
where 1 denotes the identity matrix. The inverse matrix itself is given by
R(α, β, γ)−1 = R(−γ ± π, β, −α ± π)
The sign has to be chosen such that the angles are in agreement with (2.13).
The Euler angles can be determined from the components of a rotation matrix.
For a given matrix R,
R = ⎜r21
⎝ 31
r13 ⎞
r23 ⎟
r33 ⎟⎠
the Euler angles can be deduced from (2.14),
( )
β = arccos r33 , γ = − arctan 32 ,
α = arctan
2 Symmetry Operations and Transformations of Fields
Figure 2.5 Obtaining EULER angles from a matrix or a symbol using GTGetEulerAngles. The
symbol C3z denotes a counterclockwise rotation about the angle 2∕3π about the z-axis.
Example 2 (Euler angles and GTPack). Within GTPack symmetry elements can
be represented by symbols, matrices, Euler angles, or quaternions. Starting from
symbols, matrices, or quaternions GTGetEulerAngles can be used to obtain an associated set of Euler angles. The application of the command is shown in Figure 2.5.
EULER–RODRIGUES Parameters and Quaternions
Historically, alternatives to 3 × 3 orthogonal matrices were introduced to describe
rotations. Clearly, a rotation can be represented by a rotation axis and an angle
𝜑 (−π ≤ 𝜑 ≤ +π), where the axis is represented by a unit vector n, the so-called
pole. Per definition, a positive value of 𝜑 is assigned in the active formulation, if
the rotation is counterclockwise, with respect to the direction of n. The rotation
̂ n), where the following relation holds,
parametrized by 𝜑 and n is written as R(𝜑,
̂ n) ≡ R(−𝜑,
−n) .
In the following, the relationship between rotation matrices and a representation
by means of a pole and a rotation angle will be discussed (for more details see [38]).
a) Obtaining a rotation matrix from 𝜑 and n:
A skew-symmetric (or antisymmetric) matrix has the property
ST = −S .
2.1 Rotations and Translations
Therefore, the matrix A = exp(S) is orthogonal. By choosing a skew-symmetric
matrix Z in terms of the components of the pole n,
⎛ 0
Z = ⎜ nz
⎝ y
−n z
ny ⎞
−n x ⎟ ,
0 ⎟⎠
̂ n) can be written as
the rotation R(𝜑,
̂ n) = exp (𝜑Z) = I + sin 𝜑 Z + (1 − cos 𝜑) Z2
= I + sin 𝜑 Z + 2 sin2 (𝜑∕2) Z2 .
Substituting (2.22) in (2.23) leads to the matrix representation of R(𝜑, n),
⎛1 − 2 n2 + n2 c −n s + 2n n c
n y s + 2n z n x c ⎞
x y
R(𝜑, n) = ⎜ n z s + 2n x n y c
1 − 2 n2z + n2x c
−n x s + 2n y n z c ⎟
) ⎟
⎜ −n y s + 2n z n x c
n x s + 2n y n z c
1 − 2 n2x + n2y c ⎟
where the abbreviations s = sin 𝜑, c = sin2 𝜑∕2 are used. Furthermore, from
equation (2.22) it follows Z ⋅ r = n × r. Hence, the rotation of a vector r can be
expressed as
̂ n)r = r + sin 𝜑 n × r + 2 sin2 (𝜑∕2) n × (n × r) .
This transformation is known as the conical transformation, where the vector
r rotates on a cone around the vector n.
b) Obtaining 𝜑 and n from a rotation matrix:
In general, a rotation matrix R can be written as
R = ⎜r21
⎝ 31
r13 ⎞
r23 ⎟ .
r33 ⎟⎠
In the following, proper rotations are considered, i.e., det R = 1. The transformation of the rotation axis from an arbitrary axis to the z-axis represents a
similarity transformation, S R S−1 . However, the trace of a matrix is invariant
under similarity transformations. Hence, it is possible to verify
Tr R = 2 cos 𝜑 + 1
cos 𝜑 = (Tr R − 1) .
The rotation axis itself can be identified by finding an eigenvector with eigenvalue +1. Finally, the components of n can be expressed by the components
of R, as follows from (2.24),
r − r23
r − r31
r − r12
n x = 32
, n y = 13
, n z = 21
2 sin 𝜑
2 sin 𝜑
2 sin 𝜑
2 Symmetry Operations and Transformations of Fields
Task 2 (Binary rotation). The rotation defined by the matrix
⎛cos 𝜑
R = ⎜ sin 𝜑
⎜ 0
sin 𝜑
− cos 𝜑
is a binary rotation. Find the rotation axis and proof that R describes a rotation
of π around that axis. (The rotation axis is given by the eigenvector to eigenvalue
Definition 2 (Euler–Rodrigues parameters). The Euler–Rodrigues parameters are the following combinations of 𝜑 and n:
λ = cos
Λ = sin
A rotation expressed in terms of Euler–Rodrigues parameters will be designated
̂ Λ). The multiplication of two rotations is expressed as
as R(λ,
̂ 1 , Λ1 )R(λ
̂ 2 , Λ2 )
̂ 3 , Λ 3 ) = R(λ
λ3 = λ1 λ2 − Λ1 ⋅ Λ2 ,
Λ 3 = λ1 Λ2 + λ2 Λ 1 + Λ1 × Λ2
Furthermore, the relations
̂ Λ) = R(−λ,
2 𝜑
λ + Λ = cos
+ sin2 n ⋅ n = 1
Within GTPack rotations can be represented in terms of quaternions. The following definition of the quaternion algebra shows the similarity to the formulation of
rotations by means of Euler–Rodrigues parameters.
Definition 3 (Quaternion algebra). A quaternion 𝔸 is an object [[a, A]] consisting
of a real scalar quantity a and a vector A. Quaternions fulfill the noncommutative
multiplication rule
𝔸𝔹 = [[a, A]][[b, B]] = [[ab − A ⋅ B, aB + bA + A × B]] .
∙ The quaternion multiplication is associative.
∙ Quaternions of the form [[a, 0]] are named real quaternions, because they act
like real numbers:
[[a, 0]][[b, 0]] = [[ab, 0]]
a[[b, B]] = [[a, 0]][[b, B]] = [[ab, aB]]
∙ A quaternion [[0, A]] is named a pure quaternion.
[[0, A]][[0, B]] = [[−A ⋅ B, A × B]]
2.1 Rotations and Translations
∙ A pure quaternion with a unit vector as the vectorial part is named a unit
A = |A|n ,
[[0, A]] = [[0, An]] = A[[0, n]]
=def A 𝕟
∙ An additive formulation of quaternions is in agreement with the multiplication
rule (2.34).
[[a, A]] = [[a, 0]] + [[0, A]]
∙ It is possible to write quaternions in a binary form:
[[a, A]] = a + A 𝕟,
A = |A|
∙ A quaternion may be expressed using the quaternion units
𝕚 = [[0, ex ]] ,
𝕛 = [[0, e y ]] ,
𝕜 = [[0, ez ]]
𝕚 = 𝕛 = 𝕜 = −1
𝕚𝕛 = 𝕜,
𝕛𝕚 = −𝕜
Therefore, the quaternion 𝔸 can be written as
𝔸 = [[a, A]] = a + A x 𝕚 + A y 𝕛 + A z 𝕜 .
The relationship between Euler–Rodrigues parameters and quaternions is given by
𝜑 ]]
[[λ, Λ]] = cos , sin n .
Because of this connection a separate implementation of the Euler–Rodrigues
formalism is not necessary. If a rotation is represented by a quaternion 𝔸 = [[a, A]]
the corresponding matrix form is given by
⎛a2 + A 2x − A 2y − A 2z
R = ⎜ 2(A x A y + aA z )
⎜ 2(A A − aA )
x z
2(A x A y − aA z )
a − A 2x + A 2y − A 2z
2(A y A z + aA x )
2(A x A z + aA y ) ⎞
2(A y A z − aA x ) ⎟ . (2.47)
a2 − A 2x − A 2y + A 2z ⎟⎠
This result corresponds to the representation of the rotation matrix in terms of
Euler–Rodrigues parameters in (2.24).
{ {
Within GTPack quaternions are implemented in the form 𝔸 ≡ λ, Λ1 , Λ2 , Λ3
as lists. With the help of GTQuaternionQ, it is possible to check if a given input is
a quaternion as it is defined in GTPack. As shown in Figure 2.6, the multiplication
of two quaternions according to equation (2.34) can be calculated by means of
GTQMultiplication or by using the symbolic operator ⋄. 1)
1) In Mathematica this symbol can be obtained by typing \[Diamond] or by ESC dia ESC.
2 Symmetry Operations and Transformations of Fields
Figure 2.6 The usage of GTQuaternionQ and GTQMultiplication in GTPack.
Figure 2.7 The application of GTQInverse.
For the multiplication operation an identity quaternion can be introduced, having the property 𝔸𝕀 = 𝔸. It can be verified easily (see Task 3), that this quaternion
has to be
𝕀 = [[1, 0]] .
The inverse of a quaternion, satisfying the equation
𝔸−1 =
= 𝕀, is given by
In the √
above equation, 𝔸∗ = [[a, −A]] is called the conjugate quaternion of 𝔸 and
|𝔸| = a2 + A ⋅ A is the absolute value. With the help of GTPack the inverse of a
2.1 Rotations and Translations
Figure 2.8 The calculation of the conjugate quaternion, the absolute value of a quaternion,
and the polar angle of a quaternion by means of GTQConjugate, GTQAbs and GTQPolar, respectively.
quaternion can be obtained by using the command GTQInverse. The application
of GTQInverse is demonstrated in Figure 2.7.
The related commands to calculate the conjugate quaternion and the absolute
value are GTQConjugate and GTQAbs, respectively. In equation (2.46), it is illustrated that the scalar part of a quaternion can be expressed by λ = cos(𝜑∕2). Here,
the angle 𝜑 is called the polar angle of the quaternion and can be calculated with
GTQPolar. An example for GTQConjugate, GTQAbs, and GTQPolar can be found in
Figure 2.8.
Task 3 (Working with quaternions). Use GTPack to verify
1. the identity quaternion is given by 𝕀 = [[1, 0]],
2. the inverse quaternion is given by 𝔸−1 = |𝔸|
Translations and General Transformations
Previously, rotations and the corresponding rotation matrices were introduced.
Subsequently, the product of two symmetry operations will be discussed. Here,
T1 and T2 denote two abstract symmetry transformations and R(T1 ) and R(T2 )
their rotation matrices. By means of matrix multiplication, the product T1 ⋅ T2
can be discussed via
T = T2 ⋅ T1
R(T) = R(T2 ) ⋅ R(T1 ) ,
2 Symmetry Operations and Transformations of Fields
Figure 2.9 The symmetry transformation
the beginning; (b) two-fold rotation (rotation
IC2y applied to an equilateral triangle in the
by an angle π) about the y-axis; (c) application
xy plane (Numbers are introduced to illustrate of inversion I.
the impact of the operations.) (a) Triangle at
where T = T2 ⋅ T1 can be understood as the application of T2 after T1 . For the
transformation of a vector it follows
r′ = R(T1 )r ,
r′′ = R(T2 )r′ = R(T2 ) ⋅ R(T1 )r = R(T)r .
The following example will help to illustrate the concept.
Example 3 (Product of symmetry operations). A rotation of a triangle (cf. Figure 2.9) about the angle π about the y-axis (C2 y ) is considered. Afterwards, an
inversion I is applied. In terms of rotation matrices the operation IC2 y = I ⋅ C2 y
is given as:
R(IC2 y )
0⎞ ⎛−1
⎟ ⎜
0⎟ = ⎜ 0
1⎟⎠ ⎜⎝ 0
R(C2 y )
0 ⎞ ⎛−1
⎟ ⎜
0 ⎟⋅⎜ 0
−1⎟⎠ ⎜⎝ 0
The two-fold rotation about the y-axis maps x → −x, y → y, and z → −z. Hence,
C2 y together with the inversion (x → −x, y → − y, and z → −z) gives x → x, y →
− y, and z → z. The result is a reflection at the x−z plane (cf. also Table 2.1).
Theorem 1 (General transformation). A general symmetry transformation T consists of a rotation and a translation:
r′ = R(T) ⋅ r + t(T) ,
r′ = {R(T) ∣ t(T)} r .
By means of the short notation in (2.53) the product of two operations and the
inverse can be formulated. If T = T2 ⋅ T1 , then
{R(T) ∣ t(T)} = R(T2 ) ⋅ R(T1 ) ∣ R(T2 ) ⋅ t(T1 ) + t(T2 ) .
The inverse transformation is given by
} {
{R(T) ∣ t(T)}−1 = R(T −1 ) ∣ t(T −1 ) = R−1 (T) ∣ −R−1 (T)t(T) . (2.55)
2.2 Transformation of Fields
Similarly to Theorem 1, a formulation in terms of the so-called augmented matrix
can be used. The augmented matrix W corresponding to a transformation {R(T) ∣
t(T)} is given (cf. [40]) by
⎟ .
1 ⎟⎠
In GTPack general transformations are represented by means of angle brackets.
For example, ⟨C4z , {−4, 1, 2}⟩ is a general transformation consisting of the 4-fold
rotation C4z and the translation vector t = {−4, 1, 2}. 2) The set of all transformations, consisting of all rotations of O(3) together with all possible translations in
ℝ3 , forms the so-called Euclidean group E3 .
Task 4 (Augmented matrices). Proof that the augmented matrix (2.56) leads to
the same transformation as defined in Theorem 1.
Transformation of Fields
Vectors, tensors, spinors
Gives a rotation matrix in spin space for a given angle and a given
rotation axis
Gives a rotation matrix in spin space for a given symmetry element
Physical theories like quantum mechanics or electrodynamics are field theories.
The use of symmetry arguments requires the investigation of the properties of
such fields under symmetry transformations. Electrodynamics represents the
prototype for a classical field theory, where electric and magnetic fields are vector
fields. The notation E(r, t) for the electric field means that at time t a vector E is
assigned to each point r in configuration space. Nonrelativistic quantum theory
without spin is a scalar field theory. In terms of a general tensor field formulation
the scalar field is a tensor field of rank 0, whereas the vector field is a tensor field of
rank 1. If the quantum mechanical spin is included, the transformation properties
of spinors have to be considered.
2) In Mathematica, the symbol ⟨ is obtained by \[LeftAngleBracket] or ESC<ESC. Analogously,
the symbol ⟩ is obtained.
2 Symmetry Operations and Transformations of Fields
Transformation of Scalar Fields and Angular Momentum
In the following, a scalar field ψ(r), e.g., a solution of the stationary Schrödinger
equation, is considered. A rotation r′ = R ⋅ r in configuration space leads to the
function ψ̃(r), related to ψ by
̃ (R ⋅ r) = ψ(r) ,
̃ (r) = ψ(R−1 ⋅ r) .
In first order, an infinitesimal rotation δ𝜑 about the axis n (cf. (2.25)) is approximated by
n)r = r + sin(δ𝜑) n × r + 2 sin2 (δ𝜑) n × n × r ≈ r + δ𝜑 n × r . (2.58)
In terms of a Taylor series, an expansion of (2.58) with respect to the small change
δ𝜑 n × r of the position vector leads to
̃ (r) = ψ(r) − δ𝜑 (n × r) ⋅ ∇ψ(r) ,
( (
∇ ψ(r) ,
= ψ(r) − δ𝜑 n ⋅ r ×
̂ ψ(r) .
̂ ψ(r) = ψ(r) − δ𝜑 n ⋅ L
= ψ(r) − δ𝜑 n ⋅ r × p
̂ is the angular momentum operator. A rotation by a finite angle 𝜑 is
Here, L
expressed as a sum of infinitesimal rotations, giving the equation
̂ n)ψ(r) = e−i𝜑 n⋅L∕ℏ
̃ (r) = P(𝜑,
ψ(r) .
The result can be summarized by the following theorem.
Theorem 2 (Rotation of a scalar field). A transformation of the scalar field ψ(r) to
̃ (r) with respect to a rotation around the angle 𝜑 about the axis n (with
the field ψ
rotation matrix R) is given by
̃ (r) = ψ(R−1 ⋅ r) = ̂
P(𝜑, n)ψ(r) ,
with the rotation operator
P(𝜑, n) = e−i𝜑n⋅L∕ℏ
̂ is Hermitian, it follows that the rotaSince the angular momentum operator L
tion operator P(𝜑, n) is a unitary operator. The components of the angular moLy, ̂
L z are the generators of rotations about the coordinate
mentum operator ̂
Lx , ̂
2.2 Transformation of Fields
Transformation of Vector Fields and Total Angular Momentum
A photon is described by a vector field in terms of a corresponding vector potential A(r, t). Vector mesons are particles with rest mass larger than zero described
also by vector fields, i.e., transformation of vector fields play a role in classical and
quantum theories.
A general vector field ψ(r) is considered in the following. In comparison to the
transformation of scalar fields, a rotation of a vector field has to take into account
a change of the direction of the vector field itself. The relation similar to (2.57)
̃ (r) = R ⋅ ψ(R−1 ⋅ r) .
Considering an infinitesimal rotation (2.58) δ𝜑 and an expansion up to linear order gives,
̃ (r) = ψ(R−1 ⋅ r) + δ𝜑 n × ψ(R−1 ⋅ r)
= ψ(r − δ𝜑 n × r) + δ𝜑 n × ψ(r)
+ δ𝜑 n × ψ(r) .
= ψ(r) − δ𝜑(n ⋅ L)ψ(r)
The last term of (2.63) can be expressed by means of the skew symmetric matrix
Z (cf. (2.22)),
⎛ 0
Z = ⎜ nz
⎝ y
−n z
ny ⎞
−n x ⎟ ,
0 ⎟⎠
⎛0 0 0 ⎞
⎛ 0 0 1⎞
Z = n x ⎜0 0 −1⎟ + n y ⎜ 0 0 0⎟ + n z ⎜1
⎜0 1 0 ⎟
⎜−1 0 0⎟
Sx + n y ̂
S y + nz ̂
Sz = − n ⋅ ̂
nx ̂
The spin operator ̂
S was introduced in (2.64). Hence, for an infinitesimal rotation
of a vector field it follows from (2.63),
̂ − i δ𝜑(n ⋅ ̂
̃ (r) = 1 − i δ𝜑(n ⋅ L)
S) ψ(r)
= 1 − δ𝜑(n ⋅ ̂J) ψ(r) .
̂J = L
S represents the total angular momentum. 3) The result can be generalized
to finite angles 𝜑.
3) In a pure classical consideration the operators ̂
L, ̂
S, ̂J might be defined with ℏ = 1.
2 Symmetry Operations and Transformations of Fields
Theorem 3 (Rotation of a vector field). A transformation of the vector field ψ(r)
̃ (r) with respect to a rotation about the angle 𝜑 about the axis n (with
to the field ψ
rotation matrix R) is given by
̃ (r) = R ⋅ ψ(R−1 ⋅ r) = ̂
P(𝜑, n)ψ(r) ,
P(𝜑, n) = e−i𝜑 n⋅J∕ℏ ,
̂J = L
The components of the total angular momentum operator ̂J are the generators of
rotations around the coordinate axes. The components of ̂
S are given by
S x = iℏ ⎜0
⎟ ̂
−1⎟ , S y = iℏ ⎜ 0
0 ⎟⎠
⎟ ̂
0⎟ , S z = iℏ ⎜1
0⎟ . (2.68)
̂ and ̂
The total angular momentum operator ̂J is unitary, because L
S are Hermitian.
The Hermiticity of S follows directly from (2.68). Vector fields carry a spin S = 1.
To investigate the transformation properties of spinors, the transformation of the
solution of the Pauli equation
p − eA)2
ℏ 𝜕ψ(r, t) ̂
= Hψ(r, t) =
+ V (r) − μB σ ⋅ B ψ(r, t)
has to be analyzed. Here, σ = ex σ x + e y σ y + ez σ z is the vector of the Pauli matrices,
0 1
0 −i
1 0
, ̂
σy =
, σz =
σx =
1 0
i 0
0 −1
As before, the transformation of the spinor ψ(r) is mediated by a unitary operator
P(𝜑, n),
̂ n)ψ(r)
̃ (r) = P(𝜑,
The unitarity of ̂
P guarantees that the probability density ψ † ψ is invariant under
coordinate transformations,
̃ (r) = (̂
̃ † (r)ψ
P† ̂
P ψ = ψ† ψ .
Pψ)† (̂
Pψ) = ψ† ̂
If a transformation in configuration space of the spinor components is considered,
the following relation is assumed to hold,
̃ (r) = ψ † (R−1 ⋅ r)ψ(R−1 ⋅ r) .
̃ † (r)ψ
2.2 Transformation of Fields
The components of a spinor are scalar functions of r and transform according to
Theorem 2. This results in
)† (
̂ L ψ = ψ † (r) ̂
̃ (r) = ̂
̃ † (r)ψ
PL ψ
P†L ̂
P L ψ(r)
PL = e
Here, ̂
P L denotes a transformation of the argument only (cf. Theorem 2). Comparing the general transformation (2.72) and (2.74) leads to the relation
̂† P
̂† P
̂ ψ(r) = ψ † (r) P
̂ ψ(r) .
ψ † (r) P
̂ consists of two parts. First, the unitary operator ̂
Hence, P
P L describes the transformation of the coordinates in the spinor components. The second part u(𝜑, n)
is a rotation in spin space (ℂ2 ),
−i𝜑 n⋅σ∕2
̂ n) = P
̂ L (𝜑, n) u(𝜑, n) = ̂
PL (𝜑, n) e
The concrete form of u(𝜑, n) follows from the transformation properties of the
Pauli equation. To guarantee that physical properties calculated from the Pauli
equation are invariant under a coordinate system change,
̂ ′ (r′ )ψ
̃ (r′ ) = ψ † (r)H(r)ψ(r)
̃ † (r′ )H
̂ has to be
has to be fulfilled. To determine u(𝜑, n), the spin-dependent part of H
investigated. An infinitesimal rotation leads to (cf. [41], p. 46ff for details)
u(𝜑, n) = e−i𝜑 n⋅̂σ ∕2 .
Expanding equation (2.78) in terms of a Taylor series, restricting the calculation
to |n| = 1, and keeping in mind the algebra of the Pauli matrices, 4) it is possible
to reformulate u as,
cos 𝜑2 − in z sin 𝜑2 − n y + in x sin 𝜑2
u(𝜑, n) = (
cos 𝜑2 + in z sin 𝜑2
n y − in x sin 𝜑2
Matrices like in equation (2.80) are elements of the group of 2-dimensional unitary matrices with determinant 1, called SU(2). As will be discussed in Section 8.2,
there is a 2-to-1 mapping between the rotations in spin space and the rotations
in real space. That means, vice versa, to each rotation in real space R, there correspond two matrices in spin space that differ from each other by a minus sign,
u(R) and −u(R). Within GTPack it is possible to evaluate (2.80) for a certain angle
and rotation axis using the command GTSU2Matrix. However, the matrix −u is revealed as a standard. An example for a rotation with an angle 23 π about the z-axis
is shown in Figure 2.10.
4) The Pauli matrices fulfill the following multiplication rules,
σ i ⋅ σ j = δ i, j 1 + i 𝜖i jk σ k .
2 Symmetry Operations and Transformations of Fields
Figure 2.10 Calculation of a rotation matrix in spin-space with the help of GTSU2Matrix.
Analogously to equation (2.14), rotation matrices in spin-space can be evaluated from three Euler angles. Using the z yz-convention and calculating the related matrices according to equation (2.80), the following rotation matrix can be
u(α, β, γ) =
( i
β)( −iγ
cos 2 − sin 2
e 2
e− 2 α
sin 2
cos 2
( i
e− 2 (α+γ) cos 2 −e− 2 (α−γ) sin 2
e 2 (α−γ) sin 2
e 2 (α+γ) cos 2
The above equation can be evaluated using GTGetSU2Matrix. In contrast to GTSU2Matrix, which is designed analogously to the Mathematica command RotationMatrix, GTGetSU2Matrix is capable of transforming matrices, quaternions,
symbols, or Euler angles into a rotation matrix in spin-space. The usage of
GTGetSU2Matrix is demonstrated in Figure 2.11.
To construct improper rotations in spin-space, a representation for the inversion operator needs to be established. Historically, two different matrices were
suggested, denoted as the Pauli gauge and the Cartan gauge (cf. [38, p. 106,
p. 194]). Since the angular momentum is an axial vector, which is invariant under inversion, Pauli argued that also the spin has to be invariant under inversion.
Therefore, the matrix representation of the inversion in the Pauli gauge is given
by the identity matrix,
1 0
Pauli ̂
(I) =
0 1
For rotations in three-dimensional space defined by a rotation matrix R, proper
rotations have a positive determinant det R = 1, whereas improper rotations have
2.2 Transformation of Fields
Figure 2.11 SU (2) matrices can be obtained from EULER angles, quaternions, or symbols of
symmetry elements with the help of GTGetSU2Matrix. The symbol C3z denotes a counter2
clockwise rotation by π about the z-axis.
a negative determinant det R = −1. It can be seen that this behavior is not reflected
by the Pauli gauge for rotation matrices in spin-space. A second possibility with
a negative determinant is given by the Cartan gauge, where the inversion matrix
is represented by
−i 0
Cartan ̂
(I) =
0 −i
Within GTPack the Pauli gauge is used. The results of this section are summarized
in the following theorem.
Theorem 4 (Rotation of a spinor field). A transformation of the spinor field ψ(r)
to the field ψ
rotation matrix R and spin-rotation matrix u) is given by
̃ (r) = u ⋅ ψ(R−1 ⋅ r) = ̂
P(𝜑, n)ψ(r) ,
̂ n) = e
−i𝜑 n⋅̂J∕ℏ
̂J = L
̂ + ̂s .
The components of the total angular momentum operator are the generators of
rotations around the coordinate axes. The components of the spin vector s are given
2 Symmetry Operations and Transformations of Fields
̂s x =
σ ,
2 x
̂s y =
σ ,
2 y
̂s z =
σ .
2 z
Task 5 (Rotation in spin-space). Derive equation (2.80) from equation (2.78). Try
to use Mathematica and the commands PauliMatrix and MatrixPower.
Basics Abstract Group Theory
Il suffit, pour cela, de choisir une fonction symétrique des diverses valeurs
que prend, par toutes les permutations de l’un des groupes partiels, une
fonction qui n’est invariable pour aucune substitution.
Évariste Galois
(Journal de mathématiques pures et appliquées, 11, 417–444, (1846))
The chapter provides the definition of a group as well as related algebraic
structures. Groups itself can be classified by their order as finite and infinite
groups. Additionally, infinite groups can be discrete or continous. Within this
book, the focus is on crystallographic point groups which are finite and space
groups which are discrete infinite groups. Groups itself are sets and contain
subsets of special relevance, such as classes of mutually equivalent elements,
subgroups and cosets. Such structures help classifying specific groups and also provide physically relevant information as will be important later. Maps between groups, especially structure preserving maps are an important concept
which will be introduced during this chapter. Among others, it introduces the
notion of similar groups and provides the basic framework for representation
Basic Definitions
Installation of groups and basic group properties
Installation of point and space groups
Determines the generators of a group
Installs a group from a set of generators
Constructs a matrix representation from a given multiplication table
3 Basics Abstract Group Theory
Calculates the multiplication table of a group
Estimates the order of a group
Estimates the order of a symmetry element
Finds subgroups of a group
Tests if a list of elements is a group
Tests if ′ is a subgroup of
Tests if a group is Abelian
Groups are sets of distinct elements together with an operation called ‘multiplication.’ In applications of group theory to problems of solid-state physics the elements represent the symmetries of crystals or atomic clusters, i.e., they are given
by translations and rotations that transform these objects into itself.
Definition 4 (Group). A set together with an operation (called multiplication)
is called a group, if the following conditions are satisfied.
a) If a, b ∈ , then also the product c = a ◦ b is a member of the group, i.e., c ∈ .
b) The associative law holds, i.e., if a, b, c ∈ then a ◦ (b ◦ c) = (a ◦ b) ◦ c.
c) There exists an identity element E ∈ with the property E ◦ a = a ◦ E = a for
all a ∈ .
d) For each element a ∈ there is an inverse element a−1 ∈ such that a ◦ a−1 =
a−1 ◦ a = E.
The following example will show the installation of groups within GTPack.
Example 4 (Installation of point groups in GTPack). Point groups denote the finite crystallographic groups that represent the local symmetry of an atom (point)
within a crystal. In the 3-dimensional space 32 point groups can be defined. Within GTPack they can be installed using the command GTInstallGroup. Figure 3.1
shows an example for the point group C4v , which contains all planar symmetries
that leave a square invariant. After installing the group, GTGroupQ is applied to
check if the four conditions of Definition 4 are satisfied.
Groups containing a finite number of elements are called finite groups. Crystallographic point groups are typical examples of finite groups. An infinite group
contains an infinite number of elements. Examples are the translational group of
a crystal which is infinite, but discrete. All 3 × 3 real orthogonal matrices, related to rotations in three-dimensional Euclidean space, form an infinite continuous
Definition 5 (Continuous Group). A group whose elements g = g(α1 , α2 , … , α r )
depend continuously on the independent parameters α1 , α2 , … , α r is called a continuous group. The number r of independent parameters is called the dimension of
the group.
3.1 Basic Definitions
Figure 3.1 Installation of the point group C4v . To verify that C4v forms a group GTGroupQ is
applied. In a second step, the group conditions are checked manually.
According to this definition, a 3 × 3 matrix as a representation of a rotation, will be
parameterized by the three Euler angles. Lie groups are special continuous groups,
where the elements are differentiable functions of the parameters. The continuous
groups will play a minor role throughout this book. Special properties of such
groups are mentioned if necessary.
In the following, some minor definitions are given. The order of the group
– ord() denotes the number of elements of a finite group (GTGroupOrder).
Commutativity is not a basic property of the multiplication (Definition 4). However, in cases where a ◦ b = b ◦ a ∀ a, b ∈ holds, the group is called Abelian
(GTAbelianQ). The properties of a group can be depicted in form of a multiplication table, where the result of the multiplication of all possible combinations of group elements is presented. In Figure 3.2, the multiplication table
of the point group C4v is shown (GTMultTable). The multiplication table of
an Abelian group is symmetric. As can be verified, this is not the case for C4v
(IC2a ◦ IC2 y ≠ IC2 y ◦ IC2a ).
Per definition, a group is closed under its multiplication. For finite groups it
follows that a finite number of multiplications p of a group element a with itself
leads to the identity element E, a p = E. The smallest natural number p is called
the order of the element p = ord(a) (GTOrderOfElement).
1) In Mathematica the symbol ◦ is obtained by \[SmallCircle] or ESC sc ESC. You will find it also
on the GTPack palette.
3 Basics Abstract Group Theory
Figure 3.2 Multiplication table of the non-Abelian point group C4v calculated using GTMultTable.
Theorem 5 (Fermat’s theorem). For every element a ∈ and g = ord() the relation a g = E holds.
A set of group elements of is called the generators of , if all group elements can
be found by a finite number of multiplications of the generators. The minimal
number of generators is called the basis of the group. The rank of the basis is the
number of generators in the basis. The choice of the basis is not unique. GTPack
offers commands to find the generators of a group or to construct the group from a
set of generators (GTGenerators,GTGroupFromGenerators). A group is called cyclic
group (GTCyclicQ), if it can be generated from one element only. An example of the
application of GTGenerators and GTGroupFromGenerators is shown in Figure 3.3.
Task 6 (Generators of a group). Install the point group D4h . Demonstrate that
the choice of generators is not unique. Find three different sets of generators.
A subset ′ ⊂ that satisfies the definition of a group is called a subgroup of . The
subsets {E} and are trivial subgroups of . Within GTPack, subgroups of a group
can be found using GTGetSubGroups (see Example 3.6). A test of the relation ′ ⊂
can be performed with GTSubGroupQ.
Theorem 6 (Lagrange’s theorem). If ′ is a subgroup of (′ ⊂ ) then its order
g ′ is a factor of the order g = ord() of . The integer g∕g′ is called the index of ′
in .
Example 5 (Installation of a group from its multiplication table). Alternatively to
GTInstallGroup, GTTableToGroup can be used to install a finite group from its mul-
3.1 Basic Definitions
Figure 3.3 Calculating the generators of the group C4v using GTGenerators. The group was
installed in Figure 3.1.
Figure 3.4 Installation of KLEIN’s four group from the multiplication table.
tiplication table within GTPack. This approach is especially useful if a group is not
implemented within GTPack. As an example, the Klein four group is considered.
With four elements, it represents the smallest noncyclic group. An example of the
application of GTTableToGroup to the Klein four group is shown in Figure 3.4. By
3 Basics Abstract Group Theory
applying GTTableToGroup, the group elements are represented by permutation
matrices. The defined symbols of the elements can be used afterwards within the
GTPack commands.
Task 7 (Properties of groups). A set {E, C2x , C2 y , C2z } is given. Proof that the set
forms a group. Is the group Abelian? Proof that the multiplication table of this
group has the same structure as the multiplication table of the Klein four group.
Isomorphism and Homomorphism
A group is well defined by its inner structure, which is independent of the nature of the elements (matrices, operators, permutations, . . . ) or the kind of group
multiplication. The investigation of mappings between groups allows one to study
their relationship, differences, or similarity.
Definition 6 (Homomorphism). A mapping f : → ′ of a group (, ∙) to a group
(′ , ◦) is called homomorphism if
f (a1 ∙ a2 ) = f (a1 ) ◦ f (a2 ) ,
for all a1 , a2 ∈ .
The set of elements of mapped to E′ ∈ ′ (the identity of ′ ) is called the kernel
K of the homomorphism.
If is a subgroup of ( ⊂ ) and f : → ′ is a homomorphism, then ′ is also
a subgroup of ′ . A homomorphic mapping of a group to one of its subgroups
⊂ is called endomorphism.
Definition 7 (Isomorphism). A bijective homomorphism f : → ′ is called isomorphism, ≅ ′ . If ≡ ′ , the homomorphism is called automorphism.
Example 6 (Subgroups of C4v). The multiplication table of{C4v is given in Fig}
, C2z is
ure 3.2. It can be concluded that the set of elements C4 = E, C4z , C4z
a subgroup
of}C4v , C4 ⊂ C4v . Other subgroups of C4v are C2 = E, C2z and
C s = E, IC2 y . Since they have a similar multiplication table, C s and C2 are isomorphic, C s ≅ C2 . An endomorphic mapping f : C4v → C s is given by
, C2z → E ,
IC2x , IC2 y , IC2a , IC2b → IC2 y . (3.2)
f : E, C4z , C4z
The subgroups can be found with GTGetSubGroups. The relationships can be
checked by means of GTSubGroupQ.
3.2 Structure of Groups
Structure of Groups
Structure of groups
Finds the conjugate element
Constructs the classes of a group
Gives class multiplication coefficients
Calculates the class multiplication table
Finds the center of the group
Calculates left cosets of with respect to ⊂
Calculates right cosets of with respect to ⊂
Calculates the normalizer of ′ with respect to a supergroup
Constructs conjugacy classes for ′ ⊂
Gives all normal divisors of
Gives true, if ⊂ is a normal divisor of
Equivalence relations are important to classify elements of a given set . One can
think of assigning a specific property (e.g., “length” or “color”) to every element of
. The element a is said to be equivalent to b (a ∼ b), if the elements share this
Definition 8 (Equivalence relation). A relation ∼ on a set is called the equivalence relation if the following conditions are satisfied for all elements a, b, c ∈
1. reflexivity: a ∼ a
2. symmetry: if a ∼ b then also b ∼ a
3. transitivity: if a ∼ b and b ∼ c, then also a ∼ c
By means of an equivalence relation a set is subdivided into subsets called
classes or equivalence classes. An equivalence class [[a]] of a ∈ contains all
elements a i ∈ , which are equivalent to a:
[[a]] := a i ∣ a i ∈ ∧ a ∈ ∧ a i ∼ a .
Because of transitivity of the equivalence relation, every element of the equivalence class may be used as a representant,
a ∼ b ⇒ [[a]] = [[b]] .
All equivalence classes are disjoint,
a ≁ b ⇒ [[a]] ∩ [[b]] = ∅ .
3 Basics Abstract Group Theory
The set can be obtained by a set sum of all inequivalent equivalence classes,
[[a i ]] , [[a i ]] ∩ [[a j ]] = ∅ , a i ≁ a j .
A valid equivalence relation of a group is given by the so-called conjugation operation.
Definition 9 (Conjugate element). An element a ∈ is said to be the conjugate
element of an element b ∈ if there exists an element c ∈ such that
a = c ◦ b ◦ c−1 .
The equivalence relation in terms of conjugate elements will be important throughout the book. If not stated otherwise, the term classes will be used in this connection.
Definition 10 (Classes). A class is a complete set of conjugate elements:
a = a i ∣ a, b ∈ ∧ a i = b ◦ a ◦ b −1 ∧ a fixed .
The associated commands for deriving conjugate elements and classes in GTPack
are GTConjugateElement and GTClasses. Classes exhibit the following properties.
Theorem 7 (Classes of groups). For a group the following statements about
classes hold.
1. Every element of belongs to exactly one class of , i.e., an element of cannot
be a member of two different classes.
2. The identity element E forms a class on its own.
3. In an Abelian group each element forms a class on its own.
4. A class contains only elements of the same order.
Proof. The conjugation is an equivalence relation. Thus, symmetry and transitivity lead to the fact that classes are disjoint subsets of a group. That the identity
E forms a class on its own follows from
c ◦ E ◦ c−1 = c ◦ c−1 = E .
Similarly, for Abelian groups, where the multiplication is commutative, it follows
c ◦ b ◦ c−1 = c ◦ c−1 ◦ b = b .
The last consequence follows from the fact that if a ∼ b then also a k ∼ b k . Assuming a = c ◦ b ◦ c−1 , this can be verified via,
a k = (c ◦ b ◦ c−1 )k = (c ◦ b ◦ c−1 ) ◦(c ◦ b ◦ c−1 ) ◦ ⋯ = c ◦ b k ◦ c−1 .
3.2 Structure of Groups
Thus, if ord(a) = n all elements of the class have order n.
A special subgroup of a group is given by the set of all self-adjoint elements.
Definition 11 (Center). An element of that commutes with all elements of the
group a = z i ◦ a ◦ z−1
∀ z i ∈ is said to be self-adjoint. The set of all self-adjoint
elements forms a subgroup, called the center of the group.
Within GTPack the center of a group can be obtained using GTCenter. An example
for the group C4v is shown in Example 7.
Definition 12 (Class multiplication). The multiplication of two classes is given by
the multiplication of all elements of the first class with all elements of the second
class, by taking into account the order of the operation,
Ki ⋅ K j =
h i j,l K l .
The coefficients h i j,l are called class-multiplication coefficients. The result of class
multiplication consists of complete classes.
An example for class multiplication within GTPack is presented in Example 7.
In connection to the term subgroup, the group is called the supergroup or
covering group of ′ if ⊃ ′ .
Definition 13 (Normalizer). The normalizer of a group with respect to one
of its supergroups is the set of elements g ∈ mapping onto itself.
( |) := g ∈ | g g −1 = .
The group is a subgroup of the normalizer and its covering group ⊆
( |) ⊆ . GTNormalizer calculates the normalizer of a group with respect
to its covering group (cf. Figure 3.6). Another structure, based on the definition
of conjugate elements is the so-called conjugacy class of a subgroup.
Definition 14 (Conjugated subgroups). Let be a subgroup of ( ⊂ ). The
set of all subgroups conjugated to is given by
[ ] := g g −1 , g ∈ .
Conjugated subgroups are isomorphic to .
The concept of conjugated subgroups will be used, e.g., in the discussion of sitesymmetry in Section 4.2.3.
Example 7 (Classes and class multiplication in C4v ). Figure 3.5 shows how to
calculate the center, the classes, and the class multiplication table for the point
group C4v by means of the GTPack commands GTCenter, GTClasses, and GTClassMultTable. C4v was introduced in Example 4. It has five classes {E}, {C2z }, {IC2x ,
}. The center is given by {E, C2z }.
IC2 y }, {IC2a , IC2b }, and {C4z , C4z
3 Basics Abstract Group Theory
Figure 3.5 The application of GTCenter, GTClasses, and GTClassMultTable for calculating the
center, the classes, and the class multiplication table of the point group C4v .
Example 8 (Calculating the normalizer within GTPack). Figure 3.6 illustrates how
to calculate the normalizer of the group C S with respect to the covering group
D3h . In total, D3h exhibits 14 subgroups, as can be checked with GTGetSubGroups.
One of the subgroups is the group C S = {Ee, IC2 y }. The normalizer (C S |D3h )
is given by the group C2v = {Ee, C2x , IC2 y , IC2z }.
Cosets and Normal Divisors
With respect to a particular subgroup, a group can be subdivided into left and
right cosets. Elements within a coset can be regarded as equivalent in terms of an
equivalence relation.
3.2 Structure of Groups
Figure 3.6 Calculating the normalizer of C s with respect to the covering group D 3h .
Definition 15 (Cosets). is a subgroup of ( ⊆ ). The following subsets of
are denoted as left cosets a and right cosets a, respectively,
a = {a ◦ p ∣ p ∈ ⊆ } ,
a ∈,
a = { p ◦ a ∣ p ∈ ⊆ } ,
In general, the left and right cosets with respect to a particular element a ∈ do
not necessarily contain the same elements. However, the following theorems hold
for both, left and right cosets. They will be given without proof.
Theorem 8 (Theorems on left and right cosets). Let ⊂ be a subgroup of .
Then, the following statements on left and right cosets hold.
1. For a ∈ , it follows a = a = .
2. For a ∉ , a and a do not form a group.
3. Every element of belongs to exactly one left (right) coset, i.e., two cosets are
either identical or disjoint.
4. Every coset has the same number of elements n = ord( ).
5. Is b ∈ a then b is identical to a . Every element of the coset can be used
as a representant of the coset.
6. The number of left (right) cosets of in is called index and denoted by
ind( ). The index is given by ind( ) = ord()∕ ord( ).
3 Basics Abstract Group Theory
Task 8 (Theorems on cosets). Install the point group O. Proof that the group
D4 is a proper subgroup. Show, using D4 as the subgroup, that all statements in
Theorem 8 are correct.
By taking into account all elements of a group for the decomposition into left
(right) cosets, the set is subdivided into disjoint sets. A special case is given if
the decomposition into left cosets is equal to the decomposition into right cosets.
Definition 16 (Invariant subgroup). A subgroup ⊂ is called invariant subgroup ⊲ (normal subgroup, normal divisor) if x ◦ ◦ x−1 = for all x ∈ . In
other words, the right and the left cosets of an invariant subgroup of the group
are identical.
Theorem 9 (Classes and invariant subgroups). A subgroup ∈ forms an invariant subgroup if and only if consists of complete classes of .
Proof. ⇒: If consists of complete classes of it follows that next to s ∈ also
g ◦ s ◦ g −1 ∈ , which agrees with the definition of an invariant subgroup.
⇐: Conversely, let’s assume that is an invariant subgroup but does not consist
of complete classes of . That means it is possible to find an element g ∈ and
Figure 3.7 Calculating the left and right cosets of the subgroups C s and C2v with respect to
C4v by applying GTLeftCosets and GTRightCosets.
3.2 Structure of Groups
s ∈ with the property g ◦ s ◦ g −1 ∉ , which is in contradiction to the assumption
that is an invariant subgroup.
A group is said to be simple, if it does not contain a proper invariant subgroup. If
the group does not possess a proper Abelian invariant subgroup it is called semisimple. A composite group has a proper invariant Abelian subgroup.
The center of a group is an invariant subgroup. The group ′ is a normal divisor
of its normalizer with respect to a corresponding supergroup : ′ ⊲ (′ |).
Example 9 (Coset decomposition of C4v ). The decomposition of a group into
left and right cosets can be calculated using GTLeftCosets and GTRightCosets. In
Figure 3.7, an example for the group C4v of the two subgroups C s = {Ee, IC2a } and
C2v = {Ee, C2z , IC2x , IC2 y } is shown. For the group C s with group order ord(C s ) =
2 four left and four right cosets can be found. As can be verified, the left and right
cosets differ from each other. In the case of C2v with ord(C2v ) = 4 the two left and
two right cosets are the same. Hence, C2v is an invariant subgroup of C4v .
Example 10 (Invariant subgroups of C4v ). Using GTInvSubGroupQ, it can be
checked if S ⊂ is an invariant subgroup of . To obtain all invariant subgroups
of a group , GTInvSubGroups can be used. An example for the group C4v is shown
in Figure 3.8. In total, four invariant subgroups are found.
Figure 3.8 Determining the invariant subgroups of C4v using GTInvSubGroups.
3 Basics Abstract Group Theory
Task 9 (Properties of normal divisors). From the definition of normal divisors the
following statements follow:
∙ All subgroups ⊆ with index 2 are normal divisors.
∙ All Abelian subgroups of are normal divisors.
Proof the two statements.
Quotient Groups
Quotient groups
Checks if ⊂ can form a quotient group ∕
Calculates the multiplication table of the quotient group ∕
In terms of left cosets a group can be decomposed with respect to a subgroup
⊂ , as follows,
= g1 + g2 + ⋯ + g N ,
N = ind() .
The elements g i ∈ are the coset representatives. The set ∕ = {g1 , g2 , … ,
g N } is called the quotient set. Considering the quotient set with respect to an
invariant subgroup , it is possible to induce a group structure. Since is an
invariant subgroup, left and right cosets with respect to are similar and the
following equation holds,
ai ◦ a j = ai ◦ a j .
The above relation defines a group multiplication between cosets in terms of the
coset representatives. In this case, ∕ forms a group, called the quotient group. 2)
The identity element of the quotient group is the normal divisor . The order of
the quotient group is g∕g = ord( ).
Example 11 (The quotient group C4v ∕C4 ). The group C4 = {Ee, C4z , C4z
, C2z }
is an invariant subgroup of the group C4v . Since ord(C4 ) = 4, there are two cosets
containing four elements each. In terms of the coset decomposition into left
cosets, the group C4v can be written as
C4v = Ee C4 + IC2a C4 .
A GTPack example is shown in Figure 3.9. The quotient group is constructed using
GTQuotientGroup. The group C4v ∕C4 is isomorphic to the group C2 .
2) Quotient groups are also often called factor groups.
3.3 Quotient Groups
Figure 3.9 Calculate the multiplication table of the quotient group C4v ∕C4 using GTQuotientGroup.
By means of a homomorphism f : → it is possible to map a group to a
smaller group . The kernel (i.e., the set of all elements g ∈ mapped to the identity element of ) of the homomorphism is an invariant subgroup of the original
group. The relation between and ∕ is determined by the homomorphism
Theorem 10 (Homomorphism theorem). Let f : → be a homomorphism
with kernel ( ⊲ ). Then, the homomorphism f : ∕ → , defined by f (g i ) =
f (g i ), is an isomorphism and ∕ ≅ .
Example 12 (Homomorphism theorem). Choosing = D3h and = C s , a homomorphism can be defined by mapping all proper rotations within D3h to Ee
and all improper rotations to IC2 y . Hence, D3 (the group containing all the proper rotations of D3h ) is a normal divisor of D3h (D3 ⊲ D3h ) and represents the kernel
of the mapping. The respective left and right cosets in ∕ are
) (
E, C2x , C2A , C2B , C3z , C3z
, IC2 y , IC2z , IC2C , IC2D , IC6z , IC6z
This quotient group is isomorphic to C s , i.e., D3h ∕D3 ≅ C s .
3 Basics Abstract Group Theory
Product Groups
Product groups
Checks if = 1 × 2 or = 1 ⋊ 2
Forms the product of the groups 1 and 2
In the previous section, the notion of a quotient group was introduced. Subsequently, the concept of product groups is discussed, where two groups are combined to form a larger group.
Definition 17 (Direct product group). Consider the groups 1 and 2 together
with the corresponding multiplications ◦ and ∙. The set of pairs (g1 , g2 ) ∈ 1 × 2
with g1 ∈ 1 and g2 ∈ 2 together with the multiplication relation ⋅ : 1 × 2 →
1 × 2 defined by
(g1 , g2 ) ⋅ (g1′ , g2′ ) = (g1 ◦ g1′ , g2 ∙ g2′ )
g1 , g1′ ∈ 1 ,
g2 , g2′ ∈ 2 ,
is called the direct product group.
From the structure of direct product groups it follows that the group order is given
ord(1 × 2 ) = ord(1 ) ord(2 ) .
Furthermore, 1 }× 2 contains the subgroups 1 = (g1 , E2 )|g1 ∈ 1 and 2 =
(E1 , g2 )|g2 ∈ 2 , which are isomorphic to the groups 1 and 2 , respectively.
The elements of the two subgroups 1 and 2 commute and have only the identity
element in common, 1 2 = (E1 , E2 ). Consequently, every element of is a
product of an element of 1 and 2
(g1 , g2 ) = (g1 , E2 )(E1 , g2 ) .
The direct product group of two finite groups is again finite. Vice versa, in some
cases finite groups can be expressed in terms of the direct product of two subgroups. For example, the crystallographic point group O h , which is isomorphic to
the direct product of T d and C s .
Theorem 11 (Direct product). A group is isomorphic to the direct product 1 ×
2 if the following conditions are satisfied:
1 and 2 are two proper subgroups of ,
for all g1 ∈ 1 and g2 ∈ 2 the relation g1 ◦ g2 = g2 ◦ g1 holds,
1 2 = {E},
every element of can be written as an element of 1 with an element of 2 .
3.4 Product Groups
If the group is isomorphic to 1 × 2 , we simply write = 1 × 2 in the following. Furthermore, if 1 is an invariant subgroup (1 ⊲ ) the product is called
a semidirect product group and the notation = 1 ⋊ 2 is used. Among others, the structure of semidirect product groups occurs for symmorphic space
groups, where the space groups can be written as the semidirect product of the
group of pure translations and a point group. Space groups will be introduced in
more detail in Section 4.2. The semidirect product structure allows for the application of the so-called method of induction for the calculation of matrices for
irreducible representations as will be discussed in Section 5.7. The representation theory for symmorphic space groups as a consequence will be introduced in
Section 6.4.
Example 13 (Product groups in GTPack). In Example 11 it was shown that the
quotient group D3h ∕D3 is isomorphic to the group C s . Analogously, it is possible
to construct the group D3h as a direct product of D3 and C s . An example by means
of GTProductGroup is shown in Figure 3.10. Since D3 is an invariant subgroup of
D3h , it is possible to express D3h as the semidirect product D3h = D3 ⋊ C s .
Figure 3.10 Application of GTProductGroup to represent D 3h as the semidirect product
D 3h = D 3 ⋊ C s .
Discrete Symmetry Groups in Solid-State Physics and Photonics
Die Kristallographie, als Wissenschaft betrachtet, gibt zu ganz eigenen Ansichten Anlaß. Sie ist nicht produktiv, sie ist nur sie selbst und hat keine
Folgen, besonders nunmehr, da man so manche isomorphische Körper
angetroffen hat, die sich ihrem Gehalte nach ganz verschieden erweisen. Da
sie eigentlich nirgends anwendbar ist, so hat sie sich in dem hohen Grade in
sich selbst ausgebildet.
J.W. von Goethe
(Wilhelm Meisters Wanderjahre, 3. Buch, Aus Makaries Archiv 105)
Condensed matter systems exhibit a specific symmetry. Throughout this chapter an overview about several symmetry groups present in such systems is given. Point groups describe the local symmetry of an atom or impurity within a
crystal or an atomic cluster. In three-dimensional crystals, for example, only
32 distinct point groups can be found. Some of them are isomorphic to each
other. In 2 dimensions this number reduces to 10. Space groups describe the
symmetry of lattice-periodic systems, i.e., crystals. Besides rotational symmetries these groups contain an infinite set of translational symmetries, mediated by the underlying lattice. Point and space groups can be extended by incorporating additional degrees of freedom, such as a magnetic moment. This
leads to the concept of color or Shubnikov groups. The chapter closes with
a discussion of noncrystallographic groups present in carbon nanotubes and
With respect to the structure of crystalline materials and the corresponding group
theoretical tools a large amount of software and databases are freely available.
Prominent examples are given by:
Bilbao crystallographic server [44]
International Tables for Crystallography [45]
Crystallography Open Database [46]
Inorganic Crystal Structure Database (ICSD) FIZ Karlsruhe [47]
The Cambridge Structural Database (CSD) [48].
4 Discrete Symmetry Groups in Solid-State Physics and Photonics
The strength of GTPack lies in the connection between provided group theory
commands and the programmable Mathematica environment. Interfaces between available databases and GTPack are developed. Structures can be imported
and exported, e.g., to allow for the visualization of structures by means of tools
like VESTA (Visualization for Electronic and STructural Analysis) [49]. The central part of this chapter is to provide the structural basics for Part Two and Three
of this book.
Point Groups
Point groups
Gives a graphical representation of symmetry elements
Gives a list of all currently installed symmetry elements
Installs a new axis together with the related symmetry elements
Plots the subgroup relation within a list of groups
Plots all sub- and supergroups of a group
Switches notation between Schönflies and Hermann–
Plots the subgroup relationships of the 32 crystallographic
point groups
Notation of Symmetry Elements
Two notations are commonly used for the symmetry elements of point groups.
The Schönflies 1) symmetry notation is frequently applied in connection to molecules and within solid-state physics. In contrast, the Hermann–Mauguin 2) notation is used within the International Tables For X-ray Crystallography [40, 50].
The basics of both notations are summarized in Table 4.1.
The notation used in GTPack is based on the Schönflies notation. In the cube
shown in Figure 4.1a, twofold rotational axes connect the origin of the coordinate system with the edges of the cube (a, b, c, d, e, f ). Additionally, threefold
rotational axes passing through the origin and the corners of the cube (α, β, γ, δ),
while the coordinate axes are passing through the faces of the cube are fourfold
rotational axes. For the hexagon shown in Figure 4.1b, twofold rotational axes are
given by the coordinate axes and the axes connecting the origin and the points
A, B, C, and D.
1) Arthur Moritz Schönflies (1853–1928) was a Mathematician. He became known by his
contributions to the field of crystallography.
2) Carl Hermann (1898–1961) was a German physicist and crystallographer. Charles-Victor
Mauguin (1878–1958) was a French mineralogist.
4.1 Point Groups
Table 4.1 SCHÖNFLIES (SFL) notation and HERMANN–MAUGUIN (HM) notation of point group
elements. A correspondence of the labeling of rotations is also given in Table 4.2.
Rotations through 2π∕n
C −1
n n−1
Rotations through −2π∕n
Reflection plane
Horizontal reflection plane, the reflection plane is perpendicular to the axis
of highest rotational symmetry
Vertical reflection plane, the reflection plane contains the axis of highest rotational symmetry
Inversion: x → −x, y → − y, z → −z
I Cn
Rotoinversion, a rotation is followed by an inversion (I C ni = C ni I)
Figure 4.1 Illustration of high symmetry points and axes for a cube and for a hexagon. The
labels are used for the denotation of symmetry elements in GTPack. (a) High symmetry points
of a cube; (b) high symmetry points of a hexagon.
In GTPack reflection planes and improper rotations are expressed by a combination of proper rotations and inversion. Vice versa, the inversion itself can be
expressed as a combination of a twofold rotation and a reflection. Consider for
example a mirror operation σ h with the mirror plane perpendicular to the z-axis.
The resulting coordinate transformation is x → x, y → y, and z → −z. A twofold
rotation C2z transforms x → −x, y → − y, and z → z. Hence, a composition of
both transformations leads to the inversion I transforming x → −x, y → − y, and
z → −z. Thus, the element σ h can also be written as IC2z . The concept can be extended to rotoreflections, combinations of rotations and reflections. For fourfold
and threefold rotoreflections it follows,
= IC4z
S 4z = σ h C4z = IC2z C4z = IC4z
= IC6z
= IC6z
S 3z = σ h C3z = IC2z C3z = IC6z
4 Discrete Symmetry Groups in Solid-State Physics and Photonics
Table 4.2 Comparison of proper and improper rotations in SCHÖNFLIES (SFL), HERMANN–
MAUGUIN notation (HM), and the notation of GTPack (z-axis used as the reference).
Proper rotations
Improper rotations
On the basis of the relations, derived in (4.1), Table 4.2 compares the notation
of proper and improper rotations in the different notation schemes.
The following basic operations occur in the 32 crystallographic point groups
(powers of the elements are not listed). The notation is chosen with respect to
Figure 4.1.
∙ C3α , C3β , C3γ , C3δ : 2π∕3 rotations. The elements are of order 3. The inverse
elements are C3α
∙ C2x , C2 y , C2z , … C2 f : π rotations. The elements are of order 2 and therefore
inverse to itself.
∙ C4x , C4 y , C4z : π∕2 rotations. The elements are of order 4. The inverse elements
, C4−1y , C4z
are C4x
∙ C6z , C3z : π∕3 and 2π∕3 rotations about the z-axis of the hexagon. The inverse
and C3z
, respectively.
elements are C6z
∙ C2A , C2B , C2C , C2D : π rotations of a hexagon. The elements are inverse to
The improper rotations are defined by multiplication with I. All symmetry elements C n and IC n are automatically installed in GTPack for n = 2, 3, … , 9 and
the axes according to Figure 4.1. Additional symmetry elements for additional axes can be installed using GTInstallAxis. All currently installed axes can be shown
with GTAllSymbols. The elements can also be found on the palette of GTPack. Although, symmetry elements C nm , n = 2, … , 9, m = 1, … , (n − 1) are installed in
GTPack, only 2-, 3-, 4-, and 6-fold rotation axes can be members of crystallographic point groups due to the so-called crystallographic restriction. Consider as an
example a regular pentagon, having a 5-fold rotation axis. It is not possibl |
4bf9d14f4481be7e | Energy Shift of H-Atom Electrons Due to Gibbons-Hawking Thermal Bath
The electromagnetic shift of energy levels of H-atom electrons is determined by calculating an electron coupling to the Gibbons-Hawking ectromagnetic field thermal bath. Energy shift of electrons in H-atom is determined in the framework of non-relativistic quantum mechanics.
Share and Cite:
Pardy, M. (2016) Energy Shift of H-Atom Electrons Due to Gibbons-Hawking Thermal Bath. Journal of High Energy Physics, Gravitation and Cosmology, 2, 472-477. doi: 10.4236/jhepgc.2016.24041.
The Gibbons-Hawking effect is the statement that a temperature can be associated to each solution of the Einstein field equations that contains a causal horizon. It is named after Gary Gibbons and Stephen William Hawking.
Schwarzschild spacetime contains an event horizon and so can be associated with temperature. In the case of Schwarzschild spacetime this is the temperature T of a blackhole of mass M, satisfying T/M.
De Sitter space which contains an event horizon has the temperature T proportional to the Hubble parameter H. We consider here the influence of the heat bath of the Gibbons-Hawking photons on the energy shift of H-atom electrons.
The considered problem is not in the scientific isolation, because some analogical problems are solved in the scientific respected journals. At present time it is a general conviction that there is an important analogy between black hole and the hydrogen atom. The similarity between black hole and the hydrogen atom was considered for instance by Corda [1] , who discussed the precise model of Hawking radiation from the tunnelling mechanism. In this article an elegant expression of the probability of emission is given in terms of the black hole quantum levels. So, the system composed of Hawking radiation and black hole quasi-normal modes introduced by Corda [2] is somewhat similar to the semiclassical Bohr model of the structure of a hydrogen atom.
The time dependent Schrödinger equation was derived for the system composed by Hawking radiation and black hole quasi-normal modes [3] . In this model, the physical state and the correspondent wave function are written in terms of a unitary evolution matrix instead of a density matrix. Thus, the final state is a pure quantum state instead of a mixed one and it means that there is no information loss. Black hole can be well defined as the quantum mechanical systems, having ordered, discrete quantum spectra, which respect ’t Hooft’s assumption that Schrödinger equations can be used universally for all dynamics in the universe.
Thermal photons by Gibbons and Hawking form so called blackbody, which has the distribution law of photons derived in 1900 by Planck [4] - [6] . The derivation was based on the investigation of the statistics of the system of oscillators inside of the blackbody. Later Einstein [7] derived the Planck formula from the Bohr model of atom where electrons have the discrete energies and the energy of the emitted photons which are given by the Bohr formula ħω =, where are the initial and final energies of electrons.
Now, let us calculate the modified Coulomb potential due to blackbody. The starting point of the determination of the energy shift in the H-atom is the potential, which is generated by nucleus of the H-atom. The potential at point, evidently is [8] [9] :
If we average the last equation in space, we can eliminate so called the effective potential in the form
where is the average value of the square coordinate shift caused by the thermal photon fluctuations. The potential shift follows from Equation (2):
The corresponding shift of the energy levels is given by the standard quantum mechanical formula [8]
In case of the Coulomb potential, which is the case of the H-atom, we have
Then for the H-atom we can write
where we used the following equation for the Coulomb potential
Motion of an electron in electric field is evidently described by elementary equation
which can be transformed by the Fourier transformation into the following equation
where the index ω concerns the Fourier component of above functions.
On the basis of the Bethe idea of the influence of vacuum fluctuations on the energy shift of electron [10] , the following elementary relations were used by Welton [9] and Akhiezer [8] and Berestetzkii et al. [11] :
and in case of the thermal bath of the blackbody, the last equation is of the following form [12] :
because the Planck law in (11) was written as
where the term
is the average energy of photons in the blackbody and
is the number of electromagnetic modes in the interval ω, ω + dω.
where involves the number of frequencies in the interval (ω, ω + dω).
So, after some integration, we get
where F(ω) is the primitive function of the omega-integral
which cannot be calculated by the elementary integral methods and it is not involved in the tables of integrals.
Frequencies and will be determined with regard to the existence of the fluc- tuation field of thermal photons. It was determined in case of the Lamb shift [9] [10] by means of the physical analysis of the interaction of the Coulombic atom with the surrounding fluctuation field. We suppose here that the Bethe and Welton arguments are valid and so we take the frequencies in the Bethe-Welton form. In other words, electron cannot respond to the fluctuating field if the frequency which is much less than
the atom binding energy given by the Rydberg constant [13] . So, the
lower frequency limit is
where α ≈ 1/137 is so called the fine structure constant.
The specific form of the second frequency follows from the elementary argument, that we expect the effective cutoff, since we must neglect the relativistic effect in our nonrelativistic theory. So, we write
If we take the thermal function of the form of the geometric series
and the first thermal contribution is thermal contribution
Then, with Equation (6)
where [14]
Let us only remark that the numerical form of Equation (23) has deep experimental astrophysical meaning.
In article by author [15] , which is the continuation of author articles on the finite- temperature Čerenkov radiation and gravitational Čerenkov radiation [16] [17] , the temperature Green function in the framework of the Schwinger source theory was derived in order to determine the Coulomb and Yukawa potentials at finite-temperature using the Green functions of a photon with and without radiative corrections, and then by considering the processes expressed by the Feynman diagrams.
The determination of potential at finite temperature is one of the problems which form the basic ingredients of the quantum field theory (QFT) at finite temperature. This theory was formulated some years ago by Dolan and Jackiw [18] , Weinberg [19] and Bernard [20] in 1974 and some of the first applications of this theory were the calculations of the temperature behavior of the effective potential in the Higgs sector of the standard model.
Information on the systematic examination of the finite temperature effects in quantum electrodynamics (QED) at one-loop order was given by Donoghue, Holstein and Robinett [21] . Partovi [22] in 1994 discussed the QED corrections to Planck’s radiation law and photon thermodynamics,
A similar discussion of QED was published by Johansson, Peressutti and Skagerstam [23] and Cox et al. in 1984 [24] .
Serge Haroche [25] and his research group in the Paris microwave laboratory used a small cavity for the long life-time of photon quantum experiments performed with the Rydberg atoms. We considered here the thermal gas corresponding to the Gibbons- Hawking theory of space-time (at temperature T) as the preamble for new experiments for the determination of the energy shift of H-atom electrons interacting with the Gibbons-Hawking thermal gas. It is not excluded, that the observations performed by the well educated astro-experts will be the Nobelian ones.
Conflicts of Interest
The authors declare no conflicts of interest.
[1] Corda, Ch. (2015) Precise Model of Hawking Radiation from the Tunneling Mechanism. Classical and Quantum Gravity, 32, Article ID: 195007.
[2] Corda, Ch. (2015) Quasi-Normal Modes: The “Electrons” of Blak Holes as “Gravitational Atoms”? Implications for the Black Hole Information Puzzle. Advances in High Energy Physics, 2015, Article ID: 867601.
[3] Corda, Ch. (2015) Time Dependent Schrödinger Equation for Black Hole Evaporation: No Information Loss. Annals of Physics, 353, 71.
[4] Planck, M. (1900) Zur Theorie des Gesetzes der Energieverteilung im Normalspektrum. Verhandlungen der Deutschen Physikalischen Gesellschaft, 2, 237.
[5] Planck, M. (1901) Ueber das Gesetz der Energieverteilung im Normalspectrum. Annals of Physics, 4, 553.
[6] Schöpf, H.-G. (1978) Theorie der Wärmestrahlung in historisch-kritischer Darstellung. Alademie/Verlag, Berlin.
[7] Einstein, A. (1917) Zur Quantentheorie der Strahlung. Physikalische Zeitschrift, 18, 121.
[8] Akhiezer, A.I. and Berestetzkii, V.B. (1953) Quantum Electrodynamics. GITTL, Moscow.
[9] Welton, Th. (1948) Some Observable Effects of the Quantum-Mechanical Fluctuations of the Electromagnetic Field. Physical Review, 74, 1157.
[10] Bethe, H.A. (1947) The Electromagnetic Shift of Energy Levels. Physical Review, 72, 339.
[11] Berestetzkii, V.B., Lifshitz, E.M. and Pitaevskii, L.P. (1999) Quantum Electrodynamics. Butterworth-Heinemann, Oxford.
[12] Isihara, A. (1971) Statistical Mechanics. Academic Press, Pittsburgh.
[13] Rohlf, J.W. (1994) Modern Physics from α to Z0. John Wiley & Sons Ltd., Hoboken.
[14] Sokolov, A.A., Loskutov, Y.M. and Ternov, I.M. (1962) Quantum Mechanics. State Pedago- gical Edition, Moscow. (In Russian)
[15] Pardy, M. (1994) The Two-Body Potential at Finite Temperature. CERN.TH.7397/94.
[16] Pardy, M. (1989) Finite-Temperature Cerenkov Radiation. Physics Letters A, 134, 357-359.
[17] Pardy, M. (1989) Finite-Temperature Gravitational Cerenkov Radiation. International Journal of Theoretical Physics, 34, 951-959.
[18] Dolan, L. and Jackiw, R. (1974) Symmetry Behavior at Finite Temperature. Physical Review D, 9, 3320-3341.
[19] Weinberg, S. (1974) Gauge and Global Symmetries at High Temperature. Physical Review D, 9, 3357-3378.
[20] Bernard. C.W. (1974) Feynman Rules for Gauge Theories at Finite Temperature. Physical Review D, 9, 3312-3320.
[21] Donoghue, J.F., Holstein, B.R. and Robinett, R.W. (1985) Quantum Electrodynamics at Finite Temperature. Annals of Physics, 164, 233-276.
[22] Partovi, H.M. (1994) QED Corrections to Plancks Radiation Law and Photon Thermo- dynamics. Physical Review D, 50, 1118-1124.
[23] Johansson, A.E., Peressutti, G. and Skagerstam, B.S. (1986) Quantum Field Theory at Finite Temperature: Renormalization and Radiative Corrections. Nuclear Physics B, 278, 324-342.
[24] Cox, P.H., Hellman, W.S. and Yildiz, A. (1984) Finite Temperature Corrections to Field Theory: Electron Mass, Magnetic Moment, and Vacuum Energy. Annals of Physics, 154, 211-228.
[25] Haroche, S. (2012) The Secrets of My Prizewinning Research. Nature, 490, 311.
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0879c17dc719dd48 | Harvard University,FAS
Fall 2003
Mathematics Math21b
Fall 2003
Linear Algebra and Differential Equations
Course Head: Oliver knill
Office: SciCtr 434
Email: knill@math.harvard.edu
The harmonic oscillator
Classical Harmonic Oscillator
Energy levels of the classical harmonic oscillator
d2/dx2 f(x) = c2 f(x)
is an ordinary differential equation with solution
f(x)= f(0) cos(c x) + (f'(0)/c) sin(c x).
It can be realized by masspoint attached to a spring. The constant c depends on the gravitational strength, as well as the spring constant. The differential equations can be rewritten as the linear system
dx/dt = y
dy/dt = - x
The flow leaves the level curves of the energy function H(x,y) = x2 + y2 invariant, which are circles.
Quantum Harmonic Oscillator
Eigenfunctions of the quantum harmonic oscillator
Probability densities of the quantum harmonic oscillator
Eigenfunctions fk and probability densities |fk|2.
A[g_]:=Function[y,-D[g[x],x]+x g[x] /. x->y];
f[0_]:=f0; f[k_]:=S[f[k-1]]; f[7][x]
The partial differential equation
d/dt f(t) = i T(f) = i (-d2/dx2 f(x) + x2 f(x) )
is called the Schrödinger equation for the quantum harmonic oscillator. For each eigenvalue Ln with eigenfunction fn of T, the time evolution is fn(t) = exp(i Ln t) fn(0). If f(0) = a1 f1 + a2 f2 + ... then
f(t) = exp(i L1 t) f1(0) + exp(i L2 t) f2(0) + ...
We can write T(f) = P2 + Q2, where P(f)(x) = i f'(x) and Q(f) = x f(x)(x). The Hamiltonian T has the same structure as the Hamiltonian of the harmonic oscillator giving the quantum system the name "quantum harmonic oscillator".
T has the eigenvalues Ln = 1+2n, with eigenvectors fn, which are recursively be defined by fn+1 = (x-D) fn starting with f0(x)=exp(-x2/2). Simlilar as in Fourier theory, it is possible to write any function f for which |f|2 has a finite integral over the real line as a sum of such functions and so solve the Schrödinger evolution explicitely.
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860acb53841c93ef | {\displaystyle m'} where the probability density is zero. is in units of {\displaystyle r} 0 The energy of a photon is equal to Planck’s constant, h=6.626*10-34m2kg/s, times the speed of light in a vacuum, divided by the wavelength of emission. θ This explains also why the choice of p Bohr derived the energy of each orbit of the hydrogen atom to be:[4]. {\displaystyle \alpha } ψ {\displaystyle a_{0}} r {\displaystyle M} {\displaystyle \delta } electrons in greater orbits of an atom have greater velocities. . ϕ ( {\displaystyle p} ℓ The assumptions included: Bohr supposed that the electron's angular momentum is quantized with possible values: and {\displaystyle \psi _{n\ell m}} Atomic hydrogen constitutes about 75% of the baryonic mass of the universe.[1]. ψ state is most likely to be found in the second Bohr orbit with energy given by the Bohr formula. When an electron moves from a higher energy level to a lower one, a photon is emitted. If this were true, all atoms would instantly collapse, however atoms seem to be stable. 2 − , The factor in square brackets in the last expression is nearly one; the extra term arises from relativistic effects (for details, see #Features going beyond the Schrödinger solution). {\displaystyle \mu =m_{e}M/(m_{e}+M)} {\displaystyle \ell =0,1,2,\ldots } ′ It is often alleged that the Schrödinger equation is superior to the Bohr–Sommerfeld theory in describing hydrogen atom. ℓ but different 0 Before we go to present a formal account, here we give an elementary overview. states: An electron in the [15] There are: There are several important effects that are neglected by the Schrödinger equation and which are responsible for certain small but measurable deviations of the real spectral lines from the predicted ones: Both of these features (and more) are incorporated in the relativistic Dirac equation, with predictions that come still closer to experiment. Furthermore, the spiral inward would release a smear of electromagnetic frequencies as the orbit got smaller. {\displaystyle 2\mathrm {p} } ( Coulomb potential enter (leading to Laguerre polynomials in = is the Kronecker delta function. {\displaystyle r_{0}} The most common isotope of hydrogen, termed protium (name rarely used, symbol H), has one proton and no neutrons. 0 π states. , ) -axis. These figures, when added to 1 in the denominator, represent very small corrections in the value of R, and thus only small corrections to all energy levels in corresponding hydrogen isotopes. These are cross-sections of the probability density that are color-coded (black represents zero density and white represents the highest density). d {\displaystyle n} but different The lowest energy equilibrium state of the hydrogen atom is known as the ground state. is. 2 is also indicated by the quantum numbers However, although the electron is most likely to be on a Bohr orbit, there is a finite probability that the electron may be at any other place 2. In everyday life on Earth, isolated hydrogen atoms (called "atomic hydrogen") are extremely rare. α α π {\displaystyle 1/r} , C z wavefunction. "On the Constitution of Atoms and Molecules, Part II.". is the mass of the atomic nucleus. (More precisely, the nodes are spherical harmonics that appear as a result of solving Schrödinger equation in spherical coordinates.). ℓ Exact analytical answers are available for the nonrelativistic hydrogen atom. The amount of energy in each level is reported in eV, and the maxiumum energy is the ionization energy of 13.598eV. electrons in the orbits of an atom have negative energies. R m it failed to predict other spectral details such as, it could only predict energy levels with any accuracy for single–electron atoms (hydrogen–like atoms), the predicted values were only correct to, Although the mean speed of the electron in hydrogen is only 1/137th of the, This page was last edited on 15 November 2020, at 10:50. , . . − Sommerfeld has however used different notation for the quantum numbers. }, The exact value of the Rydberg constant assumes that the nucleus is infinitely massive with respect to the electron. This introduced two additional quantum numbers, which correspond to the orbital angular momentum and its projection on the chosen axis. Experiments by Ernest Rutherford in 1909 showed the structure of the atom to be a dense, positive nucleus with a tenuous negative charge cloud around it. This article is about the physics of the hydrogen atom. {\displaystyle m_{\text{e}}/M,} For all pictures the magnetic quantum number m has been set to 0, and the cross-sectional plane is the xz-plane (z is the vertical axis). It turns out that this is a maximum at These 0 {\displaystyle r} If a neutral hydrogen atom loses its electron, it becomes a cation. The energy levels of hydrogen, including fine structure (excluding Lamb shift and hyperfine structure), are given by the Sommerfeld fine structure expression:[12].
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b49f0e6c957a3204 | : The Electronic Structure Package for Quantum Computers
Jarrod R. McClean Google Inc., Venice, CA 90291 Ian D. Kivlichan Google Inc., Venice, CA 90291 Department of Physics, Harvard University, Cambridge, MA 02138 Damian S. Steiger Google Inc., Venice, CA 90291 Theoretische Physik, ETH Zurich, 8093 Zurich, CH Kevin J. Sung Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109 Yudong Cao Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138 Chengyu Dai Department of Physics, University of Michigan, Ann Arbor, MI 48109 E. Schuyler Fried Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138 Rigetti Computing, Berkeley CA 94710 Craig Gidney Google Inc., Santa Barbara, CA 93117 Thomas Häner Theoretische Physik, ETH Zurich, 8093 Zurich, CH Vojtěch Havlíček Department of Computer Science, Oxford University, Oxford OX1 3QD, UK Cupjin Huang Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109 Zhang Jiang QuAIL, NASA Ames Research Center, Moffett Field, CA 94035 Matthew Neeley Google Inc., Santa Barbara, CA 93117 Jhonathan Romero Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138 Nicholas Rubin Rigetti Computing, Berkeley CA 94710 Nicolas P. D. Sawaya Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138 Kanav Setia Department of Physics, Dartmouth College, Hanover, NH 03755 Sukin Sim Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138 Wei Sun Google Inc., Cambridge, MA 02142 Fang Zhang Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109 Ryan Babbush Google Inc., Venice, CA 90291
August 2, 2021
Quantum simulation of chemistry and materials is predicted to be an important application for both near-term and fault-tolerant quantum devices. However, at present, developing and studying algorithms for these problems can be difficult due to the prohibitive amount of domain knowledge required in both the area of chemistry and quantum algorithms. To help bridge this gap and open the field to more researchers, we have developed the OpenFermion software package (www.openfermion.org). OpenFermion is an open-source software library written largely in Python under an Apache 2.0 license, aimed at enabling the simulation of fermionic models and quantum chemistry problems on quantum hardware. Beginning with an interface to common electronic structure packages, it simplifies the translation between a molecular specification and a quantum circuit for solving or studying the electronic structure problem on a quantum computer, minimizing the amount of domain expertise required to enter the field. The package is designed to be extensible and robust, maintaining high software standards in documentation and testing. This release paper outlines the key motivations behind design choices in OpenFermion and discusses some basic OpenFermion functionality which we believe will aid the community in the development of better quantum algorithms and tools for this exciting area of research.
matrix \xyoptionframe \xyoptionarrow \xyoptionarc \xyoptionps \xyoptiondvips \entrymodifiers=!C\entrybox
Recent strides in the development of hardware for quantum computing demand comparable developments in the applications and software these devices will run. Since the inception of quantum computation, a number of promising application areas have been identified, ranging from factoring Shor (1997) and solutions of linear equations Harrow et al. (2009) to simulation of complex quantum materials. However, while the theory has been well developed for many of these problems, the challenge of compiling efficient algorithms for these devices down to hardware realizable gates remains a formidable one. While this problem is difficult to tackle in full generality, significant progress can be made in particular areas. Here, we focus on what was perhaps the original application for quantum devices, quantum simulation.
Beginning with Feynman in 1982 Feynman (1982), it was proposed that highly controllable quantum devices, later to become known as quantum computers, would be especially good at the simulation of other quantum systems. This notion was later formalized to show how in particular instances, we expect an exponential speedup in the solution of the Schrödinger equation for chemical systems Lloyd (1996); Abrams and Lloyd (1997, 1999); Ortiz et al. (2001); Somma et al. (2002); Aspuru-Guzik et al. (2005). This opens the possibility of understanding and designing new materials, drugs, and catalysts that were previously untenable. Since the initial work in this area, there has been great progress developing new algorithms Kassal et al. (2008); Ward et al. (2008); Whitfield et al. (2011); Cody Jones et al. (2012); Toloui and Love (2013); Hastings et al. (2015); Babbush et al. (2015a, 2016, 2014); Kivlichan et al. (2017a); Sugisaki et al. (2016); Babbush et al. (2017a); Motzoi et al. (2017); Kivlichan et al. (2017b); Babbush et al. (2017b), tighter bounds and better implementation strategies Veis and Pittner (2010, 2014); Cody Jones et al. (2012); McClean et al. (2014); Wecker et al. (2014); Poulin et al. (2015); Babbush et al. (2015b); Reiher et al. (2017); Kivlichan et al. (2017b); Babbush et al. (2017b), more desirable Hamiltonian representations Seeley et al. (2012); Whitfield (2013); Tranter et al. (2015); Moll et al. (2016); Whitfield et al. (2016); Barkoutsos et al. (2017); Havlíček et al. (2017); Bravyi et al. (2017); Zhu et al. (2017); Setia and Whitfield (2017), and proof-of-concept experimental demonstrations Lanyon et al. (2010); Li et al. (2011); Wang et al. (2015); Santagati et al. (2016). Moreover, with mounting experimental evidence, variational and hybrid-quantum classical algorithms Peruzzo et al. (2014); Yung et al. (2014); McClean et al. (2016); Shen et al. (2017); Wecker et al. (2015); Sawaya et al. (2016); McClean et al. (2017); O’Malley et al. (2016); Colless et al. (2017); Kandala et al. (2017); Romero et al. (2017) for these systems have been identified as particularly promising approaches when one has limited resources; some speculate these quantum algorithms may even solve classically intractable instances without error-correction.
However, despite immense progress, much work is left to be done in optimizing algorithms in this area for both near-term and far-future devices. Already this field has seen instances where the difference between naive bounds for algorithms and expected numerics for real systems can differ by many orders of magnitude Wecker et al. (2014); Hastings et al. (2015); Poulin et al. (2015); Babbush et al. (2015b); Reiher et al. (2017). Unfortunately, developing algorithms in this area can require a prohibitive amount of domain expertise. For example, quantum algorithms experts may find the chemistry literature rife with jargon and unknown approximations while chemists find themselves unfamiliar with the concepts used in quantum information. As has been seen though, both have a crucial role to play in developing algorithms for these emerging devices.
For this reason, we introduce the OpenFermion package, designed to bridge the gap between different domain areas and facilitate the development of explicit quantum simulation algorithms for quantum chemistry. The goal of this project is to enable both quantum algorithm developers and quantum chemists to make contributions to this developing field, minimizing the amount of domain knowledge required to get started, and aiding those with knowledge to perform their work more quickly and easily. OpenFermion is an open-source Apache 2.0 licensed software package to encourage community adoption and contribution. It has a modular design and is maintained with strict documentation and testing standards to ensure robust code and project longevity. Moreover, to maximize usefulness within the field, every effort has been made to design OpenFermion as a modular library which is agnostic with respect to quantum programming language frameworks. Through its plugin system, OpenFermion is able to interface with, and benefit from, any of the frameworks being developed for both more abstract quantum software and hardware specific compilation Steiger et al. (2016); Cross et al. (2017a); Smith et al. (2016); Green et al. (2013); Selinger (2004); Wecker and Svore (2014); Valiron et al. (2015); Heckey et al. (2015); JavadiAbhari et al. (2014); Fried et al. (2017).
This technical document introduces the first release of the OpenFermion package and proceeds in the following way. We begin by mapping the standard workflow of a researcher looking to implement an electronic structure problem on a quantum computer, giving examples along this path of how OpenFermion aids the researcher in making this process as painless as possible. We aim to make this exposition accessible to all interested readers, so some background in the problem is provided as well. The discussion then shifts to the core and derived data structures the package is designed around. Following this, some example applications from real research projects are discussed, demonstrating real-world examples of how OpenFermion can streamline the research process in this area. Finally, we finish with a brief discussion of the open source philosophy of the project and plans for future developments.
I Quantum workflow pipeline
In this section we step through one of the primary problems in entering quantum chemistry problems to a quantum computer. Specifically, the workflow for translating a problem in quantum chemistry to one in quantum computing. This process begins by specifying the problem of interest, which is typically finding some molecular property for a particular state of a molecule or material. This requires one to specify the molecule of interest, usually through the positions of the atoms and their identities. However, this problem is also steeped in some domain specific terminology, especially with regards to symmetries and choices of discretizations. OpenFermion helps to remove many of these particular difficulties. Once the problem has been specified, several computational intermediates must be computed, namely the bare two-electron integrals in the chosen discretization as well as the transformation to a molecular orbital basis that may be needed for certain correlated methods. From there, translation to qubits may be performed by one of several mappings. The goal of this section is to both detail this pipeline, and show at each step how OpenFermion may be used to help perform it with ease.
i.1 Molecule specification and input generation
Electronic structure typically refers to the problem of determining the electronic configuration for a fixed set of nuclear positions assuming non-relativistic energy scales. To begin with, we show how a molecule is defined within OpenFermion, then continue on to describe what each component represents.
1from openfermion.hamiltonians import MolecularData
2geometry = [[’H’, [0, 0, 0]],
3 [’H’, [0, 0, 0.74]]]
4basis = ’sto-3g’
5multiplicity = 1
6charge = 0
7h2_molecule = MolecularData(geometry, basis, multiplicity, charge)
Listing 1: Defining a simple H instance in OpenFermion.
The specification of a molecular geometry implicitly assumes the Born-Oppenheimer approximation, which treats the nuclei as fixed point charges, and the ground state electronic energy is a parametric function of their positions. When the positions of the nuclei are specified, the electronic structure problem can be restated as finding the eigenstates of the Hamiltonian operator
where we have used atomic units (i.e. ), represent the positions of electrons, represent the positions of nuclei, and are the charges of nuclei. In our example from OpenFermion, we see that the nuclear positions may be specified by a set of coordinates along with atom labels as
1geometry = [[AtomLabel1, [x_1, y_1, z_1]],
2 [AtomLabel2, [x_2, y_2, z_2]],
3 …]
A reasonable set of nuclear positions for a given molecule can often be given from experimental data, empirical models, or a geometry optimization. A geometry optimization minimizes the lowest eigenvalue of the above Hamiltonian as a function of the nuclear position, to some local, stable optimum. These nuclear configurations are sometimes called equilibrium structures with respect to the level of theory used to calculate them.
As we focus on the non-relativistic Hamiltonian, there is no explicit dependence on electron spin. As a result, the total electronic spin and one of its components (canonically chosen to be the direction) form good quantum numbers for the Hamiltonian. That is, the Hamiltonian can be made block diagonal with separate spin labels for each block. Explicitly including this symmetry offers a number of computational advantages, including smaller spaces to explore, as well as the ability to access excited states that are the lowest energy state within a particular spin manifold using ground state methods Helgaker et al. (2002). In particular, we parameterize these manifolds by the spin multiplicity defined by
where the eigenstates of the operator have, by definition, eigenvalues of . In our example, we have used a singlet state with to specify we are looking for the lowest singlet energy state. This was specified simply by
1multiplicity = 1
and we note that a multiplicity of (a triplet state) might have also been of interest for this particular system.
Given a particular set of nuclei, it is often the case that a system with the same number of electrons as protons (or electronically neutral system) is the most stable. However, sometimes one wishes to study systems with a non-neutral charge such as the cation or anion of the system. In that case, one may specify the charge, which is defined to be the number of electrons in the neutral system minus the number of electrons in the system of interest. In our example, we specified the neutral hydrogen atom, with
1charge = 0
Finally, to specify the computational problem to be solved, rather than simply the molecule itself, it is necessary to define the basis set. This is analogous to, in some loose sense, how one might define a grid to solve a a simple partial differential equation such as the heat-equation. In chemistry, much thought has gone into optimizing specialized basis sets to pack the essential physics of the problems into functions that balance cost and accuracy. A list of some of the most common basis sets expressed as sums of Gaussians can be found in the EMSL basis set exchange Schuchardt et al. (2007). In this case, we specify the so-called minimal basis set “sto-3g”, which stands for 3 Gaussians (3G) used to approximate Slater-type orbitals (STO). This is done through the line
1basis = ’sto-3g’
In future implementations, the code may additionally support parametric or user defined basis sets with similar syntax. At present, the code supports basis sets that are implemented within common molecular electronic structure packages that will be described in more detail in the next section. With the above specifications, the chemical problem is now well defined; however, several steps remain in mapping to a qubit representation.
i.2 Integral generation
After the specification of the molecule as in the previous section, it becomes necessary to do some of the numerical work in generating the problem to be solved. In OpenFermion we accomplish this through plugin libraries that interface with existing electronic structure codes. At present there are supported interfaces to Psi4 Parrish et al. (2017) and PySCF Sun et al. (2017), with plans to expand to other common packages in the future. One needs to install these plugin libraries separately from the core OpenFermion library (instructions are provided at www.openfermion.org that includes a Docker alternative installing all these packages). We made the decision to support these packages as plugins rather than directly integrating them for several reasons. First, some of these packages have different open source licenses than OpenFermion. Second, these packages may require more intricate installations and do not run on all operating systems. Finally, in the future one may wish to use OpenFermion in conjunction with other electronic structure packages which might support methods not implemented in Psi4 and PySCF. The plugin model ensures the modularity necessary to maintain such compatibilities as the code evolves.
Once one has chosen a particular basis and enforced the physical anti-symmetry of electrons, the electronic structure problem may be written exactly in the form of a second quantized electronic Hamiltonian as
The coefficients and are defined by the basis set that was chosen for the problem (sto-3g in our example); however the computation of these coefficients in general can be quite involved. In particular, in many cases it makes sense to perform a Hartree-Fock calculation first and transform the above integrals into the molecular orbital basis that results. This has the advantage of making the mean-field state easy to represent on both a classical and quantum computer, but introduces some challenges with regards to cost of the integral transformation and convergence of the Hartree-Fock calculation for challenging systems. For example, it is necessary to specify an initial guess for the orbitals of Hartree-Fock, the method used to solve the equations, and a number of other numerical parameters that affect convergence.
In OpenFermion we address this problem by choosing a reasonable set of default parameters and interfacing with well developed electronic structure packages through a set of plugin libraries to supply the information desired. For example, using the OpenFermion-Psi4 plugin, one can obtain the two-electron integrals for this molecule in the MO basis in the Psi4 electronic structure code by simply executing
1from openfermionpsi4 import run_psi4
3h2_molecule = run_psi4(h2_molecule,
4 run_mp2=True,
5 run_cisd=True,
6 run_ccsd=True,
7 run_fci=True)
8two_electron_integrals = h2_molecule.two_body_integrals
where the h2_molecule is that defined in the previous section and when run in this way, one may easily access the MP2, CISD, CCSD, and FCI energies. This will return the computed two-electron integrals in the Hartree-Fock molecular orbital basis. Moreover, one has direct access to other common properties such as the molecular orbital coefficients through simple commands such as
1orbitals = h2_molecule.canonical_orbitals
One may also read the - and -electron reduced density matrices from CISD and FCI and the converged coupled cluster amplitudes from CCSD. These values are all conveniently stored to disk using an HDF5 interface that only loads the properties of interest from disk for convenient analysis of data after the fact.
The plugins are designed to ideally function uniformly without exposing the details of the underlying package to the user. For example, the same computation may be accomplished using PySCF through the OpenFermion-PySCF plugin by executing the commands
1from openfermionpyscf import run_pyscf
3h2_molecule = run_pyscf(h2_molecule,
4 run_mp2=True,
5 run_cisd=True,
6 run_ccsd=True,
7 run_fci=True)
9h2_filename = h2_molecule.filename
This allows the user to prepare a data structure representing the molecule that is agnostic to the package with which it was generated. In the future, additional plugins will support a wider range of electronic structure packages to meet the growing demands of users. In the last line of this example, we introduce the save feature of the MolecularData class. We note that this is called by default by the plugins generating the data, but introduce it here to emphasize the fact that this data is conveniently stored for future use. This allows one to retrieve any data about this molecule in the future without an additional calculation through
1from openfermion.hamiltonians import MolecularData
3h2_molecule = MolecularData(filename=h2_filename)
where h2_filename is the filename one chose to store the H data under. By using decorated Python attributes and HDF5 storage, the loading of data in this way is done on-demand for any array-like object such as integrals. That is, loading the molecule in this way has a minimal memory footprint, and large quantities such as density matrices or integrals are only loaded when the attribute is accessed, for example
1one_body_integrals = h2_molecule.one_body_integrals
and is performed seamlessly in the background such that no syntax is required beyond accessing the attribute.
i.3 Mapping to qubits
After the problem has been recast in the second quantized representation, it remains to map the problems to qubits. Electrons are anti-symmetric indistinguishable particles, while qubits are distinguishable particles, so some care must be taken in mapping between the two. There are now many maps that respect the correct particle statistics, and several of the most common are currently implemented within OpenFermion. In particular, the Jordan-Wigner (JW) Jordan and Wigner (1928), Bravyi-Kitaev (BK) Bravyi and Kitaev (2002); Seeley et al. (2012), and Bravyi-Kitaev super fast (BKSF) Setia and Whitfield (2017) transformations are currently supported in OpenFermion. Each of these has different properties with regard to the Hamiltonians that are produced, which may offer benefits to different types of algorithms or experiments. OpenFermion attempts to remain agnostic to the particular transformation preferred by the user.
To give a concrete example, the above Hamiltonian may be mapped to a qubit Hamiltonian through the use of the Jordan-Wigner transformation by
1from openfermion.transforms import get_fermion_operator, jordan_wigner
3h2_qubit_hamiltonian = jordan_wigner(get_fermion_operator(h2_molecule.get_molecular_hamiltonian()))
which returns a qubit operator representing the Hamiltonian after the Jordan-Wigner transformation. This data structure will be explored in more detail later in this paper, but it provides the complete specification of the Hamiltonian acting on qubits in a convenient format.
i.4 Numerical testing
While the core functionality of OpenFermion is designed to provide a map between the space of electronic structure problems and qubit-based quantum computers, some functionality is provided for numerical simulation. This can be helpful for debugging or prototyping new algorithms. For example, one could check the spectrum of the above qubit Hamiltonian after the Jordan-Wigner transformation by performing
1from openfermion.transforms import get_sparse_operator
2from scipy.linalg import eigh
4h2_matrix = get_sparse_operator(h2_qubit_hamiltonian).todense()
5eigenvalues, eigenvectors = eigh(h2_matrix)
which yields the exact eigenvalues and eigenvectors of the H Hamiltonian in the computational basis.
i.5 Compiling circuits for quantum algorithms
The core elements of OpenFermion work toward producing the QubitOperators associated with the problem of interest. These operators are the inputs for a number of quantum algorithms that then translate to quantum circuits; however the specifics of how to best translate an algorithm to a quantum device can be both device and algorithm specific. For this reason, many groups continue to implement their own compiler for this final translation from algorithm to device, and this may be the case for some time. To support this model, OpenFermion has been designed to be platform agnostic with respect to compilation and hardware, and instead we support plugins for this final translation to specific frameworks. At the moment, plugins are supported for the ProjectQ framework Steiger et al. (2016) and the Rigetti Forest Framework; plugins are denoted OpenFermion-ProjectQ and Forest-OpenFermion, respectively. We also provide a module for compiling fermionic circuits to QASM (quantum assembly) strings Cross et al. (2017b). We have plans to expand this support to all major quantum frameworks. We provide an example within these two frameworks below which we hope will illustrate more generally how OpenFermion can be used with other quantum programming platforms.
i.5.1 OpenFermion-ProjectQ Example
Here we walk through a simple example of using the OpenFermion-ProjectQ plugin with OpenFermion and ProjectQ to output a quantum circuit. We will create a simple quantum circuit that is designed to prepare a unitary coupled cluster wavefunction for H on qubits in the Jordan-Wigner encoding.
1from openfermionprojectq import uccsd_trotter_engine, uccsd_singlet_evolution
2from projectq.backends import CommandPrinter
3from projectq.ops import X
5compiler_engine = uccsd_trotter_engine(compiler_backend=CommandPrinter())
6wavefunction = compiler_engine.allocate_qureg(h2_molecule.n_qubits)
This calls the plugin’s pre-built setup for a first order Trotter decomposition of the unitary coupled cluster singles and doubles (UCCSD) operator, and uses a backend that will print the circuit rather than simulate it numerically on a quantum register. This is a convenience routine for assigning rules to the compiler such that it will decompose it into a circuit using only 1- and 2-qubit gates from a standard decomposition set. One can also specialize those decomposition rules for a more restricted gate set associated with a particular hardware platform. With the compiler engine set up, we initialize a specific unitary coupled cluster evolution operator as
1test_amplitudes = [-1.03662149e-08, 5.65340580e-02]
2evolution_operator = uccsd_singlet_evolution(test_amplitudes,
3 h2_molecule.n_qubits,
4 h2_molecule.n_electrons)
which constructs a UCCSD operator that conserves spin symmetry (singlet operator) for the corresponding number of electrons and spin-orbitals acting on a single reference. From here, one need only act the evolution operator on the wavefunction using ProjectQ’s shorthand of “Operator | qubit_register” to denote “Operatorqubit_register”,
1print('Sample Output:')
2for i in range(h2_molecule.n_electrons):
3 X | wavefunction[i]
4evolution_operator | wavefunction
and the CommandPrinter backend we initialized will print the quantum circuit in a format similar, but not identical to, the OpenQASM specification, e.g.
1Sample Output:
2X | Qureg[0]
3X | Qureg[1]
4Rx(1.57079632679) | Qureg[1]
5H | Qureg[3]
6CX | ( Qureg[1], Qureg[2] )
7CX | ( Qureg[2], Qureg[3] )
8Rz(12.566370604) | Qureg[3]
9CX | ( Qureg[2], Qureg[3] )
10CX | ( Qureg[1], Qureg[2] )
11H | Qureg[3]
where we have truncated to the first few lines.
i.5.2 Forest-OpenFermion Example
In this section we describe the interface between OpenFermion and Rigetti’s quantum simulation environment called Forest. The interface provides a method of transforming data generated in OpenFermion to a similar representation in pyQuil. For this example we use OpenFermion to build a four-site single-band periodic boundary Hubbard model and apply first-order Trotter time-evolution to a starting state of two localized electrons of opposite spin.
The Forest-OpenFermion plugin provides the routines to inter-convert between the OpenFermion QubitOperator data structure and the synonymous data structure in pyQuil called a PauliSum.
1from openfermion.ops import QubitOperator
2from forestopenfermion import pyquilpauli_to_qubitop, qubitop_to_pyquilpauli
The FermionOperator in OpenFermion can be used to translate the mathematical expression of the Hamiltonian directly to executable code. While we show how this model can be built in one line using the OpenFermion hamiltonians module below, here we take the opportunity to demonstrate the ease of creating such models for study that could be easily modified as desired. Given the Hamiltonian of the Hubbard system
where indicates nearest-neighbor spatial lattice positions and takes on values of and signifying spin-up or spin-down, respectively, the code to build this Hamiltonian is as follows:
1from openfermion.transforms import jordan_wigner
2from openfermion.ops import FermionOperator, hermitian_conjugated
4hubbard_hamiltonian = FermionOperator()
5spatial_orbitals = 4
6for i in range(spatial_orbitals):
7 electron_hop_alpha = FermionOperator(((2 * i, 1), (2 * ((i + 1) % spatial_orbitals), 0)))
8 electron_hop_beta = FermionOperator(((2 * i + 1, 1), ((2 * ((i + 1) % spatial_orbitals) + 1), 0)))
9 hubbard_hamiltonian += -1 * (electron_hop_alpha + hermitian_conjugated(electron_hop_alpha))
10 hubbard_hamiltonian += -1 * (electron_hop_beta + hermitian_conjugated(electron_hop_beta))
11 hubbard_hamiltonian += FermionOperator(((2 * i, 1), (2 * i, 0),
12 (2 * i + 1, 1), (2 * i + 1, 0)), 4.0)
In the above code we have implicitly used even indexes as spin-orbitals and odd indexes as spin-orbitals. The same model can be built using the OpenFermion Hubbard model builder routine in the hamiltonians module with a single function call.
1from openfermion.hamiltonians import fermi_hubbard
3x_dim = 4
4y_dim = 1
5periodic = True
6chemical_potential = 0
7tunneling = 1.0
8coulomb = 4.0
9of_hubbard_hamiltonian = fermi_hubbard(x_dim, y_dim, tunneling, coulomb,
10 chemical_potential=None,
11 spinless=False)
Using the Jordan-Wigner transform functionality of OpenFermion, the Hubbard Hamiltonian can be transformed to a sum of QubitOperators which are then transformed to pyQuil PauliSum objects using routines in the Forest-OpenFermion plugin imported earlier.
1hubbard_term_generator = jordan_wigner(hubbard_hamiltonian)
2pyquil_hubbard_generator = qubitop_to_pyquilpauli(hubbard_term_generator)
With the data successfully transformed to a pyQuil representation, the pyQuil exponentiate routine is used to generate a circuit corresponding to first-order Trotter evolution for .
1from pyquil.quil import Program
2from pyquil.gates import X
3from pyquil.paulis import exponentiate
4localized_electrons_program = Program()
5localized_electrons_program.inst([X(0), X(1)])
6pyquil_program = Program()
7for term in pyquil_hubbard_generator.terms:
8 pyquil_program += exponentiate(0.1 * term)
9print(localized_electrons_program + pyquil_program)
1Sample Output:
2X 0
3X 1
4X 0
5PHASE(-0.4) 0
6X 0
7PHASE(-0.4) 0
8H 0
9H 2
10CNOT 0 1
11CNOT 1 2
12RZ(-0.1) 2
13CNOT 1 2
14CNOT 0 1
The output is the first few lines of the Quil Smith et al. (2016) program that sets up the two-localized electrons on the first spatial site and then applies the time-propagation circuit. This Quil program can be sent to the Forest cloud API for simulation or for execution on hardware.
Ii Core data structures
In this section, we describe in more depth two of the core data structures used in the OpenFermion package, the FermionOperator and QubitOperator classes. As their names are meant to imply, they represent general operators acting on fermions and qubits. They are central to essentially all routines and other derived data structures within OpenFermion and thus deserve specific attention.
ii.1 FermionOperator data structure
In the above examples, we saw that an intermediate representation for the molecular problems were objects known as FermionOperators. Fermionic systems are often treated in second quantization, where anti-symmetry requirements are stored in the operators rather than in explicit wavefunction anti-symmetrization and arbitrary operators can be expressed using fermionic creation and annihilation operators and . Supposing that , , such operators could be represented within OpenFermion simply as
1from openfermion.ops import FermionOperator
2a_p_dagger = FermionOperator('1^')
3a_q = FermionOperator('0')
These operators enforce fermionic statistics in the system by satisfying the fermionic anti-commutation relations,
where . The raising operators act on the fermionic vacuum state, , to create fermions in spin-orbitals, which are single-particle spatial density functions. The connection to first quantization and explicit anti-symmetrization in Slater determinants can be seen if electron is represented in a space of spin-orbitals . Then and populate fermions in Slater determinants through the equivalence,
which instantiates a system of fermions.
Arbitrary fermionic operators on the space of spin-orbitals can be represented by weighted sums of products of these raising and lowering operators. The following is an example of one such “fermion operator”,
In the second equality above we have used the anti-commutation relations of Eq. (5) to reorder the ladder operators in into a unique “normal-ordered” form, defined so that raising operators always come first and operators are ordered in descending order of the fermionic mode on which they act. These rules are all handled transparently within OpenFermion so that essential physics are not violated. For example, the operator could be defined within OpenFermion as
1from openfermion.ops import FermionOperator
2W = (1 + 2j) * FermionOperator('4^ 3 9 3^') - 4 * FermionOperator('2')
and the “normal-ordering” can be simply performed by
1from openfermion.ops import normal_ordered
2W_normal_ordered = normal_ordered(W)
So long as ladder operators are manipulated in a fashion that is consistent with Eq. (5), addition, multiplication, and integer exponentiation are well defined for fermion operators. For instance, and are also examples of fermion operators, and are readily available in OpenFermion through standard arithmetic manipulations of the operators. For example
1W_4 = W ** 4
where in the second line, we find if we further use the normal ordered function on this seemingly complicated object, that it in fact evaluates to zero.
Internally, the FermionOperator data structure uses a hash table (currently implemented using a Python dictionary). The keys of the dictionary encode a sequence of raising and lowering operators and the value of that entry stores the coefficient. The current implementation of this class encodes the sequence of ladder operators using a tuple of 2-tuples where the 2-tuples represent ladder operators. The first element of each 2-tuple is an int specifying which fermionic mode the ladder operator acts on and the second element of each 2-tuple is a Boolean specifying whether the operator is raising (True) or lowering (False); thus, the encoding of the ladders operators can be expressed as
The sequence of ladder operators is thus specified by a sequence of the 2-tuples just defined. Some examples of the ladder operator sequence encodings are shown below,
which can also be used to initialize a FermionOperator as a user as
1O_1 = FermionOperator()
2O_2 = FermionOperator( ((2, 0), ) )
3O_3 = FermionOperator( ((4, 1), (9, 0)) )
4O_4 = FermionOperator( ((4, 1), (3, 0), (9, 0), (3, 1)) )
While this is the internal representation of sequences of ladder operators in the FermionOperator data structure, OpenFermion also supports a string representation of ladder operators that is more human-readable. One can initialize FermionOperators using the string representation and when one calls print() on a FermionOperator, the operator is printed out using the string representation. The carat symbol “ ^” represents raising, and its absence implies the lowering operator. Below are some self-explanatory examples of our string representation,
that translate to code as
1O_1 = FermionOperator('')
2O_2 = FermionOperator('2')
3O_3 = FermionOperator('4^ 9')
4O_4 = FermionOperator('4^ 3 9 3^')
A hash table data structure was chosen to facilitate efficient combination of large FermionOperators through arithmetic operations. This is preferred over an unstructured array of terms due to its native implementation in Python as well as the fact that duplicate terms are nearly automatically combined at a cost that is constant time for modestly sized examples. Similar scaling could be achieved through other data structures, but at increased complexity of implementation in the Python ecosystem. As motivation for this choice, we include in the example section two important use cases of the FermionOperator: the computation of Trotter error operators and the symbolic Fourier transformation.
ii.2 QubitOperator data structure
Continuing from the example above, once the intermediate FermionOperators have been produced, they must be mapped to the language of quantum computers, or qubits. This is handled within OpenFermion through the QubitOperator data structure. Fundamentally this operator structure is based off the Pauli spin operators and the identity, defined by
Tensor products of Pauli operators form a basis for the space of Hermitian operators; thus, it is possible to express any Hermitian operator of interest using just these few operators and their products. If one indexes the qubit a particular operator acts on, such as , that defines action by on the qubit (and implicitly action by on all others). In OpenFermion one may wish to express an operator such as
that could be initialized as
1from openfermion.ops import QubitOperator
2O = QubitOperator('Z1 Z2') + QubitOperator('X1') + QubitOperator('X2')
Similar to FermionOperator, QubitOperator is implemented internally using a hash table data structure through the native Python dictionary. This choice allows a good level of base efficiency for most arithmetic operators while harnessing the features of the Python dictionary to ease implementation details. The keys used in this implementation are similarly tuples of tuples that define the type of operator and the qubit it acts on,
This internal representation may be used to initialize QubitOperators such as the operator as
1O = QubitOperator( ((1, 'X'), (2, 'X')) )
or alternatively as seen above, a convenient string initializer is also available, which for the same operator could be used as
1O = QubitOperator('X1 X2')
ii.3 The MolecularData data structure
While the FermionOperator and QubitOperator classes form the backbone of many of the internal computations for OpenFermion, the data that defines a particular physical problem is more conveniently stored within a separate well-defined object. Namely the MolecularData object. This defines the schema by which the intermediate quantities calculated for the electronic structure of a molecule are stored, such as the two-electron integrals, basis transformation from the original basis, energies from correlated methods, and meta data related to the computation. For an exhaustive list of the current information stored within this class, it is recommended to see the documentation, as quantities are added as needed.
Importantly, this information is stored to disk in an HDF5 format using the h5py package. This allows for seamless data access to the files for only the quantities of interest, without the need for loading the whole file or complicated interface structures. Internally this is performed through the use of getters and setters using Python decorators, but this is transparent to the user, needing only to get or set in the normal way, e.g.
1two_body_integrals = h2_molecule.two_body_integrals
where the data is read from disk on the access to two_body_integrals rather than when the object is instantiated in the first line. This controls the memory impact of larger objects such as the two-electron integrals. Compression functionality is also enabled through gzip to minimize the disk space requirements to store larger molecules.
Functionality is also built into MolecularData class to perform simple activate space approximations to the problem. An active space approximation isolates a subset of the total orbitals and treats the problem within that orbital set. This is done by modifying the one- and two-electron integrals as well as the count of active electrons and orbitals within the molecule. We assume a single reference for the active space definition and the occupied orbitals are integrated out according to the integrals while inactive virtual orbitals are removed. In OpenFermion this is as simple as taking a calculated molecule data structure, forming a list of the occupied spatial orbital indices (those which will be frozen in place) and active spatial orbital indices (those which may be freely occupied or unoccupied), and calling for some molecule larger than H in a minimal basis
1from openfermion.ops import InteractionOperator
3core_constant, one_body_integrals, two_body_integrals = (
4 molecule.get_active_space_integrals(occupied_indices, active_indices))
5active_space_hamiltonian = InteractionOperator(core_constant,
6 one_body_integrals,
7 two_body_integrals)
where active_space_hamiltonian can now be used to build quantum circuits for the reduced size problem.
Iii Derived Operators
Here we detail some of the operators that derive from or utilize the FermionOperator, QubitOperator, and MolecularData classes to help facilitate computations. These include the InteractionOperator that specializes to FermionOperators with a particular structure and InteractionRDMs that utilize FermionOperators as a convenience wrapper to store reduced -electron density matrices.
iii.1 The InteractionOperator data structure
As OpenFermion deals primarily with the interactions of physical fermions, especially electrons, the Hamiltonian we have already introduced above
is ubiquitous throughout OpenFermion.
Note that even for fewer than qubits, the coefficients of the terms in the Hamiltonian of Eq. (14) can require a large amount of memory. Since common Gaussian basis functions lead to the full term scaling, instances with less than a hundred qubits can already have tens of millions to hundreds of millions of terms requiring on the order of ten gigabytes to store. Such large Hamiltonians can be expensive to generate (requiring nearly as many integrals as there are terms) so one would often like to be able to save these Hamiltonians. While good for general purpose symbolic manipulation, the FermionOperator data structure is not the most efficient way to store these Hamiltonians, or to manipulate them with efficient numerical linear algebra routines, due to the extra overhead of storing each of the associated operators.
Towards this end we introduce the InteractionOperator data structure. This structure has already been seen in passing in this document in the first example given. For example, with a MolecularData object, one may extract the Hamiltonian in the form of an InteractionOperator through
1h2_hamiltonian = h2_molecule.get_molecular_hamiltonian()
where the Hamiltonian will be returned as an InteractionOperator. The InteractionOperator data structure is a class that stores two different matrices and a constant. In the notation of Eq. (14), the InteractionOperator stores the constant , an by array representing and an by by by array representing . Note that at present this is not a spatially optimal representation if the goal is specifically to store the integrals of molecular electronic structure systems, but it is a good compromise between space efficiency, simplicity, and ease of performing other numerical operations. The reason it is suboptimal is that there may exist symmetries within the integrals that are not currently utilized. For example, there is an eight-fold symmetry in the integrals for real basis functions, . Note that for complex basis functions there is only a four-fold symmetry. While we provide methods to iterate over these unique elements, we do not exploit this symmetry in our current implementation for the reason that space efficiency of the InteractionOperator has not yet been a bottleneck in applications. This is consistent with our general design philosophy which is to maintain an active cycle of develop, test, profile, and refine. That is, rather than guess bottlenecks, we analyze and identify the most important ones for problems of interest, and focus time there.
iii.2 The InteractionRDM data structure
As discussed in the previous section, since fermions are identical particles which interact pairwise, their energy can be determined entirely by reduced density matrices which are polynomial in size. In particular, these energies depend on the one-particle reduced density matrix (1-RDM), denoted here by and the two-particle reduced density matrix (2-RDM), denoted here by . The 1-RDM and 2-RDM of a fermionic wavefunction are defined through expectation values with one- and two-body local FermionOperators;
Thus, the 1-RDM is an by matrix and the 2-RDM is an by by by tensor. Note that the 1-RDM and 2-RDMs may (in general) represent a partial tomography of mixed states, in which case Eq. (15) should involve traces over that mixed state instead of expectation values with a pure state. If one has performed a computation at the CISD level of theory, it is possible to extract that density matrix from a molecule using the following command
1cisd_two_rdm = h2_molecule.get_molecular_rdm()
We can see that the energy of with a Hamiltonian expressed in the form of Eq. (14) is given exactly as
where , and are the integrals stored by the InteractionOperator data structure.
In OpenFermion, the InteractionRDM class provides an efficient numerical representation of these reduced density matrices. Both InteractionRDM and InteractionOperator inherit from a similar parent class, the PolynomialTensor, reflecting the close parallels between the implementation of these data structures. Due to this parallel, the exact same code which implements integral basis transformations on InteractionOperator also implements integral basis transformations on the InteractionRDM data structure. Despite their similarities, they represent conceptually distinct concepts, and in many cases should be treated in a fundamentally different way. For this reason the implementations are kept distinct.
iii.3 The QuadraticHamiltonian data structure
The general electronic structure Hamiltonian Eq. (14) contains terms that act on up to 4 sites, or is quartic in the fermionic creation and annihilation operators. However, in many situations we may fruitfully approximate these Hamiltonians by replacing these quartic terms with terms that act on at most 2 fermionic sites, or quadratic terms, as in mean-field approximation theory. These Hamiltonians have a number of special properties one can exploit for efficient simulation and manipulation of the Hamiltonian, thus warranting a special data structure. We refer to Hamiltonians which only contain terms that are quadratic in the fermionic creation and annihilation operators as quadratic Hamiltonians, and include the general case of non-particle conserving terms as in a general Bogoliubov transformation. Eigenstates of quadratic Hamiltonians can be prepared efficiently on both a quantum and classical computer, making them amenable to initial guesses for many more challenging problems.
A general quadratic Hamiltonian takes the form
where is a Hermitian matrix, is an antisymmetric matrix, is the Kronecker delta symbol, and is a chemical potential term which we keep separate from so that we can use it to adjust the expectation of the total number of particles. In OpenFermion, quadratic Hamiltonians are conveniently represented and manipulated using the QuadraticHamiltonian class, which stores , , and the constant from Eq. (17). It is specialized to exploit the properties unique to quadratic Hamiltonians. Examples showing the use of this class for simulation are provided later in this document.
iii.4 Some examples justifying data structure design choices
Here we describe and show a few example illustrating that the above data structures are well designed for efficient calculations that one might do with OpenFermion. These examples are motivated by real use case examples encountered in the authors’ own research. We do not provide code examples in all cases here but routines for these calculations exist within OpenFermion.
iii.4.1 FermionOperator example: computation of Trotter error operators
Suppose that one has a Hamiltonian where the are single-term FermionOperators. Suppose now that one decides to effect evolution under this Hamiltonian using the second-order Trotter formula, as investigated in works such as Wecker et al. (2014); Hastings et al. (2015); Poulin et al. (2015); Babbush et al. (2015b, 2017a). A single second-order Trotter step of this Hamiltonian effects evolution under where is the Trotter error operator which arises due to the fact that the do not all commute. In Poulin et al. (2015) it is shown that a perturbative approximation to the operator can be expressed as
Because triangle inequality upper-bounds to this operator provide an estimate of the Trotter error which can be computed in polynomial time, symbolic computation of this operator is crucial for predicting how many Trotter steps one should take in a quantum simulation. However, when using conventional Gaussian basis functions, Hamiltonians of fermionic systems contain terms, suggesting that the number of terms in Eq. (18) could be as high as . But in practice, numerics have shown that there is very significant cancellation in these commutators which leads to a number of nontrivial terms that is closer to after normal ordering Babbush et al. (2015b). If one were to use an array or linked list structure to store FermionOperators then one has two choices. Option (i) is that new terms from the sum are appended to the list and then combined after the sum is completed. Under this strategy the space complexity of the algorithm would scale as and one would likely run out of memory for medium sized systems. Option (ii) is that one normal orders the commutators before adding to the list and then loops through the array or list before adding each term. While this approach has average space complexity of , the time complexity of this approach would then be as one would need to loop through entries in the list after computing each of the terms in the sum. Using our hash table implementation, the average space complexity is still but one does not need to loop through all entries at each iteration so the time complexity becomes .
Though still quite expensive, one can use Monte Carlo based sampling or distribute this task on a cluster (it is embarrassingly parallel) to compute the Trotter error for medium sized systems. Using the Hamiltonian representations introduced in Babbush et al. (2017a), Hamiltonians have only terms, which brings the number of terms in the sum down to in the worst case. In that case, computation of would have time complexity using the hash table implementation instead of complexity using either a linked-list or an array. The complexity enables us to compute for hundreds of qubits, well into the regime of instances that would be classically intractable to simulate. The task is even more efficient for Hubbard models which have terms.
iii.4.2 FermionOperator example: symbolic Fourier transformation
A secondary goal for OpenFermion is that the library can be used to as a tool for the symbolic manipulation of fermionic Hamiltonians in order to analyze and develop new simulation algorithms and Hamiltonian representations. To give a concrete example of this, in the recent paper Babbush et al. (2017a), authors were able to demonstrate Trotter steps of the electronic structure Hamiltonian with significantly reduced complexity: depth as opposed to depth. A critical component of that improvement was to represent the Hamiltonian using basis functions which are a discrete Fourier transform of the plane wave basis (the plane wave dual basis) Babbush et al. (2017a). The appendices of that paper begin by showing the Hamiltonian in the plane wave basis:
To obtain the Hamiltonian in the plane wave dual basis, one applies the Fourier transform of the mode operators,
Using OpenFermion one can easily generate the plane wave Hamiltonian (either manually or by using the plane wave module) and then apply the discrete Fourier transform of the mode operators (either manually or by using the discrete Fourier transform module) to verify the correct form of the plane wave dual Hamiltonian shown in Babbush et al. (2017a),
While Eq. (21) turns out to have a very compact representation, it requires a careful derivation to show that application of Eq. (20) to Eq. (19) leads to Eq. (21). However, this task is trivial for OpenFermion since the Fourier transform can be applied symbolically and the output FermionOperator can be simplified automatically using a normal-ordering routine. This example demonstrates the utility of OpenFermion for verifying analytic calculations.
iii.4.3 InteractionOperator example: fast orbital basis transformations
An example of an important numerical operation which is particularly efficient in this representation is a rotation of the molecular orbitals . This unitary basis transformation takes the form
For and , the quantities and respectively correspond to the matrix elements of these by matrices. We see then that the elements of the matrix define a unitary transformation on all orbitals. Often, one would like to apply this transformation to Hamiltonian in the orbital basis defined by in order to obtain a new Hamiltonian in the orbital basis defined by . Since computation of the integrals is extremely expensive, the goal is usually to apply this transformation directly to the Hamiltonian operator.
When specialized to real, orthogonal rotations (which is often the case in molecular electronic structure), the most straightforward expression of this integral transformation for the two-electron integrals is
Since there are integrals, the entire transformation of the Hamiltonian would take time , which is extremely onerous. A more efficient approach is to rearrange this expression to obtain the following,
which can be evaluated for each term at cost since each summation can be carried out independently. This brings the total cost of the integral transformation to . While such a transformation would be extremely tedious using the FermionOperator representation, this approach is implemented for InteractionOperators in OpenFermion using the einsum function for numerical linear algebra from numpy.
This functionality is readily accessible within OpenFermion for basis rotations. For example, the one may rotate the basis of the molecular hamiltonian of H to a new basis with some orthogonal matrix of appropriate dimension as
1from numpy import array, eye, kron
3U = kron(array([[0, 1], [1, 0]]), eye(2))
4h2_hamiltonian = h2_molecule.get_molecular_hamiltonian()
We now provide an example of where such fast basis transformations may be useful. Previous work has shown that the number of measurements required for a variational quantum algorithm to estimate the energy of a Hamiltonian such as Eq. (14) to precision scales as
Since the are determined by the orbital basis, one can alter this bound by rotating the orbitals under angles organized into the matrix of Eq. (22). Thus, one may want to perform an optimization over these angles in order to minimize . This task would be quite unwieldy or perhaps nearly impossible using the FermionOperator data structure but is viable using the InteractionOperator data structure with fast integral transformations.
iii.4.4 QuadraticHamiltonian example: preparing fermionic Gaussian states
As mentioned above, eigenstates of quadratic Hamiltonians, Eq. (17), can be prepared efficiently on a quantum computer, and OpenFermion includes functionality for compiling quantum circuits to prepare these states. Eigenstates of general quadratic Hamiltonians with both particle-conserving and non-particle conserving terms are also known as fermionic Gaussian states. A key step in the preparation of a fermionic Gaussian state is the computation of a basis transformation which puts the Hamiltonian of Eq. (17) into the form
where the and are a new set of fermionic creation and annihilation operators that also obey the canonical fermionic anticommutation relations. In OpenFermion, this basis transformation is computed efficiently with matrix manipulations by exploiting the special properties of quadratic Hamiltonians that allow one to work with only the matrices and from Eq. (17) stored by the QuadraticHamiltonian class, using simple numerical linear algebra routines included in SciPy. The following code constructs a mean-field Hamiltonian of the d-wave model of superconductivity and then obtains a circuit that prepares its ground state (along with a description of the starting state to which the circuit should be applied):
1from openfermion.hamiltonians import mean_field_dwave
2from openfermion.transforms import get_quadratic_hamiltonian
3from openfermion.utils import gaussian_state_preparation_circuit
5x_dimension = 2
6y_dimension = 2
7tunneling = 2.
8sc_gap = 2.
9periodic = True
11mean_field_model = mean_field_dwave(x_dimension, y_dimension, tunneling, sc_gap, periodic)
13quadratic_hamiltonian = get_quadratic_hamiltonian(mean_field_model)
15circuit_description, start_orbitals = gaussian_state_preparation_circuit(quadratic_hamiltonian)
The circuit description follows the procedure explained in Jiang et al. (2017). One can also obtain the ground energy and ground state numerically with the following code:
1from openfermion.utils import jw_get_gaussian_state
2ground_energy, ground_state = jw_get_gaussian_state(quadratic_hamiltonian)
Iv Models and utilities
OpenFermion has a number of capabilities that assist with the creation and manipulation of fermionic and related Hamiltonians. While they are too numerous and growing too quickly to list in their entirety, here we will briefly describe a few of these examples in an attempt to ground what one would expect to find within OpenFermion.
One broad class of functions found within OpenFermion is the generation of fermionic and related Hamiltonians. While the MolecularData structure discussed above is one example for molecular systems, we also include utilities for a number of model systems of interest as well. For example, there are currently supported routines for generating Hamiltonians of the Hubbard model, the homogeneous electron gas (jellium), general plane wave discretizations, and -wave models of superconductivity.
To give a concrete example, if one wished to study the homogeneous electron gas in 2D, which is of interest when studying the fractional quantum hall effect, then one could use OpenFermion to initialize the model as follows
1from openfermion.hamiltonians import jellium_model
2from openfermion.utils import Grid
4jellium_hamiltonian = jellium_model(Grid(dimensions=2,
5 length=10,
6 scale=1.0))
where jellium_hamiltonian will be a fermionic operator representing the spinful homogeneous electron gas discretized into a grid of plane waves in two dimensions. One is then free to transform this Hamiltonian into qubit operators and use it in the algorithm of choice.
Similarly, one may use the utilities provided to prepare a Fermi-Hubbard model for study. For example
1from openfermion.hamiltonians import fermi_hubbard
3t = 1.0
4U = 4.0
6hubbard_hamiltonian = fermi_hubbard(x_dimension=10, y_dimension=10,
7 tunneling=t, coulomb=U,
8 chemical_potential=0.0, periodic=True)
creates a Fermi-Hubbard Hamiltonian on a square lattice in 2D that is sites with periodic boundary conditions.
Besides Hamiltonian generation, OpenFermion also includes methods for outputting Trotter-Suzuki decompositions of arbitrary operators, providing the corresponding quantum circuit in QASM format Cross et al. (2017b). These methods were included in order to simplify porting the resulting quantum circuits to other simulation packages, such as LIQUi Wecker and Svore (2014), qTorch Fried et al. (2017), Project-Q Steiger et al. (2016), and qHipster Smelyanskiy et al. (2016). For instance, the function pauli_exp_to_qasm takes a list of QubitOperators and an optional evolution time as input, and outputs the QASM specification as a string. As an example,
1from openfermion.ops import QubitOperator
2from openfermion.utils import pauli_exp_to_qasm
4for line in pauli_exp_to_qasm([QubitOperator('X0 Z1 Y3', 0.5), QubitOperator('Z3 Z4', 0.6)]):
5 print(line)
outputs a QASM specification for a circuit corresponding to , which in this case is given by
1H 0
2Rx 1.5707963267948966 3
3CNOT 0 1
4CNOT 1 3
5Rz 0.5 3
6CNOT 1 3
7CNOT 0 1
8H 0
9Rx -1.5707963267948966 3
10CNOT 3 4
11Rz 0.6 4
12CNOT 3 4
OpenFermion additionally supports a number of other tools including the ability to evaluate the Trotter error operator and construct circuits for preparing arbitrary Slater determinants on a quantum computer using the linear depth procedure described in Jiang et al. (2017). Some support for numerical simulation is included for testing purposes, but the heavy lifting in this area is delegated to plugins specialized for these applications. In the future, we imagine these utilities will expand to include more Hamiltonians and specializations that add in the creation of simulation circuits for fermionic systems.
V Open Source Management and Project Philosophy
OpenFermion is designed to be a tool for both its developers and the community at large. By maintaining an open-source and framework independent library, we believe it provides a useful tool for developers in industry, academia, and research institutions alike. Moreover, it is our hope that these developers will, in turn, contribute code they found to be useful in interacting with OpenFermion to the benefit of the field as a whole. Here we outline some of our philosophy, as well as the ways in which we ensure that OpenFermion remains a high quality package, even as many different developers from different institutions contribute.
v.1 Style and testing
The OpenFermion code is written primarily in Python with optional C++ backends being introduced as higher performance is desired. Stylistically, this code follows the PEP8 guidelines for Python and demands descriptive names as well as extensive documentation for all functions. This enhances readability and allows developers to more reliably build off of contributed code.
At present, the source code is managed through GitHub where the standard pull request system is used for making contributions. When a pull request is made, it must be reviewed by at least one member of the OpenFermion team who has experience with the library and is able to comment on both the style and integration prospects with the library. As contributors become more familiar with the code, they may become approved reviewers themselves to enhance their contribution to the process. All contributors are welcome to assist with reviews but reviews from new contributors will not automatically enable merging into the master branch.
Tests in the code are written in the python unittest framework and all code is required to be both Python 2 and Python 3 compliant so that it continues to be as useful as possible for all users in the future. The tests are run automatically when a pull request is made through the Travis continuous integration (CI) framework, to minimize the chance that code that will break crucial functionality is accidentally merged into the code base, and ensure a smooth development and execution experience.
v.2 Distribution
Several options for code distribution are available to obtain OpenFermion. The code may be pulled and used directly from the GitHub repository if desired (which one can find at www.openfermion.org), and the requirements may be manually fulfilled by the user. This option offers maximum control and access to bleeding edge code and developments, but minimal convenience. A full service option is offered through the Python Package Index (PyPI) so that installation for most users can be as simple as
1python -m pip install openfermion
A middle ground between these options which is popular with many of the lead developers of this project is to pull the latest code directly from GitHub but to install with pip using the development install command in the OpenFermion directory:
1python -m pip install -e .
In the future, a version of the code may be supported for other distribution platforms as well.
The OpenFermion plugins (and the packages on which they rely) need to be installed independently using a similar procedure. One can install OpenFermion-Psi4 using pip under the name “openfermionpsi4”, OpenFermion-ProjectQ under the name “openfermionprojectq” and OpenFermion-PySCF under the name “openfermionpyscf”. These plugins can also be installed directly from GitHub (we link to repositories from the main OpenFermion page).
In addition to the traditional installation models of either installing from the PyPI registry, or downloading the source from GitHub and installing, the project also supports a Docker container. Docker containers offer a compact virtualization environment that is portable between all systems where Docker is supported. This is a convenient option for both first time users, and those who want to deploy OpenFermion to non-standard architectures (or on Windows). Moreover, it offers easy access to some of the electronic structure packages that our plugins are inter-operable with, which allows users convenient access to the full tool chain. At present, the Dockerfile is hosted on the repository, which can be used to build a Docker image as detailed in full in the Readme of the Docker folder.
Closing Remarks
The rapid development of quantum hardware represents an impetus for the equally rapid development of quantum applications. Development of these applications represents a unique challenge, often requiring the expertise or domain knowledge from both the application area and quantum algorithms. OpenFermion is a bridge between the world of quantum computing and materials simulation, which we believe will act as a catalyst for development in this critical area. Only with such software tools can we start to explore the explicit costs and advantages of new algorithms, and push forward with practical advancements. It is our hope that OpenFermion not only leads to developments in the field of quantum computation for quantum chemistry and materials, but also sparks the development of similar packages for other application areas.
The authors thank Hartmut Neven for encouraging the initiation of this project at Google as well as Alán Aspuru-Guzik, Yaoyun Shi, Matthias Troyer, and James Whitfield for supporting graduate student and postdoc developers who contributed code to OpenFermion. I. D. K. acknowledges partial support from the National Sciences and Engineering Research Council of Canada. T. H. and D. S. S. have been supported by the Swiss National Science Foundation through the National Competence Center in Research QSIT. S. S. is supported by the DOE Computational Science Graduate Fellowship under grant number DE-FG02-97ER25308.
For everything else, email us at [email protected]. |
88dc95a69a4e97d0 | Photoionization with Orbital Angular Momentum Beams
A. Picón, J. Mompart, J. R. Vázquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso Grup d’Òptica, Universitat Autònoma de Barcelona, E-08193 Bellaterra (Barcelona), Spain Servicio Láser, Universidad de Salamanca, E-37008 Salamanca, Spain Departamento de Matemáticas, ETSI Industriales & IMACI-Instituto de Matemática Aplicada a la Ciencia y la Ingeniería, Universidad de Castilla-La Mancha, E-13071 Ciudad Real, Spain JILA, University of Colorado, Boulder 80309-0440, USA (actual address)
June 26, 2021
Intense laser ionization expands Einstein s photoelectric effect rules giving a wealth of phenomena widely studied over the last decades. In all cases, so far, photons were assumed to carry one unit of angular momentum. However it is now clear that photons can possess extra angular momentum, the orbital angular momentum (OAM), related to their spatial profile. We show a complete description of photoionization by OAM photons, including new selection rules involving more than one unit of angular momentum. We explore theoretically the interaction of a single electron atom located at the center of an intense ultraviolet beam bearing OAM, envisaging new scenarios for quantum optics.
03.67.Mn, 42.50.Dv, 42.65.Lm
During the history of Physics, Light-Matter interaction has been the fundamental path towards understanding new phenomena and testing some essential theories, as in the case of photoionization. Approximately until 1992, all physical theories have described light according to three features: energy, linear momentum and polarization. The latter, which is purely related to the electric field direction of the propagating light, yields an effective light angular momentum. But, as recognized by Allen and coworkers [1], light possess another degree of freedom: the orbital angular momentum (OAM), which rather than being associated with polarization, it is related to the spatial profile of light. This newborn degree of freedom has kindled a huge activity in different lines of research, ranging from micro and nanoparticle trapping Padgett ; Bhattacharya ; Rev_Padgett_2008 to quantum state engineering in Bose-Einstein condensates Andersen , multiphoton entanglement Mair ; Kozuma ; CalvoPRA07 for quantum information applications and molecular spectroscopy Nulty . Recently, both femtosecond and high-power OAM beams have been generated experimentally using holographic plates Mariyenko_Creation_2005 ; Sola_High_2008 . In this work, we revisit the Einstein photoionization scenario Einstein , but now taking into account the orbital angular momentum of light. This allows us to unveil new phenomena beyond the standard photoionization.
Photoionization has attracted a broad interest both from a fundamental theoretical point of view and from the standpoint of applications. Laser photoionization, particularly strong field photoionization, has been a very active research topic over the last decades. Many new effects have been reported, such as ATI (Above Threshold Ionization), tunnel ionization, high-order harmonic generation, etc. Blaga ; Corkum Besides, much effort has been devoted in the recent years to the possible inhibition of photoionization at high frequencies and for very strong fields. Experimental activity on ultra-strong field ionization has recently reached the relativistic domain Mourou . All this photoionization literature can be classified into two regimes: the electric-dipole regime (where the magnetic field of the light can be neglected) and the non-dipole regime. In the electric-dipole regime, light beams carry the standard angular momentum (one unit), and the selection rules avoid the possibility of one-photon excitation of atomic transitions with angular momentum variation larger than one . However, one can overcome this limitation by considering multi-photon effects with very intense lasers. In the non-dipole regime, the light-atom interaction is more complex, exciting not only atomic transitions with angular momentum change equal to one unit , but also atomic transitions with larger angular momentum variation. In this work we present for the first time photoionization with light beams carrying OAM which give rise to new selection rules out of both the electric-dipole and the non-dipole regime, opening new perspectives for atomic transition excitations.
Much numerical work has been devoted to 3D photoionization studies. Strong field photoionization implies ionized electrons that escape at high energies, thus requiring large and dense numerical networks (to describe both electrons far away from the atomic core and energetic electrons driven by the laser). When a linearly polarized laser light is used, numerical solutions are relatively simple due to the cylindrical symmetry of the problem and only two dimensional numerical grids are needed. However, there exist relevant examples where such cylindrical symmetry is not possible Javi , and a true three dimensional numerical grating is needed. This demands a much larger complexity of the numerical simulation. Here, we present an accurate description of the atomic photoionization induced by a beam carrying OAM. This scenario requires a true 3D simulation to fully account for both the three-spatial dimensions of the electron quantum state and the 3D-spatial profile of the beam (including the transverse profile related to the OAM).
In this paper we address the interaction of a pulse beam carrying OAM with the simplest atom: hydrogen. This scenario provides the paradigm for fundamental questions: Is the OAM of light transferred to the electron quantum state? There are still open questions about the angular momentum transfer between matter and light Padgett ; Barnett ; VanEnk ; Jauregui ; Alexandrescu . The angular momentum of light can be separated in OAM and spin momentum in the paraxial regime, but how are the OAM and the spin momentum of light transferred to the matter? In some previous works VanEnk ; Jauregui ; Alexandrescu , where an ensemble of atoms was considered, it was shown how the OAM can be coupled to the center of mass of the atomic ensemble. The same interaction description out of the paraxial regime Jauregui is more plentiful. In contrast with previous works, we are considering the interaction of a Laguerre-Gaussian intense laser beam with a single quantum state placed near its optical vortex, taking into account the general form of the quantum electromagnetic-interaction-field hamiltonian, without neglecting any magnetic term and without restricting to any multipolar approximation in the transverse plane, going beyond into the comprehension of the OAM light coupling with a quantum system. To clarify the picture, we investigate photoionization with OAM beams in the Schrödinger regime, using both analytical and numerical tools.
Figure 1: Light-Matter Interaction Scheme. The addressed problem consists of: A temporal pulse with a well-defined polarization and a transverse profile that takes into account the OAM. The initial state of the electron corresponds to the fundamental state of the hydrogen atom.
I Light-Matter Interaction Scheme
We begin by considering a pulse beam propagating along the -direction with a temporal envelope wave parameterized by a quadratic sinus (see Fig. 1). This temporal envelope has a frequency , where and are the cycle number and the period of the carrier wave, respectively. A hydrogen atom is assumed to be localized at the origin of the reference system and experiences a vector potential, associated to the pulse, of the form
where is the step function, the Bohr radius, the speed of light, the carrier wave frequency, and the amplitude of the wave (it includes the polarization state). The transverse spatial structure of the pulse beam is accounted by the functions ; the Laguerre-Gaussian modes Allen . They are characterized by a width at (the beam waist), and by the indices and , representing the, so-called, winding (or topological charge) and the number of nonaxial radial nodes of the mode, respectively. Laguerre-Gaussian modes contain an azimuthal phase which gives rise to a discrete OAM of units per photon along their propagation direction. The complete spatial spectrum of scalar wave fields prepared in arbitrary superpositions of Laguerre-Gaussian (or other paraxial) modes can be measured by using a simple interferometric scheme Calvo08 . The fact that the associated electric field amplitude now depends on the transverse position, in contrast with plane waves, will be shown below to give rise to unexpected phenomena.
We assume that the hydrogen nucleus is unaffected by the electromagnetic field. If so, the electron-field coupling evolution will be described by the following Schrödinger equation
where is the electron quantum state, the Coulomb potential originated by the hydrogen nucleus, the electron mass, the electron charge, the vector potential, which in our case is given by expression (1), and the linear momentum operator, satisfying the canonical commutation relation .
Ii Selection Rules with OAM
We first extract a new set of selection rules for the interaction of atoms with OAM beams and point out the essential differences with pulse beams consisting of plane waves. The Hamiltonian (I) of the system can be expressed in the usual form; , where is the free part, whereas and refer to the interaction parts. Representing the quantum state in a spherical basis (all radial dependence is in the functions , while the angular dependence remains in the spherical harmonic functions instead), the first interaction contribution can be written as
Here, and are the unperturbed energies of the initial and final states, respectively. Taking into account the vector potential given by (1), and within the dipolar () and the transverse spatial () approximations, we derive the following set of selection rules for beams carrying any arbitrary units of OAM (see the Appendix):
where we have assumed that the quantization axis is along the beam propagation direction. In contrast with plane waves (for which and ), significant variations of the angular momentum are to be expected. In terms of photons, the selection rules (4) can be conceived as the absorption of photons carrying a total angular momentum in the propagation direction, where indicates the polarization part (or spin momentum, for right- and left-circular polarization). We would like to remark that these selection rules originate from the transverse profile, despite the dipolar approximation. Moreover, the second contribution for the interaction Hamiltonian yields, in the case of plane waves, a constant term, producing a ponderomotive force Hilbert09 . In our case, due to the transverse profile of the beam, this Hamiltonian part produces two contributions. One acting as a ponderomotive force, while the other one, remarkably, gives rise to new selection rules (see the Appendix):
We should remark that the domain of applicability of selection rules (4) and (5) extends beyond the photoionization problem.
Figure 2: Carrier Envelope Phase for a Laguerre-Gaussian Beam. In the main figure, the polarization of the pulse beam is represented for a transverse plane at , just when the pulse is maximum. The atom is centered at the origin (coincident with the beam vortex), where the electric field is zero. The arrow size is proportional to the electric field amplitude strength, which increases linearly in the radial direction up to amplitudes of about 5 au in the boundary distance, 20 au (1nm), and it is harmonically modulated by the azimuthal position. The blue line is the nodal line of the electric field, and rotates with the carrier wave frequency , setting the electric field distribution during the pulse interaction. The temporal dependence of the pulse beam is also shown at four different positions of the transverse plane. Notice that the carrier envelope phase (CEP) differs from the azimuthal dependence, not the radial.
Iii Hydrogen Simulations
Selection rules (4) and (5) constitute our first main result. To gain a more complete picture, we now proceed with the exact description of the hydrogen electron state ionization by beams carrying OAM. Prior to the interaction, we assume the electron to be in the ground state, . We simulate the evolution of the electron quantum state when the incoming pulse has , , , an angular frequency au (atomic units where , and s, ultraviolet), a period au ( as), and two possible polarizations: linear (in the -direction) and left-handed. We choose a beam waist to satisfy the paraxial regime, au ( m), which is much larger than the characteristic size of the atom (). Our atom is centered at the beam vortex, interacting with its vicinity, where the electric field amplitude is much weaker than the maximum one (reached at a distance ). This imposes the need for very intense lasers; in the proximity of the vortex singularity the electric field amplitude increases linearly (when ). For example, an electric field of about au ( V/cm), would give rise to 5 au amplitudes (during the pulse peak) at distances of about 20 au (1 nm) from the singularity. In order to clarify the structure of the electric field, in Fig. 2 we represent the polarization of the electric field in the transverse plane , when the pulse achieves its maximum value. Figure 2 also plots the pulse beam with respect to time at four different positions in the transverse plane. Notice the variation of the carrier envelope phase (CEP) depending on the azimuthal position. In fact, all the possible CEPs are encompassed in a circle around the singularity, in contrast with standard few cycle pulses technology Brabec , where much effort has been done to lock the carrier-envelope offset.
Figure 3: Initial Ionization of the Electron Quantum State. (a) In the upper row, projection of the excited state onto the plane () at four different times when the pulse beam is linearly polarized. The electron is beginning to be ionized in the first cycle of the pulse beam (with a period of =152 as). In the middle row, the superposition of spherical harmonics with widths given by the numerical simulation is represented; they show perfect agreement with the selection rules (4,5). (b) Same as upper row in (a) but for a left-circularly polarized pulse beam.
Since the electric field has a frequency au (larger than the bound hydrogen energy 0.5 au), ionization is to be expected. In fact, after the first pulse cycle we verify that 52% of the quantum electron state is ionized when the beam is linearly polarized, and 31% when the beam is circularly polarized. Note that Eq. (I) yields a total photoionization probability that depends non-linearly on the light polarization. We can express the electron quantum state at each time as , where is the excited state part. The excited state function can be decomposed in an unbound spherical basis . Our initial electron state has full spherical symmetry, belonging to the spherical harmonic . However, after the interaction with the pulse, the electron state excites different spherical harmonics. In Fig. 3 the projection of the excited state onto the -plane is depicted during the first cycle of the pulse beam, being a superposition of spherical harmonics obeying the selection rules (4). Also, depending on the input polarization, the electron evolution varies noticeably.
Figure 4: Final Quantum Electron State and their Spherical Harmonic Spectrum. Projection of the excited state () onto the plane after the interaction with the pulse beam for: (a) a Gaussian mode, (b) a Laguerre-Gaussian mode linearly polarized and (c) a Laguerre-Gaussian mode circularly polarized. (d) Spectrum of spherical harmonics for the three different cases (a), (b) and (c).
By resorting to numerical approaches, we could accurately evaluate the projections of the excited states onto the spherical harmonics and extract the corresponding probabilities . Using this numerical method, the widths of the spherical harmonic superpositions at different times have been derived, showing excellent agreement with the electron state evolution, as represented in Fig. 3(a). Moreover, we have analyzed the lowest spherical harmonic content of the final excited electron state (see Fig. 4) in three scenarios: (i) with a Gaussian pulse beam (in the transverse spatial approximation, it could be considered as a plane wave) linearly polarized in the -direction (the electron state is ionized about 30%); (ii) the case for a pulse spatially modulated by a Laguerre-Gaussian mode, linearly polarized in the -direction; (iii) when the Laguerre-Gaussian pulse is left-circularly polarized. The main remark is that the spherical harmonics and are most efficiently excited by the plane-wave-like pulse, in striking contrast with the Laguerre-Gaussian scenario, where no such excitation exists. If the Laguerre-Gaussian pulse is linearly polarized, then, , and are the most occupied states, whereas if it is circularly polarized, and are the most relevant. There is a small contribution from , as the electron is less ionized. We emphasize that these results are in accordance with the derived selection rules (4) and (5).
Figure 5: Temporal Evolution of the Electron Angular Momentum. Solid red curve represents the OAM of the electron in the -direction during interaction, parameters as in Fig. 3(a). Notice that the electron begins with null OAM and after the pulse beam it gains up to 1.53 au (1.53). Dash gray curve represents the population of the ground state. As time increases, it is ionized up to 0.52 of the initial population.
It is interesting to examine the total angular momentum transferred to the quantum electron state. First, we calculate the mean value of the electron OAM, , during its evolution. Figure 5 shows the time evolution of the orbital angular momentum along the -direction in the same case of Fig. 3(a), while a depopulation of the fundamental state occurs. The electron starts in the ground state, with zero OAM. As the pulse begins to interact with the electron, the OAM of the latter in the -direction oscillates, but notice that at the end of the pulse, the electron quantum state gains a finite amount of OAM: 1.53 au (). There is no OAM contribution in other directions. We expect, as the ground state has no OAM in the absence of a field, that the electron excited states belong to unbound states bearing OAM. On the other hand, when the pulse is left-circularly polarized, as the case of Fig. 3(b), the OAM in the -direction is negligible, as it is expected. Regarding the excited state position components, mean values are zero except for the -component, where a small shift of au is present, caused by a non-vanishing magnetic field at the origin.
Iv Discussion
We present the first work addressing the photoionization process induced by beams carrying OAM. We have found novel selection rules (4) and (5) for a pulse beam characterized by a topological charge . In addition, other interesting effects have been revealed. If the electron excited state after interaction is let to evolve, it goes away from the origin. In contrast, by introducing a pulse beam with , we notice that the excited electron state remains confined, within a radius of 10 au, due to the ponderomotive force induced by the Laguerre-Gaussian profile. We also simulate the case where the atom is displaced 2 au from the origin in the and -directions, with . In both cases, the atom is ionized much faster since the electric field is more intense now, and the excited electron state remains trapped. From evaluations of the mean values of the position component, we have observed an electron motion around the vortex. Varying the polarization of the beam, in particular for left-circular polarization, an ionized state with a ring structure () is predicted, see Figs. 3(b) an 4(c). Furthermore, modifying the initial phase, different ionized state structures can be generated. Thus, by tunning the phase and the polarization, one expects a manipulation of the ionized state. There still remain many open questions, such as the feasibility to achieve high-harmonic generation Courtial ; Patchkovskii with OAM, the experimental challenges to extend this phenomena to more complex systems (such as Rydberg atoms) and the possibility to use OAM for nuclear quantum optics applications Ledingham ; Keitel . For instance, by exploiting the transverse profile, one could address the M1 transition at 3.5 eV of Th-229 Peik_Nuclear_2003 .
V Appendix
We derive in this section the selection rules for the interaction of a light beam possessing orbital angular momentum with matter. From now on, it is convenient to take the quantization axis in the beam propagation direction, the position vector can thus be written as
Equation (6) can be divided into two parts; one depends exclusively on the radial contribution and the other one on the angular contribution. If we expand the angular contribution of Eq. (6),
we can notice basically two terms that are repeated and . These two terms can be decomposed in spherical harmonics functions, getting then the selection rules. In order to obtain the spherical harmonic decomposition, we must consider two cases, when and . Let us begin with case , where the angular terms read as
where we have used and . The first term (V), in integral (6), gives rise to the straightforward selection rule: , is even and . In equation (9) there are two contributions, the first one yields also straightforwardly the selection rule: , is even and , but the second one is a bit subtle. First of all, we need to decompose the product into spherical harmonics. Using the following formula;
where are the corresponding Clebsch-Gordan coefficients, we know that it is possible to write
Hence, the second contribution from Eq. (9) to the selection rules can be summarized as: , is even and .
Now, let us consider the case when . Proceeding in an analogous way than before, we can now write expressions (V) and (9) as:
where we have used and . The second term (12), in integral (6), yields: , is even and . On the other hand, in Eq. (V) there are two contributions, the first one giving rise to: , is even and . In the second one, the product must be decomposed into spherical harmonics using formula (10). As before, we can write
implying that: , is even and .
Generalizing all the last calculations, the selection rules derived from the matrix element (6) for beams carrying any unit of orbital angular momentum are summarized in Eq. (4).
At variance with the plane waves (), we can expect larger exchange of angular momentum. Of course, playing with the beam polarization ( and ), as we can note in equation (4), we can modify the selection rules. For example, for a right-circular polarization ( and ), the only surviving terms are given by expressions (V) and (V), restricting . Analogously, for a left-circular polarization ( and ) the only surviving terms are given by expressions (9) and (12), restricting . Selection rules (4), in photon terms, can be thought as the absorption of a photon carrying a total angular momentum in the propagation direction , where indicates the polarization part (spin momentum, for right-circular polarization and for left-circular polarization ). We would like to remark that these selection rules are exclusive to the transverse profile. Moreover, the second interaction hamiltonian , in the case of plane waves, is just a constant term, yielding a ponderomotive force. In our case, it is quite different. Analogously to equation (3), we can write
in order to extract the selection rules. Again, considering the vector potential (1), in which the dipolar and transverse spatial approximation has been taken into account, we can write
Eq. (14), if we expand the quadratic potential, two distinguishable terms appear:
where the second term does not depend on time, it acts as a well-potential. On the other hand, the first term gives rise to new selection rules as
yielding the selection rules of Eq. (5).
Vi Acknowledgments
We acknowledge support by the Spanish Ministry of Education and Science under contracts FIS2005-01369, FIS2008-02425, FIS2006-04151, FIS2007-29091-E, and Consolider projects SAUUL and QOIT, CSD2007-00013, CSD2006-00019, and the Catalan, Junta de Castilla y León, and Junta de Castilla-La Mancha Governments under contracts SGR2005-00358, SA146A08, and PCI08-0093. We also acknowledge R. Corbalán for fruitful discussions.
• (1) L. Allen, M. V. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
• (2) A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and Extrinsic Nature of the Orbital Angular Momentum of a Light Beam,” Phys. Rev. Lett. 88, 053601 (2002).
• (3) S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser & Photonics Review 2, 299 (2008).
• (4) M. Bhattacharya and P. Meystre, “Using a Laguerre-Gaussian Beam to Trap and Cool the Rotational Motion of a Mirror,” Phys. Rev. Lett. 99, 153603 (2007).
• (5) M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized Rotation of Atoms from Photons with Orbital Angular Momentum,” Phys. Rev. Lett. 97, 170406 (2006).
• (7) R. Inoue, N. Kanai, T. Yonehara, Y. Miyamoto, M. Koashi, and M. Kozuma, “Entanglement of orbital angular momentum states between an ensemble of cold atoms and a photon,” Phys. Rev. A 74, 053809 (2006).
• (8) G. F. Calvo, A. Picón, and A. Bramon, “Measuring two-photon orbital angular momentum entanglement,” Phys. Rev. A 75, 012319 (2007).
• (9) M. van Veenendaal and I. McNulty, “Prediction of Strong Dichroism Induced by X Rays Carrying Orbital Momentum,” Phys. Rev. Lett. 98, 157401 (2007).
• (10) I. G. Mariyenko, J. Strohaber, and C.J.G.J. Uiterwaal, “Creation of optical vortices in femtosecond pulses,” Opt. Express 13, 7599 (2005).
• (11) I. Sola et al., “High power vortex generation with volume phase holograms and non-linear experiments in gases,” Appl. Phys. B 91, 115-118 (2008).
• (12) A. Einstein, “Concerning an Heuristic Point of View Toward the Emission and Transformation of Light,” Ann. Phys. 17, 132 (1905).
• (13) P. B. Corkum and F. Krausz, “Attosecond science,” Nature Phys. 3, 381 (2007).
• (14) C. I. Blaga et al., “Strong-field photoionization revisited,” Nature Phys. 5, 335 (2009).
• (15) G. A. Mourou, “Optics in the relativistic regime,” Rev. Mod. Phys. 78, 309 (2006).
• (16) J. R. Vázquez de Aldana and L. Roso, “Magnetic field effects in strong field ionization of single-electron atoms: Three-dimensional numerical simulations,” Laser and Part. Beams 20 185-193 (2002).
• (17) S. M. Barnett, “Optical Angular-Momentum Flux,” J. Opt. B: Quantum Semiclass. Opt. 4 S7 (2002).
• (18) S. J. Van Enk, “Selection rules and centre-of-mass motion of ultracold atoms,” Quantum Opt. 6, 445 (1994).
• (19) R. Jáuregui, “Rotational effects of twisted light on atoms beyond the paraxial approximation,” Phys. Rev. A 70, 033415 (2004).
• (20) A. Alexandrescu, D. Cojoc, and E. Di Fabrizio, “Mechanism of Angular Momentum Exchange between Molecules and Laguerre-Gaussian Beams,” Phys. Rev. Lett. 96, 243001 (2006).
• (21) G. F. Calvo, A. Picón, and R. Zambrini, “Measuring the Complete Transverse Spatial Mode Spectrum of a Wave Field,” Phys. Rev. Lett. 100, 173902 (2008).
• (22) Shawn. A. Hilbert et al., “Temporal lenses for attosecond and femtosecond electron pulses,” Proc. Natl. Acad. Sci. 106, 10558 (2009).
• (24) S. Patchkovskii, Z. Zhao, T. Brabec, and D. M. Villeneuve, “High Harmonic Generation and Molecular Orbital Tomography in Multielectron Systems: Beyond the Single Active Electron Approximation,” Phys. Rev. Lett. 97, 123003 (2006).
• (25) J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Second-harmonic generation and the conservation of orbital angular momentum with high-order Laguerre-Gaussian modes,” Phys. Rev. A 56, 4193 (1997).
• (26) K. W. D. Ledingham, P. McKenna, and R. P. Singhal, “Applications for Nuclear Phenomena Generated by Ultra-Intense Lasers,” Science 300, 1107 (2003).
• (27) T. J. Bürvenich, J. Evers, and C. H. Keitel, “Nuclear Quantum Optics with X-Ray Laser Pulses,” Phys. Rev. Lett. 96, 142501 (2006).
• (28) E. Peik and Chr. Tamm, “Nuclear laser spectroscopy of the 3.5 eV transition in Th-229,” Europhys. Lett. 61, 181 (2003).
For everything else, email us at [email protected]. |
8203ecb92f0567bd | Session 17 July 2021
The Living Force
FOTCM Member
Thank you all for another Brilliant session!
(Joe) In the US, there are certain states - notably Florida and Texas - which have been exemplary in their pushback against this Covid nonsense. They didn't lock down, they opened and returned to normal more quickly, etc. That's kind of strange in a sense because you wouldn't expect that to come from the US - or at least we wouldn't. My point is that right now and over the past year, Florida and Texas were good places to live given what people in most places in the Western world were subjected to. So my question is: What's the prognosis for places like Florida and Texas?
A: Suppression will be attempted...
Q: (L) Alright, I think we've covered what we had to cover, right?
(Joe) "Suppression will be attempted..."
(Andromeda) But not necessarily successful.
(Joe) Revolution?
A: Exactly. Remember the Alamo!!
Q: (Joe) I hope so. I wanna see SOMEBODY do something. If it's gonna happen anywhere, it's gonna happen in America.
(Niall) Trump's coming back!
(Joe) 1776 will rise again!... [laughter]
(L) Is there anything we need to know or ask? Consider it asked to help us out through this turmoil...
A: Things will get worse before they get better. Stay alert and use knowledge!!! Goodbye.
Chaos in California:
Padawan Learner
Thanks for the good session Laura & Crew!
It doesn’t surprise me that the building collapse in Miami was due to cheap construction. A good amount of houses and buildings in Miami are from the mid 90s and no one bothers to do any real renovations. It’s sad that they charge people $1,500+ for rent for living in those houses/apartments.
Being a Florida native, many people here see through the BS and no one really wears masks to go inside stores or anything, it’s actually hardly enforced. Although, there are a good amount of people that have gotten the vaccine so let’s see how it goes when they attempt the suppression. Let’s see how people are going to react when the truth becomes too obvious ( food shortages, economic collapse, earth changes, UFO’s, etc).
We just have to keep on hanging on and dodge the craziness that these psychopaths throw at us.
Dagobah Resident
Seem like the lizzies try to do the trick again. In the past the Cs talked about the use of light waves (strange waves?)to cancel DNA sectors.
Seems like the Mark of Cain all over again¡¡¡
The unvaxxed are the Abel...or the able.
Just read the commets on Twitter. The vaxxed people seems very jealous
Aha!! It turns out that graphene is nothing more than a substance composed of pure carbon, with atoms arranged in a regular hexagonal pattern, similar to graphite:
Each atom in a graphene sheet is connected to its three nearest neighbors by a σ-bond, and contributes one electron to a conduction band that extends over the whole sheet. This is the same type of bonding seen in carbon nanotubes and polycyclic aromatic hydrocarbons, and (partially) in fullerenes and glassy carbon.[6][7] These conduction bands make graphene a semimetal with unusual electronic properties that are best described by theories for massless relativistic particles.[2] Charge carriers in graphene show linear, rather than quadratic, dependence of energy on momentum, and field-effect transistors with graphene can be made that show bipolar conduction. Charge transport is ballistic over long distances; the material exhibits large quantum oscillations and large and nonlinear diamagnetism.[8] Graphene conducts heat and electricity very efficiently along its plane. The material strongly absorbs light of all visible wavelengths,[9][10] which accounts for the black color of graphite; yet a single graphene sheet is nearly transparent because of its extreme thinness. The material is also about 100 times stronger than would be the strongest steel of the same thickness.[11][12]
Graphene is a zero-gap semiconductor, because its conduction and valence bands meet at the Dirac points. The Dirac points are six locations in momentum space, on the edge of the Brillouin zone, divided into two non-equivalent sets of three points. The two sets are labeled K and K'. The sets give graphene a valley degeneracy of gv = 2. By contrast, for traditional semiconductors the primary point of interest is generally Γ, where momentum is zero.[52] Four electronic properties separate it from other condensed matter systems.
However, if the in-plane direction is no longer infinite, but confined, its electronic structure would change. They are referred to as graphene nanoribbons. If it is "zig-zag", the bandgap would still be zero. If it is "armchair", the bandgap would be non-zero.
Graphene's hexagonal lattice can be regarded as two interleaving triangular lattices. This perspective was successfully used to calculate the band structure for a single graphite layer using a tight-binding approximation.[52]
Electronic spectrum
Electrons propagating through graphene's honeycomb lattice effectively lose their mass, producing quasi-particles that are described by a 2D analogue of the Dirac equation rather than the Schrödinger equation for spin-1⁄2 particles.[62][63]
Single-atom wave propagation
The wikipedia article discusses more properties than I put here, but one can see that graphene has all the electronic and quantum properties and uses for the effects we are seeing in vaccines.
Now, in the C's comment when the lizzies burned that DNA, that same DNA if I am not mistaken is what connected us somehow to our higher centers and densities.
Think about this for a moment... what if now for the purpose of connecting and bridging densities and doing an invasion you need to reestablish what was there in the past but with the difference of having control over the on and off? That is, they can turn the bridge on and off at will...
FOTCM Member
The C's have relayed that message more than once.
What is your interpretation of it? My take is that nothing can get better in 3D, can it?
Even if the PTB botch things up and their influence wanes it can only get worse because of the expected earth changes.
So our existence can only improve in 5D or 7D, or so I think...
I guess it depends on how one defines „better“. From a certain perspective things were clearly „better“ before Covid and many other times in 3D history. The same applies to „worse“ times within 3D. So I don’t think that what you are suggesting necessarily has to be the case.
Ursus Minor
The Living Force
FOTCM Member
If I understand you correctly we still might regain our rights and freedoms here in 3D (which would mean things are getting better) albeit with scarce food and without central heating.
The Living Force
FOTCM Member
Yesterday, Dr. Malone tweeted the following: “So I hope this is hyperbole and an overreaction, but last night an experienced journalist told me that I needed to get security because I was at risk of being assassinated. I do not know even how to begin to think about this. I am just a middle class person. Security??!??”
The meaning of Zombie has changed with Covid - It is not like in popular culture. With those nano particles, more people can be made manipulated to do the PTB's bidding. The reality is all beyond science fiction now a days.
The Living Force
That's an interesting question. There are reports about magnetic after-vaccination effects in Russia. I saw a video from a pretty reliable source about a woman in St. Petersburg who got hear left arm hold coins at the place of a shot after vaccination (Sputnik V probably). After several days the effect wore off. It still could be a hoax of course.
Russian authorities have been quoted on stating that they can track those peeps, who have taken the vaccine. Also this article says there are Russian Covid vaccines based on nano-particles.
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abf801718a8f5765 | Category Archives: 3D complex numbers
De Padé approximations; why are they so good?
Already a few years back I wanted to write a post on the so called de Padé apprximations because they are so good at taking the logarithm of a matrix. For me the access to an internet application that calculated those logs from matrix representations was a very helpful thing to speed things up. It would have taken me much much longer to find the first exponential circle on the 3D complex numbers if I could not use such applets.
But in the year 2018 pure evil struck the internet: the last applets or websites having them disappeared to never come back. Ok by that time I had perfected my method of simply using matrix diagonalization for finding such logs of matrices. You can still find it easily if you do an internet search on ‘Calculation of the 7D number tau‘.
Yet in the beginning I only had such applets as found on the internet and I soon found out that using the so called de Padé approximation always gave much better results compared to say a Taylor approximation.
It is not very hard to understand how to perform such a de Padé approximation. Much harder to understand is how de Padé found them, after all it looks like a strike of genius if this works. The genius part is of course found in the stuff you can simply neglect in such approximations, at first it baffles the mind and later you just accept it that you are more stupid as de Padé was…;)
Anyway this week I stumbled upon a cute video and as such I decided to write a small post upon this de Padé stuff. (On the shelf are still a possible new way of making an antenna based on the 3D exponential circles and some updates on magnetism.)
So let us first take a look at the video, here we go:
As you see the basic idea is pretty simple: you use those two polynomials to ‘approximate’ that Taylor series and as a bonus you have a much better approximation of the original function. All in all this is amazing and it makes you wonder if there are methods out there that are even better compared to this de Padé approximation.
Now you can choose beforehand what degree polynomial you use in the nominator and denominator. There are plenty of situations where this brings a big benefit like in the video they point out the divergence problems of say the sine function that is bounded between +1 and -1 on the real axis. The Taylor approximations always go completely beserk outside some interval where they fit quite well. With the appropiate choice of the degrees of the polynomials in the de Padé approximation you can avoid this kind of stuff.
In my view the maker of the video should also have pointed out that a de Padé approximation can have it’s own troubles when you divide by zero. And when the original function never has such a pole at that point, the de Padé approximation also goes very bad. These de Padé approximations
are indeed much better compared to the average Taylor approximation but they are not from heaven. You still have to use your own brain and may be that is a good thing.
De Padé still rules in the year 2021, likely it was a good idea to start with.
In this post I did not cover the matrices and why a de Padé approximation of the logarithm of a matrix is good or bad. If you want to find exponential circles and curves for yourself, use those applets mostly on imaginary units who’s matrix representation has a determinant of +1. In case you want to find your very first exponential circle, solve the next problem:
Ok, it is late at night so let me hit that button ‘publish website’ and see you around.
Three video’s to kill the time in case you are bored to the bone…
A couple of days ago I started on a new post, it is mostly about elliptic curves and we will go and see what exactly happens if you plug in one of those counter examples to the last theorem of Pierre de Fermat. There is all kinds of weird stuff going on if you plug such counter example in such a ‘Frey elliptic curve’. I hope next week it will be finished.
In this post I would like to show you three video’s so let’s start that: In the first video a relatively good introduction to the last theorem of Fermat is given. One of the important details of that long proof is the relation between elliptic curves and so called modular forms. And now I understand a bit better as why math professors go bezerk on taking such an elliptic curve modulo a prime number; the number of solutions is related to a coefficient of such an associated modular form. It boggles the mind because what do those other coefficients mean? As always just around the corner is a new ocean of math waiting to get explored.
Anyway, I think that I can define such modular forms on the 3D complex and circular numbers too so may be that is stuff for a bunch of future posts. On the other hand the academic community is never ever interested in my work whatsoever so may be I will skip that whole thing too. As always it is better to do what you want and not what you think other people would like to see. The more or less crazy result is shown in the picture below and after that you can see the first video.
Yet it might be this does not work on the 3D complex numbers…
Next video: At MIT they love to make a fundamental fool of themselves by claiming that their version of a nuclear fusion reactor will be the first that puts power on the electricity grid… Ok ok, after five or six years I have terminated the magnetic pages on the other website because it dawned on me that the university people just don’t want to read my work. I have explained many many times that it is just impossible that electrons are magnetic dipoles but as usual nothing happens.
Oops, wasn’t it some years ago that Lockheed Martin came bragging out they would make mobile nuclear fusion reactors and by now (the year 2021) there would be many made already? Of course I would never work properly because at Lockheed Martin they to refuse to check if the idea’s of electron spin are actually correct. If electrons are magnetic monopoles all fusion reactors based on magnetic confinement will never work. Just look at Lockheed Martin: So much bragging but after all those years just nothing to show. Empty headed arrogant idiots is whart they are.
And now MIT thinks it is their time to brag because they have mastered much stronger magnetic fields with their new high temperature superconducting magnets. Yes well you can be smart on details like super conducting magnets but if you year in year out refuse to take a look at electron spin and is that Pauli matrix nonsense really true in experiments? If you refuse that year in year out, you are nothing but a full blown arrogant overpaid idiot. And you truly deserve the future failure that will be there: A stronger magnetic field only makes the plasma more turbulent faster. And your fantasies of being the first to put electricity on the grid? At best you are a pathetic joke.
MIT & me, are we mutual jokes to each other?
Just like ITER and the Wendelstein 7X this will not work!
It is very difficult to make a working nuclear fusion reactor on earth if you just don’t want to study the magnetic properties of electrons while you try to contain the plasma with magnetic fields. Oh the physics imbeciles and idiots think they understand plasma? They even do not understand why the solar corona is so hot and if year in year out I say that magnetic fields accelerate particles with a net magnetic charge, the idiots and imbeciles just neglect it because they are idiots and imbeciles.
The third video is about a truly Hercules task: Making a realistic model of the sun so that can run in computer simulations… If humanity is still around 10 thousand years from now may be they have figured it out but the sun is such a complicated thing it just cannot be understood in a couple of decades. There is so much about the sun that is hard to understand. For example a number of years ago using the idea that electrons are magnetic monopoles, it thought that rotating plasma like in some tornado kind of structure is all you need to get extremely strong magnetic fields. But I never ever wrote down only one word in that direction. Anyway about a full year later I learned about the rotational differential for the sun: at the equator it spins much faster as it does on the poles. And that would definitely give rise to a lot of those tornade like structurs that must be below the sun spots.
Of course nothing happens because of ‘university people’ and at present day I do not give a shit any longer. I am 100% through with idiots and imbeciles like that. For me it only counts that I know, that I have figured out something and trying to communicate that to a bunch of overpaid highly absorbed in their giant ego’s idiots and imbeciles is a thing I just stopped doing. If it is MIT, ITER or Max Planck idiots and imbeciles, why should I care?
Ok, that was it for this post. If you are not related to a university or academia thanks for your attention. And to the university shitholes: please go fuck yourselves somewhere we don’t have to watch it.
Oversight of all counter examples to the last theorem of Pierre de Fermat, Part 3.
It is late at night, my computer clock says it is 1.01 on a Sunday night. But I am all alone so why not post this update? This post does not have much mathematical depth, it is all very easy to understand if you know what split complex numbers are.
In the language of this website, the split complex numbers are the 2D circular numbers, In the past I named a particular set of numbers complex or circular. I did choose for circular because the matrix representations of circular numbers are the so called circulant matrices. It is always better to give mathematical stuff some kind of functional name so people can make sense of what the stuff is about. For me no silly names like ‘3D Venema positive numbers’ or ‘3D Venema complex numbers’. In math the objects should have names that describe them, the name of a person should not be hanged on such an object. For example the Cayley-Hamilton theorem is a total stupid name, the names of the humans who wrote it out are not relevant at all. Further reading on circulant matrices: Circulant matrix.
I also have a wiki on split complex numbers for you, but like all common sources they have the conjugate completely wrong. Professional math professors always think that taking a conjugate is just replacing a + by a – but that is just too simplistic. That’s one of the many reasons they never found 3D complex numbers for themselves, if you do that conjugate thing in the silly way all your 3D complex math does not amount to much…
Link: Split-complex number.
This is the last part on this oversight of counter examples to the last theorem of Pierre de Fermat and it contains only the two dimensional split complex numbers. When I wrote the previous post I realized that I had completely forgetten about the 2D split numbers. And indeed the math results as found in this post are not very deep, it’s importance lies in the fact that the counter examples now are unbounded. All counter examples based on modular arithmetic are always bounded, periodic to be precise, so professional math professors could use that as a reason to declare that all a bunch of nonsense because the real integers are unbounded. And my other counter examples that are unbounded are only on 3D complex & circular number spaces and the 4D complex numbers so that will be neglected and talked into insignificance because ‘That is not serious math’ or whatever kind of nonsense those shitholes come up with.
All in all despite the lack of mathematical depth I am very satisfied with this very short update. The 2D split numbers have a history of say 170 years so all those smart math assholes can think a bit about why they never formulated such simple counter examples to the last theorem of Fermat… May be the simplicity of the math results posted is a good thing in the long run: compare it to just the natural numbers or the counting numbers. That is a set of numbers that is very simple too, but they contain prime numbers and all of a sudden you can ask thousands and thousands of complicated and difficult questions about natural numbers. So I am not ashamed at all by the lack of math depth in this post, I only point to the fact that over the course of 170 years all those professional math professors never found counter examples on that space.
This post is just 3 pictures long although I had to enlarge the lastest one a little bit. The first two pictures are 550×825 pixels and the last one is 550×975 pixels. Here we go:
That was it for this post, one of the details as why this post is significant is the use of those projector numbers. You will find that nowhere on the entire internet just like the use of 3D complex numbers is totally zero. Let’s leave it with that, likely the next post is about magnetism and guess what? The physics professors still think there is no need at all to give experimental proof to their idea of the electron having two magnetic poles. So it are not only the math professors that are the overpaid idiots in this little world of monkeys that think they are the masters of the planet.
Oversight of all counter examples to the last theorem of Pierre de Fermat, Part 2.
Post number 191 already so it will be relatively easy to make it to post number 200 this year. If you think about it, the last 190 posts together form a nice bunch of mathematics.
In this post we will pick on where we left it in the last post; we start with the three dimensional complex and circular numbers. In the introduction I explain how the stuff with a pair of divisors of zero works and from there it is plain sailing so to say. When back in Jan of this year I constructed the first counter example to the last theorem of Pierre de Fermat I considered it a bit ‘non math’ because it was so easy. And when one or two days later I made the first counter example using modular arithmetic I was really hesitant to post it because it was all so utterly simple…
But now half a year later it has dawned on me that all those professional math professors live up to their reputation of being overpaid under performers because in a half year of time I could find not one counter example on our beloved internet. And when these people write down some calculations that could serve as a counter example, they never say so and use it only for other purposes like proving the little theorem of Fermat. It has to be remarked however that in the past three centuries of time, when people tried to find counter examples, they likely started with the usual integers from the real line and as such tried to find counter examples. Of course that failed and this is not because they are stupid or so. It is the lack of number spaces they understand or know about that prevented them in finding counter examples to the last theorem of Pierre de Fermat.
If you do not know anything about 3D complex or circular numbers, you are not a stupid person if you cannot find counter examples to the last theorem. But you are definitely very very stupid if you do not want to study 3D complex numbers, if you refuse that it proves you have limited mathematical insights and as such likely all your other math works will be limited in long term value too.
While writing this post all of a sudden I realized I skipped at least one space where counter examples are to be found: It is on the space of so called split complex numbers. I did not invent that space, that was done by the math professors. The split complex numbers are a 2D structure just like the complex plane but instead of i^2 = -1, on the split complex plane the multiplication is ruled by i^2 = 1. Likely I will write a small post about the split complex number space. (Of course in terms of the language of this website, the 2D split complex numbers are the 2D circular numbers.)
This post is 8 pictures long, I kept on to number them according to the previous post so we start at picture number 11. They are all in the size of 825×550 pixels. I hope it is worth of your time. Here we go:
In this post I used only ‘my own spaces’ like 3D complex and circular numbers and the 4D complex numbers. As such it will be 100% sure the math professionals will 100% not react on it. Even after 30 years these incompetents are not able to judge if there is any mathematical value in spaces like that. Why do we fork out so much tax payer money to those weirdo’s? After all it is a whole lot of tax payer money for a return of almost nothing. Ok ok a lot of math professors also give lectures in math to other studies like physics so not all tax payer money is 100% wasted but all in all the math professors are a bunch of non-performers.
I think I will write a small post about the 2D split complex numbers because that is a space discovered by the math pro’s. So for them we will have as counter examples to the last theorem of Pierre de Fermat all that modulo calculus together with the future post on the split complex numbers. Not that this will give a reaction from the math pro’s but it will make clear you just cannot blame me for the non reactive nature of the incompetents; the blame should go to those who deserve it… Or not?
May be the next post is about magnetism and only after that I will post the split complex number details. We’ll see, anyway if you made it untill here thanks for your attention and I hope you learned a bit from the counter examples to the last theorem of Pierre de Fermat.
Why can’t I find counter examples to Fermat’s last theorem on the internet?
After a few weeks it is finally dawning on me that it might very well be possible that the professional math people just do not have a clue about how easy it is to find counter examples to the FLT. (FLT = Fermat’s Last Theorem.) That is hard to digest because it is so utterly simple to do and understand on those rings of integers modulo n.
But I did not search long and deep and I skipped places like the preprint archive and only used a bit of the Google thing. And if you use the Google thing of course you get more results from extravert people. That skews the results of course because for extraverts talking is much more important compared to the content of what you are talking or communicating. That is the problem with extraverts; they might be highly social but they pay a severe price for that: their thinking will always be shallow and never some stuff deeply thought through…
As far as I know rings of the integers modulo n are not studied very much. Of course the additive groups modulo n are studied and the multiplicative groups modulo n are studied but when it comes to rings all of a sudden it is silent always everywhere. And now I am looking at it myself I am surprised how much similarity there is between those kind of rings and the 3D complex & circular numbers. Of course they are very different objects of study but you can all chop them in two parts: The numbers that are invertible versus the set of non-invertibles. For example in the ring of integers modulo 15 the prime factors of 15 are 3 and 5. And those prime factors are the non-invertibles inside this ring. This has all kinds of interesting math results, for example take the (exponential) orbit of 3. That is the sequence of powers of 3 like in: 3, 3^2 = 9, 3^3 = 27 = 12 (mod 15), 3^4 = 36 = 6 (mod 15) and 3^5 = 18 = 3. As you see this orbit avoids the number 1 because if it would pass through 1 you would have found an inverse of 3 inside our ring and that is not possible because 3 is a non invertible number…
Likely my next post will be about such stuff, I am still a bit hesitant about it because it is all so utterly simple but you must never underestimate how dumb the overpaid math professors can be: Just neglecting rings modulo n could very well be a common thing over there while in the meantime they try to act as a high IQ person by stating ‘We are doing the Langlands program’ & and more of that advanced blah blah blah.
Anyway it is getting late at night so from all that nonsense weird stuff you can find on Google by searching for counter examples to the last theorem of Fermat I crafted 3 pictures. Here is the first one:
I found this retarded question on quora. For me it is hard to process what the person asking this question was actually thinking. Why would the 2.999…. be important? What is this person thinking? Does he have integer solutions to say 2.9 and 2.99 and is this person wondering what would happen if you apply those integer solutions to 2.99999999…..???????
It is retarded, or shallow, on all levels possible. So to honor the math skills of the average human let’s make a new picture of this nonsense:
We will never be intimidated by the stupidity of such questions and simply observe these are our fellow human beings. And if ok, if you are a human being running into tons of problems, in the end you can always wonder ‘Am I a problem myself because I am so stupid?’
If you have figured out that question, you are getting more solid & you look more like a little cube:
I want to end this post on a positive note: Once you understand how stupid humans are you must not view that as a negative. On the contrary, that shows there is room for improvement.
The last theorem of Fermat does not hold for the 3D so called Gaussian integers.
On the one hand it is a pity I have to remove the previous post from the top position. Never ever I would have thought that the Voyager probes would be a big help in my quest of proving that electrons are not magnetic dipoles. Electrons are magnetic monopoles, if your local physics professor thinks otherwise why not ask you local physics professor for the experimental evidence there is for the electron magnetism dipole stuff?
On the other hand this post is about Gaussian intergers for the 3D complex and circular numbers and it is with a bit of pride that I can say we have a bunch of beautiful results because the last theorem of Fermat does not hold in these spaces.
The last theorem of Fermat is a kind of negative result, it says that it is impossible for three integers x, y and z that x^n + y^n = z^n, this for integer values of n greater than 2 of course. (For n = 2 I think most readers know it is possible because those are the Pythagoras triples.)
Anyway I succeeded into writing the number 3 as the sum of two Gaussian 3D integers that are also divisors of zero. So this pair of integers, in this post I name them A and T because they are related to the famous 3D numbers alpha and tau, are divisors of zero so as such AT = 0. As such as a denial of the Fermat theorem, an important result as posted here is that A^n + T^n = 3^n. So on the 3D complex & circular numbers this result is possible while if you use only the 2D complex plane and the real line this is not possible…
But there are plenty of spaces where the Fermat conjecture or the last theorem does not hold. A very easy to understand space is the ring of integers modulo 15. In this ring there are numbers that do not have a multiplicative inverse, say 3 and 5. And if inside this ring you multiply 3 and 5 you get 15 and 15 = 0 in this ring… Hence inside this ring we have that 8^n = 3^n + 5^n (mod 15) also contradicting the Fermat stuff.
I did some internet searches like ‘Fermat last theorem and divisors of zero’ but weirdly enough nothing popped up. That was weird because I view the depth of the math results related to this divisor of zero as the depth of a bird bath. It is not a deep result or so, just a few centimeters deep. But sometimes just a few centimeters can bring a human mind into another world. For example a long time ago when I still was as green as grass back in the year 1986 I came across the next excercise: Calulate the rest of 103 raised to 103 and divided by 13. I was puzzled, after all 103^103 is a giant number so how can you find the rest after dividing it by 13? But if you give that cute problem a second thought, after all that is also bird bath deep because you can solve it with your human brain…
This post is 11 pictures long, all of the standard size of 550×775 pixels. Because I could not find anything useful about the last Fermat theorem combined with divisors of zero I included a small addendum so all in all this post is 12 pictures long.
After so much Gaussian integer stuff, there is only one addendum about the integers modulo 30. In that ring you can also find some contradictions to the standard way of presenting the last theorem of Fermat.
Ok, if you are still fresh after all that modulo 30 stuff, for reasons of trying to paint an overall picture let me show you a relatively good video on the Kummer stuff. Interesting in this video is that Kummer used the words `Ideal numbers´ and at present stuff like that is known as an ideal. For myself speaking I never use the word ´ideal´ for me these are ´multiplicative attractors´ because if a number of such an ideal multiplies a number outside that ideal, the result is always inside that ideal. Here is a relatively good video:
And now you are at the end of this post. Till updates.
The total differential for the complex plane & the 3D and 4D complex numbers.
I am rather satisfied with the approach of doing the same stuff on the diverse complex spaces. In this case the 2D complex plane and the 3D & 4D complex number systems. By doing it this way it is right in your face: a lot of stuff from the complex plane can easily be copied to higher dimensional complex numbers. Without doubt if you would ask a professional math professor about 3D or higher dimensional complex numbers likely you get a giant batagalization process to swallow; 3D complex numbers are so far fetched and/or exotic that it falls outside the realm of standard mathematics. “Otherwise we would have used them since centuries and we don’t”. Or words of similar phrasing that dimishes any possible importance.
But I have done the directional derivative, the factorization of the Laplacian with Wirtinger derivatives and now we are going to do the total differential precisely as you should expect from an expansion of the century old complex plane. There is nothing exotic or ‘weird’ about it, the only thing that is weird are the professional math professors. But I have given up upon those people years ago, so why talk about them?
In the day to day practice it is a common convention to use so called straight d‘s to denote differentiation if you have only one variable. Like in a real valued function f(x) on the real line, you can write df/dx for the derivative of such a function. If there are more then one variable the convention is to use those curly d’s to denote it is partial differentiation with respect to a particular variable. So for example on the complex plane the complex variable z = x + iy and as such df/dz is the accepted notation while for differentiation with respect to x and y you are supposed to write it with the curly d notation. This practice is only there when it comes to differentiation, the opposite thing is integration and there only straight d‘s are used. If in the complex plane you are integrating with respect to the real component x you are supposed to use the dx notation and not the curly stuff.
Well I thought I had all of the notation stuff perfectly figured out, oh oh how ultrasmart I was… Am I writing down the stuff for the 4D complex numbers and I came across the odd expression of dd. I hope it does not confuse you, in the 4D complex number system I always write the four dimensional numbers as Z = a + bl + cl^2 + dl^3 (the fourth power of the imaginary unit l must be -1, that is l^4 = -1, because that defines the behavior of the 4D complex numbers) so inside Z there is a real variable denoted as d. I hope this lifts the possible confusion when you read dd
More on the common convention: In the post on the factorization of the Laplacian with Wirtinger derivatives I said nothing about it. But in case you never heard about the Wirtinger stuff and looked it up in some wiki’s or whatever what, Wirtinger derivatives are often denoted with the curly d‘s so why is that? That is because Wirtinger derivatives are often used in the study of multi-variable complex analysis. And once more that is just standard common convention: only if there is one variable you can use a straight d. If there are more variable you are supposed to write it with the curly version…
At last I want to remark that the post on the factorization of the Laplacian got a bit long: in the end I needed 15 pictures to publish the text and I worried a bit that it was just too long for the attention span of the average human. In the present years there is just so much stuff to follow, for most people it is a strange thing to concentrate on a piece of math for let’s say three hours. But learning new math is not an easy thing: in your brain all kind of new connections need to be formed and beside a few hours of time that also needs sleep to consolidate those new formed connections. Learning math is not a thing of just spending half an hour, often you need days or weeks or even longer.
This post is seven pictures long, have fun reading it and if you get to tired and need a bit of sleep please notice that is only natural: the newly formed connetions in your brain need a good night sleep.
Here we go with the seven pictures:
Yes, that’s it for this post. Sleep well and think well & see you in the next post. (And oh oh oh a professional math professor for the first time in his or her life they calculate the square Z^2 of a four dimensional complex number; how many hours of sleep they need to recover from that expericence?)
See ya in the next post.
Factorization of the Laplacian (for 2D, 3D and 4D complex numbers).
Originally I wanted to make an oversight of all ways the so called Dirac quantization condition is represented. That is why in the beginning of this post below you can find some stuff on the Dirac equation and the four solutions that come with that equation. Anyway, Paul Dirac once managed to factorize the Laplacian operator, that was needed because the Laplacian is part of the Schrödinger equation that gives the desired wave functions in quantum mechanics. Well I had done that too once upon a time in a long long past and I remembered that the outcome was highly surprising. As a matter of fact I consider this one of the deeper secrets of the higher dimensional complex numbers. Now I use a so called Wirtinger derivative; for example on the space of 3D complex numbers you take the partial derivatives into the x, y and z direction and from those three partial derivatives you make the derivative. And once you have that, if you feed it a function you simply get the derivative of such a function.
Now such a Wirtinger derivative also has a conjugate and the surprising result is that if you multiply such a Wirtinger derivative against it’s conjugate you always get either the Laplacian or in the case of the 3D complex numbers you get the Laplacian multiplied by the famous number alpha.
That is a surprising result because if you multiply an ordinary 3D number X against it’s conjugate you get the equation of a sphere and a cone like thing. But if you do it with parital differential operators you can always rewrite it into pure Laplacians so there the cones and spheres are the same things…
In the past I only had it done on the space of 3D numbers so I checked it for the 4D complex numbers and in about 10 minutes of time I found out it also works on the space of 4D complex numbers. So I started writing this post and since I wanted to build it slowly up from 2D to 4D complex numbers it grew longer than expected. All in all this post is 15 pictures long and given the fact that people at present day do not have those long timespan of attention anymore, may be it is too long. I too have this fault, if you hang out on the preprint archive there is just so much material that often after only five minutes of reading you already go to another article. If the article is lucky, at best it gets saved to my hard disk and if the article has more luck in some future date I will read it again. For example in the year 2015 I saved an article that gave an oversight about the Dirac quantization condition and only now in 2020 I looked at it again…
The structure of this post is utterly simple: On every complex space (2D, 3D and 4D) I just give three examples. The examples are named example 1, 2 and not surprising I hope, example 3. These example are the same, only the underlying space of complex numbers varies. In each example number 1 I define the Wirtinger derivative, in example 2 I take the conjugate while in the third example on each space I multiply these two operators and rewrite the stuff into Laplacians. The reason this post is 15 pictures long lies in the fact that the more dimensions you have in your complex numbers the longer the calculations get. So it goes from rather short in the complex plane (the 2D complex numbers) to rather lengthy in the space of 4D complex numbers.
At last I would like to remark that those four simultanious solutions to the Dirac equation it once more shouts at your face: electrons carry magnetic charge and they are ot magnetic dipoles! All that stuff like the Pauli matrices where Dirac did build his stuff upon is sheer difficult nonsense: the interaction of electron spin with a magnetic field does not go that way. The only reason people in the 21-th century think it has some merits is because it is so complicated and people just loose oversight and do not see that it is bogus shit from the beginning till the end. Just like the math professors that neatly keep themselves stupid by not willing to talk about 3D complex numbers. Well we live in a free world and there are no laws against being stupid I just guess.
Enough of the blah blah blah, below are the 15 pictures. And in case you have never ever heard about a thing known as the Wirtinger derivative, try to understand it and may be come back in five or ten years so you can learn a bit more…
As usual all pictures are 550×775 pixels in size.
Oh oh the human mind and learning new things. If a human brain learns new things like Cauchy-Riemann equations or the above factoriztion of the Laplacian, a lot of chages happen in the brain tissue. And it makes you tired and you need to sleep…
And when you wake up, a lot of people look at their phone and may be it says: Wanna see those new pictures of Miley Cyrus showing her titties? And all your new learned things turn into insignificance because in the morning what is more important compared to Miley her titties?
Ok my dear reader, you are at the end of this post. See you in the next post.
The directional derivative (for 3D & 4D complex numbers).
A couple of days ago all of a sudden while riding my bicycle I calculated what the so called directional derivative is for 3D & 4D complex numbers. And it is a cute calculation but I decided not to write a post about it. After all rather likely I had done stuff like that many years ago.
Anyway a day later I came across a few Youtube video’s about the directional derivative and all those two guys came up with was an inner product of the gradient and a vector. Ok ok that is not wrong or so, but that is only the case for scalar valued functions on say 3D space. A scalar field as physics people would say it. The first video was from the Kahn academy and the guy from 3Blue1Brown has been working over there lately. It is amazing that just one guy can lift such a channel up in a significant manner. The second video was from some professional math professor who went on talking a full 2.5 hour about the directional derivative of just a scalar field. I could not stand it; how can you talk so long about something that is so easy to explain? Now I do not blame that math professor, may be he was working in the USA and had to teach first year math students. Now in the USA fresh students are horrible at math because in the USA the education before the universities is relatively retarded.
Furthermore I tried to remember when I should have done the directional derivative. I could not remember it and in order to get rid of my annoyance I decided to write a small post about it. Within two hours I was finished resulting in four pictures of the usual 550×775 pixel size. So when I work hard I can produce say 3 to 4 pictures in two hours of time. I did not know that because most of the time I do not work that fast or hard. After all this is supposed to be a hobby so most of my writing is done in a relaxed way without any hurry. I have to say that may be I should have taken a bit more time at the end where the so called Cauchy-Riemann equations come into play. I only gave the example for the identiy function and after that jumped to the case of a general function. May be for the majority of professional math professors that is way to fast, but hey just the simple 3D complex numbers are ‘way to fast’ for those turtles in the last two centuries…
Anyway, here is the short post of only 4 pictures:
Should I have made the explanation longer? After all so often during the last years I have explained that the usual derivative f'(X) is found by differentiating into the direction of the real numbers. At some point in time I have the right to stop explaining that 1 + 1 = 2.
Also I found a better video from the Kahn academy that starts with a formal definition of the directional derivative:
At last let me remark that this stuff easily works for vector valued functions because in the above limit you only have to subtract two vectors and that is always allowed in any vector space. And only if you hang in a suitable multiplication like the complex multiplication of 3D or 4D real space you can tweak it like in the form of picture number 4 above.
That was all I had for you today, this is post number 166 already so I am wondering if this website is may be becoming too big? If people find something, can they find what they are searching for or do they get lost in the woods? So see you in another post, take care of yourself & till the next post.
On the work of Shlomo Jacobi & a cute more or less new Euler identity.
For a couple of years I have a few pdf files in my possession written by other people about the subject of higher dimensional complex and circular numbers. In the post we will take a look at the work of Shlomo Jacobi, the pdf is not written by him because Shlomo passed away before it was finished. It is about the 3D complex numbers so it is about the main subject of this website.
Let me start with a link to the preprint archive:
On a novel 3D hypercomplex number system
Link used:
Weirdly enough if you search for ‘3D hypercomplex number’ the above pdf does not pop up at all at the preprint archive. But via his name (Shlomo Jacobi) I could find it back. Over the years I have found three other people who have written about complex numbers beyond the 2D complex plane. I consider the work of Mr. Jacobi to be the best so I start with that one. So now we are with four; four people who have looked at stuff like 3D complex numbers. One thing is directly curious: None of them is a math professional, not even a high school teacher or something like that. I think that when you are a professional math professor and you start investigating higher dimensional complex numbers; you colleagues will laugh about it because ‘they do not exist’. And in that manner it are the universities themselves that ensure they are stupid and they stay stupid. There are some theorems out there that say a 3D complex field is not possible. That is easy to check, but the math professionals make the mistake that they think 3D complex numbers are not possible. But no, the 2-4-8 theorem of say Hurwitz say only a field is not possible or it says the extension of 2D to 3D is not possible. That’s all true but it never says 3D complex numbers are not possible…
Because Shlomo Jacobi passed away an unknown part of the pdf is written by someone else. So for me it is impossible to estimate what was found by Shlomo but is left out of the pdf. For example Shlomo did find the Cauchy-Riemann equations for the 3D complex numbers but it is only in an epilogue at the end of the pdf.
The content of the pdf can be used for a basic introduction into the 3D complex numbers. It’s content is more or less the ‘algebra approach’ to 3D complex numbers while I directly and instantly went into the ‘analysis approach’ bcause I do not like algebra that much. The pdf contains all the basic stuff: definition of a 3D complex number, the inverse, the matrix representation and stuff he names ‘invariant spaces’. Invariant spaces are the two sets of 3D complex numbers that make up all the non-invertible numbers. Mr. Jacobi understands the concept of divisors of zero (a typical algebra thing that I do like) and he correctly indentifies them in his system of ‘novel hypercomplex numbers’. There is a rudimentary approach towards analysis found in the pdf; Mr. Jacobi defines three power series named sin1, sin2 and sin3 . I remember I looked into stuff like that myself and somewhere on this website it must be filed under ‘curves of grace’.
A detail that is a bit strange is the next: Mr. Jacobi found the exponential circle too. He litarally names it ‘exponential circle’ just like I do. And circles always have a center, they have a midpoint and guess how he names that center? It is the number alpha…
Because Mr. Jacobi found the exponential circle I applaud him long and hard and because he named it’s center the number alpha, at the end I included a more or less new Euler identity based on a very simple property of the important number alpha: If you square alpha it does not change. Just like the square of 1 is 1 and the square of 0 is 0. Actually ‘new’ identity is about five years old, but in the science of math that is a fresh result.
The content of this post is seven pictures long, please read the pdf first and I hope that the mathematical parts of your brain have fun digesting it all. Most pictures are of the standard size of 550×775 pixels.
Yes all you need is that alpha is it’s own square.
Ok ok, may be you need to turn this into exponential circles first in order to craft the proof that a human brain could understand. And I am rolling from laughter from one side of the room to the other side; how likely is it that professional math professors will find just one exponential circle let alone higher dimensional curves?
I have to laugh hard; that is a very unlikely thing.
End of this post, see you around & see if I can get the above stuff online. |
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Quantum billiards with correlated electrons confined in triangular transition metal dichalcogenide monolayer nanostructures
Forcing systems through fast non-equilibrium phase transitions offers the opportunity to study new states of quantum matter that self-assemble in their wake. Here we study the quantum interference effects of correlated electrons confined in monolayer quantum nanostructures, created by femtosecond laser-induced quench through a first-order polytype structural transition in a layered transition-metal dichalcogenide material. Scanning tunnelling microscopy of the electrons confined within equilateral triangles, whose dimensions are a few crystal unit cells on the side, reveals that the trajectories are strongly modified from free-electron states both by electronic correlations and confinement. Comparison of experiments with theoretical predictions of strongly correlated electron behaviour reveals that the confining geometry destabilizes the Wigner/Mott crystal ground state, resulting in mixed itinerant and correlation-localized states intertwined on a length scale of 1 nm. The work opens the path toward understanding the quantum transport of electrons confined in atomic-scale monolayer structures based on correlated-electron-materials.
A laser-induced quench through a first-order structural transition can create small domain structures with atomically precise shapes that are beyond the reach of current nanofabrication technologies. Viewed from a quantum physics perspective, such structures represent a fruitful playground for investigating the quantum behavior of particles in geometrically confined systems. Regular shapes, and equilateral triangles (ETs) in particular, are of special interest because they allow the study of the crossover from periodic limit cycles to chaotic trajectories1,2,3. Quantum scars—quantum interference (QI) patterns that follow traces of the paths of classical particles4,5,6,7—were investigated until now in fabricated mesoscopic semiconductor heterostructures and graphene by scanning gate microscopy8,9. However, with strongly interacting electrons such patterns are not expected due to their tendency for localization, and more elaborate structures are expected that require new theoretical approaches beyond the noninteracting electron approach. Recently, oscillating patterns were observed in interacting Rydberg atom arrays and theoretically motivated by the existence of many-body scars—manifold of low entangled excited states10,11. Correlation effects may be expected to give rise to perturbed trajectories in which the entanglement dynamics show a dependence on the nature of the perturbation. The observation of QI in confined correlated electron systems would be of both fundamental interest, and also of practical importance for designing coherent electron devices with correlated materials.
Here we use scanning tunneling microscopy to investigate QI in ET-shaped monolayer nanostructures of TaS2 as small as ~2.6 to ~12.5 nm wide (8–38 unit cells) (Fig. 1). TaS2 is a prototypical electronically correlated quasi-2D material, which is prone to carrier localization and the formation of different charge orders at different temperatures that become commensurate at ‘magic’ filling fractions12.
Fig. 1: Monolayer 1T-TaS2 structure in the shape of an ET bounded by a 1H-TaS2 monolayer.
figure 1
a A schematic picture of the left-handed packing of polarons in a C superlattice within an ET of outer dimensions 16a (inner dimensions l = 11a), where a = 0.33 Å is the lattice constant. The CCDW superlattice vector \({\boldsymbol{A}}\) is shown in terms of the lattice vectors a and b. The unit cells of the 1T and 1H crystal structures are shown in the insert. b The schematic structure of ETs on top of a 1T-TaS2 substrate. c Ideal polaron packing in L and R-handed CCDW superlattices. d A phase diagram showing the charge density wave ordering transitions of the 1T- and 2H- polytypes as a function of temperature. The measurement temperature and phase transition temperatures are indicated. e A band diagram of the 1T-1H boundary. An edge state of width w forms within the 1T phase as a result of band bending (see Supplementary information). f An STM image of different sized a 1T-TaS2 \(ETs\) embedded laterally within a 1H-TaS2 monolayer on the surface of a 1T-TaS2 single crystal. Note the ubiquitous presence of the edge state. The insert (bottom-left) shows a high-resolution image of a small ET with 6 polarons. More images are shown in the Supplementary information. g, h Fourier transforms of the 1H and 1T regions respectively. The \(\sqrt{13} \;\times\sqrt{13}\) CCDW SL and CL peaks are indicated. A peak attributed to the polarons ordered parallel to the edge of the ETs is also indicated (E).
The orthorhombic 1T polytype of 1T-TaS2 below ~180 K, has a commensurate charge-density-wave (CCDW) with a large modulation amplitude of \(\sim 1\) electron per 13 Ta sites, localized in a \(\sqrt{13}\;\times \sqrt{13}\) superlattice structure, which can exhibit either left (L) or right (R) chirality with respect to the crystal lattice13 (Fig. 1a). The lattice surrounding each 13th Ta atom is distorted by the extra charge14 resulting in the formation of a polaron. Due to Coulomb interactions between such polarons, the system is correlated and thought to be susceptible to the formation of a Mott state14,15, whence it is often discussed in terms of a polaronic Wigner crystal12,16,17. The 2H (trigonal) polytype is metallic above 75 K, but forms a commensurate CCDW below this temperature, which—in contrast to the 1 T polytype—is metallic down to 1 K, below which it becomes superconducting18.
The ET nanostructures are created by a laser pulse-induced quench through an inversion-symmetry-breaking polytype transformation of the surface atomic monolayer of a 1T polytype TaS2 single crystal. The resulting ET domains are embedded laterally by a 1H-TaS2 crystalline layer (the 1H signifies it is a monolayer). The entire structure is epitaxial on a 1T-TaS2 single crystal (Fig. 1a, b)).
Understanding the behavior of correlated electrons confined in such small ETs theoretically represents a substantial challenge. Considering CDWs as standing wave interferences formed by counter-propagating Fermi electrons19,20,21 suggests an investigation using the quantum billiards (QB) approach with Fermi electrons. On the other hand, the 2D polaronic Wigner crystal picture, where the polarons are subject to Mott localization14 suggests a correlated electron picture. Here we compare conventional QB calculations, a (classical) charged lattice gas strongly correlated electron model, and a fully quantum many-body correlated electron model using exact diagonalization methods with the aim to understand the rich variety of QI textures in both classical and quantum regimes observed by STM.
The ET structures (Fig. 1f) are created by a controlled exposure of a freshly exfoliated 1T-TaS2 single crystal to laser pulses in ultrahigh vacuum at 80 K22, where the majority of the top surface is transformed to the 1H polytype, but ET structures of 1T polytype remain structurally unchanged23,24. The domains have atomically defined sides parallel to the crystal axes of the 1T layer, matching the lattice structure of the surrounding 1H layer, forming a perfect ET shape with edges at 60° to each other (Fig. 1a, b). The work functions for the 1T-TaS2 and 2H-TaS2 polytypes are \(\phi =5.6\) and \(5.2\,{\mathrm{eV}}\) respectively, so the H phase acts as a barrier for electrons in the 1T ET nanostructure25. At 80 K, where the measurements are preformed, the 1H polytype is metallic, while the 1T polytype is nominally in the insulating CCDW phase14. (For reference, low-temperature scans showing the appearance of the 3 × 3 CDW of the 1T layer are shown in the Supplementary information). The H-polytype layer thus defines a confining potential barrier \({\phi }_{B}\) to the electrons inside the ET (Fig. 1e). As a result of self-organization of charges at the \(1T-1H\) interface, an edge state is formed which is clearly visible in ETs of all sizes (Fig. 1f). An interfacial band diagram based on a conventional metal-semiconductor junction is shown in Fig. 1e (see Supplementary information for details). The width \(w\) of the edge state (ES) is approximately equal to the screening length in 1T-TaS2, \(\zeta =1 \sim 2\, {\mathrm{nm}}\)26,27.
The presented STM images in Figs. 1f and 2 are measurements of the local density of states (LDOS), which may be considered either in the local state approximation as \({\rho }_{local}(E,{\bf{r}})\) at the tip position \({\bf{r}}\), for states without long-range translation invariance, \({\rho }_{local}(E,{\bf{r}})\propto\) \({\Sigma }_{i=1,N}|{\psi }_{i}({E}_{i},{\bf{r}}){|}^{2}\delta (E-{E}_{i})\); or in the quasiparticle approximation \({\rho}_{QP}(E,{\bf{r}})\propto {\Sigma}_{k}|{{{{\psi }}}_{k}({\bf{r}})}|^{2}\delta (E - \varepsilon ({\bf{k}}))\), where \(\varepsilon ({\bf{k}})\) is the energy of all the electrons with different wavevector \({\bf{k}}\) that interfere locally at position \({\bf{r}}\). The latter case is relevant when we discuss the electrons in the unconfined 1H layer. It applies also to the interfering itinerant ET-confined electron eigenstates with different \({{\bf{k}}}_{{\boldsymbol{i}}}\) but the same energy \(E({\bf{k}})\). In either case, the resulting LDOS patterns inside the ET can be observed by STM topography at constant current. The QI is superimposed on the fine sub-nm structure of the atomic orbitals (see, for example, the high-resolution image in the insert to Fig. 1f). This detailed orbital structure of surface atoms is related to structural effects and is not of present concern, so we shall focus primarily on the mesoscopic QI patterns.
Fig. 2: STM images of different sized ETs with l = 8…38a at 80 K with a bias voltage of 0.8 V.
figure 2
Each STM image is accompanied by a schematic figure indicating the appearance of L or R chirality of the C order (yellow and red respectively). Multiple localized QI features do not conform to the C order (black dots). a A single dot is observed for l = 8a. b For l = 10a, L and R chiralities of C order appear. c l = 13a with 5 polarons, d l = 13a with 6 polarons of L and R chiralities. e l = 28a with L and R chiralities created nearby with the same laser pulse exposure. Similar domain wall patterns (dashed lines) are observed in both cases, as schematically shown. f An ET with l = 38a showing a single C domain at the center, surrounded by QPI at the edges (white dots). g The registry of the CCDW orders in the ETs and the 1T-TaS2 layer below. The grid pattern is centered on the CCDW in the 1T layer below the 1H layer. All ETs show phase shifts of the polaron order with respect to the substrate, and to each other.
Large-area Fourier transforms of the inside and outside of the ETs, shown in Fig. 1g, h respectively, reveal both crystal lattice (CL) peaks and CCDW superlattice (SL) peaks. The areas outside the ETs give sharp FFT peaks corresponding to the 1H CL and weak SL peaks from the 1T-TaS2 CCDW layer underneath (Fig. 1h). Inside the ETs, we see an additional intensity that is almost uniformly distributed between the CCDW FFT peaks. This is attributed to QI features, and the periodic ordering along the inside edges of the ETs (labeled E).
In Fig. 2 we show a representative set of STM images of the inside of ETs, with inner ET dimensions ranging from \(l=8a\) (2.64 nm) to \(38a\) (12.35 nm), where \(a\) is the CL constant of 1T-TaS2. Universally, we observe that the electrons try to form commensurate order at the center. The CCDW superlattice structure is significantly distorted, however, particularly in small ETs. For the smallest ETs such as the one with \(l=\,8\,a\) (Fig. 2a), a single, deformed dot is visible at the center of the ET. Already with \(l=10\,a\) (Fig. 2b), the pattern with 3 maxima adjusts to the preferred CCDW order of either L or R chirality. In Fig. 2c, d, we clearly see very different QI textures in ETs with the same \(l\), which is likely caused by small geometrical imperfections and/or initial conditions that results in different electron trajectories within ETs of equal size. In the same vein, two ETs with \(l=28a\) (Fig. 2e) created simultaneously nearby to each other show quite complicated but closely matching, mirror images of opposite chirality28. A remarkable feature of these ETs, particularly well visible in the R structure in Fig. 2e is the nontrivial QI pattern in the corner which does not fit the CCDW order. As the size of the ETs further increases, the polaron pattern approaches the CCDW superstructure. Thus, for \(l=38a\) the CCDW fills almost the entire ET, with QI distortions visible only at the edges.
Remarkably, the CCDW charge order within the ETs is not in register with respect to the CCDW order of the layer below. (The latter is visible through the surrounding 1H monolayer, and is emphasized by the mesh in Fig. 2h.) Moreover, neighboring ETs also have different register (Fig. 2f). This implies that the effect of the underlying CCDW and lattice potential is not sufficiently strong to force the CCDW order in the top layer. The QI patterns appear to be determined by the ET boundary, not by the inter-layer coupling as has been suggested in bulk crystals29.
The tunneling spectra inside, outside and across the edge of an ET at 80 K are presented as the normalized differential conductance (NDC), \(({\rm{dI}}/{\rm{dV}})/({\rm{I}}/{\rm{V}})\) in Fig. 3. The recorded positions are color coded. For comparison, we also show a 1T-TaS2 CCDW bulk crystal spectrum at 4 K showing the characteristic upper and lower Hubbard bands (UHB and LHB) at \(-0.15\) and \(+0.25\) V, and CCDW-derived bands at −0.28 and +0.42 V30.
Fig. 3: NDC curves of the 1H layer outside, the 1T layer inside, and the ES of an ET.
figure 3
a An STM image at \(V=-0.8\,{\rm{V}}\) with indicated positions where STS curves are recorded, using color coded dots. b The STS curves (in corresponding color) for the 1H layer, the edge state (ES) inside, and outside the ET boundary, and the 1T phase inside the ET. The zero level is indicated in each case by a line in the same color code. The STS of the CCDW state of a 1T-TaS2 monocrystal at 4 K is shown for comparison (dotted). c An STS line scan showing the LDOS (color scale) across the ET boundary. The opening of a pseudogap in the 1H polytype is clearly observed, which increases in the 1T layer inside the ET.
Overall, the states within \(\pm 0.32\,{\mathrm{eV}}\) of the Fermi level correspond to Ta-\(5d\) zone-folded sub-bands of the 1T-TaS2 CCDW phase31, and pristine Ta-5d bands of the 1H monolayer which give rise to the observed NDC. A line scan across the ES (Fig. 3c) shows the spatial variation of LDOS across the ET boundary, which can be compared with the naive band diagram in Fig. 1e. As the STM tip moves across the ES, the screening length 1–2 nm limits the sharpness of the features. Concurrent with the band alignment between 1T and 1H phases, charges will self-organize at the boundary, resulting in the observed ES. On the 1H side, we observe a uniform pseudogap, which we associate with the CDW state nearby (below 75 K).
A most striking feature of the data is the high degree of spatial homogeneity of the NDC curves outside the ET (the 1T layer) and of the edge state itself. This is contrasted by the huge spatial spectral variations inside the ET occurring on a scale of 1 nm. Outside the ETs, beside the broad bands, a small, asymmetric gap-like feature is visible around \({\pm}\!20\) mV with peaks at \(\pm 50\) mV. This feature is similar to the one reported in single layer 1H-TaSe2 on epitaxial bi-layer graphene substrates at 5 K32 and monolayer 2H-TaS2 on graphene18, where it is was attributed to the CDW gap of the 1H monolayer. Considering that the data were recorded at 80 K, this would suggest that in the monolayer, the CDW gap is already present above the \({T}_{c}=75\) K of the bulk material. However, no modulation is visible in the FFTs corresponding to the \(3\times 3\) period of the 1H-TaS2 CCDW (Fig. 1g), implying that there is no long-range CDW order in the 1H layer at 80 K. In some positions we also observe signatures of the characteristic peak at +0.25 V that corresponds to the UHB of bulk 1T-TaS2, which are attributed to the CCDW in the layer below. On the 1T side of the ES, inside the ET, the pseudogap size increases significantly.
Analysing the STS map in more detail at selected points, we note that the ES shows remarkable homogeneity along both sides of the boundary (green and pink dots respectively). The NDCs on the outside edge (the bright border, green) are very similar to the 1H monolayer (yellow). However, the broad peak at −0.21 V (yellow) splits into two, at −0.15 and −0.25 V (green). Inside the ET, the ES peak at −0.25 V shifts further to −0.28 V (pink), but no other significant differences are observed.
Inside the ET, the NDC curves vary significantly from spot to spot. A number of sharp peaks are observed that have no parallel in bulk 1T or 2H-TaS2. Such large spatial variations of LDOS are indeed expected for a confined system with no long-range order. For example, the observation of nodal domains in space is one of the expected features for a QB33 that modify the CCDW pattern. The sharp features on the energy scale \(\pm 0.1\,{\mathrm{V}}\) are thus attributed to eigenstates of the confined nanostructure.
Investigating the effect of different bias voltages, a set of images at the ET boundary are shown in the Supplementary information. The detailed pattern and contrast change with V, but the correlation-localized polarons remain in place, in accordance with modeling of the correlated electron state. The contrast changes most when the scanning bias voltage is close to zero, as the contribution of the occupied polaronic states to the LDOS is small and we only see the states very close to the Fermi level.
Within a noninteracting picture, electrons confined within ETs are expected to exhibit canonical QB behavior. On the other hand, the presence of strong correlations localizes electrons in a commensurate structure12, filling all available space. The noninteracting and strongly interacting pictures are at odds with each other. To investigate the dichotomy, we will compare the observed LDOS patterns with theoretical modeling ranging from noninteracting quantum billiard description to strongly interacting classical gas. We find that the strongly correlated approach is more appropriate but fails to fully describe the QI patterns. Finally, in an attempt to combine the itinerant electron picture with correlations, we calculate electron density patterns that show spatial localization textures using full quantum correlated electron calculation using exact diagonalization methods, finding a remarkable propensity for localization as commensurate filling is approached.
The quantum billiard with inter-layer interactions12. For an ideal ET with multiple noninteracting electrons, the levels are filled subject to the Pauli principle. The calculated integrated LDOS based on solutions of the Schrödinger equation are compared with the STM images for different ETs in Fig. 4, column A). The integrated LDOS is given by \({\Sigma }_{N = {n..m}}{|{\psi }_{N}|}^{2}\), where the integers \(n\) and \(m\) indicate the range of eigenstates for the summation. By choosing appropriate \(n\) and \(m\) (by inspection), the predicted integrated LDOS patterns show the correct number of maxima within the ET, but the pattern is always symmetric with respect to the ET shape, and parallel with the edges, which the experimental patterns are not. While qualitatively describing the integrated LDOS patterns, the QB approach fails to describe the unusual features of the experimentally observed integrated LDOS (column D of Fig. 4).
Fig. 4: A comparison of QB, QB + V and CLG model predictions for integrated LDOS with experiment for ETs of different sizes with different sides \(l\).
figure 4
a Column A represents the QB solution (V = 0) for different summed states (N), corresponding to \({{V}_{B}}\simeq {0.8}\,{\mathrm{V}}\). b Column B shows the QB + V model eigenstates with \(=-1\), and c column C is the CLG MC calculation result at 1/13 filling. d Column D shows experimental images for \(l=8\,a\), \(10\,a\), \(28\,a\) and \(38\,a\). Note the domain wall predicted for \({l} = {28}\,{a}\) (col. C), and corresponding STM image (col. D) shown by dashed lines.
The energy scale of the states is given by the confinement energy \(E \sim \frac{{N}^{2}{\hslash }^{2}{\pi }^{2}}{2{m}^{\ast }{l}^{2}}\). Assuming \({m}^{\ast }\sim {m}_{e}\), for the smallest ET with side \(l=2.15\) nm (8a), the first four levels \(N=1\ldots 4\) have energies ranging from \({E} \approx {0.08}-{1.26}\,{\mathrm{eV}}\), while in the largest ET shown here, with \(l=12.8\,{\mathrm{nm}}\,(38a)\), for \(N=1\ldots 21,\) \({E} \approx {0.0022-0.97}\,{\mathrm{eV}}\). A more realistic effective mass \(\sim 3\,{m}_{e}\) would compress the energy scale, increasing the number of experimentally observed levels in the STS range \(\pm 0.5\,{\mathrm{V}}\). The temperature broadening at 80 K is ~7 meV, so the levels are likely too closely spaced to be experimentally resolved. Nevertheless, the energy scale of the observed spectral features (Fig. 3b) is in qualitative agreement with the experiment.
Next, we introduce a periodic potential \(V(x,y)\) with magnitude VB in the QB calculation\(,-\frac{{\hslash }^{2}}{2m}[{\Delta }_{{\rm{x}},{\rm{y}}}+V(x,y)]\psi (x,y)=E\psi (x,y)\), to account for the periodic lattice distortion created by interaction of the top layer with the CCDW in the layer below. The results are shown in Column B of Fig. 4. Now the 2D pattern corresponding to the CCDW is tilted at 13° with respect to the ET edges, and the crystal axes. As expected, for a significantly large \(V({\bf{r}})\), the calculated integrated LDOS pattern follows the CCDW ordering at the center of the ETs quite well. The periodic potential also introduces a gap in the integrated LDOS which bears resemblance to the observed STS gap (Fig. 3). However, the calculation fails to reproduce the complex features and malleable nature of the integrated LDOS patterns at the edges. It also completely fails to reproduce the observed domain walls. (The predicted integrated LDOS patterns and spectra for the QB, and QB + V models are described in the Supplementary information).
Classical correlated electrons confined within an ET. A correlated electron model is needed to account for the malleable features in the electronic order within ETs. Within a charge-lattice gas (CLG) Monte-Carlo (MC) calculation, classical point charges, subject to screened Coulomb repulsion can move via thermal hopping on an atomic lattice shaped in the form of an ET (Fig. 4c). A crucial parameter in the presented modeling is the filling, defined as the number of electrons divided by the number of lattice sites \(f\). At various magic fillings, such as \(f=\frac{1}{13}\) for the case of bulk 1T-TaS2, the model predicts an electronic superlattice which is perfectly commensurate with the underlying atomic lattice12. Confining the system size to a small triangle inevitably introduces edge effects and distortions into the configurational ordering of electrons. This is simply due to the fact that a \(\frac{1}{13}\) electronic lattice does not match the edges of the triangle as shown in Fig. 4c, requiring the particles to accommodate. (For more details concerning the model refer to the Supplementary information.) Here we outline the similarities between experimental observations and the simulated electronic configurations (Fig. 4, column C): (1) The model predicts the spontaneous formation of two chiralities of the \(1/13\) electronic lattice at angles \({\pm}\!13,9^\circ\) with respect to the ET edges, which are both experimentally observed. In contrast to previous calculations where the external potential fixed the angle of the QI pattern, here it emerges as a nontrivial consequence of many-body correlations. (2) Smaller triangles induce stronger edge effects into the electron configuration, to the point of entirely breaking up the expected \(1/13\) electronic lattice. Electrons at the triangle’s edge align with the edge and since the edges are close one to another, there is no room for a proper \(1/13\) lattice to emerge. The resulting configuration is still largely influenced by strong correlations, as electrons on average are \(\sqrt{13}\) atomic lattice spacings apart. This is shown by the connecting lines in Fig. 4 (column C). (3) Edge effects in larger triangles are diminished towards the center, as shown for \(l=38a\) in Fig. 4 (column D), and the \(1/13\) Wigner crystal lattice emerges. However, near the edges the electrons still align with the edge of the ET.
In the simple classical calculation above, all charges are equivalent and should ideally have the same LDOS and thus appear identical under a tunneling microscope. However, the experimental STM images show various irregular, elongated and triangular shapes within the triangles that are sometimes aligned in rows (e.g. Fig. 4) which such modeling cannot describe.
Quantum correlated electrons confined within an ET. For an understanding of the departures from the simple CLG model, we need to consider the itinerant correlated electrons which follow quantum billiard trajectories inside the triangle. To account for the interplay of the strong electronic repulsions and their itinerant nature we perform a quantum many-body calculation on small ET using the exact diagonalization of the spin-less fermions with the long-range interaction:
$$H=-t{\sum }_{\langle i,j\rangle }{c}_{i}^{\dagger }{c}_{j}+{\sum }_{i\ne j}V(|i-j|)[{n}_{i}-\bar{n}][{n}_{j}-\bar{n}],$$
where\(\,{c}_{i}\) (\({n}_{i}\)) is the annihilation (density) operator at site \(i\), \(t\) is the hopping parameter, \((V (|i - j|))\) is the Yukawa interaction. To ensure charge neutrality, we have subtracted the uniform background charge density n (see Methods for details). While in the classical limit the relevant parameter is filling \(N\), the quantum extension introduces another dimensionless parameter \(t/V\) that governs the transition between the localized (\(t/V=0\)) and the delocalized regime (\(t/V\gg 1\)). In agreement with the classical simulation, there exist special fillings where the electrons form commensurate fillings of the ET. Due to the size restrictions in the many-body simulation, we consider an ET with the lattice size \(L=8\) and the number of electrons \(N=10\) corresponding to the special filling f = 1/3, see Fig. 5a). In this commensurate situation, the state is extremely robust against the delocalization which can be observed by comparing the density distribution in the quantum \(t/V=0\) and the classical case \(t/V=0.01\), see Fig. 5a). The analogous comparison of the density distribution for the incommensurate filling exhibits a strong redistribution of charges forming nontrivial QPI and their delocalization tendency is driven by lowering the kinetic energy via the closed loops QPI, see Fig. 5c, d.
Fig. 5: Density plot of the charge distribution for spin-less particles on a triangle with \(L=7\).
figure 5
The first row shows the commensurate case with \(N=10\) electrons. The second row shows the incommensurate case with \(N=12\) electrons. The first column represents the classical case with \(t/V=0\). The second column (a, c) shows the quantum case with \(t/V=0.01\). The size of the dot is a measure of the electron density. The last column (b, d) represents the quantum \(t/V=0.01\) solution in a graph representation where the thickness of the bond represents a relative contribution to the total kinetic energy and a size of the dot the electronic density. e The norm of the density difference distribution D (see the main text) for various number of electrons N versus the ratio between the hopping integral and the interaction \(t/V\) or the Wigner-Seitz radius \({r}_{s}=1.92{e}^{2}/\hslash {v}_{F}\), and \({v}_{F}\) is the Fermi velocity.
As a more quantitative measure of the electron delocalization with respect to the classical limit we introduce the parameter \(D=\sqrt{{\Sigma }_{i}{[\langle {n}_{i}\rangle -{\langle {n}_{i}\rangle }^{t=0}]}^{2}}\). A comparison for different fillings shows that incommensurate states (\(N\ne 10\)) are orders of magnitude more susceptible to delocalization than commensurate ones (Fig. 5e). The geometrical constrains of ET can therefore dramatically shift the delocalization transitions and this resolves the apparent contradiction how we can observe delocalized QPI patterns in small structures while the corresponding bulk situation would be localized.
In larger ETs, electron trajectories are modified by correlations within the center, and boundary conditions at the barriers which cannot be understood solely on the basis of semi-classical correlated polaron packing, or free-electron QB. However, correlated electron modeling quite successfully predicts the appearance of domain walls in the CCDW structure forced by the confinement, which cannot be explained by any free-electron QB model. The observation of mirrored, but slightly different QPI patterns within ETs of the same size reveals that the QPI patterns are emergent, self-organized many-body electronic states. The fact that very small changes in geometry and/or barely perceptible imperfections on the atomic level give rise to dramatic changes of the QPI patterns is consistent with chaotic behavior. However, the imperfections in the ET construction (Fig. 1) are expected to change the detailed QI patterns, but not the observed generic features. We showed that for special fillings the system is robust toward these small imperfections, and it would be interesting to understand the implications of these correlations on the many-body spectrum and the time-evolution in light of recently observed many-body scars in quantum simulators11,34. The understanding of behavior and interaction of itinerant electrons with correlation-localized polarons in intertwined textures opens the way to microscopic electronics devices with correlated quantum materials. Moreover, the intertwined orders visible in the ET QPI patterns may help in understanding quantum materials in which a coexistence of itinerant and polaronic correlation-localized carriers is observed, reconciling the seeming dichotomy of different experiments that observe itinerant states (e.g. ARPES, quantum oscillations35) and localized states (e.g. optics36,37, STM38) within the same material under seemingly identical conditions.
Data availability
All of the data supporting the conclusions are available within the article and the Supplementary Information. Additional data are available from the corresponding author upon reasonable request.
Code availability
The code used in this article is available from the corresponding author upon reasonable request.
1. Heller, E. J. Bound-state eigenfunctions of classically chaotic hamiltonian systems: scars of periodic orbits. Phys. Rev. Lett. 53, 1515–1518 (1984).
MathSciNet Article ADS Google Scholar
2. Casati, G. & Prosen, T. Mixing property of triangular billiards. Phys. Rev. Lett. 83, 4729–4732 (1999).
CAS Article ADS Google Scholar
3. Marcus, C. M., Rimberg, A. J., Westervelt, R. M., Hopkins, P. F. & Gossard, A. C. Conductance fluctuations and chaotic scattering in ballistic microstructures. Phys. Rev. Lett. 69, 506–509 (1992).
CAS Article ADS Google Scholar
4. Wilkinson, P. B. et al. Observation of ‘scarred’ wavefunctions in a quantum well with chaotic electron dynamics. Nat. Nanotechnol. 380, 608–610 (1996).
CAS Google Scholar
5. Linke, H., Christensson, L., Omling, P. & Lindelof, P. Stability of classical electron orbits in triangular electron billiards. Phys. Rev. B—Condens. Matter Mater. Phys. 56, 1440–1446 (1997).
CAS Article ADS Google Scholar
6. Fromhold, T. M. et al. Magnetotunneling spectroscopy of a quantum well in the regime of classical chaos. Phys. Rev. Lett. 72, 2608–2611 (1994).
CAS Article ADS Google Scholar
7. Ponomarenko, L. A. et al. Chaotic dirac billiard in graphene quantum dots. Science 320, 356–358 (2008).
CAS Article ADS Google Scholar
8. Crook, R. et al. Imaging fractal conductance fluctuations and scarred wave functions in a quantum billiard. PRL 91, 730–734 (2003).
Article Google Scholar
9. Cabosart, D. et al. Recurrent quantum scars in a mesoscopic graphene ring. Nano Lett. 17, 1–6 (2017).
10. Ho, W. W., Choi, S., Pichler, H. & Lukin, M. D. Periodic orbits, entanglement, and quantum many-body scars in constrained models: matrix product state approach. Phys. Rev. Lett. 122, 040603 (2019).
MathSciNet CAS Article ADS Google Scholar
11. Turner, C. J., Michailidis, A. A., Abanin, D. A., Serbyn, M. & Papić, Z. Weak ergodicity breaking from quantum many-body scars. Nat. Phys. 14, 745–749 (2018).
CAS Article Google Scholar
12. Vodeb, J. et al. Configurational electronic states in layered transition metal dichalcogenides. N. J. Phys. 21, 083001 (2019).
CAS Article Google Scholar
13. Wilson, J. A., Disalvo, F. J. & Mahajan, S. Charge-density waves and superlattices in metallic layered transition-metal dichalcogenides. Adv. Phys. 24, 117–201 (1975).
CAS Article ADS Google Scholar
14. Sipos, B., Berger, H., Forro, L., Tutis, E. & Kusmartseva, A. F. From Mott state to superconductivity in 1T-TaS2. Nat. Mater. 7, 960–965 (2008).
CAS Article ADS Google Scholar
15. Fazekas, P. & Tosatti, E. Charge carrier localization in pure and doped 1t-TaS2. Phys. B C. 99, 183–187 (1980).
CAS Article ADS Google Scholar
16. Klanjsek, M. et al. A high-temperature quantum spin liquid with polaron spins. Nat. Phys. 13, 1130–1134 (2017).
CAS Article Google Scholar
17. Karpov, P. & Brazovskii, S. Modeling of networks and globules of charged domain walls observed in pump and pulse induced states. Sci. Rep. 8, 1–7 (2018).
Google Scholar
18. Hall, J., Ehlen, N., Berges, J., Loon, E. van & ACS, C. van E. Environmental control of charge density wave order in monolayer 2H-TaS2. ACS Appl. Nano 13, 10210–10220 (2019).
19. Bardeen, J. Classical versus quantum models of charge-density-wave depinning in quasi-one-dimensional metals. Phys. Rev. B, Condens. matter 39, 3528–3532 (1989).
CAS Article ADS Google Scholar
20. Miller, J. H., Wijesinghe, A., Tang, Z. & Guloy, A. Coherent quantum transport of charge density waves. PRB 87, 115127 (2013).
Article ADS Google Scholar
21. Miller, J. H., Wijesinghe, A. I., Tang, Z. & Guloy, A. M. Correlated quantum transport of density wave electrons. PRL 108, 036404 (2012).
Article ADS Google Scholar
22. Ravnik, J. et al. A time-domain phase diagram of metastable states in a charge ordered quantum material. Nat. Commun. 12, 2323 (2021).
CAS Article ADS Google Scholar
23. Ravnik, J., Vaskivskyi, I. & Gerasimenko, Y. Strain-Induced metastable topological networks in laser-fabricated TaS2 polytype heterostructures for nanoscale devices. ACS Appl. Nano 2, 3743–3751 (2019).
CAS Article Google Scholar
24. Wang, Z. et al. Surface-limited superconducting phase transition on 1T -TaS2. ACS Nano 12, 12619–12628 (2018).
CAS Article Google Scholar
25. Shimada, T., Ohuchi, F. S. & Parkinson, B. A. Work function and photothreshold of layered metal dichalcogenides. Jpn. J. Appl. Phys. 33, 2696–2698 (1994).
CAS Article ADS Google Scholar
26. Ma, L. et al. A metallic mosaic phase and the origin of Mott-insulating state in 1T-TaS2. Nature. Communications 7, 1–8 (2016).
Article Google Scholar
27. Cho, D. et al. Correlated electronic states at domain walls of a Mott-charge-density-wave insulator 1 T -TaS 2. Nature. Communications 8, 392 (2017).
Google Scholar
28. Gerasimenko, Y. A., Karpov, P., Vaskivskyi, I., Brazovskii, S. & Mihailovic, D. Intertwined chiral charge orders and topological stabilization of the light-induced state of a prototypical transition metal dichalcogenide. npj Quant. Mater. 4, 1–9 (2019).
CAS Article Google Scholar
29. Stahl, Q. et al. Collapse of layer dimerization in the photo-induced hidden state of 1T-TaS2. Nature. Communications 11, 1–7 (2020).
Google Scholar
30. Cho, D. et al. Nanoscale manipulation of the Mott insulating state coupled to charge order in 1T-TaS2. Nat Commun. 7, 10453 (2016).
CAS Article ADS Google Scholar
31. Rossnagel, K. On the origin of charge-density waves in select layered transition-metal dichalcogenides. J. Phys.-Condens. Matter 23, 213001 (2011).
CAS Article ADS Google Scholar
32. Ryu, H. et al. Persistent charge-density-wave order in single-layer TaSe2. ACS Appl. Nano 18, 689–694 (2018).
CAS Google Scholar
33. Samajdar, R. & Jain, S. R. Nodal domains of the equilateral triangle billiard. J. Phys. A: Math. Theor. 47, 195101–195123 (2014).
MathSciNet Article ADS Google Scholar
34. Bernien, H. et al. Probing many-body dynamics on a 51-atom quantum simulator. Nature 551, 579–584 (2017).
CAS Article ADS Google Scholar
35. Vignolle, B. et al. Quantum oscillations and the Fermi surface of high-temperature cuprate superconductors. Comptes Rendus Physique. 12, 446–460 (2011).
CAS Article ADS Google Scholar
36. Mihailovic, D. et al. Application of the polaron-transport theory to sigma (omega) in Tl2Ba2Ca1-xGdxCu2O8, YBa2Cu3O7- d, and La2-xSrxCuO4. Phys. Rev. B, Condens. matter 42, 7989–7993 (1990).
CAS Article ADS Google Scholar
37. Mertelj, T., Demsar, J., Podobnik, B., Poberaj, I. & Mihailovic, D. Photoexcited carrier relaxation in YBaCuO by picosecond resonant Raman spectroscopy. Phys. Rev. B 55, 6061–6069 (1997).
CAS Article ADS Google Scholar
38. Hoffman, J. E. et al. Imaging quasiparticle interference in Bi2Sr2CaCu2O8+d. Science 297, 1148–1151 (2002).
CAS Article ADS Google Scholar
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We wish to acknowledge discussions with Tomaž Prosen, single crystals grown for this work by Petra Sutar, funding from the Slovenian Research Agency (ARRS), projects P1-0040, N1-0092 and young researcher grants, P17589 and P08333. D.G. acknowledges the support by ARRS under Programs No. J1-2455 and P1-0044. The Flatiron Institute is a division of the Simons Foundation. This project has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 701647.
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J.R., I.V., Y.G., and D.M. conceived the experiments. J.R., Y.V., I.V., P.A., and Y.G. conducted the STM measurements and analyzed the data. J.V., D.G., V.K., and D.M. performed the theoretical calculations. D.M. wrote the paper and supervised the project.
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Correspondence to Jan Ravnik or Dragan Mihailovic.
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Ravnik, J., Vaskivskyi, Y., Vodeb, J. et al. Quantum billiards with correlated electrons confined in triangular transition metal dichalcogenide monolayer nanostructures. Nat Commun 12, 3793 (2021).
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8dba11577bf783d0 | Algorithm 919
Algorithm 919: A Krylov Subspace Algorithm for Evaluating the ϕ-Functions Appearing in Exponential Integrators. We develop an algorithm for computing the solution of a large system of linear ordinary differential equations (ODEs) with polynomial inhomogeneity. This is equivalent to computing the action of a certain matrix function on the vector representing the initial condition. The matrix function is a linear combination of the matrix exponential and other functions related to the exponential (the so-called ϕ-functions). Such computations are the major computational burden in the implementation of exponential integrators, which can solve general ODEs. Our approach is to compute the action of the matrix function by constructing a Krylov subspace using Arnoldi or Lanczos iteration and projecting the function on this subspace. This is combined with time-stepping to prevent the Krylov subspace from growing too large. The algorithm is fully adaptive: it varies both the size of the time steps and the dimension of the Krylov subspace to reach the required accuracy. We implement this algorithm in the matlab function phipm and we give instructions on how to obtain and use this function. Various numerical experiments show that the phipm function is often significantly more efficient than the state-of-the-art.
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d2ab1bab3ce3f9a6 | May 15, 2019
How to describe nuclear properties ab initio at a low computational cost?
Figure 1: Reach of ab initio methods: quasi-exact methods (orange), valence-space methods (green) and wave-function expansion methods (red).
The predictions of nuclear properties based on a realistic description of the strong interaction is at the heart of the ab initio endeavour in low-energy nuclear theory. Ab initio calculations have long been limited to light nuclei or to nuclei with specific proton and neutron numbers. Theoreticians from Irfu/DPhN have developed novel ab initio methods that led to a significantly increase of the number of nuclei that can be accessed. The most recent one, called Bogoliubov many-body perturbation theory (BMBPT), provides a light-weighted alternative capable of providing the same accuracy as competing methods at a computational cost that is lowered by two orders of magnitude. This has been achieved by allowing symmetries of the nuclear Hamiltonian to spontaneously break in the calculation. This exciting new development, paving the way for precise computations of heavier nuclei using reasonable computing resources, has recently been published in Physics Letter B [1].
Atomic nuclei are systems composed of nucleons, i.e. protons and neutrons, interacting via inter-nucleon forces. These forces emerge from strong interactions between constituent quarks and gluons, whose dynamics is described by the quantum field theory of Quantum Chromo Dynamics (QCD). Unfortunately, QCD displays a non-perturbative character at low energies characterising the realm of nuclear structure. In this context, the systematic and controlled description of the atomic nucleus poses a formidable task. This longstanding (and still unanswered) problem is at the heart of the so-called ab initio approach to the nuclear quantum many-body problem. It requires:
i) modelling the inter-nucleon interactions entering the A-body Schrödinger equation (an eigenvalue equation for the Hamiltonian where A is the number of nucleons composing the nucleus) with a sound connection to QCD
ii) developing mathematical methods allowing for accurate and controlled approximations of the exact solutions of the A-body Schrödinger equation.
Three decades ago, the seminal work of S. Weinberg paved the way for a systematic theory of inter-nucleon interactions anchored into QCD. He created a mathematical framework, called chiral effective field theory (EFT), which allows for constructing systematically improvable nuclear Hamiltonians1. Such Hamiltonians have nowadays replaced previous phenomenological models and have become the standard input to the A-body Schrödinger equation. Nevertheless, finding the solution of the Schrödinger equation for a large range of nuclei remains a highly non-trivial problem, both from a formal and a computational perspective. Therefore, such calculations have long been limited to light systems with mass number A ? 12.
Figure 2: Neutron states of 16O (doubly closed shell) and 18O (singly open shell) in the standard non-interacting shell model. Filled (open) circles correspond to occupied (unoccupied) single-particle states in the ground-state reference state.
Expanding the exact solution
Over the past 15 years, mathematical methods expanding the exact solution with respect to a simple mean-field reference state have been designed and, thus, enabled the description of heavier nuclei up to tin isotopes (A =50). However, these methods have remained limited until recently to nuclei with specific numbers of protons and neutrons: the so-called doubly closed-shell nuclei. Indeed, to first approximation, a nucleus can be described by a state obtained by filling up protons and neutrons on two sets of shells that can only accept a specific number of them each. When the proton and neutron numbers are such that the upper neutron and proton shells are entirely filled, the corresponding nucleus is coined as ‘doubly closed-shell’ (see Fig. 2). These nuclei are relatively more stable and simpler to describe than their neighbours as they authorise the use of standard Slater determinants as reference states. In the past years, theoreticians from Irfu/DPhN have developed several different expansion methods allowing one to perform ab initio calculations of singly open-shell nuclei, i.e. nuclei whose upper proton or neutron shell is not fully occupied and that are relatively more challenging to solve for. This has extended the reach of ab initio calculations from a few tens to several hundreds of nuclei.
Breaking the symmetry of the Hamiltonian
The key idea behind these approaches is to allow the reference state to break a symmetry of the underlying Hamiltonian. For semi-magic nuclei, the relevant symmetry to be broken is the so-called U(1) global gauge symmetry, an abstract symmetry associated with the simple fact that nuclei are made of specific numbers of protons and neutrons. In these approaches, the system is first allowed to not have exactly Z protons or N neutrons in order to handle the complexity associated with the partially filled character of the upper shell. This idea leads to employing a so-called Bogoliubov reference state (solving the Hartree-Fock-Bogoliubov mean-field equations) that generalises the use of a simpler Slater determinant (solution of the well-celebrated Hartree-Fock mean-field theory). This allows to capture from the outset the superfluid character of singly open-shell nuclei. While the breaking of U(1) symmetry is a standard tool in simple mean-field descriptions, it had never been applied in beyond-mean-field methods aiming at an accurate solution of the A-body Schrödinger equation and, thus, allowing for ab initio calculations. The most recent formalism developed by theoreticians from Irfu/DPhN consists of a perturbative expansion around the particle-number-breaking Bogoliubov reference state and is, thus, coined as Bogoliubov many-body perturbation theory (BMBPT). In Fig. 3, a systematic comparison of BMBPT results with other state-of-the-art methods, among which one has also been developed by the same group ( ADC(2) ), is shown for three different isotopic chains. While it is obvious that BMBPT performs extremely well against existing methods for both binding energies and two-neutron separation energies, it does so for a computational price that is two orders of magnitude lower. This makes BMBPT an extremely useful candidate for performing large survey calculations across the nuclear chart, which enables in-depth testing of next-generation nuclear Hamiltonians. At the same time future extensions to even more challenging doubly open-shell nuclei are much simpler than in other frameworks.
Figure 3: Ground-state binding energies (top) and two-neutron separation energies (bottom) computed within second-order BMBPT along O, Ca and Ni isotopic chains. Results using other many-body methods are shown for comparison. Experimental values are shown as black bars. Larger deviations from experiment in mid-mass systems are due to approximations made in the construction of the input Hamiltonian that do not affect the conclusions of the benchmark.
In summary, the theoreticians from Irfu/DPhN have added a new ab initio quantum many-body method dedicated to the ab initio description of mid-mass open shell nuclei that can compete with all previously available methods at a much lower computational cost. Being almost entirely developed at Irfu/DPhN, this newly designed method marks the increasing significance of the CEA theory group in the sector of ab initio nuclear structure theory.
[1] A. Tichai, P. Arthuis, T. Duguet, H. Hergert, V. Somà, R. Roth, Phys. Lett. B786 (2018) 195
Contact: Alexander TICHAI CEA-Saclay/Irfu/DPhN/LENA
1. Hamiltonian: mathematical operator describing the dynamics of interacting particles. In the case of the nuclear Hamiltonian, it is written as the sum of the kinetic energies of the A nucleons and the sum of the two-body, three-body, … interactions between the nucleons. Contrary to simpler cases, like Coulomb repulsion in electromagnetism, the strong interaction does not allow for writing down a closed analytical form of the corresponding potential in terms of spatial, spin and isospin degrees of freedom
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6e601b3d0be985f1 | Leticia Corral, Mexican Astrophysicist, Recognized For Correcting Stephen Hawking’s Theory
Discussion in 'News and Views' started by Allisiam, Jan 10, 2016.
1. Allisiam
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Leticia Corral, Mexican Astrophysicist, Recognized For Correcting Stephen Hawking’s Theory
By Latin Times | Jan 07 2016, 11:32AM EST
Mexican astrophysicist, Dr. Leticia Corral, has been rewarded for her discovery about the universe’s origins. Twitter
Mexican astrophysicist from Chihuahua, Dr. Leticia Corral, just caught the attention of the entire scientific community around the world by correcting Stephen Hawking on his hypothesis of the universe’s origin. For this, Corral presented a mathematical model, which apparently proves that Hawking’s mistake was to not take time's asymmetry into consideration. “I believe that the universe is cyclical; that it all started from one point and returns to it,” she explained.
The International Association of Engineers recognized Dr. Corral for her work. “This prize is like a recognition to my spirit; it makes me feel like my thoughts go further than time and space,” she said. “It’s for doing what I love and am passionate about, every day.” However, the piece was initially written for Entropy magazine, which rejected it, but Corral was convinced her work was correct so she approached the Association.
Dr. Corral’s work indicates that the universe will come to an end once all movement stops. You can see more on her work here and here.
2. admin
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Roger Penrose and the Big Bang Curvature
Shiloh Za-Rah - Posted May 5th 2010
Hi Mike!
There are a number of points, which align Arp with the mainstream. Now I know you rather accept the prevailing cosmological standard models of the Big Bang Cosmology and the various attempts (barring the multiverses, the anthropic principle and related topics perhaps).
For about 20 years now, I have supported Alan Sandage's measurements of the Hubble Constant. He for long set it at the 55 km/Mpc.s mark and only recently, with the pressure of the WMAP data, has he 'relented' to somewhere around 65 km/Mpc.s.
In my decade long analysis and study of the cosmology, I found the following.
1. The standard model describes the thermodynamic evolution of the cosmos very accurately. So you can reanalyse the WMAP data in their description of the Cosmic Microwave Background BlackBody Radiation (CMBBR) and use this CMBBR as a basis for the emerging parameters of the cosmoevolution.
2. The standard model has 'misinterpreted' the Guth-inflation in the context of the now prevalent membrane physics of the spacetime metrics.
The standard model postulates the Big Bang singularity to become a 'smeared out' minimum spacetime configuration (also expressible as quantum foam or in vertex adjacency of Smolin's quantum loops). This 'smearing out' of the singularity then triggers the (extended) Guth-Inflation, supposedly ending at a time coordinate of so 10-32 seconds after the Big Bang.
Without delving into technical details; the Guth-Inflation ended at a time coordinate of 3.33x10-31 seconds and AT THAT coordinate, the Big Bang became manifest in the emergence of spacetime metrics in the continuity of classical general relativity and the quantum gravitational manifesto.
This means, that whilst the Temperature background remains classically valid, the distance scales for the Big Bang will become distorted in the standard model in postulating a universe the scale of a 'grapefruit' at the end of the inflation.
The true size (in Quantum Relativity) of the universe at the end of the inflation was the size of a wormhole, namely at a Compton-Wavelength (Lambda) of 10-22 meters and so significantly smaller, than a grapefruit.
Needless to say, and in view of the CMBR background of the temperatures, the displacement scales of the standard model will become 'magnified' in the Big Bang Cosmology of the very early universe in the scale ratio of say 10cm/10 -20cm=1021 i.e. the galactic scales in meter units.
If you study the inflation cosmology more closely, you will find that many cosmologists already know, that the universe had to be 'blown up' to the Hubble Horizon instantaneously (so this is not popularised, as it contradicts the 'grapefruit' scale of Alan Guth).
3. A result of this is that the 'wormhole' of the Big Bang MUST be quantum entangled (or coupled) to the Hubble Horizon. And from this emerges the modular duality of the fifth class of the superstrings in the Weyl-String of the 64-group heterosis.
Again, without technical detail, the Big Bang wormhole becomes a hologram of the Hubble Horizon and they are dimensionally separated by the Scale-parameter between a 3-dimensional space and a 4-doimensional space. This is becoming more and more mainstream in the 5-dimensional spacetime of Kaluza-Klein-Maldacena in de Sitter space becoming the BOUNDARY for the 4D-Minkowski-Riemann-Einstein metrics of the classical cosmology. Of course the Holographic Universe of Susskind, Hawking, Bekenstein and Maldacena plays a crucial part in this, especially as M-Theory has proven, (YES PROVEN in scientific terms), the entropic equivalence of the thermodynamics of Black Holes in the quantum eigenstates of the classical Boltzmann-Shannon entropy.
So your 'speculative' status of string theory is a little 'out of date'. The trouble with the Susskind googolplex solutions is that they (if just Witten would have access to my data) fail to take into account the superstring selftransformations of the duality-coupled five classes. They think that all five classes manifest at the Planck-scale (therefore the zillions of solutions), they do not and transform into each other to manifest the Big Bang in a minimum spacetime configuration at the Weylian wormhole of class HE(8x8).
Roger Penrose has elegantly described the link of this to classical General Relativity in his "Weyl Curvature Hypothesis".
Quote from:'The large, the Small and the Human Mind"-Cambridge University Press-1997 from Tanner Lectures 1995"; page 45-46:
"I want to introduce a hypothesis which I call the 'Weyl Curvature Hypothesis'. This is not an implication of any known theory. As I have said, we do not know what the theory is, because we do not know how to combine the physics of the very large and the very small. When we do discover that theory, it should have as one of its consequences this feature which I have called the Weyl Curvature Hypothesis. Remember that the Weyl curvature is that bit of the Riemann tensor which causes distortions and tidal effects. For some reason we do not yet understand, in the neighbourhood of the Big Bang, the appropriate combination of theories must result in the Weyl tensor being essentially zero, or rather being constrained to be very small indeed.
The Weyl Curvature Hypothesis is time-asymmetrical and it applies only to the past type singularities and not to the future singularities. If the same flexibility of allowing the Weyl tensor to be 'general' that I have applied in the future also applied to the past of the universe, in the closed model, you would end up with a dreadful looking universe with as much mess in the past as in the future. This looks nothing like the universe we live in. What is the probability that, purely by chance, the universe had an initial singularity looking even remotely as it does?
The probability is less than one part in (1010)123. Where does this estimate come from? It is derived from a formula by Jacob Bekenstein and Stephen Hawking concerning Black Hole entropy and, if you apply it in this particular context, you obtain this enormous answer. It depends how big the universe is and, if you adopt my own favourite universe, the number is, in fact, infinite.
What does this say about the precision that must be involved in setting up the Big Bang? It is really very, very extraordinary, I have illustrated the probability in a cartoon of the Creator, finding a very tiny point in that phase space which represents the initial conditions from which our universe must have evolved if it is to resemble remotely the one we live in. To find it, the Creator has to locate that point in phase space to an accuracy of one part in (1010)123. If I were to put one zero on each elementary particle in the universe, I still could not write the number down in full. It is a stupendous number". End of Quote
4. Then of course I claim, that the Theory of Quantum Relativity represents a kind of 'Newtonian Approximation' to the 'Theory we have yet to find', mentioned by Roger Penrose in the above.
Then the 'phase spaced' de Broglie inflation is in moduar quantum entanglement with the Weyl-Wormhole of the Zero-Curvature of Roger Penrose's hypothesis and this solves the 'Riddle of Space' in somewhat the manner Allen Francom has postulated.
The Hubble-Universe consists of 'adjacent' Weyl-wormholes, discretisizing all physical parameters in holofractal selfsimilarity.
Penrose's Weyl-tensor is zero as the quasi-reciprocal of the infinite curvature of the Hubble Event Horizon - quasi because the two scales (of the wormhole and Hubble Universe) are dimensionally separated in the modular coupling of the 11D supermembrane boundary to the 10D superstring classical cosmology of the underpinning Einstein-Riemann-Weyl tensor of the Minkowski (flat) metric.
5. Finally then, the Hubble Law as applied in the standard model becomes a restricted case, applicable ONLY at the Node of the 11D asymptotic limit/boundary also BEING the Initial condition, Penrose writes of.
Then and there the Hubble Constant is truly Constant at 58.03 km/MPc.s; vindicating both Alan Sandage and Halton Arp, the latter in his questioning of the Hubble Law to characterise the cosmic distance scales.
6. Because of the duality coupling between the wormhole and the Hubble horizon, the Hubble-Horizon in 10D is always smaller than the Hubble Horizon in 11D (the first is defined in a 4D Minkowski spacetime and the second in a 5D Kaluza-Klein hypersphere). So the standard cosmology will measure an 'accelerating universe' where there is actually an 'electromagnetic intersection' of the 11D- Big Bang Light having reflected from the 11D boundary and recoupling with the 10D expansion.
Halton Arp's redshifts are also dual in that the special relativistic doppler formulation is absolutely sufficient to relate the cosmological redshift to cosmic displacement scales (and without the Hubble Law Ho=vrec/D). So the redshift measurement is the true parameter and must then be correlated with the expansion factor of General Relativity to ascertain the lowerD coordinates of the observed phenpomena encompassed by the higherD coordinates (through the values of the expansion parameter).
Briefly, the expanding universe presently moves at 0.22c with a deceleration of about 0.01 nanometers per second squared. But because the Hubble Horizon ITSELF recedes presently at 0.22c particular 'redshift corrections' must be applied to the VALID measurements of the latter to ascertain the cosmological distance scales of the lightemitters.
John Shadow
--- In Christianity_Debate@yahoogroups.com, "MikeA" <atomicbohr@...> wrote:
--- In Christianity_Debate@yahoogroups.com, drcsanyi drcsanyi@ wrote:
It is you who do not recognize the vast difference between Arp and the "Big Bang" model.
[MIKE] I see Arp has been busy since he retired. This is what I gather. Arp disagrees as he has his whole career that quasars' redshifts are due to distance. He now apparently feels that they are being ejected from certain very active galaxies and that because of this the universe the Hubble constant should be about 55 not the 70 something it now is calculated to be. I should point out that there is nothing very unusual in this. Allan Sandage who took over Hubble's task when Hubble died thinks the number is closer to 55 than 70. And Thomas Matthews who discovered these ubiquitous quasars with Sandage has also found some quasars that are nearby. Please note I said some.
Arp is now of the opinion that Hoyle was right and is fooling with a cyclic steady state universe if that is not an oxymoron. From what i can discern one is going to get the same observations with either model for the forseeable future.
Post last edited May 5th 2010
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• Post n°39
empty. shiloh on Thu Oct 03, 2013 2:21 pm
shiloh wrote:
SUSAN - Posted Dec 1st 2010
Have Physicists Found Echoes From Before the Big Bang?
The Big Bang was not the beginning, Roger Penrose believes.
However, Davies says, even if the finding is correct, this explanation for it —a black hole smash-up from before the Big Bang leaving its fingerprint after the Big Bang —isn’t the only possibility. “People have been looking for concentric rings in the cosmic microwave background for a while,” he says, because such a finding could support several different ideas.
“Unfortunately, the paper does not provide the necessary detail on how they performed their calculations.
Shiloh - Posted Dec 2nd 2010
Roger Penrose, whose work I often use in Quantum Relativity; is basically on the right path of reconstructing the cosmogenesis.
However the cyclic universe is built on its own protoversal seed and the 'Big Crunches' are electromagnetic and not inertial.
This means, that there will be no gravitational contraction in a shrinking of the protoverse; but the electromagnetic lightpath becomes multidimensional and multivalued. One can so model this on a cyclic electromagnetic cosmology with a 'Hubble heartbeat' of a semibeat of so 16.9 Billion years. This also allows a cosmogenetic Black Hole-White Hole evolution, which resets the wormhole singularity every 4 trillion years or so to eschew any theorized 'heat death' of the universe, due to the stellar generations 'running out' of their nuclear fuel of the nucleosynthesis of the primordial elements, based on hydrogen, helium and lithium.
Indeed, the Inflation PRECEDED the Big Bang and this is the simple solution for the 'inflation paradoxes' as some might term it.
The WMAP data in the picture in this post actually is descriptive for the wavequark model in Quantum Relativity with an inner gluonic (anti)neutrino kernel or core, an Inner Mesonic (down quark) Ring and an Outer Leptonic (strange quark) Ring.
It is just the 'New Standard Model' of Unitary Symmetry for the quarkian waves as "matter waves". The smallest quantum as microcosmic reality written in the galactic sky of the macrocosm.
Discover Interview: Roger Penrose Says Physics Is Wrong, From String Theory to Quantum Mechanics
By Susan Kruglinski, Oliver Chanarin|Tuesday, October 06, 2009
Roger Penrose could easily be excused for having a big ego. A theorist whose name will be forever linked with such giants as Hawking and Einstein, Penrose has made fundamental contributions to physics, mathematics, and geometry. He reinterpreted general relativity to prove that black holes can form from dying stars. He invented twistor theory—a novel way to look at the structure of space-time—and so led us to a deeper understanding of the nature of gravity. He discovered a remarkable family of geometric forms that came to be known as Penrose tiles. He even moonlighted as a brain researcher, coming up with a provocative theory that consciousness arises from quantum-mechanical processes. And he wrote a series of incredibly readable, best-selling science books to boot.
And yet the 78-year-old Penrose—now an emeritus professor at the Mathematical Institute, University of Oxford—seems to live the humble life of a researcher just getting started in his career. His small office is cramped with the belongings of the six other professors with whom he shares it, and at the end of the day you might find him rushing off to pick up his 9-year-old son from school. With the curiosity of a man still trying to make a name for himself, he cranks away on fundamental, wide-ranging questions: How did the universe begin? Are there higher dimensions of space and time? Does the current front-running theory in theoretical physics, string theory, actually make sense?
Because he has lived a lifetime of complicated calculations, though, Penrose has quite a bit more perspective than the average starting scientist. To get to the bottom of it all, he insists, physicists must force themselves to grapple with the greatest riddle of them all: the relationship between the rules that govern fundamental particles and the rules that govern the big things—like us—that those particles make up. In his powwow with DISCOVER contributing editor Susan Kruglinksi, Penrose did not flinch from questioning the central tenets of modern physics, including string theory and quantum mechanics. Physicists will never come to grips with the grand theories of the universe, Penrose holds, until they see past the blinding distractions of today’s half-baked theories to the deepest layer of the reality in which we live.
You come from a colorful family of overachievers, don’t you?
My older brother is a distinguished theoretical physicist, a fellow of the Royal Society. My younger brother ended up the British chess champion 10 times, a record. My father came from a Quaker family. His father was a professional artist who did portraits—very traditional, a lot of religious subjects. The family was very strict. I don’t think we were even allowed to read novels, certainly not on Sundays. My father was one of four brothers, all of whom were very good artists. One of them became well known in the art world, Sir Roland. He was cofounder of the Institute of Contemporary Arts in London. My father himself was a human geneticist who was recognized for demonstrating that older mothers tend to get more Down syndrome children, but he had lots of scientific interests.
How did your father influence your thinking?
The important thing about my father was that there wasn’t any boundary between his work and what he did for fun. That rubbed off on me. He would make puzzles and toys for his children and grandchildren. He used to have a little shed out back where he cut things from wood with his little pedal saw. I remember he once made a slide rule with about 12 different slides, with various characters that we could combine in complicated ways. Later in his life he spent a lot of time making wooden models that reproduced themselves—what people now refer to as artificial life. These were simple devices that, when linked together, would cause other bits to link together in the same way. He sat in his woodshed and cut these things out of wood in great, huge numbers.
So I assume your father helped spark your discovery of Penrose tiles, repeating shapes that fit together to form a solid surface with pentagonal symmetry.
It was silly in a way. I remember asking him—I was around 9 years old—about whether you could fit regular hexagons together and make it round like a sphere. And he said, “No, no, you can’t do that, but you can do it with pentagons,” which was a surprise to me. He showed me how to make polyhedra, and so I got started on that.
Are Penrose tiles useful or just beautiful?
My interest in the tiles has to do with the idea of a universe controlled by very simple forces, even though we see complications all over the place. The tilings follow conventional rules to make complicated patterns. It was an attempt to see how the complicated could be satisfied by very simple rules that reflect what we see in the world.
The artist M. C. Escher was influenced by your geometric inventions. What was the story there?
In my second year as a graduate student at Cambridge, I attended the International Congress of Mathematicians in Amsterdam. I remember seeing one of the lecturers there I knew quite well, and he had this catalog. On the front of it was the Escher picture Day and Night, the one with birds going in opposite directions. The scenery is nighttime on one side and daytime on the other. I remember being intrigued by this, and I asked him where he got it. He said, “Oh, well, there’s an exhibition you might be interested in of some artist called Escher.” So I went and was very taken by these very weird and wonderful things that I’d never seen anything like. I decided to try and draw some impossible scenes myself and came up with this thing that’s referred to as a tri-bar. It’s a triangle that looks like a three-dimensional object, but actually it’s impossible for it to be three-dimensional. I showed it to my father and he worked out some impossible buildings and things. Then we published an article in the British Journal of Psychology on this stuff and acknowledged Escher.
Escher saw the article and was inspired by it?
He used two things from the article. One was the tri-bar, used in his lithograph called Waterfall. Another was the impossible staircase, which my father had worked on and designed. Escher used it in Ascending and Descending, with monks going round and round the stairs. I met Escher once, and I gave him some tiles that will make a repeating pattern, but not until you’ve got 12 of them fitted together. He did this, and then he wrote to me and asked me how it was done—what was it based on? So I showed him a kind of bird shape that did this, and he incorporated it into what I believe is the last picture he ever produced, called Ghosts.
You have called the real-world implications of quantum physics nonsensical. What is your objection?
In quantum mechanics an object can exist in many states at once, which sounds crazy. The quantum description of the world seems completely contrary to the world as we experience it.
It doesn’t make any sense, and there is a simple reason. You see, the mathematics of quantum mechanics has two parts to it. One is the evolution of a quantum system, which is described extremely precisely and accurately by the Schrödinger equation. That equation tells you this: If you know what the state of the system is now, you can calculate what it will be doing 10 minutes from now. However, there is the second part of quantum mechanics—the thing that happens when you want to make a measurement. Instead of getting a single answer, you use the equation to work out the probabilities of certain outcomes. The results don’t say, “This is what the world is doing.” Instead, they just describe the probability of its doing any one thing. The equation should describe the world in a completely deterministic way, but it doesn’t.
Erwin Schrödinger, who created that equation, was considered a genius. Surely he appreciated that conflict.
Schrödinger was as aware of this as anybody. He talks about his hypothetical cat and says, more or less, “Okay, if you believe what my equation says, you must believe that this cat is dead and alive at the same time.” He says, “That’s obviously nonsense, because it’s not like that. Therefore, my equation can’t be right for a cat. So there must be some other factor involved.”
So Schrödinger himself never believed that the cat analogy reflected the nature of reality?
Oh yes, I think he was pointing this out. I mean, look at three of the biggest figures in quantum mechanics, Schrödinger, Einstein, and Paul Dirac. They were all quantum skeptics in a sense. Dirac is the one whom people find most surprising, because he set up the whole foundation, the general framework of quantum mechanics. People think of him as this hard-liner, but he was very cautious in what he said. When he was asked, “What’s the answer to the measurement problem?” his response was, “Quantum mechanics is a provisional theory. Why should I look for an answer in quantum mechanics?” He didn’t believe that it was true. But he didn’t say this out loud much.
That’s right. People don’t want to change the Schrödinger equation, leading them to what’s called the “many worlds” interpretation of quantum mechanics.
That interpretation says that all probabilities are playing out somewhere in parallel universes?
It says OK, the cat is somehow alive and dead at the same time. To look at that cat, you must become a superposition [two states existing at the same time] of you seeing the live cat and you seeing the dead cat. Of course, we don’t seem to experience that, so the physicists have to say, well, somehow your consciousness takes one route or the other route without your knowing it. You’re led to a completely crazy point of view. You’re led into this “many worlds” stuff, which has no relationship to what we actually perceive.
The idea of parallel universes—many worlds—is a very human-centered idea, as if everything has to be understood from the perspective of what we can detect with our five senses.
The trouble is, what can you do with it? Nothing. You want a physical theory that describes the world that we see around us. That’s what physics has always been: Explain what the world that we see does, and why or how it does it. Many worlds quantum mechanics doesn’t do that. Either you accept it and try to make sense of it, which is what a lot of people do, or, like me, you say no—that’s beyond the limits of what quantum mechanics can tell us. Which is, surprisingly, a very uncommon position to take. My own view is that quantum mechanics is not exactly right, and I think there’s a lot of evidence for that. It’s just not direct experimental evidence within the scope of current experiments.
In general, the ideas in theoretical physics seem increasingly fantastical. Take string theory. All that talk about 11 dimensions or our universe’s existing on a giant membrane seems surreal.
You’re absolutely right. And in a certain sense, I blame quantum mechanics, because people say, “Well, quantum mechanics is so nonintuitive; if you believe that, you can believe anything that’s nonintuitive.” But, you see, quantum mechanics has a lot of experimental support, so you’ve got to go along with a lot of it. Whereas string theory has no experimental support.
I understand you are setting out this critique of quantum mechanics in your new book.
The book is called Fashion, Faith and Fantasy in the New Physics of the Universe. Each of those words stands for a major theoretical physics idea. The fashion is string theory; the fantasy has to do with various cosmological schemes, mainly inflationary cosmology [which suggests that the universe inflated exponentially within a small fraction of a second after the Big Bang]. Big fish, those things are. It’s almost sacrilegious to attack them. And the other one, even more sacrilegious, is quantum mechanics at all levels—so that’s the faith. People somehow got the view that you really can’t question it.
A few years ago you suggested that gravity is what separates the classical world from the quantum one. Are there enough people out there putting quantum mechanics to this kind of test?
No, although it’s sort of encouraging that there are people working on it at all. It used to be thought of as a sort of crackpot, fringe activity that people could do when they were old and retired. Well, I am old and retired! But it’s not regarded as a central, as a mainstream activity, which is a shame.
After Newton, and again after Einstein, the way people thought about the world shifted. When the puzzle of quantum mechanics is solved, will there be another revolution in thinking?
It’s hard to make predictions. Ernest Rutherford said his model of the atom [which led to nuclear physics and the atomic bomb] would never be of any use. But yes, I would be pretty sure that it will have a huge influence. There are things like how quantum mechanics could be used in biology. It will eventually make a huge difference, probably in all sorts of unimaginable ways.
In your book The Emperor’s New Mind, you posited that consciousness emerges from quantum physical actions within the cells of the brain. Two decades later, do you stand by that?
I think it will be beautiful.
[4:41:49 AM-Friday, October 4th, 2013 +10UCT]
[2:52:23 AM] Shiloh: Yes
[2:53:06 AM] Shiloh: yes and our job is done, so we are getting more 'normal' and irritated with all things including our own human stuff
[2:53:43 AM] Shiloh: In the timewarp, we were banned from Camelot around April 6th, 2011
[2:54:02 AM] Shiloh: This is October 1st, 2013
[2:54:29 AM] Shiloh: So we feeling the 'war' with them for the next month or so now
[2:54:38 AM] Sirius 17: oh i see
[2:55:04 AM] Shiloh: November 1st 2013 will be like March 5th, 2011
[2:55:10 AM] Sirius 17: yes too much time to feel and think about our own mortality
[2:55:58 AM] Shiloh: And so the Fukushima tsunami is mapping in the mirror of October 27th or so
[2:56:24 AM] Shiloh: We went there around February 22nd or so
[2:56:42 AM] Shiloh: This becomes around November 12th
[2:56:44 AM] Sirius 17: i kept thinking if the Logos wants me dead it should do it quicker lol, then of course i felt bad thinking this way and that i am not going to die, well not the way i think i guess
[2:56:53 AM] Sirius 17: hmmm wow that is interesting
[2:57:11 AM] Shiloh: I dont know; but I think by age 60 there must be some change in me
[2:57:24 AM] Shiloh: I dont see myself becoming an old man
[2:57:42 AM] Shiloh: Yes this Ison thing is here now
[2:57:47 AM] Sirius 17: yes
[2:58:00 AM] Sirius 17: and it is from outside our solar system
[2:58:12 AM] Sirius 17: i really think its means 'invasion' time
[2:58:20 AM] Shiloh: In many forms
[2:58:23 AM] Sirius 17: yes
[2:58:34 AM] Shiloh: Try to watch those videos with my last post
[2:58:49 AM] Sirius 17: i almost finished the nabs one about dulce
[2:58:56 AM] Shiloh: It drags on
[2:59:10 AM] Shiloh: Watch the shorter ones with Kaku and Hawking
[2:59:15 AM] Sirius 17: and while it did have some syncro it was pretty much BS of the highest order at about the half way point
[2:59:22 AM] Shiloh: Yes
[2:59:49 AM] Sirius 17: but i have seen clips and interviews about the same subject and years ago i did read the dulce papers
[2:59:59 AM] Sirius 17: but this reminds me of Bill's Serpo project
[3:00:09 AM] Sirius 17: they just make this shit up at will to make money
[3:00:22 AM] Sirius 17: sell books, create controversy you know
[3:00:32 AM] Sirius 17: spinning it all on half truths and rumors
[3:00:44 AM] Shiloh: Yes there is a big component of this and Nabsers like Carol cant see it; blaming Dacos instead
[3:00:48 AM] Sirius 17: like you said the human imagination is so powerful at weaving incredible stories
[3:01:30 AM] Shiloh: See the videos of the galaxies merging?
[3:01:33 AM] Sirius 17: yes well like we said before, she cannot get past the Dragon symbol and archetype
[3:01:40 AM] Shiloh: Idiotic
[3:01:48 AM] Shiloh: Proud and vain actually
[3:02:05 AM] Sirius 17: to her the mere mention of Draco conjures images of a horney horned Devil in her mind lol, forked tongue and all
[3:02:26 AM] Shiloh: http://www.themistsofavalon.net/t6759p30-the-factuals-versus-the-nabs
[3:02:29 AM] Shiloh: Yes
[3:02:51 AM] Shiloh: Mindfuc11ed most of the chicks there
[3:03:04 AM] Shiloh: This Jenetta is a nabser too
[3:03:08 AM] Shiloh: I like Sanicle though
[3:03:54 AM] Sirius 17: oh i like this milky way and andromeda merger
[3:04:15 AM] Sirius 17: looks like the triangulum galaxy comes into play too
[3:04:48 AM] Sirius 17: yeah i like Sanicle too, she seems at least open to logical discourse somewhat
[3:06:10 AM] Shiloh: Yes she is the sort of Nabs which i can accomodate as not being Nabs
[3:09:37 AM] Sirius 17: oh yeah the first video is better, explains the time factor
[3:09:51 AM] Shiloh: I found 3 of them
[3:10:02 AM] Sirius 17: lol billions and billions of years...."carl sagan's voice"
[3:10:04 AM] Shiloh: All relevant to my deeper exposition following
[3:10:37 AM] Shiloh: It includes the ET factor see
[3:10:44 AM] Shiloh: Kaku in the 3rd
[3:10:56 AM] Sirius 17: yes the eternal dance...you romantic devil
[3:11:06 AM] Shiloh: Well Abba and Baab
[3:11:12 AM] Shiloh: One day I will dance like that again
[3:11:26 AM] Sirius 17: yes you will
[3:11:41 AM] Shiloh: Young virile bodies for us dear
[3:11:56 AM] Shiloh: I am still virile lol but cant move like that
[3:12:21 AM] Shiloh: Amazing though, that noone appreciates those posts
[3:12:43 AM] Shiloh: Reading them must surely indicate it is well informed
[3:12:58 AM] Shiloh: So science lover-freak Carol has no words for it
[3:13:18 AM] Shiloh: Showing her fakeness in regards to science
[3:13:25 AM] Sirius 17: i know it is mind blowing that no one at least comments and says, wow Shiloh, this must of taken you quite some time and nice graphics ect ect...
[3:13:36 AM] Sirius 17: something
[3:13:44 AM] Sirius 17: no they fuc11ing hate it apparently
[3:13:51 AM] Shiloh: They are 'afraid' lol very afraid of what it means for them should it be true
[3:14:02 AM] Sirius 17: yes likely
[3:14:07 AM] Shiloh: I know they are
[3:14:26 AM] Shiloh: Why idiot 44 still tries to bring dragons into her BS
[3:14:52 AM] Sirius 17: she misses us lol
[3:14:58 AM] Sirius 17: can't get us out of her head
[3:15:46 AM] Shiloh: I dont know
[3:15:58 AM] Shiloh: i dont miss her babble though
[3:16:10 AM] Sirius 17: me neither
[3:16:51 AM] Sirius 17: but that last post where she mentions us as 'copy cats' was so incoherent that i just shook my head and wondered wtf is going on inside her brain
[3:17:35 AM] Sirius 17:
View: http://www.youtube.com/watch?feature=player_embedded&v=uabNtlLfYyU#t=85[3:17:54
AM] Sirius 17: time marker 1:58 or so, curiosity killed the cats
[3:18:11 AM] Sirius 17: yes where the hell is the curiousity of the moabytes
[3:18:19 AM] Sirius 17: of the why and how
[3:18:56 AM] Shiloh: Lol
[3:19:20 AM] Shiloh: If it is above their head, like real science, they dismiss it as conspiracies
[3:20:02 AM] Shiloh: Other people labeled us as copycats and plagiarisers of others works remember?
[3:20:43 AM] Shiloh: As other people know better than Thuban, DD and Carol and lot then use their 'critical thinking' to make their 'choices'
[3:20:48 AM] Sirius 17: time marker 3.36...there it is, the solution is outside of time and space, so simple yet i doubt any of them even watched these videos
[3:20:59 AM] Sirius 17: yes i know, parrots we are
[3:21:28 AM] Sirius 17: and all your science papers that YOU wrote must be copied from elsewhere....where i wonder lol
[3:21:47 AM] Shiloh: Yes how often did we say space and time are required for physicality
[3:21:57 AM] Sirius 17: god forbid you actually wrote something on your own Tony....its unthinkable
[3:22:20 AM] Shiloh: Oh yes, I am too stupid to create anything by myself
[3:22:47 AM] Shiloh: So I am deceiving people as evil draco in stealing other human's ideas
[3:22:59 AM] Shiloh: Lawlessline said this or such
[3:23:42 AM] Shiloh: But you see in all 3 videos that at the end they have no answers
[3:23:51 AM] Shiloh: So I gave the answers
[3:23:56 AM] Sirius 17: yes
[3:24:16 AM] Shiloh: For our testimony not for human judgements on it
[3:24:40 AM] Shiloh: This is the last thing to do imo
[3:24:58 AM] Shiloh: Just to put it there for their bebafflements and ignorance
[3:25:19 AM] Shiloh: This Penrose pic is nice
[3:25:34 AM] Shiloh: It is just the New Standard Model of the quarkian waves
[3:25:38 AM] Shiloh: Matter waves
[3:26:04 AM] Shiloh: The smallest quantum written in the galactic sky
[3:26:58 AM] Sirius 17:
View: http://www.youtube.com/watch?feature=player_embedded&v=wHHz4mB9GKY#t=75
[3:27:01 AM] Sirius 17: 1:15 lol
[3:27:20 AM] Sirius 17: relativity is perfectly intelligible to anyone who is able to think
[3:29:47 AM] Shiloh: Not many apparently
[3:31:57 AM] Sirius 17: i am watching the kaku video now
[3:32:31 AM] Shiloh: Yes he mentions the ET civilisations as a consequence of evolution
[3:43:30 AM] Sirius 17: hahaha Bambie vs Godzilla
[3:43:36 AM] Sirius 17: oh dear
[3:49:34 AM] Sirius 17: god kaku is so smart, i really like this guy and his rationality
[3:50:38 AM] Shiloh: He is ok yes
[3:50:45 AM] Shiloh: I added Penrose to the message
[3:50:50 AM] Shiloh: Discover Interview: Roger Penrose Says Physics Is Wrong, From String Theory to Quantum Mechanics
For further details, a consultation of the Thuban archives on http://cosmosdawn.com is suggested.
[3:51:30 AM] Sirius 17: yes i have not gotten to that part yet, i am on the last kaku video
[3:52:02 AM] Sirius 17: but Michio is right, we are on the verge of either taking off onto a new civilization or destroying ourselves
[3:52:14 AM] Sirius 17: precarious times to witness
[3:54:22 AM] Shiloh: Yes
[3:54:30 AM] Shiloh: This is it
[3:54:43 AM] Shiloh: Why our 'death' is pointless in the cosmic sense
[4:15:13 AM] Sirius 17: humm yes i like this penrose article
[4:16:03 AM] Shiloh: I am posting it
[4:30:21 AM] Shiloh: Done
[4:30:37 AM] Shiloh: Of course he is wrong about strings as we understand them
[4:30:42 AM] Sirius 17:
1. Calculate the Phasevelocity (Vph) of the matterwave for a constant sourcesink frequency (fmax=f*) parameter using Vph=R.f* for a chosen displacement coordinate R.
2. Calculate the total inertia as a density function for the object to be matterwaved or 'teleported' in de Broglie parameters.
3. Convert this density function into the masseigen-frequency selfstate (fmin=1/f*) using the magnetopolic currentflow from the selfdual Planckian superstring transform.
[4:30:55 AM] Shiloh: But he is right about them in terms of parallel universe crapola
[4:31:02 AM] Sirius 17: lol Tony this is way too deep for anyone to understand your talking first principles
[4:31:18 AM] Shiloh: Yes it is not for humans
[4:31:27 AM] Sirius 17: i didn't paste all the 'instructions' here but its funny to me
[4:31:38 AM] Shiloh: And I cant do those calculations in the required detail either
[4:32:02 AM] Sirius 17: well the ETs can
[4:32:18 AM] Shiloh: But the new physics can use this somewhat in the form of summation integrals like in the consciousness paper
[4:32:36 AM] Sirius 17: yeah once they can 'see' it
[4:32:52 AM] Sirius 17: problem is they have to get out of the current paradigm limitations to be able to see it
[4:33:01 AM] Sirius 17: they are so very much box thinkers
[4:33:04 AM] Shiloh: Those 10 instructions are however as you said the basic principles and those I can explain as I did in the rest of the message
[4:33:47 AM] Shiloh: It is the micro-macro quantization and this is what the resurrection physics is
[4:34:09 AM] Shiloh: The intro in bold is the general rundown
[4:34:30 AM] Shiloh: Mass to magnetics to electricity then reversed
[4:34:37 AM] Sirius 17: yeah i figured this, transformation physics yes
[4:34:54 AM] Shiloh: But I put this there for Carol and co to see how limited they are
[4:35:00 AM] Sirius 17: how to trap the hubble heart beat within the finitum
[4:35:02 AM] Shiloh: They always ask for details see?
[4:35:27 AM] Shiloh: So there are some they wont get from the quacks like MacCanney
[4:35:33 AM] Shiloh: Or Keshe etc
[4:36:00 AM] Sirius 17: it just kills me, if they only read through this stuff a little slowly and methodically they could see how beautiful it is
[4:36:17 AM] Sirius 17: or maybe i am dreaming
[4:36:26 AM] Shiloh: No it is all selfconsistent
[4:36:42 AM] Shiloh: The Legacy post shows this and tries to prove it
[4:36:44 AM] Sirius 17: but really that is all i did is read it slowly, sometimes over and over until i could see it
[4:37:06 AM] Sirius 17: of course we have had hours of discussion and so on
[4:37:37 AM] Sirius 17: i don't know , it seems that anyone interested even slightly could see this
[4:37:49 AM] Shiloh: Well imo you are the only person, who can comment on the overviews
[4:37:59 AM] Shiloh: They are fakes
[4:38:08 AM] Shiloh: Their Nabs science is pure fake
[4:38:10 AM] Sirius 17: i am still not entirely sure how it is i understand some things
[4:38:31 AM] Shiloh: Floyd is no fake, but he has no science education to understand it
[4:38:53 AM] Sirius 17: no and some understanding of the sciences is required
[4:39:11 AM] Shiloh: I feel Carol is rather annoyed with those posts
[4:39:17 AM] Shiloh: Unsettling
[4:39:29 AM] Sirius 17: you cannot dismiss for example what physics already knows, such as relativity, quantum theory ect, it all goes together with the omni science
[4:39:53 AM] Sirius 17: because it requires deep thought to understand it
[4:39:57 AM] Shiloh: Because all her conspiracy science crap is right there in a novel form, which would even surprise the conspirators
[4:39:59 AM] Sirius 17: and to learn a new language
[4:40:14 AM] Sirius 17: they are not familar with the language of science, let alone omni-science
[4:40:30 AM] Sirius 17: its work
[4:40:40 AM] Sirius 17: takes effort
[4:40:40 AM] Shiloh: Can you see Jenetta or Burgundia read science words and not to call Devakas?
[4:40:51 AM] Sirius 17: oh hell no
[4:41:00 AM] Shiloh: Easier to call us names and evil dragons
[4:41:16 AM] Shiloh: Let them sit in their judgement seats
[4:41:21 AM] Sirius 17: yes and this is a microscopic example of the world, far easier to point fingers then to solve problems
[4:41:25 AM] Shiloh: But Carol pretends to like science
[4:41:33 AM] Sirius 17: yes as does Brook
[4:41:35 AM] Shiloh: This is the hypocrisy
[4:41:39 AM] Shiloh: Yes
[4:41:42 AM] Shiloh: They run
[4:41:49 AM] Sirius 17: but when shi20 gets too heavy even for her she goes total NABS
[4:42:17 AM] Sirius 17: i like how Penrose thinks
[4:42:26 AM] Sirius 17: he is a pragmatic fellow
[4:42:39 AM] Sirius 17: interesting his history that he struggled with math
[4:42:47 AM] Shiloh: It is a Nabs forum and only differs from Ryan and Cassidy in having a dislike for the formers and latters
[4:42:52 AM] Shiloh: The info is just as bad and biased towards their agendas of 'love and light' sillinesses and their schisms
[4:43:11 AM] Shiloh: Yes I saw this and this was just like me in grade 8; when I was more interested in Sonja than algebra and trigonometry
[4:43:18 AM] Shiloh: I wanted it slowly or not at all
[4:43:26 AM] Sirius 17: and i agree with him, no one should be timed on tests, its like you finish when you have solved it, each person has their own timing in this and with practice comes speed
[4:43:40 AM] Sirius 17: and confindence
[4:43:55 AM] Sirius 17: this is what happend to me, they ripped my confidence when i was little
[4:44:05 AM] Sirius 17: because i was slow i was told i was too dumb
[4:44:09 AM] Shiloh: Yes, me too; but I was simply disinterested in mathematics and the calculations, algebras in those days
[4:44:12 AM] Sirius 17: and after a while you believe it
[4:44:20 AM] Shiloh: I have Mercury in Taurus lol
[4:44:31 AM] Sirius 17: i have to work twice as hard as anyone else to get math
[4:44:34 AM] Sirius 17: and to keep it
[4:44:38 AM] Shiloh: I always hated time pressure
[4:44:42 AM] Sirius 17: yes me too
[4:44:57 AM] Sirius 17: but with unlimited time i do quite well
[4:45:19 AM] Shiloh: But this slowness allows thoroughness and this is Roger Penrose
[4:45:32 AM] Sirius 17: yes i can appreciate that
[4:45:33 AM] Shiloh: Pedantic Virgo Moon
[4:45:44 AM] Sirius 17: hehe yeah like you
[4:45:49 AM] Sirius 17: lover of details
[4:45:56 AM] Shiloh: I dont know where his planets are
[4:46:04 AM] Shiloh: I need his birthdate
[4:46:36 AM] Sirius 17: born 8 August 1931),
[4:46:44 AM] Sirius 17: http://en.wikipedia.org/wiki/Roger_Penrose
[4:46:47 AM] Shiloh: Leo
[4:46:52 AM] Shiloh: yes I check
[4:47:05 AM] Sirius 17: Born in Colchester, Essex, England,
[4:48:15 AM] Shiloh: You would not believe this
[4:48:26 AM] Shiloh: He has a Virgo Mercury and a Taurus Moon
[4:48:35 AM] Shiloh: The exact Mirror of me
[4:48:37 AM] Sirius 17: i found it interesting of his relationship to Escher
[4:48:46 AM] Sirius 17: lol
[4:48:52 AM] Shiloh: Yes the patterns
[4:48:58 AM] Sirius 17: i do believe it
[4:49:06 AM] Shiloh: Lol
[4:49:19 AM] Sirius 17: not too much suprises me anymore with us
[4:49:36 AM] Sirius 17: of course its all too incredible anyhow
[4:49:42 AM] Shiloh: Mars in Libra and Venus in Cancer
[4:49:52 AM] Shiloh: I have Mars in Cancer
[4:50:03 AM] Sirius 17: lol strange he is a mirror
[4:50:06 AM] Shiloh: Detrimental makes us peaceful
[4:50:10 AM] Shiloh: Nonviolent
[4:50:48 AM] Shiloh: Mars in water is antiwar
[4:51:20 AM] Shiloh: Saturn in Capricorn is native and so a father figure
[4:51:31 AM] Shiloh: He has authority and knows it
[4:51:40 AM] Sirius 17:
[4:52:07 AM] Shiloh: Yes I agree with him there
[4:52:18 AM] Sirius 17: yes i can see that
[4:52:21 AM] Sirius 17: and i agree too
[4:53:03 AM] Sirius 17: this harkens back to the first principles as well and the underpinning metaphysical algorithmic universe...Logos intelligence
[4:53:15 AM] Sirius 17: i mean it is so logical
[4:53:24 AM] Sirius 17: it just makes sense to me
[4:53:57 AM] Shiloh: Leo Jupiter and Cancer Pluto and Aries Uranus and Virgo Neptune
[4:54:08 AM] Sirius 17: all these people have fear of AI, i dont, because it don't feel it is possible
[4:54:46 AM] Shiloh: Dragon Node and Lilith in Aries, the latter right at the cusp with Pisces
[4:54:52 AM] Sirius 17: if there is to be anything like AI it must come from the human brain period and then remove the 'artificalness'
[4:55:01 AM] Shiloh: Chiron in Taurus
[4:55:14 AM] Sirius 17: oh his lilith is in Aries
[4:55:17 AM] Sirius 17: interesting
[4:55:32 AM] Shiloh: Right in the transit
[4:55:42 AM] Shiloh: One age changes into another age at 0° Aries
[4:55:55 AM] Sirius 17: yes he is before his time lol
[4:56:59 AM] Shiloh: Those are 9 year cycles
[4:58:49 AM] Shiloh: 40° per sign as 9x40°=360°
[4:59:40 AM] Sirius 17: interesting
Last edited by shiloh on Fri Oct 04, 2013 3:54 am; edited 1 time in total
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6a43fc321c61cf34 | You are here
Nuclear theory
Nuclear-reaction theory
The physics of low-energy nuclear reactions is essential to explain the evolution of the Universe. Nuclear reactions characterize the different phases in a variety of astrophysical environments, from hydrogen burning in main-sequence stars to explosive nucleosynthesis during the last stages of stellar evolution. In this context, the properties of nuclear systems away from stability provide invaluable insight. Moreover, understanding the mechanisms of nuclear collisions has implications beyond nuclear astrophysics, with applications in energy, medical and material sciences.
At ECT* (Jesús Casal), research on nuclear-reaction theory is carried out to answer questions regarding the structure and reaction dynamics of weakly bound systems. For exotic nuclei at the limit of nuclear stability, coupled-channels methods are used to incorporate continuum effects. This includes the description of i) radiative capture reactions, ii) low-energy breakup, transfer and proton-target knockout reactions, and iii) nucleon-nucleon correlations in two-nucleon decays.
Nuclear many-body theory and infinite matter
The study of nuclear systems, from finite to infinite ones, requires the knowledge of the nuclear interaction and the choice of a many-body approach to solve the Schrödinger equation.
On one side, the nuclear interaction can be constructed as an effective interaction which is fit to reproduce the properties of either finite nuclei along the nuclear chart or even infinite matter, and the Schrödinger equation can usually be solved within the mean field approximation (Hartree-Fock). This approach goes under the name of energy-density functional theory.
On the other side, one can construct a realistic interaction which is instead fit to reproduce the nucleon-nucleon scattering data or even properties of few-body systems. In this latter case on then needs to solve the many-body problem via more sophisticated approximations beyond the mean field level, in order to build in the nuclear correlations. This defines the so called ab initio nuclear theory. The main difference between the two approaches is that the former is strongly model dependent, while the latter strives to be predictive.
At ECT* we are currently investigating the properties of infinite nuclear matter employing the ab initio self-consistent Green’s function approach (Arianna Carbone). This method is based on the use of the Green’s function to calculate both microscopic and bulk properties of the nuclear system. The use of interactions derived from chiral effective field theory gives us the possibility to be the more consistent as possible with the underlying quantum theory, QCD.
We are investigating both zero and finite-temperature properties of nuclear matter, with the objective of providing model independent nuclear physics input for the determination of the neutron star equation of state, to be used in astrophysical simulations of core-collapse supernovae and binary neutron star mergers. |
40fc3ea888c378ec | Tuesday, February 07, 2012
Last week, Subir Sachdev came to Munich to give three Arnold Sommerfeld Lectures. I want to take this opportunity to write about a subject that has attracted a lot of attention in recent years, namely applying AdS/CFT techniques to condensed matter systems like trying to write gravity duals for D-wave superconducturs or strange metals (it's surprisingly hard to find a good link for this keyword).
My attitude towards this attempt has somewhat changed from "this will never work" to "it's probably as good as anything else" and in this post I will explain why I think this. I should mention as well that Sean Hartnoll has been essential in this phase transition of my mind.
Let me start by sketching (actually: caricaturing) what I am talking about. You want to understand some material, typically the electrons in a horribly complicated lattice like bismuth strontium calcium copper oxide, or BSCCO. To this end, you come up with a five dimensional theory of gravity coupled to your favorite list of other fields (gauge fields, scalars with potentials, you name it) and place that in an anti-de-Sitter background (or better, for finite temperature, in an asymptotically anti-de-Sitter black hole). Now, you compute solutions with prescribed behavior at infinity and interpret these via Witten's prescription as correlators in your condensed matter theory. For example you can read off Green functions and (frequency dependent) conductivities, densities of state.
How can this ever work, how are you supposed to guess the correct field content (there is no D-brane/string description anywhere near that could help you out) and how can you ever be sure you got it right?
The answer is you cannot but it does not matter. It does not matter as it does not matter elsewhere in condensed matter physics. To clarify this, we have to be clear about what it means for a condensed matter theorist to "understand" a system. Expressed in our high energy lingo, most of the time, the "microscopic theory" is obvious: It is given by the Schrödinger equation for $10^23$ electrons plus as similar number of noclei feeling the Coulomb potential of the nuclei and interacting themselves with Coulomb repulsion. There is nothing more to be known about this. Except that this is obviously not what we want. These are far too many particles to worry about and, what is more important, we are interested in the behavior at much much lower energy scales and longer wave lengths, at which all the details of the lattice structure are smoothed out and we see only the effect of a few electrons close to the Fermi surface. As an estimate, one should compare the typical energy scale of the Coulomb interactions, the binding energies of the electrons to the nucleus (Z times 13.6 eV) or in terms of temperature (where putting in the constants equates 1eV to about 10,000K) to the milli-eV binding energy of Cooper pairs or the typical temperature where superconductivity plays a role.
In the language of the renormalization group, the Coulomb interactions are the UV theory but we want to understand the effective theory that this flows to in the IR. The convenient thing about such effective theories is that they do not have to be unique: All we want is a simple to understand theory (in which we can compute many quantities that we would like to know) that is in the same universality class as the system we started from. Differences in relevant operators do not matter (at least to leading order).
Surprisingly often, one can find free theories or weakly (and thus almost free) theories that can act as the effective theory we are looking for. BCS is a famous example, but Landau's Fermi Liquid Theory is another: There the idea is that you can almost pretend that your fermions are free (and thus you can just add up energies taking into account the Pauli exclusion principle giving you Fermi-surfaces etc) even though your electrons are interacting (remember, there is always the Coulomb interaction around). The only effect the interactions have, is to renormalize the mass, to deform the Fermi surface away from a ball and to change the hight of the jump in the T=0 occupation number. Experience shows that this is an excellent description in more than one dimension (that has the exception of the Luttinger liquid) and can probably traced back to the fact that a four-Fermi-interaction is non-renormalizable and thus invisible in the IR.
Only, it is important to remember that the fields/particles in that effective theories are not really the electrons you started with but just quasi-particles that are build in complicated ways out of the microscopic particles carrying around clouds of other particles and deforming the lattice they move in. But these details don't matter and that is the point.
It is only important to guess the effective theory in the same universality class. You never derive this (or: hardly ever). Following an exact renormalization group flow is just way beyond what is possible. You make a hopefully educated guess (based on symmetries etc) and then check that you get good descriptions. But only the fact, that there are not too many universality classes makes this process of guessing worthwhile.
Free or weakly coupled theories are not the only possible guesses for effective field theories in which one can calculate. 2d conformal field theories are others. And now, AdS-technology gives us another way of writing down correlation functions just as Feynman-rules give us correlation functions for weakly coupled theories. And that is all one needs: Correlation functions of effective field theory candidates. Once you have those you can check if you are lucky and get evidence that you are in the correct universality class. You don't have to derive the IR theory from the UV. You never do this. You always just guess. And often enough this is good enough to work. And strictly speaking, you never know if your next measurement shows deviations from what you thought would be an effective theory for your system.
In a sense, it is like the mystery that chemistry works: The periodic table somehow pretends that the electrons in atoms are arranged in states that group together like for the hydrogen atom, you get the same n,l,m,s quantum numbers and the shells are roughly the same (although with some overlap encoded in the Aufbau principle) as for hydrogen. This pretends that the only effect of the electron-electron Coulomb potential is to shield the charge of the nucleus and every electron sees effectively a hydrogen like atom (although not necessarily with integer charge Z) and Pauli's exclusion principle regulates that no state is filled more than once. One could have thought that the effect of n-1 electrons on the last is much bigger, after all, they have a total charge that is almost the same of the nucleous, but it seems, the last electron only sees the nucleus with a 1/r potential although with reduced charge.
If you like, the only thing one should might worry about is that the Witten prescription to obtain boundary correlators from bulk configurations really gives you valid n-point functions of a quantum theory (if you feel sufficient mathematical masochism for example in the sense of Wightman) but you don't want to show that it is the quantum field theory corresponding to the material you started with.
Friday, February 03, 2012
How I Learned to Stop Worrying and Love QFT
2. Stone-von-Neumann Theorem (Dennis)
3. Pure Operations, POVMs (Mario)
4. Measurement Problem (Anupam, David)
6. Decoherence (Kostas, Cosmas)
7. Pointer Basis (Greeks again)
8. Consistent Histories (Hao)
9. Many Worlds (Max)
10. Bohmian Interpretation (Henry, Franz)
See also the seminar's wiki page.
Have fun! |
0bf46a24baac38f4 | Max Planck
Max Planck
Max planck.jpg
Max Karl Ernst Ludwig Planck
April 23, 1858
Kiel, Germany
Died October 4, 1947
Göttingen, Germany
Residence Flag of Germany.svg Germany
Nationality Flag of Germany.svg German
Field Physicist
Institutions University of Kiel
Humboldt-Universität zu Berlin
Georg-August-Universität Göttingen
Alma mater Ludwig-Maximilians-Universität München
Academic advisor Philipp von Jolly
Notable students Gustav Ludwig Hertz Nobel.svg
Erich Kretschmann
Walther Meißner
Walter Schottky
Max von Laue Nobel.svg
Max Abraham
Moritz Schlick
Walther Bothe Nobel.svg
Known for Planck's constant, quantum theory
Notable prizes Nobel.svg Nobel Prize in Physics (1918)
He was the father of Erwin Planck.
Max Karl Ernst Ludwig Planck (April 23, 1858 – October 4, 1947) was a German physicist who is widely regarded as one of the most significant scientists in history. He developed a simple but revolutionary concept that was to become the foundation of a new way of looking at the world, called quantum theory.
In 1900, to solve a vexing problem concerning the radiation emitted by a glowing body, he introduced the radical view that energy is transmitted not in the form of an unbroken (infinitely subdivisible) continuum, but in discrete, particle-like units. He called each such unit a quantum (the plural form being quanta). This concept was not immediately accepted by physicists, but it ultimately changed the very foundations of physics. Planck himself did not quite believe in the reality of this concept—he considered it a mathematical construct. In 1905, Albert Einstein used that concept to explain the photoelectric effect, and in 1913, Niels Bohr used the same idea to explain the structures of atoms. From then on, Planck's idea became central to all of physics. He received the Nobel Prize in 1918, and both Einstein and Bohr received the prize a few years later.
Planck was also a deeply religious man who believed that religion and science were mutually compatible, both leading to a larger, universal truth. By basing his convictions on seeking the higher truth, not on doctrine, he was able to stay open-minded when it came to formulating scientific concepts and being tolerant toward alternative belief systems.
Life and work
Early childhood
Planck was born in Kiel to Johann Julius Wilhelm Planck and his second wife, Emma Patzig. He was the sixth child in the family, including two siblings from his father's first marriage. Among his earliest memories was the marching of Prussian and Austrian troops into Kiel during the Danish-Prussian War in 1864. In 1867, the family moved to Munich, and Planck enrolled in the Maximilians gymnasium. There he came under the tutelage of Hermann Müller, a mathematician who took an interest in the youth and taught him astronomy and mechanics as well as mathematics. It was from Müller that Planck first learned the principle of conservation of energy. Planck graduated early, at age 16. This is how Planck first came in contact with the field of physics.
Planck was extremely gifted when it came to music: He took singing lessons and played the piano, organ, and cello, and composed songs and operas. However, instead of music, he chose to study physics.
Munich physics professor Philipp von Jolly advised him against going into physics, saying, "in this field, almost everything is already discovered, and all that remains is to fill a few holes." Planck replied that he did not wish to discover new things, only to understand the known fundamentals of the field. In 1874, he began his studies at the University of Munich. Under Jolly's supervision, Planck performed the only experiments of his scientific career: Studying the diffusion of hydrogen through heated platinum. He soon transferred to theoretical physics.
In 1877, he went to Berlin for a year of study with the famous physicists Hermann von Helmholtz and Gustav Kirchhoff, and the mathematician Karl Weierstrass. He wrote that Helmholtz was never quite prepared (with his lectures), spoke slowly, miscalculated endlessly, and bored his listeners, while Kirchhoff spoke in carefully prepared lectures, which were, however, dry and monotonous. Nonetheless, he soon became close friends with Helmholtz. While there, he mostly undertook a program of self-study of Rudolf Clausius's writings, which led him to choose heat theory as his field.
In October 1878, Planck passed his qualifying exams and in February 1879, defended his dissertation, Über den zweiten Hauptsatz der mechanischen Wärmetheorie (On the second fundamental theorem of the mechanical theory of heat). He briefly taught mathematics and physics at his former school in Munich. In June 1880, he presented his habilitation thesis, Gleichgewichtszustände isotroper Körper in verschiedenen Temperaturen (Equilibrium states of isotropic bodies at different temperatures).
Academic career
With the completion of his habilitation thesis, Planck became an unpaid private lecturer in Munich, waiting until he was offered an academic position. Although he was initially ignored by the academic community, he furthered his work on the field of heat theory and discovered one after the other the same thermodynamical formalism as Josiah Willard Gibbs without realizing it. Clausius's ideas on entropy occupied a central role in his work.
In April 1885, the University of Kiel appointed Planck an associate professor of theoretical physics. Further work on entropy and its treatment, especially as applied in physical chemistry, followed. He proposed a thermodynamic basis for Arrhenius's theory of electrolytic dissociation.
Within four years, he was named the successor to Kirchhoff's position at the University of Berlin—presumably thanks to Helmholtz's intercession—and by 1892 became a full professor. In 1907, Planck was offered Boltzmann's position in Vienna, but turned it down to stay in Berlin. During 1909, he was the Ernest Kempton Adams Lecturer in Theoretical Physics at Columbia University in New York City. He retired from Berlin on January 10, 1926, and was succeeded by Erwin Schrödinger.
After the appointment to Berlin, the Planck family lived in a villa in Berlin-Grunewald, Wangenheimstraße 21. Several other professors of Berlin University lived nearby, among them the famous theologian Adolf von Harnack, who became a close friend of Planck. Soon the Planck home became a social and cultural center. Numerous well-known scientists—such as Albert Einstein, Otto Hahn, and Lise Meitner—were frequent visitors. The tradition of jointly playing music had already been established in the home of Helmholtz.
After several happy years, the Planck family was struck by a series of disasters: In July 1909, Marie Planck died, possibly from tuberculosis. In March 1911, Planck married his second wife, Marga von Hoesslin (1882-1948); in December his third son, Herrmann, was born.
During the First World War, Planck's son Erwin was taken prisoner by the French in 1914, and his son Karl was killed in action at Verdun in 1916. His daughter Grete died in 1917 while giving birth to her first child; her sister lost her life two years later under the same circumstances, after marrying Grete's widower. Both granddaughters survived and were named after their mothers. Planck endured all these losses with stoic submission to fate.
During World War II, Planck's house in Berlin was completely destroyed by bombs in 1944, and his youngest son, Erwin, was implicated in the attempt made on Hitler's life on July 20, 1944. Consequently, Erwin died a horrible death at the hands of the Gestapo in 1945.
Professor at Berlin University
In Berlin, Planck joined the local Physical Society. He later wrote about this time: "In those days I was essentially the only theoretical physicist there, whence things were not so easy for me, because I started mentioning entropy, but this was not quite fashionable, since it was regarded as a mathematical spook." Thanks to his initiative, the various local Physical Societies of Germany merged in 1898 to form the German Physical Society (Deutsche Physikalische Gesellschaft, DPG), and Planck was its president from 1905 to 1909.
Planck started a six semester course of lectures on theoretical physics. Lise Meitner described the lectures as "dry, somewhat impersonal." An English participant, James R. Partington, wrote, "using no notes, never making mistakes, never faltering; the best lecturer I ever heard." He continues: "There were always many standing around the room. As the lecture-room was well heated and rather close, some of the listeners would from time to time drop to the floor, but this did not disturb the lecture."
Planck did not establish an actual "school," the number of his graduate students was only about 20 altogether. Among his students were the following individuals. The year in which each individual achieved the highest degree is indicated after the person's name (outside the parentheses); the individual's year of birth and year of death are given within parentheses.
Max Abraham 1897 (1875-1922)
Moritz Schlick 1904 (1882-1936)
Walther Meißner 1906 (1882-1974)
Max von Laue 1906 (1879-1960)
Fritz Reiche 1907 (1883-1960)
Walter Schottky 1912 (1886-1976)
Walther Bothe 1914 (1891-1957)
Black-body radiation
In 1894, Planck had been commissioned by electricity companies to discover how to generate the greatest luminosity from light bulbs with the minimum energy. To approach that question, he turned his attention to the problem of black-body radiation. In physics, a black body is an object that absorbs all electromagnetic radiation that falls onto it. No radiation passes through it and none is reflected. Black bodies below around 700 K (430 °C) produce very little radiation at visible wavelengths and appear black (hence the name). Above this temperature, however, they produce radiation at visible wavelengths, starting at red and going through orange, yellow, and white before ending up at blue, as the temperature is raised. The light emitted by a black body is called black-body radiation (or cavity radiation). The amount and wavelength (color) of electromagnetic radiation emitted by a black body is directly related to its temperature. The problem, stated by Kirchhoff in 1859, was: How does the intensity of the electromagnetic radiation emitted by a black body depend on the frequency of the radiation (correlated with the color of the light) and the temperature of the body?
This question had been explored experimentally, but the Rayleigh-Jeans law, derived from classical physics, failed to explain the observed behavior at high frequencies, where it predicted a divergence of the energy density toward infinity (the "ultraviolet catastrophe"). Wilhelm Wien proposed Wien's law, which correctly predicted the behavior at high frequencies but failed at low frequencies. By interpolating between the laws of Wien and Rayleigh-Jeans, Planck formulated the now-famous Planck's law of black-body radiation, which described the experimentally observed black-body spectrum very well. It was first proposed in a meeting of the DPG on October 19, 1900, and published in 1901.
By December 14, 1900, Planck was already able to present a theoretical derivation of the law, but this required him to use ideas from statistical mechanics, as introduced by Boltzmann. So far, he had held a strong aversion to any statistical interpretation of the second law of thermodynamics, which he regarded as having an axiomatic nature. Compelled to use statistics, he noted: "… an act of despair … I was ready to sacrifice any of my previous convictions about physics …"
The central assumption behind his derivation was the supposition that electromagnetic energy could be emitted only in quantized form. In other words, the energy could only be a multiple of an elementary unit. Mathematically, this was expressed as:
E = h \nu
where h is a constant that came to be called Planck's constant (or Planck's action quantum), first introduced in 1899, and \nu is the frequency of the radiation. Planck's work on quantum theory, as it came to be known, was published in the journal Annalen der Physik. His work is summarized in two books Thermodynamik (Thermodynamics) (1897) and Theorie der Wärmestrahlung (theory of heat radiation) (1906).
At first, Planck considered that quantization was only "a purely formal assumption … actually I did not think much about it…" This assumption, incompatible with classical physics, is now regarded as the birth of quantum physics and the greatest intellectual accomplishment of Planck's career. (However, in a theoretical paper published in 1877, Ludwig Boltzmann had already been discussing the possibility that the energy states of a physical system could be discrete.) In recognition of this accomplishment, Planck was awarded the Nobel prize for physics in 1918.
The discovery of Planck's constant enabled him to define a new universal set of physical units—such as Planck length and Planck mass—all based on fundamental physical constants.
Subsequently, Planck tried to integrate the concept of energy quanta with classical physics, but to no avail. "My unavailing attempts to somehow reintegrate the action quantum into classical theory extended over several years and caused me much trouble." Even several years later, other physicists—including Lord Rayleigh, James Jeans, and Hendrik Lorentz—set Planck's constant to zero, in an attempt to align with classical physics, but Planck knew well that this constant had a precise, nonzero value. "I am unable to understand Jeans' stubbornness—he is an example of a theoretician as should never be existing, the same as Hegel was for philosophy. So much the worse for the facts, if they are wrong."
Einstein and the theory of relativity
In 1905, the three epochal papers of the hitherto completely unknown Albert Einstein were published in the journal Annalen der Physik. Planck was among the few who immediately recognized the significance of the special theory of relativity. Thanks to his influence, this theory was soon widely accepted in Germany. Planck also contributed considerably to extend the special theory of relativity.
To explain the photoelectric effect (discovered by Philipp Lenard in 1902), Einstein proposed that light consists of quanta, which he called photons. Planck, however, initially rejected this theory, as he was unwilling to completely discard Maxwell's theory of electrodynamics. Planck wrote, "The theory of light would be thrown back not by decades, but by centuries, into the age when Christian Huygens dared to fight against the mighty emission theory of Isaac Newton …"
In 1910, Einstein pointed out the anomalous behavior of specific heat at low temperatures as another example of a phenomenon that defies explanation by classical physics. To resolve the increasing number of contradictions, Planck and Walther Nernst organized the First Solvay Conference in Brussels in 1911. At this meeting, Einstein was finally able to convince Planck.
Meanwhile, Planck had been appointed dean of Berlin University. Thereby, it was possible for him to call Einstein to Berlin and establish a new professorship for him in 1914. Soon the two scientists became close friends and met frequently to play music together.
World War I and the Weimar Republic
At the onset of the First World War Planck was not immune to the general excitement of the public: "… besides of much horrible also much unexpectedly great and beautiful: The swift solution of the most difficult issues of domestic policy through arrangement of all parties… the higher esteem for all that is brave and truthful…"
He refrained from the extremes of nationalism. For instance, he voted successfully for a scientific paper from Italy to receive a prize from the Prussian Academy of Sciences in 1915, (Planck was one of its four permanent presidents), although at that time Italy was about to join the Allies. Nevertheless, the infamous "Manifesto of the 93 intellectuals," a polemic pamphlet of war propaganda, was also signed by Planck. Einstein, on the other hand, retained a strictly pacifist attitude, which almost led to his imprisonment, from which he was saved only by his Swiss citizenship. But already in 1915, Planck revoked parts of the Manifesto, (after several meetings with Dutch physicist Lorentz), and in 1916, he signed a declaration against the German policy of annexation.
In the turbulent post-war years, Planck, by now the highest authority of German physics, issued the slogan "persevere and continue working" to his colleagues. In October 1920, he and Fritz Haber established the Notgemeinschaft der Deutschen Wissenschaft (Emergency Organization of German Science), which aimed at providing support for the destitute scientific research. They obtained a considerable portion of their funds from abroad. In this time, Planck held leading positions also at Berlin University, the Prussian Academy of Sciences, the German Physical Society, and the Kaiser Wilhelm Gesellschaft (KWG, which in 1948 became the Max Planck Gesellschaft). Under such circumstances, he himself could hardly conduct any more research.
He became a member of the Deutsche Volks-Partei (German People's Party), the party of peace Nobel prize laureate Gustav Stresemann, which aspired to liberal aims for domestic policy and rather revisionist aims for international politics. He disagreed with the introduction of universal suffrage and expressed later the view that the Nazi dictatorship was the result of "the ascent of the rule of the crowds."
Quantum mechanics
At the end of the 1920s, Bohr, Werner Heisenberg, and Wolfgang Pauli had worked out the Copenhagen interpretation of quantum mechanics. It was, however, rejected by Planck, as well as Schrödinger and Laue. Even Einstein had rejected Bohr's interpretation. Planck called Heisenberg's matrix mechanics "disgusting," but he gave the Schrödinger equation a warmer reception. He expected that wave mechanics would soon render quantum theory—his own brainchild—unnecessary.
Nonetheless, scientific progress ignored Planck's concerns. He experienced the truth of his own earlier concept, after his struggle with the older views. He wrote, "A new scientific truth does not establish itself by its enemies being convinced and expressing their change of opinion, but rather by its enemies gradually dying out and the younger generation being taught the truth from the beginning."
Nazi dictatorship and World War II
When the Nazis seized power in 1933, Planck was 74. He witnessed many Jewish friends and colleagues expelled from their positions and humiliated, and hundreds of scientists emigrated from Germany. Again he tried the "persevere and continue working" slogan and asked scientists who were considering emigration to stay in Germany. He hoped the crisis would abate soon and the political situation would improve again. There was also a deeper argument against emigration: Emigrating non-Jewish scientists would need to look for academic positions abroad, but these positions better served Jewish scientists, who had no chance of continuing to work in Germany.
Hahn asked Planck to gather well-known German professors, to issue a public proclamation against the treatment of Jewish professors. Planck, however, replied, "If you are able to gather today 30 such gentlemen, then tomorrow 150 others will come and speak against it, because they are eager to take over the positions of the others." Although, in a slightly different translation, Hahn remembers Planck saying: "If you bring together 30 such men today, then tomorrow 150 will come to denounce them because they want to take their places." Under Planck's leadership, the KWG avoided open conflict with the Nazi regime. One exception was Fritz Haber. Planck tried to discuss the issue with Adolf Hitler but was unsuccessful. In the following year, 1934, Haber died in exile.
As the political climate in Germany gradually became more hostile, Johannes Stark, prominent exponent of Deutsche Physik ("German Physics," also called "Aryan Physics") attacked Planck, Arnold Sommerfeld, and Heisenberg for continuing to teach the theories of Einstein, calling them "white Jews." The "Hauptamt Wissenschaft" (Nazi government office for science) started an investigation of Planck's ancestry, but all they could find out was that he was "1/16 Jewish."
In 1938, Planck celebrated his 80th birthday. The DPG held an official celebration, during which the Max Planck medal (founded as the highest medal by the DPG in 1928) was awarded to French physicist Louis de Broglie. At the end of 1938, the Prussian Academy lost its remaining independence and was taken over by Nazis (Gleichschaltung). Planck protested by resigning his presidency. He continued to travel frequently, giving numerous public talks, such as his famous talk on "Religion and Science." Five years later, he was still sufficiently fit to climb 3,000-meter peaks in the Alps.
During the Second World War, the increasing number of Allied bombing campaigns against Berlin forced Planck and his wife to leave the city temporarily and live in the countryside. In 1942, he wrote: "In me an ardent desire has grown to persevere this crisis and live long enough to be able to witness the turning point, the beginning of a new rise." In February 1944, his home in Berlin was completely destroyed by an air raid, annihilating all his scientific records and correspondence. Finally, he was in a dangerous situation in his rural retreat during the rapid advance of Allied armies from both sides. After the end of the war, Planck, his second wife, and their son Herrmann moved to Göttingen, where he died on October 4, 1947.
Max Planck commemorated on the German 2 Mark Coin
Religious views
Max Planck was a devoted Christian from early life to death. As a scientist, however, he was very tolerant toward other religions and alternate views, and was discontent with the church organization's demands for unquestioning belief. He noted that "natural laws … are the same for men of all races and nations."
Planck regarded the search for universal truth as the loftiest goal of all scientific activity. Perhaps foreseeing the central role it now plays in current thinking, Planck made great note of the fact that the quantum of action retained its significance in relativity because of the relativistic invariance of the Principle of Least Action.
Max Planck's view of God can be regarded as pantheistic, with an almighty, all-knowing, benevolent but unintelligible God who permeates everything, manifest by symbols, including physical laws. His view may have been motivated by an opposition—like that of Einstein and Schrödinger—to the positivist, statistical, subjective universe of scientists such as Bohr, Heisenberg, and others. Planck was interested in truth and the Universe beyond observation, and he objected to atheism as an obsession with symbols.[1]
Planck was the very first scientist to contradict the physics established by Newton. This is why all physics before Planck is called "classical physics," while all physics after him is called "quantum physics." In the classical world, energy is continuous; in the quantum world, it is discrete. On this simple insight of Planck's was constructed all of the new physics of the twentieth century.
Planck had the firm conviction that religion and science are mutually compatible, both leading to a higher, universal truth that embraces everything. His convictions were based on seeking that higher truth, not on doctrine, and he was aware that science itself had just started on the quest. This allowed him to keep an open mind when young, in terms of scientific theory and to be tolerant toward alternative belief systems. His scientific views were, of course, in the classical mode of solids and forces—the quantum view of a much more sophisticated reality was not available to him. For he had just started the revolution and had second thoughts about the "reality" of his own concept of particle-like energy.
Unlike religion with its great leaps, science proceeds by baby steps. The small step taken by Planck was the first of the many needed to reach the current "internal wave and external particle" view of modern physics a century later.
Honors and medals
• "Pour le Mérite" for Science and Arts 1915 (in 1930 he became chancellor of this order)
• Nobel Prize in Physics 1918 (awarded 1919)
• Lorentz Medal 1927
• Adlerschild des Deutschen Reiches (1928)
• Max Planck medal (1929, together with Einstein)
• The asteroid 1069 was given the name "Stella Planckia" (1938)
Planck units
• Planck time
• Planck length
• Planck temperature
• Planck current
• Planck power
• Planck density
• Planck mass
1., The Religious Affiliation of Physicist Max Planck. Retrieved July 16, 2007.
Selected publications by Planck
• Gamow, George. 1966. Thirty Years That Shook Physics: The Story of Quantum Theory. Garden City, NY: Doubleday.
• Heilbron, J. L. 2000. The Dilemmas of an Upright Man: Max Planck and the Fortunes of German Science. Cambridge, MA: Harvard University Press. ISBN 0-674-00439-6
• Rosenthal-Schneider, Ilse. 1980. Reality and Scientific Truth: Discussions with Einstein, von Laue, and Planck. Wayne State University. ISBN 0-8143-1650-6
External links
All links retrieved September 6, 2018.
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0ea9bad036a23554 | Background and Motivation
The mathematical sciences play a vital part in all aspects of modern society. Without research and training in mathematics, there would be no engineering, economics or computer science; no smart phones, MRI scanners, bank accounts or PIN numbers. Mathematics is playing a key role in tackling the modern-day challenge of cyber security and in predicting the consequences of climate change, whereas in manufacturing sectors such as automotive and aerospace industries benefit from superior virtual design processes. Likewise the life sciences sector, with significant potential for economic growth, would not be in such a strong position without mathematics research and training, providing the expertise integral to the development of areas such as personalised healthcare and pharmaceuticals, and underpinning development of many medical technologies. The
emergence of truly massive datasets across most fields of science and engineering increases the need for new tools from the mathematical sciences.
One of the classic ways in which mathematical science research plays a role in the economy is through collecting data towards understanding it, by using tools and techniques, enabling the discovery of new relationships or models. Modelling of physical phenomena already dates back several centuries, and well-known systems of equations with the names of Maxwell, Navier-Stokes, Korteweg-de Vries and more recently the Schrödinger equation plus many others are now well established. But it was not until the advent of computers in the middle of the previous century, and the development of sophisticated computational methods (like iterative solution methods for large sparse linear systems) that this could be taken to a higher level, by performing computations using these models. Software tools with advanced computational mathematical techniques for the solution of the aforementioned systems of equations have become common place, and are heavily used by engineers and scientists.
Mirroring this activity is increased awareness by society and industry that mathematical simulation is ubiquitous to address the challenging problems of our times. Industrial processes, economic models and critical events like floods, power failures or epidemics have become so complicated that its realistic description does not require the simulation of a single model, but rather the cosimulation of various models. Better scientific understanding of the factors governing these will provide routes to greater innovation power and economic well-being across an increasingly complex, networked world with its competitive and strongly interacting agents. Industry, but also science, is highly dependent on the development of virtual environments that can handle the complex problems that we face today, and in the future.
For example, if the origins of life are to be explained, biologists and mathematicians need to work together, and most of the time spent will be on evaluating and simulating the mathematical models. Using the mathematics of evolutionary dynamics, the change from no life to life (referring to the self-replicating molecules dominating early Earth) can be explained. Another example is the electronics industry, which all of us rely on for new developments in virtually every aspect of our everyday life. Innovations in this branch of industry are impossible without the use of virtual design environments that enable engineers to develop and test their complex designs behind a screen, without ever having to go into the time-consuming (several months) process of prototyping.
Principles of computational science and engineering rooted in modern applied mathematics are at the core of these developments, subjects that are set to undergo a renaissance in the 21st century. Indeed, no less a figure than Stephen Hawking is on record as saying that the 21st century will be the century of complexity. Another great figure, yet young, is Fields medallist Terence Tao, who was a major contributor to the recently published document entitled “The mathematical sciences in 2025”, stating: “Mathematical sciences work is becoming an increasingly integral and essential component of a growing array of areas of investigation in biology, medicine, social sciences, business, advanced design, climate, finance, advanced materials, and many more – crucial to economic growth and societal well- being”.
Growing computing power, nowadays including multicore architectures and GPU’s, does not provide the solution to the ever growing demand for more complex and more realistic simulations. In fact, it has been demonstrated that Moore’s Law, describing the advances in computing power over the last 40 years, equally holds for mathematical algorithms. Hence, it is important to develop both faster computers and faster algorithms, at the same time. This is essential if we wish to keep up with the growing demands by science and technology for more complex simulations. Traditionally, algorithmic speed-ups have come from developments in the area of linear system solution, in which iterative algorithms developed since the 1970’s have been prominent and very effective. Since the start of the new century, however, another powerful development is seen in mathematics, as well as in the systems and control area. This field, which we term ‘model reduction’ for simplicity (we will detail this further in the next section), aims at capturing essential features of models, thereby drastically reducing the size of the problem to be solved. As it holds many promises, this will be the basis for the challenge addressed in this COST Action. |
ee94820eead78004 | principal quantum number
• atomic orbital designation
TITLE: orbital
...of numerals and letters that represent specific properties of the electrons associated with the orbitals—for example, 1s, 2p, 3d, 4f. The numerals, called principal quantum numbers, indicate energy levels as well as relative distance from the nucleus. A 1s electron occupies the energy level nearest the nucleus. A 2s electron, less...
TITLE: transition element: Atomic orbitals of the hydrogen atom
SECTION: Atomic orbitals of the hydrogen atom
...the nucleus occupy the various atomic orbitals available to them. The simplest configuration is the set of one-electron orbitals of the hydrogen atom. The orbitals can be classified, first, by principal quantum number, and the orbitals have increasing energy as the principal quantum number increases from 1 to 2, 3, 4, etc. (The sets of orbitals defined by the principal quantum numbers 1,...
• hydrogen atom
TITLE: spectroscopy: Hydrogen atom states
SECTION: Hydrogen atom states
...atom is composed of a single proton and a single electron. The solutions to the Schrödinger equation are catalogued in terms of certain quantum numbers of the particular electron state. The principal quantum number is an integer n that corresponds to the gross energy states of the atom. For the hydrogen atom, the energy state En is equal to...
TITLE: chemical bonding: Quantum numbers
SECTION: Quantum numbers
Three quantum numbers are needed to specify each orbital in an atom, the most important of these being the principal quantum number, n, the same quantum number that Bohr introduced. The principal quantum number specifies the energy of the electron in the orbital, and, as n increases from its lowest value 1 through its allowed values 2, 3, . . . , the energies of the corresponding... |
0f0d89564cd0e2b9 | About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 362985, 10 pages
Research Article
The Existence and Uniqueness Result for a Relativistic Nonlinear Schrödinger Equation
Department of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, China
Received 3 August 2013; Revised 15 December 2013; Accepted 17 December 2013; Published 16 January 2014
Academic Editor: Bernhard Ruf
We study the existence and uniqueness of positive solutions for a class of quasilinear elliptic equations. This model has been proposed in the self-channeling of a high-power ultrashort laser in matter.
1. Introduction
In this paper, we consider the following quasilinear Schrödinger equation: where , , , and . Solutions of (1) are related to standing waves for the following quasilinear Schrödinger equation: where , is a given potential, is real constant, and and are real functions. Quasilinear equations such as (2) have been accepted as models of several physical phenomena corresponding to various types of ; see [15] for physical backgrounds.
The superfluid film equation in plasma physics has this structure for (see [6]). Putting , where and is a real function, (2) turns into the following equation: where is the new potential function and is the new nonlinearity. In this case, the first existence results are due to [7]. In [7], the main existence results are obtained through a constrained minimization argument. Subsequently, a general existence result was derived in [8]. The idea in [8] is to make a change of variables and reduce the quasilinear problem to semilinear one and Orlicz space framework was used to prove the existence of positive solutions via the Mountain pass theorem. The same method of changing of variables was also used in [9] but the usual Sobolev space framework was used as the working space. Precisely, since the energy functional associated (3) is not well defined in , they first make the changing of unknown variables , where is defined by ODE as follows: and , . Then, after the changing of variable, to find the solutions of (2), it suffices to study the existence of solutions for the following semilinear equation: where By using the classical results given by [10], they proved the existence of a spherically symmetric solution. In [11], the authors give a sufficient condition for uniqueness of the ground state solutions by using the same change of variables as [9].
In the case , (2) models the self-channeling of a high-power ultrashort laser in matter (see [12]). In this case, few results are known. In [13], the authors proved global existence and uniqueness of small solutions in transverse space dimensions 2 and 3 and local existence without any smallness condition in transverse space dimension 1. But they did not study the existence of standing waves. But we have to point out that the method of change of variables as (4) cannot be generalized to treat the case . In [14], the authors made the changing of known variable (see also [15]) and proved the existence of nontrivial solution with and . In this paper, for and , we will show the existence and uniqueness result for (1) by using a change of variables due to [14, 15]. One main difficulty in dealing with this problem seems to be that of obtaining the boundedness of a (PS) sequence for the corresponding functional. We overcome this difficulty by using Jeanjean’s result [16].
Our main result is the following.
Theorem 1. Assume that , , , and . There exists such that if , then the positive solution of (1) is unique.
In this paper, C denotes positive (possibly different) constant, denotes the usual Lebesgue space with norm , , and denotes the Sobolev space with norm .
2. Preliminaries
We note that the solutions of (1) are the critical points of the following functional: Since the functional may not be well defined in the usual Sobolev spaces , we make a change of variables as where . Since is monotonous with , the inverse function of exists. Then after the change of variables, can be written as By Lemma 2 listed below, we have and , so is well defined in and .
If is a nontrivial solution of (1), then for all it should satisfy
We show that (11) is equivalent to Indeed, if we choose in (11), then we get (12). On the other hand, since , if we let in (12), we get (11). Therefore, in order to find the nontrivial solutions of (1), it suffices to study the existence of the nontrivial solutions of the following equation:
Before we close this section, we give some properties of the change of variables.
Lemma 2. for all ;
for all ;
for all ;
for all .
Proof . Since and , so , for ; that is, , for , which proves (1).
Since and is increasing, so properties (2) and (3) are obvious.
For (4), the result is obvious since is an increasing bounded function.
Since which proves (5).
For (6), since is a increasing function, then , which implies that . On the other hand, by (1) and , we get .
3. Existence
At first, we give two Lemmas.
Lemma 3. There exist such that for all .
Proof. Let Then, by Lemma 2 and , we have Thus, for sufficiently small, there exists a constant such that Then, we have Thus, by choosing small, we get the result when .
Lemma 4. There exists such that .
Proof. Given with , we will prove that as , which will prove the result if we take with large enough. By Lemma 2, we have as , so as . Thus, we get the result.
We will use the following Theorem which is due to Jeanjean [16].
Theorem 5. Let be a Banach space equipped with the norm and let be an interval. One considers a family of -functionals on of the form where , for all , and such that either or as . One assumes that there are two points in such that setting there hold, for all , Then, for almost every , there is a subsequence such that (i) is bounded;(ii);(iii) in the dual of .
We consider the functional where .
Let . We find that for all . On the other hand, if , then either , which implies , or ; in this case, to verify that , we start splitting since and by Lemma 2 (6), we have , so so .
For defined above with , using Lemma 4, we get a such that . Also from Lemma 2 we know that as . Thus setting we have, for all , Therefore, using Theorem 5, for almost all , there exists a subsequence such that (i) is bounded in ;(ii); (iii) in .
Lemma 6. Assume that is a bounded Palais-Smale sequence of the functional for . Then there exists a nontrivial critical point of .
Proof. We first note that satisfies and, for any , Since is a bounded Palais-Smale sequence, there exists such that in and in for . By the Lebesgue Dominated Theorem, we have Hence, is a weak solution of (1). If , then we get the result.
Otherwise, if , we claim that for all , cannot occur. Suppose by contradiction that (33) occurs, that is, vanish; then, by the Lions compactness Lemma (see [17, 18]), in for any . Since , then by the proof of Lemma 2, we get which implies that Since and , then On the other hand, note by Lemma 2 (5) that Combing Lemma 2, we have . In fact, we only need to show that ; let ; then by Lemma 2 (5), we have so for and for , which implies that . Thus, since is dense in , by choosing in (31), we deduce that So Combing (36) and (34), we have so we get a contradiction since . Thus, does not vanish and there exist , and such that Define and . Since is a Palais-Smale sequence for , is also a Palais-Smale sequence for with if in . Since does not vanish, we have that is a nontrivial solution of (1).
From Lemma 6, we see that, for almost all , there exists a solution to the following Schrödinger equation: where
Therefore, we can choose such that . Setting , we have . We can deduce that is a solution to (13) if we show that . To prove this, in view of Lemma 6, we first check that is bounded in .
Notice that the Pohozaev identity implies that the solutions of (45) satisfy
Lemma 7. The sequence is bounded.
Proof . Since is a solution to (45) with , by (47), we have which implies that is bounded. On the other hand, together with (41), we have Since is bounded, so is bounded. To verify that is bounded in , we start splitting since so is bounded.
Lemma 8. Assume that , , and . Then (1) has a nontrivial solution.
Proof. The boundedness of in follows from Lemma 7; we have that is bounded in for . Then for any , we have since so as ; thus we have as . By knowing that since so , and we distinguish two cases. Either or . In the first case, we get and the result follows from Lemma 6.
In the second case, we define the sequence by with satisfying (if for a , defined by (56) is not unique, we choose the smaller possible value). By construction is bounded. Moreover by the definition of (56), we have so . Then following the proof above, we have and . On the other hand, by the proof of Lemmas 3 and 2 (6), there exists a constant such that as , uniformly in . Thus, since , there is such that , . Similarly, following the proof of Lemma 3, we have with as . Then recording that , we obtain from (56) that Using Lemma 6 again, we complete the proof of Lemma 8 which implies that is a solution for (1).
Remark 9. In [14], the authors considered the existence of solutions for the following quasilinear Schrödinger equation: where the nonlinearity is Hölder continuous and satisfies the following conditions: if ; as ; there exists such that ; there exists such that for any , there holds .
If we take , , and , (59) turns into (1) with . We point out that the existence result in [14] does not cover our result.
Now, we show that is not satisfied for if . In fact, if and only if By Lemma 2 (5), we have Thus, we only need to show that is under the hypothesis . Then, by (63), we have
Remark 10. In [14], is used to prove the boundedness of (PS) sequence. In this paper, since does not satisfy our condition, we obtain the boundedness of (PS) sequence by using Jeanjean's result [16].
4. Uniqueness
In this section, we study the uniqueness of the positive radial solution of (13). We put We apply the following uniqueness result due to Serrin and Tang [19].
Theorem 11. Suppose that there exists such that (1) is continuous on , on , and for ;(2) and on .
Then the semilinear problem has at most one positive radial solution.
Now we can see that defined in (65) is of the class . Moreover, by the proof of Lemma 2, we have that is increasing and ; then . So ; then there exists a unique such that , and So (1) of Theorem 11 holds. From , we can also observe that . Since is increasing and , this implies that
Lemma 12. Suppose and . Then there exists such that if , then satisfies (2) of Theorem 11.
Proof. We observe that Thus we have only to show that , for . Since so Then by complicated computations, we have where
For , it follows that . Thus it suffices to show that , for , in order to prove that .
By (4) of Lemma 2 and , we have Thus, for sufficiently large , we obtain if and only if for .
Next, we investigate the sign of . Firstly, we express in terms of and , and since , so and Thus we obtain We note that so Moreover, we have , , and as . Then, from , we have so for if is sufficiently large; that is, for . From (68), there exists such that if , then we obtain for .
By Lemma 8, we can apply Theorem 11, Hence we obtain the uniqueness of positive radial solutions of (13).
5. Conclusion
By the discussion of Section 3, we have a nontrivial solution of (1). Then using the result of Gidas et al. [20], we know that the nontrivial solution is a positive radial solution with as and . Combined with the discussion of Section 4, we complete the proof of Theorem 1. That is, if , , , and , there exists such that if , then the positive solution of (1) is unique.
Conflict of Interests
This paper is supported by NSFC (nos. 11201154 and 11371146) and the Fundamental Research Funds for the Central Universities (nos. 2013ZM0112 and 2013ZM0113).
1. A. G. Litvak and A. M. Sergeev, “One dimensional collapse of plasma waves,” JETP Letters, vol. 27, pp. 517–520, 1978.
2. R. W. Hasse, “A general method for the solution of nonlinear soliton and kink Schrödinger equations,” Zeitschrift für Physik B, vol. 37, no. 1, pp. 83–87, 1980. View at Publisher · View at Google Scholar · View at Scopus
3. A. M. Kosevich, B. A. Ivanov, and A. S. Kovalev, “Magnetic solitons,” Physics Report, vol. 194, no. 3-4, pp. 117–238, 1990. View at Publisher · View at Google Scholar · View at Scopus
4. G. R. W. Quispel and H. W. Capel, “Equation of motion for the Heisenberg spin chain,” Physica A, vol. 110, no. 1-2, pp. 41–80, 1982. View at Publisher · View at Google Scholar · View at Scopus
5. B. Hartmann and W. Zakzewski, “Electrons on hexagonal la |
06199a387e9f894f | Wednesday, January 28, 2015
Igpay Atinlay
On page 258 of "The Annubis Gates" by Tim Powers, the hero, Brendan Doyle, is leafing through some ancient manuscript .. when he comes across this message, written in his own handwriting from his future self!
Can you read it?
I had some faint recollection of a child's (or gypsy's) speech code, where everything ends in -AY. After some Google searching I got it: Pig Latin!
There is an urban myth that Google Translate has Pig Latin as one of its languages and I checked: it's not there, but ...
Google in Pig Latin
This site, however, translates from English to Pig Latin - did you know that web becomes ebay? It turns out that translating Pig Latin back into English is hard, and not deterministically possible, as different words in English can map to the same word in Pig Latin (for instance, "oat" and "two" may both translate to "oatway").
Andway ownay erehay isway away ideovay eway ooktay esterdayyay.
* Hi Brendan, can you dig it?
Monday, January 26, 2015
A rather spectral NETGEAR N300 WiFi range extender
Saturday, January 24, 2015
Time to enter the Dark Web?
Here are three (mildly) transgressive Internet links you might or might not care to follow:
1. Recently-deceased Leon Brittan's link to that paedophile ring
2. The Sun's Page 3 website
3. Adolf Hitler's "Mein Kampf"
Let's suppose you clicked on any of the above, who knows you've done it?
Ignoring the person standing behind you, then anyone who clicks "back" on your browser, who looks at your browser history or perhaps who inspects your machine's cookies. You can address this problem, partially, by using private browsing - although any downloads will still be on your machine, and who knows about temp files buried away?
If you had logged into Google, or Amazon, or other website owners, then they certainly know where you went, keep extensive records, .. and could be subpoenaed.
They also know your location. You may be unaware that your browser can run a script asking the operating system for the WiFi SSID you're currently attached to. The big players like Google keep vast databases which link SSIDs with their geographical location: this is how Google Maps magically knows where you are. Hard to stop this happening without disabling scripts, which will stop most websites working.
Even if you were maximally careful on your own machine, your ISP - the provider of your Internet service - keeps a record of your site-visits. It can correlate your personal details (name, address, bank details) with your allocated IP address and link that with the websites you visit.
Normally this is like, who cares? These logs get to Terabyte size and no human scans them. They're expensive to keep and are wiped after some months. But the Government is pushing to legally mandate ISPs to keep these records, on everyone, for at least a year - and make them available to the security services. Is it time to get worried?
If the proposal gets through (and there's a good case for it on anti-terrorist grounds) then everyone can potentially be hoovered-up by a log-searching algorithm. Perhaps one day soon they'll start to care about 'mildly-transgressive' Internet behaviour, and your name will go down on a file somewhere. Between Google's profiling us for targeted advertising, and GCHQ tagging us for subversion, most of us might want to draw a line somewhere.
A common response is to suggest using Internet proxies (eg anonymouse, vtunnel) for any web searches beyond the most anodyne. But these are cumbersome and ad-infested - and who knows what the proxy guys are doing with the correlation between your identity and your surfing information (which they have even if your target sites don't),
The best answer is an Internet VPN service, which unfortunately involves paying some modest fee. Your traffic goes through an encrypted tunnel (eg IPsec) and is proxied at the VPN service provider's Internet breakout point. The rest of the Internet doesn't see your IP address so your web searches appear to come from the VPN service provider; meanwhile your ISP only sees your traffic going to the VPN service provider and has no idea where it's destined for afterwards. It only remains to trust the VPN service provider to not keep your transaction logs for any length of time. When 'The Man' comes asking for the last six months of your usage, there's nothing to show. This is quite a big business for a variety of reasons (watching BBC iPlayer when out of the UK is one) and the market leaders appear trustworthy enough - their business depends upon it.
They tell a good story but I somehow doubt that these VPN service providers can really evade an after-the-fact subpoena. The utility is to prevent speculative trawling.
Do we care enough? Today, probably not .. but it's nice to know we have the option going forwards.
Note: Private Internet Access was named PC Magazine's Editor's Choice in 2013. Read their review.
Colonoscopy Pathology Report
I recently wrote about my colonoscopy experience (in December 2014). Today I received the pathology report: here's the relevant text.
Always good to know you've got to go back, even if it's 2020.
Here's the story on "benign tubular adenoma".
"What is a polyp in the colon?
What is an adenoma?
What are tubular adenomas, tubulovillous adenomas, and villous adenomas?
Thursday, January 22, 2015
Diary: jury service + car paint touch-up + Christmas lights
Clare was meant to be doing jury service this week and next, at Taunton; this is what we had her doing instead.
We're Auris 2007 3J6 (Super Red III)
Driving up the narrow, steep and twisty Old Bristol Road, you get to meet stuff coming the other way and it's kinda inevitable that you gouge a little against those stone walls. That's become the narrative, anyway, despite my complete amnesia on the said event.
Clare turned up in Taunton on Monday after an early start. One of 24, but the trials from the previous week were still ongoing, so all were dispatched back home until Wednesday afternoon. Yesterday they did indeed need a jury, but selecting 12 from 23 (don't ask), Clare suffered the fate of the bottom-of-the-pack card and a poor randomisation procedure. So rejected, she was sent back home again.
There is a final, extremely low-probability opportunity - she has to call again Monday afternoon. By then, however, there will have been a new set of 24 arrivals. There are apparently rare scenarios involving extended jury deliberations when even more jurors are required to keep justice trundling along - we shall see.
All of the above is allegedly, of course.
In other banal chores, we finally took down our outside twinkling Christmas tree lights, which had indicated to a fascinated passing trade that we had hitherto taken leave of our senses.
Sunday, January 18, 2015
"Testament of Youth" (film)
This is what the estimable and amusing Camilla Long had to say about this weepie:
"Testament of Youth is far from perfect, but at least Vera Brittain’s book about her experiences growing up in England as the First World War looms is a decent starting point. This is a slightly shameless attempt to capitalise on last year’s anniversary, but on the whole it’s fairly good stuff.
"Brittain is played by the Swedish actress Alicia Vikander, who is pink and earnest, but never quite manages to be not Swedish. She also looks almost identical to Pippa Middleton, especially in a scene at the end where she poses in a nightdress exactly like the famous Bum Dress from the royal wedding. Unlike Middleton, however, she is principled and furious. She spends a lot of time being angry about really Edwardian things, like pianos.
"In the opening scene, she is horrified that her father (Dominic West) is happy to buy her a piano — amazingly not pronounced pihano — but not a place at university. (Health warning here: this is a film where everyone talks about Oxford all the time. Oxford this, Oxford that. It makes me want to vomit. I can’t work out what is this film’s worse fate: dying in the trenches or not being able to go to Oxford.)
"As it happens, Brittain only wants to go to Oxford out of sheer boredom. She dumps her place almost as soon as her brother and her fiancé, Roland (Kit Harington), sign up for the war. She is deeply in love with Roland. I know this because they meet a) in porticoes and b) amid drying laundry, and c) she doesn’t honk with laughter when he actually tries to fly a kite. Yes, this is a film that uses kites as a metaphor for love. It is the film Downton really longs to be, literary and bluestockingy and full of clichés about “big pushes” and Spanish flu and phone calls that ruin tea.
"Harington is moistly beautiful as Roland, sending Vera poems and announcing, stiltedly, that he has decided not to go to Oxford. I think one of my out-and-out ultimate fantasies is Kit Harington standing in a forest wearing white trousers and shouting “YOU MUST WRITE”. In that sense, at least, this film did not disappoint.
"This is what I call an all-orifices film: there’s bromance, romance, weeping and an awful lot of slushy clucking around field hospitals. There’s a superb cameo by Hayley Atwell as a nurse looking after “filthy Huns”. I could have spent two hours watching a bustling Atwell maliciously changing some jabbering Bavarian’s bedpan. This looks like yet another weepy teatime film. But it’s better than that, and Vikander makes a great Keira Knightley."
Here's Vera Brittain (Alicia Vikander, who as noted is from Sweden).
And here's the girl who broke up the (Time Team) band, Mary-Ann Ochota:
So that confused me for a while.
Vera was consumed by grief at twenty minute intervals as her fiancé Roland, male friend Victor and gay younger brother Edward were successively killed by the Hun (all three seemed rather dim to me). Vera does that teary, trembly-lip thing beautifully except you keep thinking: 'acting'. And then towards the end she demonstrates her all-consuming grief by art-house tropes such as decorously sliding into a freezing Buxton pond, and anguishedly smearing herself with freezing Buxton mud on the moors. I tend to imagine that the searing cold and generally unhygienic nature of these emotional excesses would bring a body down to earth pretty rapidly - but what do I know of the searing passions of a feisty 25 year old?
Vera returns to Oxford as the film ends, about to transmogrify herself into a committed pacifist and emotion-charged writer about private loss. Thoughts of self-indulgence briefly passed through my mind as I headed for the exit.
Saturday, January 17, 2015
Pretentious, moi?
XKCD: too good to be lost as an ephemeron
Years ago, I had a car sticker which read:
"Another Family for Situation Semantics"
I was delighted that no-one had any idea what this pretentious sentiment actually meant. Now I can reveal the extremely tedious truth.
Situation semantics was a non-standard logic developed by Jon Barwise and John Perry in the early 1980s at Stanford University. It was an attempt to create a better semantics for natural language than the more conventional Montague Semantics, by making the model-elements contextually-restricted 'situations' rather than whole worlds. I wrote it up as part of my Ph.D work but it was not central, as it did not lend itself to computational inference. In any event, world-wide interest subsequently slumped.
And families? In best West Coast tradition, their book was written in an irritatingly folksy style, with plenty of examples using 'family situations'.
In retrospect, I cringe.
The Economist this week
In The Economist this week, Schumpeter has a knowing piece about how to successfully network:
"The first principle for would-be networkers is to abandon all shame. Be flagrant in your pursuit of the powerful and the soon-to-be-powerful, and when you have their attention, praise them to the skies. ... "
At Schumpeter I merely cringed along with the columnist. My blood boiled, however, at the first science article: "University Challenge", a tendentious piece of wish-fulfilment fantasy with dodgy methodology, misleading and unconvincing graphs, no correlation coefficients and in one diagram no regression line, and a wholly unconvincing, nay stupid, conclusion. All leavened with a deep ignorance of the underlying science combined with the promulgation of lazy fallacies and an entrenched gullibility.
I think it's fair to say I was unimpressed!
The view down our road this chilly morning
Thursday, January 15, 2015
Predicting IQ across the world from genotypes
Early days for this - I wrote about it last November, where I tried to 'predict' my own IQ. Now a much better article has appeared, written by Anatoly Karlin. Interesting stuff, highlighting the ground-breaking research of Davide Piffer.
Wednesday, January 14, 2015
Obesity genes and me
The BBC science programme Horizon is currently running a three part series on the science of dieting. They have identified three categories of the obese and one - the 'constant cravers' - are defined by having 'obesity genes'.
It seems likely that I'm a 'constant craver'.
A little internet research provides a short list which can be cross-correlated with my 23andMe genotype download.
1. The FTO gene
As Wikipedia explains: "In 2009 variants in the FTO gene were further confirmed to associate with obesity in two very large genome-wide association studies of body mass index (BMI). It was shown that adults bearing the at-risk AT and AA alleles at rs9939609 consumed between 125 and 280 Calories per day than those carrying the protective TT genotype," (c. 5-12% of the daily allowance).
A quick search of my Excel spreadsheet for rs9939609 confirms I'm AT at this location. No wonder I was thirteen and a half stone before starting the 5:2 diet (I'm now at 11 stone = 70 kg but not without continuing maintenance). As a carrier of one of the 'A' risk alleles my disposition to obesity is 30% higher than that of baseline TT people.
2. The MC4R gene
"Mutations in the MC4R gene account for 6-8% of obesity cases. A common variant of the MC4R gene, distributed in about 22% of the population, increases the risk for weight gain by causing increased appetite and decreased satiety. Calorie restriction through portion control and smart food choices is the best strategy for weight loss for people carrying this variant."
The relevant SNP is rs17782313 where C alleles are associated with higher body mass index (BMI). The three options are CC, CT, TT - where TT is baseline normal, CT is associated with a BMI increase of 0.22 units and CC with a BMI increase of 0.44 units. As is so often the case, the allele effects are, as you see, additive.
What am I? Yep, it's bad: CC.
3 The ADIPOQ gene
The relevant allele is rs17366568. "A significant genotypic association was observed between ADIPOQ rs17366568 and obesity. The frequencies of AG and AA genotypes were significantly higher in the obese group (11%) than in the non-obese group (5%) (P=0.024). The odds of A alleles occurring among the obese group were twice those among the non-obese group (odds ratio 2.15; 95% confidence interval 1.13-4.09)." (From here).
At last some good news! I am GG at this location.
Doubtlessly I'll return to this topic when more is known, especially as the results to-date are so personally depressing!
In the deep midwinter ...
A small dusting of snow
The sun's now out and it's rapidly melting.
This afternoon we went down to the Wells Film Centre to watch the film "Testament of Youth" (Vera Brittain). But I had erred! It's not on until next week. So we arrived back early and I checked the vole trap by the side of the fridge in the kitchen. Why?
The cat had been behaving slightly oddly, patrolling with interest behind the fridge, and this morning I saw a brown blur speeding in that general direction as I entered the kitchen. "We've got a vole," I confided to Clare, "Where's the vole trap?"
After a false start with cat food, we discovered that voles particularly adore bread-and-butter and oatmeal. So suitably prepared, we left for the movies.
The vole we'd trapped was alert and boisterous, and has been released into the garden where it can vole anew. Next time a video!
Matthew Parris (mouse killer!) take note!*
* Opinion piece from The Times today (14/01/2015)
"I hate poisoning animals. Unlike their London cousins, Derbyshire mice are suckers for the traditional mousetrap so I baited two traps with Nutella and sorrowfully set them in the airing cupboard. I flinched next morning from checking. I hoped against irrational hope they would be empty. I opened the door. My heart sank. Both had sprung.
"Sadly I carried the small corpses to the dustbin. One — a mother — was a really beautiful little honey-brown creature with (unusually) a white breast. Her blind, pink babies (up to 12) would already be dead."
"I miss them, and somehow think the less of myself."
Yes indeed, Mr Parris!
Tuesday, January 13, 2015
The Collapse of Democracy
Back in the late sixties, my Politics course at Warwick University taught that the democratic state acted as an arbiter between different sectional interests. My Marxist comrades knew better: the state actually operated to reproduce the power and position of the ruling bourgeoisie, while hiding behind an obfuscated, hegemonic ideology.
Yes, we certainly knew how to do jargon in those days!
Of course, both propositions are true. Marxists from Karl onwards have agreed that bourgeois democracy is the preferred form of capitalist state. Why? Because under capitalism, economic power is decentralised (private ownership of the means of production) so some kind of inclusive politics is the best method of synthesising overall political policy. If the state achieves the political autonomy of autocracy or dictatorship we have the familiar principal-agent problem. How do we get the state to properly advance the (weighted average of the) interests of the distributed capitalist power-elite? How do we stop the state going off on some crazy project of its own?
The Nazis in Germany are the usual case study, and my analysis above broadly paraphrases Trotsky's writings about the rise of fascism there.
The democratic government is distinguished from its dictatorial cousins by its unwillingness to decisively back one faction of society over everyone else, even if such a focussed policy is objectively necessary to break some social logjam. "We all know what has to be done; it's just that none of us knows how to be re-elected afterwards."
Bourgeois democracy is like pacifism - it's an unstable equilibrium requiring all sides to show restraint and be prepared to accept being overruled. It's when a significant social force won't accept compromise and sticks to its guns come what may that you get the logjam. The inclusive speech of liberal politicians becomes strained and ineffectual - weak hand-wringing and appeasement. The logjam-party takes heart while ordinary folk begin to despair. Oppositional parties calling for effective action begin to gain traction, parties which don't much care about discredited 'democratic' ideals. What if we're rather blasé about being re-elected afterwards, anyway - or we believe that subsequent 'facts on the ground' will make all the difference, come the day?
Returning to the party-of-the-logjam, there's nothing like a sharply defined and highly-deprecated religious identity to underpin a hard-nosed refusal to compromise under any circumstances: 'our martyred dead' and so forth. You can see where this is going: bourgeois democracy can handle small to medium logjams by uniting the majority and deploying state force against the obstructionist, unyielding minority and winning - Margaret Thatcher is the textbook example. But if the logjam gets too big and/or intractable, you slide into civil war (cf Libya) and the democratic state is swept aside and is transformed, or collapses.
None of these drastic things will be happening in Western Europe any time soon; we're at the very start of a long, tortuous and only semi-slippery slope. However, to mix metaphors, when your problem is currently a small but extremely intractable hole, it's surely time to stop digging?
After the Apocalypse
The worst way for the world to end is global thermonuclear war ... because of the after effects, particularly the radiation, obviously. A large asteroid strike is nearly as bad. The third worst way, surprisingly, is the impact of a large solar Coronal Mass Ejection. This would wipe out the power grid, including the transformers; in the absence of any kind of power the transformers themselves could not be fixed so everything depending on electricity would crash - including the economy.
The problem is that our current population in England of around 53 million is sustained by our
technological base. Knock this back and we revert to the carrying capacity of the Domesday book period (around one million). If agriculture fails, however, we revert to hunter-gatherer status .. just ten thousands individuals in a country the size of England!
In the catastrophes above, trashing the infrastructure largely leaves the population intact. They fight viciously and starve over the next months, consuming much needed resources and wasting the period of grace before many supplies become unusable. This is why the 'best apocalypse' is more like a souped-up version of Ebola or The Black Death: a pandemic which is aggressively virulent, has a long incubation period (for maximum infectivity) and near 100% subsequent mortality. Yes, our civilization will crash, but the infrastructure will not be too damaged in the process.
And then you'll need Lewis Dartnell's book "The Knowledge: How to Rebuild our World from Scratch" from which the apocalypse palette above was taken.
Dartnell, a prolific science writer, organises his recovery material under the major themes of mediaeval sustenance: agriculture, food and clothing, materials (clay, lime, acids, nitrates, metal-working), medicine, power, transport and communications. There's not enough detail for anyone to actually construct (for example) a working plough - but at least we townies are told how it actually works, and what its function is - and that it therefore has to be on the list.
Well-written and full of interesting little snippets as this book is, reading it is to be reminded anew how precarious our comfortable lives actually are. If the ATMs stopped and the supermarkets failed, how scarily different things would be, and how quickly!
Sunday, January 11, 2015
When Sharia comes to town ...
Or it might just be a touch chilly ...
Nous sommes tous Charlie
In my Marxist days, I would have been penning articles about the bourgeois hypocrisy of leaders such as Hollande, Cameron and Merkel marching piously in Paris in defence of political correctness. I would have pointed out that the crass, obscene and unfunny cartoons of Charlie Hebdo posed no threat to the established order, despite the professed '68 Marxism of the authors, as the establishment never believed in any of the propositions lampooned in the first place.
In the spirit of the new diversity, the fact that apparently we are now "all Charlie", let me outline three excitingly innovative views of the recent events in Paris. Naturally, these accounts are designed both to be true and to offend.
1. The Physicist's view
Various atoms and molecules in the Paris area recently continued to conform to the predictions of the Standard Model of Quantum Mechanics and General Relativity. As expected.
2. The Psychopath's view
Various biological machines in the Paris area with conflicting goals came into conflict. Some of the machines were destroyed.
3. The Jihadi's view
Do you think I'm crazy?
Sigh: one is not meant to explain one's work .. but: (1) is meant to get you thinking about the nature of 'free will' in all this; (2) is meant to highlight the methodological 'dispassionate' approach of rational science vs. emotionalism-altruism-empathy in human affairs. As for (3), I think we've had enough secularists floundering to represent to the world at large a community which organises itself (in contradistinction to a society ordered by abstract principles and the law) as an 'honour culture'.
Saturday, January 10, 2015
Roomba's first outing (Vimeo)
Just playing with Vimeo. Back in 2007 I bought Clare a Roomba for Christmas. This rather primitive video (camera phones were so primitive back then!) shows its first outing in our bedroom, from our house in Andover, Hampshire. How nostalgic, looking back to the days when we were Roomba-naive!
Two fuzzy notions made crisp
1. Family and Friends
Family is easy; your kin group which is defined and preferred through inclusive fitness. Friends corresponds to that circle of individuals with whom one practices reciprocal altruism (qv). Since reciprocal altruism requires trust extended over time, it's not surprising that friendship tends to be psychologically regulated.
"According to Trivers, the following emotional dispositions and their evolution can be understood in terms of regulation of altruism.
• Friendship and emotions of liking and disliking.
• Moralistic aggression. A protection mechanism from cheaters acts to regulate the advantage of cheaters in selection against altruists. The moralistic altruist may want to educate or even punish a cheater.
• Gratitude and sympathy. A fine regulation of altruism can be associated with gratitude and sympathy in terms of cost/benefit and the level in which the beneficiary will reciprocate.
• Guilt and reparative altruism. Prevents the cheater from cheating again. The cheater shows regret to avoid paying too dearly for past acts.
• Subtle cheating. A stable evolutionary equilibrium could include a low percentage of mimics in controversial support of adaptive sociopathy.
• Trust and suspicion. These are regulators for cheating and subtle cheating.
• Partnerships. Altruism to create friendships."
2. Birds vs Frogs (or Lee Smolin's seers vs. master craftsmen)
Freeman Dyson calls mathematicians who take a lofty conceptual view of their subject birds and those who work in details and solve their problems consecutively frogs. Smolin has a similar division in mind for theoretical physicists.
The crisp distinction is between deduction and abduction. Deduction draws consequences from theories and boundary conditions - and is the home territory of the master craftsmen and frogs; abduction is the creative synthesis of the most parsimonious and elegant theory which can be conjured up to explain the available data - the business of seers and birds.
(Your author, to tell the truth, has always felt more avian).
Thursday, January 08, 2015
Charlie Hebdo .. and internal colonies
France, like the UK, has lots of Muslims - probably in excess of 5% of its population (exact numbers are not counted by the rigorously secular state). The Muslims tend to derive ethnically from France's ex-colonies in North Africa. As many of them are geographically concentrated, they constitute internal colonies.
Islam, like Judaism and Christianity, started out as a religion of pastoralists. Lots of references in the key texts to shepherds, and sorting out sheep from goats. Christianity was refashioned into a docile religion for the subjects of empire (another Roman achievement) while those same Romans dispersed the Jews after AD 70 and 135 so that their subsequent history was mostly within the empires of others - pastoralism was then not really their thing.
Islam, not so much.
What do we know about pastoralists? They keep animals (cattle, horses) in environments which don't support farming. Cattle are easy to steal and relatively low-maintenance. If you lose your cattle you don't get to have many descendants, so pastoralists are a might touchy about showing personal weakness, do not respond well to slights and have been known for their endemic blood feuds.
The more cattle you have, the wealthier you get so there are returns to scale. However, keeping the pastoralist "empire" together is tricky: it's just too easy for one clan or tribe to steal off the others. Islam is helpful to pastoral empire-builders, a 'glue-religion' which creates a framework for inter-tribal cooperation under a common deity and set of rules. However, as a cursory examination of Islamic world history shows, even a binding religion can't suppress the inherent centrifugal forces of pastoralism. And it always ends in violence.
So not all internal colonies are equal. A Chinese internal colony would, no doubt, be industrious, non-violent and successful; ditto the Jewish equivalent, historically the ghetto. History supports such observations. Again, Islamic internal colonies .. not so much.
This is not fundamentally a discussion at the level of values: free speech is pretty abstract and always circumscribed; it's about the norms and protocols of the way we organise society, the ways we relate to each other. People sometimes model social organisation using the game-theoretic model of "hawks vs. doves". The secularised West is ideologically a "dove" culture; Islamic culture, following its pastoral roots, is in most variants "hawk".
In the absence of any easy answers*, I would predict selective and enhanced surveillance and containment are going to be the outcome of all of this, across Europe.
* So what are the alternatives? Historically, socially-deprecated internal colonies have either been expelled (the Jews pretty much everywhere, the Muslims in Spain) with much suffering - or forcibly assimilated. The two policies were often pursued simultaneously.
Naturally any such operation in contemporary Europe would be met with the most ferocious and violent opposition from the said internal colony, together with a collapse of social cohesion on the part of the state attempting the ethnic cleansing.
That is not to say that it hasn't happened within recent history (Balkan wars, Soviet Russia under Stalin, the Nazis): just that a liberal democratic state can't do it. **
** That's not to say we won't get gradualist or 'at-the-margins' versions of these policies advocated by populist parties amongst others: a kind of 'Reconquista-lite'.
Permanent jobs are much better (still!)
Congratulations to Adrian on getting his first permanent job (in finance) yesterday. Finance, along with marketing, is one of the two core experience-areas for career progression so .. result!
Wednesday, January 07, 2015
"The Book of Strange New Things" - Michel Faber
Just finished reading "The Book of Strange New Things"by Michel Faber to my wife, Clare, in half-hour nightly chunks. Here is what The Guardian made of it:
"Beatrice Leigh is a nurse, an evangelical Christian, a cat owner and “an independent and capable woman”, not necessarily in that order. She lives in a Britain perhaps not so far in our future, in which “institutions that have been around forever are going to the wall” and a collapsing economy and deteriorating climate have become indices for one another. It would be easier for Bea if she had her husband Peter’s support, but he can’t help: he’s trillions of miles away, on a planet called Oasis, with a mission to convert its alien inhabitants. The conversations of Bea and Peter, which scaffold Michel Faber’s astonishing and deeply affecting sixth novel, are held via a kind of interstellar email. The awkwardness of this medium amplifies to screaming pitch our sense of the emotional space between them. “Sometimes,” she tells him angrily, “I feel as though your leaving caused things to fall apart.”
"Peter, meanwhile, finds it hard to focus on anything but his situation. The jump between worlds causes him to hallucinate. Oasis is too much to take in. His mission is financed by and carried out under the auspices of a shadowy corporate called Usic. They need him but won’t say why. The base personnel describe themselves as “a community”, “in partnership” with the indigenous population – “we do not use the word ‘colony’ ”. Yet many of them specialise in oil and mining technology, and Usic is already building infrastructure to support a larger population. Trade has begun, although it has taken a weirdly localised form: the Oasans produce food for the human settlement; in return, they seem to want only Earth analgesics and the Bible, the eponymous “book of strange new things”. On being shown a picture of Peter’s pet cat, they ask if it’s a Christian. When he tells them that, though he loves it anyway, the cat can’t be a Christian because it’s an animal, they respond: “We also love those who have no love for Jesus. However, they will die.” Finding a way through these mysteries requires Peter, whose Christianity is never presented as less than honest, to identify and dismantle his own deep temperament – avoidant, confused, manipulative, mistaking obsession for commitment.
"Like every fiction of Faber’s, The Book of Strange New Things is determined not to be mistaken for any other fiction written by Faber. At the same time, it’s difficult to read the description of an alien face as “a placenta with two foetuses – maybe three-month-old twins, hairless and blind – nestled head to head, knee to knee”, or an energy-saving light bulb as “a segment of radioactive intestine suspended from a wire”, without remembering the hallucinatory intensity of work such as Under the Skin. Oasis is a strange world, half paradisal, half dull, prime real estate for the imagination realised with determined sensuality. The atmosphere is full of “the sound of agitated leaves”, although there are very few leaves anywhere. The rain tastes sweet. The Oasans always wear gloves, and hooded pastel-coloured robes made of a fabric “disconcertingly like bath towel”. When they try to pronounce an “s”, they make the noise of “a ripe fruit being thumbed into two halves”.
"This is a big novel – partly because it has to construct and explain its unhomely setting, partly because it has such a lot of religious, linguistic, philosophical and political freight to deliver – but the reader is pulled through it at some pace by the gothic sense of anxiety that pervades and taints every element. Earth is becoming untenable. The more he feels at home with the Oasans, the more guilt Peter feels at abandoning his wife. The Usic personnel – who think of themselves as outcasts, members of a foreign legion – seem self-repressed to the edge of explosion. The Oasans, with their inexplicable faces and obsession with sharp objects can’t, surely, be as simple, gentle and fragile as they seem. And has their language, literalistic to the core, caused them to make a basic mistake about the Christian promise of eternal life? Even the planet’s low-diversity ecology seems to harbour some tension in need of resolution. The reader is desperate for relief, which can only come from turning another page, and then another and another.
"In Peter’s quarters at the Usic base, he finds “a red button on the wall labelled EMERGENCY, but no buttons labelled BEWILDERMENT”. Equally lost in the wild, dragged on by a mounting sense of urgency, we dread some upshot both ironic and gruesome: but while its surface finds the comic in everything from corporate architecture to the communication of taken-for-granted religious concepts, the deepest levels of the book privilege directness over irony. What you see is what you get: humans and aliens patiently trying to dismantle the very concepts of human and alien; making contact, making the best they can of a bad job. “We need a certain proportion of things to be OK,” Bea tells Peter, “in order to be able to cope with other things going wrong.” Perhaps that’s all we can ever hope for.
"Meanwhile, we have their letters, full of heartbreaking chat and a growing anger on her side, and on his a kind of restless evasiveness as he tries to find her life as interesting as his own. He misses her desperately, but he’s charmed and overwhelmed by all the strange new things; alone with everything they used to handle as a couple, she’s increasingly frustrated and desperate. The tragedy is that while we know that, Peter doesn't. If he spends the novel lagging behind the edge of the present, Bea spends it trying to stay ahead. She’s less concerned with understanding than keeping her head above the water. History is happening too fast and too completely for them. But what begins on Oasis must end on Earth and if Peter sets out as a holy fool, God requires him to finish as Orpheus. “I hear rain again. I love you and miss you. Don’t worry about anything.”
Here is what I see. A writer - Michel Faber - whose beloved wife and collaborator, Eva, is dying of cancer while he can only look on impotently. His anger and frustration channelled into a novel about a weak, superficial and fickle man who indulges his evangelical Christianity on Oasis as a psychological crutch while his pregnant wife is abandoned to global civilisation collapse back on Earth.
Faber is the ultimate 'show not tell' writer, so what might at first sight appear a rather low-key travelogue is in reality a quietly furious critique of institutional stupidity and personal inadequacy.
Saturday, January 03, 2015
"Proxima"/"Ultima" - Stephen Baxter
From the review at The Bookbag.
" In Proxima, alien hatches were discovered across the galaxy, hatches that when opened caused completely unimaginable events to occur - amongst many strange happenings, one character suddenly had a twin she didn't have previously , and one hatch led to a different earth, where the Roman Empire never died.
"It is there that Ultima begins - on a world where the Roman Empire never fell, and the technology and culture is markedly different as a result.
"This world is explored in length. Great length, and to be honest, reading about it begins to feel like quite a chore. Proxima was heavily character driven, and I went into Ultima hoping for more exploration of Yuri and Stef, but instead had to read what felt like a science fiction history book, with little tension, drive or excitement to make me want to keep turning the pages. It's a very well written exploration of a strange new culture, but given that the last book left me invested in characters and gripped to find out what would happen, I was disappointed to have to slowly read through a section that did absolutely nothing to thrill, entertain or challenge me.
"Thankfully, things do pick up - characters begin to become vivid again, and the story picks up such a pace that the finale is genuinely quite staggering - and I concede that it did make reading the book, especially the lackadaisical first two or so hundred pages, worthwhile. In addition, the ideas and concepts that Baxter is dealing with, as well as a diverse cast of voices with which he tells his story, really do come together to make for a fantastic story in the end."
Well, I beg to differ. I read these two blockbusters back to back hoping for some transcendent plot-revelation .. which failed to ever materialise. It seems that there is some Stephen Baxter conservation law:
Characterisation + Setting + Plot = constant.
Setting-wise, Baxter recycles his Roman Empire studies from previous books together with lots of scientific extrapolation about planets circling red dwarf stars. This subtracts from his characterisation, his menagerie of black-and-white-delineated characters who are merely activated stereotypes (marionettes might be a better word). The plot is all that keeps the pages ticking over, but in the end that reduces to just another tedious multiverse fantasy.
So sorry, couldn't recommend these two. Use the sparse hours of your life more productively.
Friday, January 02, 2015
The binary million
Today is the first and only time in my life I celebrate my binary millionth birthday. Naturally I got a nice card, helpfully re-purposed, from my wife.
The cover of my card
The greeting inside
I always say it's the thought that counts, don't you?
Here's the streaming video of the traditional ceremony of present-giving, in which I am eternally doomed to both joy and disappointment. Be warned that it's 800 MB with c. 6.5 minutes run-time. Sorry about the quietness of the sound - the humour, such as it is, is back-loaded.
Remember: absolutely no-one is making you watch home videos! No homework or future test involved.
Thursday, January 01, 2015
Dreamers and Doers
Sean Carroll has a guest post by Chip Sebens on the Many-Interacting-Worlds Approach to Quantum Mechanics. Here's the first part of it.
"In Newtonian physics objects always have definite locations. They are never in two places at once. To determine how an object will move one simply needs to add up the various forces acting on it and from these calculate the object’s acceleration. This framework is generally taken to be inadequate for explaining the quantum behavior of subatomic particles like electrons and protons. We are told that quantum theory requires us to revise this classical picture of the world, but what picture of reality is supposed to take its place is unclear. There is little consensus on many foundational questions: Is quantum randomness fundamental or a result of our ignorance? Do electrons have well-defined properties before measurement? Is the Schrödinger equation always obeyed? Are there parallel universes?
"Some of us feel that the theory is understood well enough to be getting on with. Even though we might not know what electrons are up to when no one is looking, we know how to apply the theory to make predictions for the results of experiments. Much progress has been made―observe the wonder of the standard model―without answering these foundational questions. Perhaps one day with insight gained from new physics we can return to these basic questions. I will call those with such a mindset the doers. Richard Feynman was a doer:
-Feynman, The Character of Physical Law (chapter 6, pg. 129)
"In contrast to the doers, there are the dreamers. Dreamers, although they may often use the theory without worrying about its foundations, are unsatisfied with standard presentations of quantum mechanics. They want to know “how it can be like that” and have offered a variety of alternative ways of filling in the details. Doers denigrate the dreamers for being unproductive, getting lost “down the drain.” Dreamers criticize the doers for giving up on one of the central goals of physics, understanding nature, to focus exclusively on another, controlling it. But even by the lights of the doer’s primary mission―being able to make accurate predictions for a wide variety of experiments―there are reasons to dream:
“Suppose you have two theories, A and B, which look completely different psychologically, with different ideas in them and so on, but that all consequences that are computed from each are exactly the same, and both agree with experiment. … how are we going to decide which one is right? There is no way by science, because they both agree with experiment to the same extent. … However, for psychological reasons, in order to guess new theories, these two things may be very far from equivalent, because one gives a man different ideas from the other. By putting the theory in a certain kind of framework you get an idea of what to change. … Therefore psychologically we must keep all the theories in our heads, and every theoretical physicist who is any good knows six or seven different theoretical representations for exactly the same physics.”
-Feynman, The Character of Physical Law (chapter 7, pg. 168)
"In the spirit of finding alternative versions of quantum mechanics―whether they agree exactly or only approximately on experimental consequences―let me describe an exciting new option which has recently been proposed by Hall, Deckert, and Wiseman (in Physical Review X) and myself (forthcoming in Philosophy of Science), receiving media attention in: Nature, New Scientist, Cosmos, Huffington Post, Huffington Post Blog, FQXi podcast… Somewhat similar ideas have been put forward by Böstrom, Schiff and Poirier, and Tipler.
"The new approach seeks to take seriously quantum theory’s hydrodynamic formulation which was developed by Erwin Madelung in the 1920s. Although the proposal is distinct from the many-worlds interpretation, it also involves the postulation of parallel universes. The proposed multiverse picture is not the quantum mechanics of college textbooks, but just because the theory looks so “completely different psychologically” it might aid the development of new physics or new calculational techniques (even if this radical picture of reality ultimately turns out to be incorrect)."
Click here for the rest of it.
The essential mystery of quantum mechanics is that the theory is built around the dynamics of a thing called the wave function (hence wave mechanics), conventionally labelled ψ. The value of the wave function at each point in space and time is given by the solution to the Schrödinger equation (with appropriate boundary conditions): you imagine the ψ wave flowing around obstacles, through slits, and interfering with itself. The trouble is, the wave function is (apparently) not a 'real entity'. For one thing its values are complex, not real (all observables are real numbers); for another, in its multi-particle mode, the wave function lives in an arbitrarily high-dimension space called configuration space, not our conventional 3 + 1 dimensional space-time.
The wave function, as mentioned, is not itself observable. But if you square the value of the wave function (e.g. in a region of space at a point in time) you get the probability of observing the attribute-value of your interest (e.g. the probability of finding the particle in that region at that time).
The theory is incredibly accurate in giving you the correct probabilities; but it does not tell you what reality is actually doing. About that, quantum mechanics is not just silent - it informs you that your prior beliefs about the world consisting of well-defined particles with defined positions and momenta cannot be true (Bell's theorem).
The doers get on and calculate .. and design the modern technological world; the dreamers wonder whether there is completely non-obvious way to reconstruct the world of appearances ('reality') such that (relativistic) quantum mechanics turns out to be true in that structure of reality.
To date, no-one ever quite succeeded. Maybe Chip Sebens is onto something; maybe the Everett many-worlds formulation of quantum mechanics (still a work-in-progress) can be made to work.
It is my birthday tomorrow (I've reached binary one million) and I expect a present which will shed further light on these perplexing issues. |
25a57bf1b38e9a39 | tisdag 20 januari 2015
Funeral of Schrödinger's Cat in Sweden
Swedish physics professors Karl Erik Eriksson and Bengt Gustavsson performed a symbolic academic funeral of (i) Schrödinger's cat along with (ii) multiversa and (iii) probabilistic dice interpretations of quantum mechanis, in a worthy ceremony at the Alma-Löv art museum in Värmland in the heart of Sweden on November 20, 2014.
I fully agree with these professors of physics that the modern physics of (i)-(iii) is dead and that the funeral thus puts an end to three tragic episodes of physics, as the Queen of sciences and tremendous success, see post of Jan 16.
From the ashes a new form of quantum mechanics may emerge, maybe in the form of a physical quantum mechanics based on a second order real-valued Schrödinger equation without cat, dice and parallel worlds, as discussed in recent posts.
Recall that the reason to introduce the dice leading to the cat and parallel worlds, was that the standard first order complex-valued form of Schrödinger's equation, does not describe any physics.
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ed73aa1d278fec3a | Research Highlights : Physics
The indecisive insulator
04 January 2008 (Volume 3 Issue 2)
Researchers are applying relativistic quantum theory to explain how graphene could switch from a metal to an insulator
Figure 1: In two dimensions, the electronic energy band in graphene follows a cone-shaped distribution, similar to the behavior of relativistic massless Dirac fermions.
enlarge image
Copyright © Akira Furusaki 2007
Graphene, which consists of single sheets of carbon atoms peeled off graphite, has recently been fabricated for the first time. Graphene has unusual electrical properties that originate from the unconventional manner in which its electrons behave. A team from the University of California, the Paul Scherrer Institute in Switzerland and the RIKEN Discovery Research Institute in Wako are gaining insight into graphene by expanding the quantum theory for relativistic particles1.
Electron transport in solids is usually non-relativistic and governed by the Schrödinger equation. However the electrons in graphene effectively behave like massless relativistic particles, which are described by a Dirac equation. This means that in two dimensions the electronic energy band is cone-shaped (Fig. 1), and gives graphene the potential to switch from a conducting metal to an insulator.
“Electrons are scattered randomly by impurities and defects in a solid,” explains project-member Akira Furusaki from RIKEN. “When such scattering happens sufficiently frequently, electrons become localized in a finite region and cannot propagate over a distance. This phenomenon is called Anderson localization.”
During Anderson localization, the wavefunction—or probability distribution of different states—of an electron is very narrow in space. If all the electrons in a solid are Anderson localized, the solid is an insulator. In contrast, electrons in a conducting metal are free to move, having wavefunctions extended over the entire system.
Furusaki and co-workers extended an aspect of quantum field theory called the nonlinear sigma model to examine Anderson localizations in graphene. The model is defined whenever electrons move by diffusion, and has been a standard tool to describe transport properties of electrons in disordered solids.
The researchers discovered that when the nonlinear sigma model is used to describe the transport of two-dimensional Dirac electrons in a random electrostatic potential, a topological term is required in the mathematical formulation (at the same time, a German and Russian team reached a similar conclusion independently2).
The topological term arises from Majorana fermions—theoretical particles that are their own antiparticles—originating in the theory of the Anderson localization in graphene. “The presence of a topological term can change low-energy (long-distance) properties of the model drastically and is responsible for metallic transport in graphene,” says Furusaki.
In future the researchers hope to generalize their theory to three dimensions. “We also plan to examine other systems such as disordered superconductors,” says Furusaki, “in which the transport of low-energy quasiparticles may be highly complex.”
1. Ryu, S., Mudry, C., Obuse, H. & Furusaki, A. Z2 topological term, the global anomaly, and the two-dimensional symplectic symmetry class of Anderson localization. Physical Review Letters 99, 116601 (2007). | article |
2. Ostrovsky, P.M., Gornyi, I.V. & Mirlin, A.D. Quantum criticality and minimal conductivity in graphene with long-range disorder. Physical Review Letters 98, 256801 (2007). |
cd60b63eabf02f10 | Notable Names: Richard Feynman
What defines genius? Real genius, not just the smart kid in the back of the class with all the answers. People like Galileo, Da Vinci, Einstein. The brilliant minds that take standard concepts, turn them upside down, and show us exactly why it never made such sense to us before. They take two dimensional images, and show us three dimensional truths.
Feynman, explaining something cool.
Or in the case of Richard Feynman, they take the most basic bits of the universe, and give us quantum electrodynamics. Feynman was a brilliant mathematician and physicist, and arguably one of the greatest science lecturers of all time. Let’s delve for a bit, via Feynman, into the wacky, weird world of energy: the stuff everything you have ever known or interacted with (including yourself, and this computer screen!) is composed of.
Now, I’m no physicist, but listening to Feynman’s lectures and interviews motivates me to learn more about the big majestic mystery of our physical universe. Born in 1918 in New York, Feynman was an intelligent student who had mastered differential and integral calculus by the time he was 15. He was turned away from Columbia University before being accepted at the famed MIT in Boston. After completing his bachelor’s, he then went on to Princeton, excelling constantly in physics, mathematics, and computational sciences. Indeed, his reputation for unprecedented thinking, clarifying lectures, and charming genius was so great that Albert Einstein himself attended his first graduate lecture. He was on his way to revolutionizing the field of physics, generating theories that are still being studied as our technology advances enough to measure it in laboratories. Feynman’s reputation even led him to the Manhattan Project, at the tender age of 24.
If you’re not into atomic or war history, the Manhattan Project was a secret project developed by the American government, that led to the creation of the first atomic bomb. The Manhattan Project operated from 1942-1946 in Los Alamos, New Mexico, and Feynman was a major contributor in the theoretical and computational division. Feynman has said that his idea of assisting on the project with the purpose of defending the US against Germany and Japan (who were supposed to be racing to develop the bomb first), should have dissipated when the threat did. He continued on with the work, stating that he was driven by solving the problem, not thinking deeply about the moral complications. He was also present at the Trinity Bomb test – the first atomic explosion, and the official inception of the Atomic Age. Shortly after, and despite the pleading of Robert Oppenheimer (head of the Los Alamos lab) to stay and continue contributing, Feynman took a post at Cornell briefly. He claimed he was uninspired by the atmosphere and close to burning out intellectually there, so he took a post at Cal Tech, where he ended up doing some of his best research. This includes:
• a model of weak decay: The ‘weak’ interaction is one of the four fundamental forces of the universe, along with the strong nuclear, electromagnetic, and gravity. The interactions of these forces control all the little bits of our universe that cannot be broken down any further; the rules that regulate our most basic building blocks (that we know of). According to the Standard Theory, these are known as quarks, leptons, gauge bosons and higg boson (You may have heard about the Higgs-Boson, as it has been appearing quite frequently in the news. It is the only undiscovered particle of these, and scientists are quite close to finding it, thanks to the Large Hadron Collider’s incredible technology). While gravity is most commonly known force to us regular folks, the weak force controls quarks and leptons – known collectively as ‘fermions’ because they are the two particles of matter, not light. Weak force controls both radioactive decay, and hydrogen fusion – the force allowing the sun to shine, and all life to live. You may not think it’s that important, but without the weak force, there is no you, because there would be no universe, no sun, no energy to get that tan in the summer! A classic example of weak decay is when a neutron breaks down into a proton, electron, and anti-neutrino. Feynman ultimately developed a new and succinctly described model for this decay factor, incorporating ideas that had been lacking before.
• physics of the superfluidity of supercooled liquid helium: Helium is the second most abundant particle in the observable universe, and its behaviour is amongst the strangest of all. It also has the unique property of having one of the lowest boiling and melting points: -269°C and -272°C respectively. In liquid form, helium had been observed to behave rather bizarrely when it was cooled slightly below the boiling point (Check out this excellent video for a visual representation). Feynman didn’t solve the whole problem, but applied the Schrödinger equation successfully to display the quantum mechanical behaviour on a macroscopic scale (I’ll try to briefly explain quantum mechanics in a moment).
• quantum electrodynamics: This is the work Feynman is best known for, and for which he won a joint Nobel Prize in 1965. The quantum world itself is a section of physics that deals in the tiniest part of matter we know about – atoms. It’s a bizarre world that breaks down all the other rules that govern our everyday life. The five main ideas behind quantum theory are:
A) Energy is not continuous, but moves in small, discreet bundles.
B) Elementary particles move like matter AND waves (excellent video explaining this crazy phenomena here).
C) This movement is intrinsically random.
D) It is impossible to know the location and momentum of a particle at the same time – the more precisely one is known, the less precise the other measurement is.
E) The quantum world is absolutely nothing like the one we live in.
Feynman was one of the founding father of the Quantum Electrodynamic Theory. While complicated, it basically describes (through mathematics) all interactions of light with matter, and of charged particles (a subatomic particle or ion with an electric charge) with one another. It was important because it was the first theory to cohesively integrate Einstein’s special relativity theory into each equation, as well as satisfying the Schrödinger equation (a problem that Paul Dirac and Norman Wiener, two scientists that had developed the theory previously, were unable to solve).
The three main concepts of Feynman’s QED theory is that: A) a photon goes from a location and time to another location and time, B) an electron goes from a location and time to another location and time, and C) an electron emits or absorbs a photon at a certain place and time. OK – what does that mean? To help explain these, Feynman came up with the self-named Feynman diagrams. Feynman Diagram Elements
Feynman Diagram (simple).
The first image shows us the symbols of parts A, B, or C of his theory. The second shows us an example of a Feynman Diagram – an ‘electron-positron annihilation’. Not to be mistaken for a Star Trek battle, this is when an negative electron (e−), and it’s opposite, a positive electron (positron [e+]) collide. This results in the annihilation of both, and photons are sent shooting out from the collision. Feynman’s theories and his well-known diagrams make ideas like this clearer, and more accessible visually to a large portion of the mathematically-disinclined population. Keep in mind, these diagrams are not set paths – just simplified suggestions representing potential quantum relationships symbolically.
It’s important to note that QED theory doesn’t tell you what will happen, but predicts the probability of what will happen. In quantum mechanics, this means that you add up the sum of all possibilities, to any given endpoint, and predict the probability of the end result based on this total sum. We can loosely think of this as taking a random walk. You’ve had a bad day at work and want to clear your mind. Without knowing your final destination, you decide to cross the road to the other side, which happens to be infinite. Your brain is (hopefully!) measuring where potholes in the road you may have to avoid are, and the probability of whether or not you will get hit by a car. Your brain then tells you when to finally move, and on what path. Your exact footsteps are not predictable, nor is where or when you will step onto the sidewalk, but your brain has calculated the possibilities. And if you were a quantum particle participating in the theory, you would end up with a path and endpoint that were the sum of all possibilities. This computational method was referred to by Feynman as the path integral formulation , and stands in contrast to previous theories that predicted a single, unique trajectory. This formula helps us to understand (or at least diversify) our understanding of the movement of the very tiny little building blocks of our universe.
Phew. If I have confused you, I’m sorry. I’m a bit confused myself at this point! Particles here, mathematics all over the chalkboard, what does that mean when I need to drag myself out of bed and go to work to feed the kids? The quantum world is difficult to grasp, and I would suspect that it’s still somewhat difficult even for the most brilliant of minds like Feynman. But that doesn’t mean its existence is irrelevant. It in fact informs everything about our lives, our composition, our beautiful planet tucked away here in this tiny corner of the universe. If our goal is to know ourselves, understanding the smallest bits is surely important, difficult as it may be. I’m sure this was one of Feynman’s motivating factors.
While working on all of these ideas and more, Feynman also dedicated a large portion of his career to teaching. While still at Cal Tech, he was asked to get the undergraduates really involved and appreciative of physics. After several years of work, this resulted in the extremely accessible, beautiful, and inspiring Feynman’s Lectures on Physics which I highly recommend if you have the remotest interest in physics. Perhaps it will clear up any confusion I may have left you floundering in today!
Now, I barely understand a percent of the incredible problems that Feynman naturally intuited, thought about deeply, and solved. However, the reason I appreciate him and his success as a physicist is due not only to his inherent genius, but also to his understanding of human nature. He was always open to new ideas and subjects, and constantly engaged his whole brain with love, academics, and artists – even creating some art himself under the pseudonym of ‘Ofey’. Watching his interviews and documentaries is always a pleasure, as he somehow manages to circumvent the common way of thinking, and present what have otherwise been very difficult concepts as clear and simple. Feynman has always managed to grasp the type of mind required to appreciate the universe – curious and humourous. As one of his colleagues best described, when you hear Feynman speak, you understand clearly the science behind physics. Once you leave the room however, you find yourself struggling to follow the same pathway that Feynman drew in your brain. I’d suspect it’s because few of us have ever taken that path before, and were so amazed by the beautiful things Feynman was showing us, that we forgot to remember the path. If we were to work hard enough though, we may be able to figure out the average probability to get back (A Feynman pun!).
Richard Fenyman continued to revolutionize and bring physics to light (another pun!) for the rest of us. He worked on the Challenger disaster of ’86, and raised awareness of the huge discrepancies between the NASA management teams and their poorly informed understanding of physics. In his rather stark review, he says quite truthfully, “For a successful technology, reality must take precedence over public relations, for Nature cannot be fooled.”
Feynman died from several forms rare cancers at the age of 69, in Los Angeles. His last words, in true humourous form, “I’d hate to die twice. It’s so boring.”
In memory of true genius, Richard P. Feynman 1918-1988.
What is necessary “for the very existence of science” and so forth, and what the characteristics of nature are, are not to be determined by pompous preconditions. They are determined always by the material with which we work, by nature herself. We look, and we see what we find, and we cannot say ahead of time -successfully- what it is going to look like.
The most reasonable possibilities turn out often not to be the situation.
What is necessary for the very existence of science is just the ability to experiment, the honesty in reporting results -the results must be reported without somebody saying what they’d like the results to have had been rather than what they are- and finally -an important thing-, the intelligence to interpret the results but an important point about this intelligence is that it should not be sure ahead of time what must be.
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There is an well-known infamous DICTUM:
-Second Quantization is a functor, First Quantization is a mystery-.
Indeed, second quantization is the "Fock functor", which builds the Fock space in a canonical way out of the Hilbert space of a single particle.
But, what about first quantization? There is probably no hope to canonically associate an Hilbert space to the manifold of states of a classical particle (mathematically, there seem to be an inherent element of choice as far as turning functions into operators).
However, there is (I suspect) some functorial description for going the other way around, FROM the quantum scenario INTO the classical one (corresponding to the limit $h\rightarrow 0$). If this is true, there maybe a "fiber" of candidate quantum descriptions, all collapsing into the same classical one.
Any place where this has been worked out clearly?
Possibly relevant question: – José Figueroa-O'Farrill Jun 10 2011 at 18:49
1 Answer
You may be interested in the answers to a question on physics.SE, "Quantum Mechanics on Manifold".
The gist of it is that there is a plethora of different quantization schemes (canonical quantization, geometric quantization, ...). As you note, they all have the same classical theory as a limit. Overview:
Quantization Methods: A Guide for Physicists and Analysts
That said, speaking as a physicist (caveat emptor), I think the paper
pretty much covers everything that is of practical relevance. Unsurprisingly, the Schrödinger equation on an embedded manifold $M \subseteq \mathbb{R}^3$ depends not only on intrinsic data, like the metric or curvature, but also on extrinsic data about the embedding of the manifold inside 3D space. That's to be expected because an electron can tunnel through the full 3D space and hence take "extrinsic short-cuts". I think that's also the reason why there is so much ambiguity in quantization schemes: it's simply not something that you can do completely intrinsically.
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35d545dcd9c6a6c5 | Interview with Research Fellow Terence Tao
September 2003
From an early age, you clearly possessed a gift for mathematics. What stimulated your interest in the subject, and when did you discover your talent for mathematical research? Which persons influenced you the most?
Ever since I can remember, I have enjoyed mathematics; I recall being fascinated by numbers even at age three, and viewed their manipulation as a kind of game. It was only much later, in high school, that I started to realize that mathematics is not just about symbolic manipulation, but has useful things to say about the real world; then, of course, I enjoyed it even more, though at a different level.
My parents were the ones who noticed my mathematical ability, and sought the advice of several teachers, professors, and education experts; I myself didn't feel anything out of the ordinary in what I was doing. I didn't really have any other experience to compare it to, so it felt natural to me. I was fortunate enough to have several good mentors during my high-school and college years who were willing to spend time with me just to discuss mathematics at a leisurely pace. For instance, there was a retired mathematics professor, Basil Rennie (who sadly died a few years ago), who I would visit each weekend and talk about recreational mathematics over tea and cakes. At the local university, Garth Gaudry also spent a lot of time with me and eventually became my masters thesis advisor; he was also the one who got me working in analysis, where I still primarily do most of my mathematics, and who encouraged me to study in the US. Once in graduate school, I of course benefitted from interaction with many other mathematicians, such as my advisor Eli Stein. But the same would be true of any other graduate student in mathematics.
You left Australia in 1992 to study mathematics at Princeton University. What inspired your decision to go there?
This was mostly at the urging of my masters thesis advisor Garth Gaudry. He felt that regardless of where I ended up eventually, in Australia, the US, or elsewhere, it would be good to get some international experience at the graduate level. There are of course several good universities in Australia also, but they unfortunately suffer somewhat from geographic isolation. I recall applying to a dozen places, and ending up with acceptance offers from both Princeton and MIT. I guess in the end I chose Princeton for its strength in harmonic analysis (in particular because of Eli Stein - I was already learning harmonic analysis in Australia out of his textbooks!)
After I completed my dissertation at Princeton, I was torn between continuing to work in the US and returning to Australia. Eventually I settled on a compromise in which I spent half the time in each; but now that I have settled down in Los Angeles with my work and family, I will probably stay in the US permanently.
What is the primary focus of your research today? Can you comment on the results of which you are most fond?
I work in a number of areas, but I don't view them as being disconnected. I tend to view mathematics as a unified subject and am particularly happy when I get the opportunity to work on a project that involves several fields at once. Perhaps the largest "connected component" of my research ranges from arithmetic and geometric combinatorics at one end (the study of arrangements of geometric objects such as lines and circles, including one of my favorite conjectures, the Kakeya conjecture, or the combinatorics of addition, subtraction and multiplication of sets), through harmonic analysis (especially the study of oscillatory integrals, maximal functions, and solutions to the linear wave and Schrödinger equations), and ends up in nonlinear PDE (especially nonlinear wave and dispersive equations).
Currently my focus is more at the nonlinear PDE end of this range, especially with regard to the global and asymptotic behavior of such evolution equations, and also with the hope of combining the analytical tools of nonlinear PDE with the more algebraic tools of completely integrable systems at some point. In addition, I work in a number of areas adjacent to one of the above fields; for instance I have begun to be interested in arithmetic progressions and connections with number theory, as well as with other aspects of harmonic analysis such as multilinear integrals, and other aspects of PDE, such as the spectral theory of Schrödinger operators with potentials or integrable systems.
Finally, with Allen Knutson, I have a rather different line of research: the algebraic combinatorics of several related problems, including the sum of Hermitian matrices problem, the tensor product muliplicities of representations, and intersections of Schubert varieties. Though we only have a few papers in this field, I still count this as one of my favorite areas to work in. This is because of all the unexpected structure and algebraic "miracles" which occur in these problems, and also because it is so technically and conceptually challenging. Of course, I also enjoy my work in analysis, but for a different reason. There are fewer miracles, but instead there is lots of intuition coming from physics and from geometry. The challenge is to quantify and exploit as much of this intuition as possible.
In analysis, many research programs do not conclude in a definitive paper, but rather in a progression of steadily improving partial results. Much of my work has been of this type (especially with regards to the Kakeya problem and its relatives, still one of my primary foci of research). But I do have two or three results of a more conclusive nature of which I feel particularly satisfied. The first is my original paper with Allen Knutson, in which we characterize the eigenvalues of a sum of two Hermitian matrices, first by reducing it to a purely geometric combinatorial question (that of understanding a certain geometric configuration called a "honeycomb"), and then by solving that question by a combinatorial argument. (There have since been a number of other proofs and conceptual clarifications, although the exact role of honeycombs remains partly mysterious). The second is my paper on the small energy global regularity of wave maps to the sphere in two dimensions, in which I introduce a new "microlocal" renormalization in order to turn this rather nonlinear problem into a more manageable semilinear evolution. While the result in itself is not yet definitive (the equation of general target manifolds other than the sphere was done afterwards, and the large energy case remains open, and very interesting), it did remove a psychological stumbling block by showing that these critical wave equations were not intractable. As a result there has been a resurgence of interest in these equations. Finally, I have had a very productive and enjoyable collaboration with Jim Colliander, Markus Keel, Gigliola Staffilani, and Hideo Takaoka, culminating this year in the establishment of global regularity and scattering for a critical nonlinear Schrödinger equation (for large energy data); this appears to be the first unconditional global existence result for this type of critical dispersive equation. The result required assembling and then refining several recent techniques developed in this field, including an induction-on-energy approach pioneered by Bourgain, and a certain interaction Morawetz inequality we had discovered a few years earlier. The result seems to reveal some new insights into the dynamics of such equations. It is still in its very early days, but I feel confident that the ideas developed here will have further application to understanding the large energy behavior of other nonlinear evolution equations. This is the topic I am still immensely interested in.
You have worked on problems quite far from the main focus of your research, e.g. Horn's conjecture. Could you comment on the motivation for this work and the challenges it presented? On your collaborations and the idea of collaboration in general? Can a mathematician in this day of specialization hope to contribute to more than one area?
My work on Horn's conjecture stemmed from discussions I had with Allen Knutson in graduate school. Back then we were not completely decided as to which field to specialize in and had (rather naively) searched around for interesting research problems to attack together. Most of these ended up being discarded, but the sum of Hermitian matrices problem (which we ended up working on as a simplified model of another question posed by another graduate student) was a lucky one to work on, as it had so much unexpected structure. For instance, it can be phrased as a moment map problem in symplectic geometry, and later we realized it could also be quantized as a multiplicity problem in representation theory. The problem has the advantage of being elementary enough that one can make a fair bit of progress without too much machinery — we had begun deriving various inequalities and other results, although we eventually were a bit disappointed to learn that we had rediscovered some very old results of Weyl, Gelfand, Horn, and others by doing so. By the time we finished graduate school, we had gotten to the point where we had discovered the role of honeycombs in the problem. We could not rigorously prove the connection between honeycombs and the Hermitian matrices problem, and were otherwise stuck. But then Allen learned of more recent work on this problem by algebraic combinatorialists and algebraic geometers, including Klyachko, Totaro, Bernstein, Zelevinsky, and others. With the more recent results from those authors we were able to plug the missing pieces in our argument and eventually settle the Horn conjecture.
Collaboration is very important for me, as it allows me to learn about other fields, and, conversely, to share what I learned about my own fields with others. It broadens my experience, not just in a technical mathematical sense but also in being exposed to other philosophies of research, of exposition, and so forth. Also, it is considerably more fun to work in groups than by oneself. Ideally, a collaborator should be close enough to one's own strengths that one can communicate ideas and strategies back and forth with ease, but far enough apart that one's skills complement rather than replicate each other.
It is true that mathematics is more specialized than at any time in its past, but I don't believe that any field of mathematics should ever get so technical and complicated that it could not (at least in principle) be accessible to a general mathematician after some patient work (and with a good exposition by an expert in the field). Even if the rigorous machinery is very complicated, often the ideas and goals of a field are often so simple, elegant, and natural that I feel it is frequently more than one's while to invest the time and effort to learn about other fields. Of course, this task is helped immeasurably if you can talk at length with someone who is already expert in those areas; but again, this is why collaboration is so useful. Even just attending conferences and seminars that are just a little bit outside your own field is useful. In fact, I believe that a subfield of mathematics has a better chance of staying dynamic, fruitful, and exciting if people in the area do make an effort to make good surveys and expository articles that try to reach out to other people in neighboring disciplines and invite them to lend their own insights and expertise to attack the problems in the area. The need to develop fearsome and impenetrable machinery in a field is a necessary evil, unfortunately, but as understanding progresses it should not be a permanent evil. If it serves to keep away other skilled mathematicians who might otherwise have useful contributions to make, then that is a loss for mathematics.
Also, counterbalancing the trend toward increasing complexity and specialization at the cutting edge of mathematics is the deepening insight and simplification of mathematics at its common core. Harmonic analysis, for instance, is a far more organized and intuitive subject than it was in, say, the days of Hardy and Littlewood; results and arguments are not isolated technical feats but put into a wider context of interaction between oscillation, singularity, geometry, and so forth. PDE also appears to be undergoing a similar conceptual organization, with less emphasis on specific techniques such as estimates and choices of function spaces, and instead sharing more in common with the underlying geometric and physical intuition. In some ways, the accumulated rules of thumb, folklore, and even just some very well chosen choices of notation can make it easier to get into a field nowadays. (It depends on the field, of course; some have made far more progress with conceptual simplification than others).
Can you describe your research in accessible terms? Does it have applications to other areas?
I guess I can start with nonlinear PDE, which is perhaps the area of my research which is closest to "real life" applications. I am interested in many nonlinear PDE which arise in physics (Korteweg-de Vries, nonlinear Schrödinger, wave maps, Yang-Mills, Einstein, ...). However, in many cases these equations are only a simplified model of the physical reality, and the arguments which justify them as a good approximation to the actual situation are usually just heuristic. Thus, it is quite important to know whether these models are robust or not, in that they can tolerate the errors and approximations used to pass from reality to the model. In order for a model to be robust, it should first be well posed; in other words, solutions should exist for all time and depend in a stable way on the initial data (or on any forcing terms). If a model predicts instead that some physical quantity (e.g. the energy) should go to infinity in finite time, then that is pretty clearly not a good model for reality.
One of my main research interests is then in understanding the global existence, regularity, and asymptotic behavior of nonlinear evolution equations (particularly nonlinear Schrödinger and wave equations). These results are already interesting in themselves, but what I find most fascinating is that the very process of discovering results in these problems, especially if one enforces the discipline of working in a low regularity setting or under minimal assumptions on the data, leads one to uncover new insights and facts about such equations, which then become useful in many other settings too. To take one example, in my joint work with Colliander, Keel, Staffilani, and Takaoka, we were studying the scattering behavior of a certain subcritical nonlinear Schrodinger equation under the assumption of infinite energy. The case of finite energy was already handled before, so this research may have seemed of purely academic interest. However, the challenge of working in the infinite energy case took away several of the tools which were available for this problem, and forced us to come up with a new tool -in this case, we discovered a certain interaction Morawetz inequality which was totally new. This inequality not only solved our problem, but led to a substantial simplification of the original argument in the finite energy case, and later was a key ingredient in our resolution of the global regularity problem for the _critical_ nonlinear Schrodinger equation, which was previously unknown even for smooth, finite energy data. It is these sorts of discoveries that make the investment into even quite technical areas of a subject quite worthwhile. It seems unlikely that this inequality would have been discovered if we were only looking at the case of smooth solutions (in which many other techniques were available).
Much of my other work is not as directly related to physical reality as nonlinear PDE, but there are some connections, and again there is always the chance that while working on a technical problem, some new insight or tool will be discovered which can then be applied to other problems, and which clarify conceptually the field as a whole. For instance, linear and nonlinear PDE often encounters the issue of how to control a superposition of oscillating waves at various frequencies, positions, and directions; in many cases this issue can be dealt with by standard harmonic analysis techniques such as the Fourier transform, but one can consider model harmonic analysis problems (such as the restriction conjecture) in which the standard techniques give only partial results. This in turn turns out (by the harmonic analysis analogue of geometric optics, which converts questions about waves into questions about light rays) to be related to another problem about incidences of light rays and other geometric objects, which in turn connects to the purely combinatorial Kakeya problem mentioned earlier. This in turn connects (via the re-interpretation of lines as arithmetic progressions) to some questions in arithmetic combinatorics, and this in turn can connect to other areas of mathematics such as number theory. I find all these connections fascinating, and also give me an excuse to learn adjacent fields of mathematics; my investment of time has almost always been rewarded with new understanding and another collection of techniques to add to my "toolbox".
What advice would you give to young people starting out in math (i.e. high school students and young researchers)?
Well, I guess they should be warned that their impressions of what professional mathematics is may be quite different from the reality. In elementary school I had the vague idea that professional mathematicians spent their time computing digits of pi, for instance, or perhaps devising and then solving Math Olympiad style problems. In high school and lower-division undergraduate mathematics, one can often get the impression that mathematics is a "solved" science, and that all that one needs to do nowadays is remember all the tools and techniques that were derived centuries earlier. Conversely, in upper-division mathematics, the subjects that seem the most beautiful, and which sound like the most fun to work on - for me, they were C-* algebras and elementary number theory - often end up being very heavily studied already, and nowadays just the foundation for even more interesting areas of research. It's really difficult to tell, until late in graduate school, what the active areas of research really are, and which ones will be most suited to your taste. I myself was lucky that the field I chose (harmonic analysis) was one that was quite fruitful and enjoyable to me, even if the things I work on now are things I would not even have contemplated thinking about as a graduate student.
So, I would not fixate too much on wanting to go into one specific subfield of mathematics (or even into mathematics altogether), based on one's experiences, say, at the undergraduate level; in many ways, the subject only just begins to get interesting at the graduate level and above. Conversely, subjects which seemed quite dry when learnt in an undergraduate context can become revitalized under the more modern perspective that one might learn later in one's career. To give just one example, the theory of matrices and determinants makes much more sense when viewed using the concepts of linear transformations and wedge products (and this in turn can be viewed even more clearly using even higher conceptual tools such as vector bundles, tensor products, categories, Lie groups, Clifford algebras, etc.). Not many people study determinants directly any more, but there are many fields that are descendants of the classical theory of determinants which are all very active and important.
The other thing is not to be too afraid (or too disdainful) of other fields, and to take the effort to at least get a little understanding of what is going on in neighboring fields of activity. You never know when it will come in handy. For instance, I have recently started encountering little algebraic geometry problems to solve in the Kakeya problem, which previously I thought to be a purely combinatorial problem involving elementary incidence geometry. This forced me to revisit my old graduate textbooks on algebraic varieties and the like, but this time with more motivation and experience I was able to learn a lot more, and eventually could use the rudiments of the subject that I had learnt to make some progress on the Kakeya problem.
You have some quite specific ideas about problem solving. Can you tell us about them?
Everyone has their own problem solving style, of course. Andrew Wiles worked on Fermat's Last Theorem more or less continuously for about seven years. I myself couldn't do that; if I don't see hints of progress within a week or two, then even if the problem is tremendously exciting I will feel inclined to shelve it and work on other problems. After a year or so I might return to the problem and hopefully with a fresh perspective and some new ideas and tools I can make further progress.
One good thing about analysis is that for every difficult unsolved problem, there is often a less difficult model problem which can be worked on first. Or if solving a certain problem requires one to resolve obstruction A and obstruction B, one can often locate toy subproblems in which only one of the two obstructions is present (or at least one may make an artificial hypothesis which suppresses one of the two obstructions). So often it makes excellent tactical sense to move away from your original target problem, and work instead on problems which are perhaps less intrinsically interesting, but which are simpler and which embody at least one of the difficulties you would also encounter in the original problem. So, a large part of my problem solving technique involves understanding the problem and its obstructions well enough that one can concoct reasonable models to work in. For instance, one might model a PDE by an ODE after making various assumptions on the solution which one could then justify by various non-rigorous heuristics. The ODE in turn might be modeled by a discrete difference scheme, and perhaps after a few more heuristic reductions of this type one may be left with a very simple problem, say involving just a finite number of quantities, which still contains the key obstruction; at this point, one should either be able to see what has to be done to resolve the obstruction, or else start concocting a counter example (which is then also interesting; or else the counter example construction, when passing back to the original problem; hits another obstruction, which you then work on again by passing to a model, and so forth). As such, my work on a problem often takes me in rather unpredictable directions, but even the directions which seem to waste a lot of time can often be quite educational, if only in a negative sense that certain techniques are unlikely to work. Of course, there are times when I cannot make progress on an obstruction even after I have simplified away all the other aspects of the problem; usually if I can't get anywhere from that point after a few days or so, I will move on to try something else instead (e.g. I will work on another problem which does not have this particular obstruction). Very occasionally, after some years pass, I work on what I think to be a completely unrelated problem, and discover (either by myself, by a collaborator, or by reading another paper) an idea which has a chance of resolving an obstruction which blocked me in an earlier problem. This, so far, has happened only a few times for me, but it is very satisfying when it does. Of course, if I had stayed focused on my original problem continuously, I might also have found the solution eventually; but I find that more difficult to sustain than the more scatter shot approach I am used to. It may work out differently for others, though.
What advice would you give to older people interested in mathematics as a hobby? What should they read? How should they proceed?
It is unfortunately difficult to get into an advanced field of mathematics these days without being in contact with people already in the field; there is only so much one can learn from books alone (although the web and internet, if used correctly, is a great resource also nowadays). Some fields change so rapidly that one can take a vacation from a field for, say, five years, and barely recognize the field when one returns. But if one already has mastered one area of mathematics, that often gives enough confidence to start exploring other areas. As a graduate student, I was daunted by analytic number theory and PDE, two subjects I was quite interested in, because both subjects also required a fair understanding of harmonic analysis, which I did not have at the time. But once I had some experience in harmonic analysis I was able to revisit those fields and learn them more effectively.
As I said earlier, the actual experience of research in a field may differ quite dramatically from what one might imagine it to be like as an outsider. Number theory, to give an example, is very accessible (at least at first) and thus attracts a lot of interest from recreational mathematicians, but they can become discouraged upon learning that modern progress in the field relies in such sophisticated tools as exponential sums, zeta functions, modular forms, elliptic curves, and so forth. This is not to say that there is no longer any place for elementary number theory - witness, for instance, the recent excitement over the first deterministic polynomial time primality testing algorithm - but the field is not what one may believe it to be from the outside.
As I have spent my entire life with mathematics, perhaps I am not the best qualified to discuss how to get into the field at a later age, but perhaps it would make sense to stay close to one's existing strengths; for instance, if one has an engineering background, then mathematical physics, dynamical systems, or ODE might make sense, or if one has a computer science background then discrete mathematics or complexity theory might make sense.
Your education and now your teaching career is divided between two countries. Is this by choice?
Partly by choice and partly by visa considerations; I had a visa requirement that required me to spend 24 months in my home country before applying for permanent residency. But now I have completed that requirement, and am settled in Los Angeles with my wife and son, so will probably be spending much more time in the US now.
How has maintaining your ties with the Australian mathematical community influenced your research?
Most directly, I have had several collaborations with Australian mathematicians, or with international mathematicians who were visiting Australia. But while Australian mathematics has a slightly different character from, say, American or French or Japanese or Russian mathematics, it is not so distinct that one can readily separate any distinctly Australian flavors from other aspects of my research. I get a lot out of visiting mathematics departments throughout the US and internationally, because every department has a slightly different focus and set of strengths in mathematics, and I have collaborators from all over the world; I think the internationality of mathematics is one of its great strengths.
Can you comment on the culture of research mathematics in Australia? How does it compare to the U.S.?
Culturally, they are fairly similar. The two major differences are firstly, that Australia is much smaller in population and more isolated geographically, which means that it is more difficult (though not impossible) to maintain an active visitor program. Still, there is enough interaction between Australia and the rest of the mathematical community that I believe the research in Australia is quite competitive by international standards. The other major difference is that there are very little sources of private funding (either corporate or philanthropic) in Australia for higher mathematics research, and as such we are almost entirely reliant on state and federal government support. This may create some difficulties for Australia in the long term, as it may be difficult to recruit and retain good talent with the limited financial resources available.
How has your Clay fellowship made a difference for you?
The Clay has been very useful in granting a large amount of flexibility in my travel and visiting plans, especially since I was also subject to certain visa restrictions at the time. For instance, it has made visiting Australia much easier. Also I was supported by CMI on several trips to Europe and on an extended stay at Princeton, both of which were very useful to me mathematically, allowing me to interact and exchange ideas with many other mathematicians (some of whom I would later collaborate with).
Recently you received two great honors: the AMS Bôcher Prize and the Clay Research Award, for results that distinguish you for your contributions to analysis and other fields. Have your findings opened up new areas or spawned new collaborations? Who else has made major contributions to this specific area of research?
The work on wave maps (the main research designated by the Bôcher prize) is still quite active; after my own papers there were further improvements and developments by Klainerman, Rodnianski, Shatah, Struwe, Nahmod, Uhlenbeck, Stefanov, Krieger, Tataru, and others. (My work in turn built upon earlier work of these authors as well as Machedon, Selberg, Keel, and others). Perhaps more indirectly, the mere fact that critical nonlinear wave equations can be tractable may have helped encourage the parallel lines of research on sister equations such as the Einstein equations, Maxwell-Klein-Gordon, or Yang-Mills. This research is also part of a larger trend where the analysis of the equations is moving beyond what can be achieved with Fourier analysis techniques and energy methods, and is beginning to incorporate more geometric ideas (in particular, to use ideas from Riemannian geometry to control geometric objects such as connections and geodesics, these in turn can be used to control the evolution of the nonlinear wave equation).
The Clay award recognized not only the work on wave maps, but also on sharp restriction theorems for the Fourier transform, which was an area pioneered by such great mathematicians as Carleson, Sjolin, Tomas, Stein, Fefferman, and Cordoba almost thirty years ago, and which has been invigorated by more recent work of Bourgain, Wolff, and others. These problems are still not solved fully; this would require, among other things, a complete solution to the Kakeya conjecture. The relationship of these problems both to geometry and to PDE has been greatly clarified however, and the technical tools required to make concrete these connections are also much better understood. Recent work by Vargas, Lee, and others continue to develop the theory of these estimates.
The Clay award also mentioned the work on honeycombs and Horn's conjecture. Horn's conjecture has now been proven in a number of ways (thanks to later work by Belkale, Buch, Weyman, Derksen, Knutson, Totaro, Woodward, Fulton, Vakil and others), and we are close to a more satisfactory geometric understanding of this problem. Lately, Allen and I have been more interested in the connection with Schubert geometry, which is connected to a discrete analogue of a honeycomb which we call a "puzzle". These puzzles seem to encode in some compact way the geometric combinatorics of Grassmannians and flag varieties, and there is some exciting work of Knutson and Vakil that seems to "geometrize" the role of these puzzles (and the combinatorics of the Littlewood-Richardson rule in general) quite neatly. There is also some related work of Speyer that may shed some light on one of the more mysterious combinatorial aspects of these puzzles, namely that they are "associative".
What research problems are you likely to explore in the future?
It's hard to say. As I said before, even five years ago I would not really have imagined working on what I am doing now. I still find the problems related to the Kakeya problem fascinating, as well as anything to do with honeycombs and puzzles. But currently I am more involved in nonlinear PDE, with an eye toward moving toward integrable systems. Related to this is a long-term joint research project with Christoph Thiele on the nonlinear Fourier transform (also known as the scattering transform) and its connection with integrable systems. I am also getting interested in arithmetic progressions and their connections with combinatorics, number theory, and even ergodic theory. I have also been learning bits and pieces of differential geometry and algebraic geometry and may take more of an interest in those fields in the future. Certainly at this point I have more interesting directions to pursue than I have time to work with!
What are your thoughts on the Millennium Prize Problems, the Navier-Stokes Equation for example?
The prize problems are great publicity for mathematics, and have made the recent possible resolution of Poincaré's conjecture—which is already an amazing and very important mathematical achievement—much more publicized and exciting than it already was. It is unclear how close the other problems are to resolution, though they all have several major obstructions that need to be resolved first. For Navier Stokes, one of the major obstructions is turbulence. This equation is "supercritical", which roughly means that the energy can interact much more forcefully at fine scales than it can at coarse scales (in contrast to subcritical equations where the coarse scale behavior dominates, and critical equations where all scales contribute equally). As yet we do not have a good large data global theory for any supercritical equation, let alone Navier Stokes, without some additional constraints on the solution to somehow ameliorate the behavior of the fine scales. A new technique that would allow us to handle very turbulent solutions effectively would be a major achievement. Perhaps one hope lies in the stochastic models of these flows, although it would be a challenge to show that these stochastic models really do model the deterministic Navier-Stokes equation properly.
Again, there are many sister equations of Navier-Stokes, and it may well be that the ultimate solution to this problem may lie in first understanding a related model equations—the Euler equations, for instance. Even Navier-Stokes is itself a model for other, more complicated, fluid dynamics. So while Navier-Stokes is certainly an important equation in fluid equations, there should not be given the impression that the Clay prize problem is the only problem worth studying there. |
e03f22be92b24bf0 | Digital Library of Mathematical Functions
About the Project
33 Coulomb FunctionsPhysical Applications
§33.22 Particle Scattering and Atomic and Molecular Spectra
§33.22(i) Schrödinger Equation
With e denoting here the elementary charge, the Coulomb potential between two point particles with charges Z_{1}e,Z_{2}e and masses m_{1},m_{2} separated by a distance s is V(s)=Z_{1}Z_{2}e^{2}/(4\pi\epsilon_{0}s)=Z_{1}Z_{2}\alpha\hbar\lightspeed/s, where Z_{j} are atomic numbers, \epsilon_{0} is the electric constant, \alpha is the fine structure constant, and \hbar is the reduced Planck’s constant. The reduced mass is m=m_{1}m_{2}/(m_{1}+m_{2}), and at energy of relative motion E with relative orbital angular momentum \ell\hbar, the Schrödinger equation for the radial wave function w(s) is given by
With the substitutions
{\sf k}=(2mE/\hbar^{2})^{{1/2}},
(33.22.1) becomes
§33.22(ii) Definitions of Variables
{\sf k} Scaling
The {\sf k}-scaled variables \rho and \eta of §33.2 are given by
At positive energies E>0, \rho\geq 0, and:
Attractive potentials: Z_{1}Z_{2}<0, \eta<0.
Zero potential (V=0): Z_{1}Z_{2}=0, \eta=0.
Repulsive potentials: Z_{1}Z_{2}>0, \eta>0.
Positive-energy functions correspond to processes such as Rutherford scattering and Coulomb excitation of nuclei (Alder et al. (1956)), and atomic photo-ionization and electron-ion collisions (Bethe and Salpeter (1977)).
At negative energies E<0 and both \rho and \eta are purely imaginary. The negative-energy functions are widely used in the description of atomic and molecular spectra; see Bethe and Salpeter (1977), Seaton (1983), and Aymar et al. (1996). In these applications, the Z-scaled variables r and \epsilon are more convenient.
Z Scaling
The Z-scaled variables r and \epsilon of §33.14 are given by
For Z_{1}Z_{2}=-1 and m=m_{e}, the electron mass, the scaling factors in (33.22.5) reduce to the Bohr radius, \BohrRadius=\hbar/(m_{e}\lightspeed\alpha), and to a multiple of the Rydberg constant,
Attractive potentials: Z_{1}Z_{2}<0, r>0.
Zero potential (V=0): Z_{1}Z_{2}=0, r=0.
Repulsive potentials: Z_{1}Z_{2}>0, r<0.
i{\sf k} Scaling
The i{\sf k}-scaled variables z and \kappa of §13.2 are given by
Attractive potentials: Z_{1}Z_{2}<0, \imagpart{\kappa}<0.
Zero potential (V=0): Z_{1}Z_{2}=0, \kappa=0.
Repulsive potentials: Z_{1}Z_{2}>0, \imagpart{\kappa}>0.
Customary variables are (\epsilon,r) in atomic physics and (\eta,\rho) in atomic and nuclear physics. Both variable sets may be used for attractive and repulsive potentials: the (\epsilon,r) set cannot be used for a zero potential because this would imply r=0 for all s, and the (\eta,\rho) set cannot be used for zero energy E because this would imply \rho=0 always.
§33.22(iii) Conversions Between Variables
r=\kappa z/2,
\epsilon=-1/\kappa^{2},Z from i{\sf k}.
z=2r/\kappa,i{\sf k} from Z.
Resolution of the ambiguous signs in (33.22.11), (33.22.12) depends on the sign of Z/\mathsf{k} in (33.22.3). See also §§33.14(ii), 33.14(iii), 33.22(i), and 33.22(ii).
§33.22(iv) Klein–Gordon and Dirac Equations
The relativistic motion of spinless particles in a Coulomb field, as encountered in pionic atoms and pion-nucleon scattering (Backenstoss (1970)) is described by a Klein–Gordon equation equivalent to (33.2.1); see Barnett (1981a). The motion of a relativistic electron in a Coulomb field, which arises in the theory of the electronic structure of heavy elements (Johnson (2007)), is described by a Dirac equation. The solutions to this equation are closely related to the Coulomb functions; see Greiner et al. (1985).
§33.22(v) Asymptotic Solutions
The Coulomb solutions of the Schrödinger and Klein–Gordon equations are almost always used in the external region, outside the range of any non-Coulomb forces or couplings.
For scattering problems, the interior solution is then matched to a linear combination of a pair of Coulomb functions, \mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right) and \mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right), or \mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right) and \mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right), to determine the scattering S-matrix and also the correct normalization of the interior wave solutions; see Bloch et al. (1951).
For bound-state problems only the exponentially decaying solution is required, usually taken to be the Whittaker function \mathop{W_{{-\eta,\ell+\frac{1}{2}}}\/}\nolimits\!\left(2\rho\right). The functions \phi_{{n,\ell}}(r) defined by (33.14.14) are the hydrogenic bound states in attractive Coulomb potentials; their polynomial components are often called associated Laguerre functions; see Christy and Duck (1961) and Bethe and Salpeter (1977).
§33.22(vi) Solutions Inside the Turning Point
The penetrability of repulsive Coulomb potential barriers is normally expressed in terms of the quantity \rho/({\mathop{F_{{\ell}}\/}\nolimits^{{2}}}\!\left(\eta,\rho\right)+{\mathop{%
G_{{\ell}}\/}\nolimits^{{2}}}\!\left(\eta,\rho\right)) (Mott and Massey (1956, pp. 63–65)). The WKBJ approximations of §33.23(vii) may also be used to estimate the penetrability.
§33.22(vii) Complex Variables and Parameters
The Coulomb functions given in this chapter are most commonly evaluated for real values of \rho, r, \eta, \epsilon and nonnegative integer values of \ell, but they may be continued analytically to complex arguments and order \ell as indicated in §33.13.
Examples of applications to noninteger and/or complex variables are as follows.
• Scattering at complex energies. See for example McDonald and Nuttall (1969).
• Searches for resonances as poles of the S-matrix in the complex half-plane \imagpart{\sf{k}}<0. See for example Csótó and Hale (1997).
• Regge poles at complex values of \ell. See for example Takemasa et al. (1979).
• Eigenstates using complex-rotated coordinates r\to re^{{i\theta}}, so that resonances have square-integrable eigenfunctions. See for example Halley et al. (1993).
• Solution of relativistic Coulomb equations. See for example Cooper et al. (1979) and Barnett (1981b).
• Gravitational radiation. See for example Berti and Cardoso (2006).
For further examples see Humblet (1984). |
536f39462760551d | Intro to Quantum Mechanics
This page is intended to give an ordinary person a brief overview of the importance and wonder of quantum mechanics. Unfortunately, most people believe you need the mind of Einstein in order to understand QM so they give up on it entirely. (Interesting side note: Einstein didn't believe QM was a correct theory!) Even some chemists fall into that category-- to represent physical chemistry our departmental T-shirts have a picture of the below atom, which is almost a century out of date. <Sigh>
So please read on, and take a dip in an ocean of information that I find completely invigorating!
Old atom {1 kB}
If the above picture is your idea of an atom, with electrons looping around the nucleus, you are about 70 years out of date. It's time to open your eyes to the modern world of quantum mechanics! The picture below shows some plots of where you would most likely find an electron in a hydrogen atom (the nucleus is at the center of each plot).
Hydrogen electron orbitals {18 kB}
What is quantum mechanics?
Simply put, quantum mechanics is the study of matter and radiation at an atomic level.
Why was quantum mechanics developed?
In the early 20th century some experiments produced results which could not be explained by classical physics (the science developed by Galileo Galilei, Isaac Newton, etc.). For instance, it was well known that electrons orbited the nucleus of an atom. However, if they did so in a manner which resembled the planets orbiting the sun, classical physics predicted that the electrons would spiral in and crash into the nucleus within a fraction of a second. Obviously that doesn't happen, or life as we know it would not exist. (Chemistry depends upon the interaction of the electrons in atoms, and life depends upon chemistry). That incorrect prediction, along with some other experiments that classical physics could not explain, showed scientists that something new was needed to explain science at the atomic level.
If classical physics is wrong, why do we still use it?
Classical physics is a flawed theory, but it is only dramatically flawed when dealing with the very small (atomic size, where quantum mechanics is used) or the very fast (near the speed of light, where relativity takes over). For everyday things, which are much larger than atoms and much slower than the speed of light, classical physics does an excellent job. Plus, it is much easier to use than either quantum mechanics or relativity (each of which require an extensive amount of math).
What is the importance of quantum mechanics?
The following are among the most important things which quantum mechanics can describe while classical physics cannot:
Discreteness of energy
If you look at the spectrum of light emitted by energetic atoms (such as the orange-yellow light from sodium vapor street lights, or the blue-white light from mercury vapor lamps) you will notice that it is composed of individual lines of different colors. These lines represent the discrete energy levels of the electrons in those excited atoms. When an electron in a high energy state jumps down to a lower one, the atom emits a photon of light which corresponds to the exact energy difference of those two levels (conservation of energy). The bigger the energy difference, the more energetic the photon will be, and the closer its color will be to the violet end of the spectrum. If electrons were not restricted to discrete energy levels, the spectrum from an excited atom would be a continuous spread of colors from red to violet with no individual lines.
Emission spectra {52 kB}
The concept of discrete energy levels can be demonstrated with a 3-way light bulb. A 40/75/115 watt bulb can only shine light at those three wattage's, and when you switch from one setting to the next, the power immediately jumps to the new setting instead of just gradually increasing.
It is the fact that electrons can only exist at discrete energy levels which prevents them from spiraling into the nucleus, as classical physics predicts. And it is this quantization of energy, along with some other atomic properties that are quantized, which gives quantum mechanics its name.
The wave-particle duality of light and matter
In 1690 Christiaan Huygens theorized that light was composed of waves, while in 1704 Isaac Newton explained that light was made of tiny particles. Experiments supported each of their theories. However, neither a completely-particle theory nor a completely-wave theory could explain all of the phenomena associated with light! So scientists began to think of light as both a particle and a wave. In 1923 Louis de Broglie hypothesized that a material particle could also exhibit wavelike properties, and in 1927 it was shown (by Davisson and Germer) that electrons can indeed behave like waves.
How can something be both a particle and a wave at the same time? For one thing, it is incorrect to think of light as a stream of particles moving up and down in a wavelike manner. Actually, light and matter exist as particles; what behaves like a wave is the probability of where that particle will be. The reason light sometimes appears to act as a wave is because we are noticing the accumulation of many of the light particles distributed over the probabilities of where each particle could be.
For instance, suppose we had a dart-throwing machine that had a 5% chance of hitting the bulls-eye and a 95% chance of hitting the outer ring and no chance of hitting any other place on the dart board. Now, suppose we let the machine throw 100 darts, keeping all of them stuck in the board. We can see each individual dart (so we know they behave like a particle) but we can also see a pattern on the board of a large ring of darts surrounding a small cluster in the middle. This pattern is the accumulation of the individual darts over the probabilities of where each dart could have landed, and represents the 'wavelike' behavior of the darts. Get it?
Quantum tunneling
This is one of the most interesting phenomena to arise from quantum mechanics; without it computer chips would not exist, and a 'personal' computer would probably take up an entire room. As stated above, a wave determines the probability of where a particle will be. When that probability wave encounters an energy barrier most of the wave will be reflected back, but a small portion of it will 'leak' into the barrier. If the barrier is small enough, the wave that leaked through will continue on the other side of it. Even though the particle doesn't have enough energy to get over the barrier, there is still a small probability that it can 'tunnel' through it!
Let's say you are throwing a rubber ball against a wall. You know you don't have enough energy to throw it through the wall, so you always expect it to bounce back. Quantum mechanics, however, says that there is a small probability that the ball could go right through the wall (without damaging the wall) and continue its flight on the other side! With something as large as a rubber ball, though, that probability is so small that you could throw the ball for billions of years and never see it go through the wall. But with something as tiny as an electron, tunneling is an everyday occurrence.
On the flip side of tunneling, when a particle encounters a drop in energy there is a small probability that it will be reflected. In other words, if you were rolling a marble off a flat level table, there is a small chance that when the marble reached the edge it would bounce back instead of dropping to the floor! Again, for something as large as a marble you'll probably never see something like that happen, but for photons (the massless particles of light) it is a very real occurrence.
The Heisenberg uncertainty principle
People are familiar with measuring things in the macroscopic world around them. Someone pulls out a tape measure and determines the length of a table. A state trooper aims his radar gun at a car and knows what direction the car is traveling, as well as how fast. They get the information they want and don't worry whether the measurement itself has changed what they were measuring. After all, what would be the sense in determining that a table is 80 cm long if the very act of measuring it changed its length!
At the atomic scale of quantum mechanics, however, measurement becomes a very delicate process. Let's say you want to find out where an electron is and where it is going (that trooper has a feeling that any electron he catches will be going faster than the local speed limit). How would you do it? Get a super high powered magnifier and look for it? The very act of looking depends upon light, which is made of photons, and these photons could have enough momentum that once they hit the electron they would change its course! It's like rolling the cue ball across a billiard table and trying to discover where it is going by bouncing the 8-ball off of it; by making the measurement with the 8-ball you have certainly altered the course of the cue ball. You may have discovered where the cue ball was, but now have no idea of where it is going (because you were measuring with the 8-ball instead of actually looking at the table).
Werner Heisenberg was the first to realize that certain pairs of measurements have an intrinsic uncertainty associated with them. For instance, if you have a very good idea of where something is located, then, to a certain degree, you must have a poor idea of how fast it is moving or in what direction. We don't notice this in everyday life because any inherent uncertainty from Heisenberg's principle is well within the acceptable accuracy we desire. For example, you may see a parked car and think you know exactly where it is and exactly how fast it is moving. But would you really know those things exactly? If you were to measure the position of the car to an accuracy of a billionth of a billionth of a centimeter, you would be trying to measure the positions of the individual atoms which make up the car, and those atoms would be jiggling around just because the temperature of the car was above absolute zero!
Heisenberg's uncertainty principle completely flies in the face of classical physics. After all, the very foundation of science is the ability to measure things accurately, and now quantum mechanics is saying that it's impossible to get those measurements exact! But the Heisenberg uncertainty principle is a fact of nature, and it would be impossible to build a measuring device which could get around it.
Spin of a particle
In 1922 Otto Stern and Walther Gerlach performed an experiment whose results could not be explained by classical physics. Their experiment indicated that atomic particles possess an intrinsic angular momentum, or spin, and that this spin is quantized (that is, it can only have certain discrete values). Spin is a completely quantum mechanical property of a particle and cannot be explained in any way by classical physics.
It is important to realize that the spin of an atomic particle is not a measure of how it is spinning! In fact, it is impossible to tell whether something as small as an electron is spinning at all! The word 'spin' is just a convenient way of talking about the intrinsic angular momentum of a particle.
Magnetic resonance imaging (MRI) uses the fact that under certain conditions the spin of hydrogen nuclei can be 'flipped' from one state to another. By measuring the location of these flips, a picture can be formed of where the hydrogen atoms (mainly as a part of water) are in a body. Since tumors tend to have a different water concentration from the surrounding tissue, they would stand out in such a picture.
What is the Schrödinger equation?
Every quantum particle is characterized by a wave function. In 1925 Erwin Schrödinger developed the differential equation which describes the evolution of those wave functions. By using Schrödinger's equation scientists can find the wave function which solves a particular problem in quantum mechanics. Unfortunately, it is usually impossible to find an exact solution to the equation, so certain assumptions are used in order to obtain an approximate answer for the particular problem.
Schrodinger equation {5 kB}
What is a wave packet?
As mentioned earlier, the Schrödinger equation for a particular problem cannot always be solved exactly. However, when there is no force acting upon a particle its potential energy is zero and the Schrödinger equation for the particle can be exactly solved. The solution to this 'free' particle is something known as a wave packet (which initially looks just like a Gaussian bell curve). Wave packets, therefore, can provide a useful way to find approximate solutions to problems which otherwise could not be easily solved.
First, a wave packet is assumed to initially describe the particle under study. Then, when the particle encounters a force (so its potential energy is no longer zero), that force modifies the wave packet. The trick, of course, is to find accurate (and quick!) ways to 'propagate' the wave packet so that it still represents the particle at a later point in time. Finding such propagation techniques, and applying them to useful problems, is the topic of my research.
1. Claude Cohen-Tannoudji, Bernard Diu, and Franck Laloë, Quantum Mechanics, Volumes 1 and 2, John Wiley & Sons, New York (1977).
2. John J. Brehm and William J. Mullin, Introduction to the Structure of Matter: A Course in Modern Physics, John Wiley & Sons, New York (1989).
3. Donald A. McQuarrie, Quantum Chemistry, University Science Books, Mill Valley, Calif. (1983).
A Few Places That Refer to This Page
Links2Go Key Resource in Quantum Physics
The American Institute of Physics, as a part of a series about the achievements of Albert Einstein.
A Hypernote(#7) in an article in Science magazine (vol. 282, 23 Oct 1998, pp. 637-638) about quantum teleportation.
Professor David Banach's Philosophy of Science homepage.
A science link at Sandhills Community College.
Phil Plait's Bad Astronomy pages, dealing with a question about electron probability.
Useful Links
Basic Ideas of Quantum Mechanics.
Quantum Mechanics Overview.
Written by Todd Stedl (
Last modified on 25 July 1996.
Minor revisions on 25 March 2000.
Reconstructed on 18 August 2004 after a server meltdown.
Moved to on 31 July 2005.
We use AN Hosting: better support and better security than our previous web host. |
ff9a9f3afcad5448 | Free chemistry software - molecular modeling
Molecular modeling is a very large and important field of chemistry. As computers have increased in raw computing power, the usage of computers to calculate molecular properties is not a specialized fields for the few. Today, every chemist can perform calculation for even large molecules.
Roughly speaking, you can divide the calculations in two separate groups: molecular mechanics and quantum chemistry. The first is based on classical mechanics, while the second group uses quantum mechanics as a underlaying model and equations. The software list at http://en.wikipedia.org/wiki/Molecular_modeling shows a wide range of offerings. Many of them are commercial and closed-source solutions.
Molecular mechanics calculations are used when no chemistry is going to. By definition, chemistry is the rearrangement of atoms - and that involves electrons. But molecular mechanics can be used to investigate how a molecule is solvent that is, how its structure changing when it is surrounded by a solvent like water.
Gromacs is one of the oldest and most successful molecular mechanics software suites. It is covered by the GNU General Public License (version 2), and most distributions like Debian GNU/Linux have a package of it. It does not come with a fancy user interface, and the user primarily interacts with Gromacs at the command line. It is a big advantage as some of the operations can take a long time. Seldomly, you sit by your computer and work with Gromacs. The typical usage is to write small shell scripts and run them as batch jobs.
Today, most supercomputers in the chemical industry and academia are Linux clusters which are build from commodity hardware. That means that a supercomputer is a distributed system where the individual processors are loosely coupled using Ethernet (or maybe InfiniBand). The MPI framework is used by Gromacs to utilize such a supercomputer (if you have a SMP system, MPI can still be used for parallellization). Queuing systems (SUN Grid Engine, OpenPBS/Torque, etc.) schedule which batch job to execute, and the command-line nature of Gromacs comes to its rights on such systems.
Explaining all details of Gromacs is not the scope here. But let us a quick tour on how to use some of the many utilities and programs of Gromacs. The assignment is to take the experimentally determined structure a small biological active molecule and create a solvated version of the molecule. The structure found at the Protein DataBank is for a crystal, and IGF-1 (as most other molecules in your body) is in a solution where water is the solvent (remember, 60 % of you body is water). For the tour, the Insulin-like growth factor 1 (IGF-1) is chosen. IGF-1 is a small protein (or peptide) which is involved of the growth and regeneration of your body. You can download a file with the experimental structure from Protein DataBank.
First, you must pre-process the downloaded file into files used by Gromacs. In that process you decided the force field. The force field is the parametrization of the interaction between the atoms, and all calculations in Gromacs (and any other molecular mechanical program) are based on Newton's second law. In the command-line below, two files are generated (2GF1.gro and 2GF1.top).
pdb2gmx -f 2GF1.pdb -o 2GF1.gro -p 2GF1.top -ignh -ff G53a6
Now you have to edit the output file (2GF1.gro) in order to change box size. You can do an energy minimization and generate a solvation box using the commands (some steps might take some time):
mdrun -v -deffnm 2GF1-EM-vacuum -c 2GF1-EM-vacuum.gro
editconf -f 2GF1-EM-vacuum.gro -o 2GF1-PBC.gro -bt dodecahedron -d 1.2
genbox -cp 2GF1-PBC.gro -cs spc216.gro -p 2GF1.top -o 2GF1-water.gro
The final file is 2GF1-water.gro which is the biological molecule solvated in water. It might not sound as a great deal, but the file can be used in further simulation involving the solvated molecule.
Other molecular modeling packages exists. NAMD is a highly scalable molecular dynamics program. It is aimed at large molecules (proteins) and can utilize very large parallel computers. But NAMD is not free software as defined by Free Software Foundation. You can download it and use it for any non-commercial purpose.
Free chemistry software - utilities
Free chemistry software - utilities
One of the major annoyances as chemists in front of computer is faced with is the vast number of file formats. The good news is that most file formats are text files so it is possible to reverse engineer them by looking at a number of examples. One open source project called OpenBabel tries to help chemists in converting between the formats (currently OpenBabel supports 113 file formats related to chemistry). Most Linux distributions have packages for OpenBabel, including Debian GNU/Linux (it's a version from 2009 you find in Debian stable). Converting a molecular structure of caffeine from one file format (SDF) to another (PDB) is simply done by the following command:babel -isdf caffeine.sdf -opdb caffeine.pdb
You can find many small molecules - with 3D structures, physical properties and toxicology data - at PubChem. For larger molecules (proteins mainly), you can go to the Protein Data Bank. The file for caffeine as used above can be found at PubChem.
OpenBabel project also includes a number of other utilities including a chemist's version of grep called obgrep (searching for molecules with a particular substructure within a database) and simple program to (energy) minimize a molecule called obminimize.
GNOME Chemistry Utils is a set of utilities developed for GNOME users. The set includes a calculator (for calculating the molecular mass of a molecule), the periodic table of the elements, and a spectrum viewer. The periodic table of the elements can give you the physical and chemical properties of all elements. Most chemists have a periodic table of elements close when working,
and having one on your desktop seems as a good idea.
Chemists do a lot of drawing: they draw structures of molecules. In can be regarded as a generic representation of a molecules 3-dimensional structure using a 2D paper. Understanding and drawing such chemical structures are an integral part of any chemist's education and chemists have used these drawing for more than 150 years (the discovery of the electron and the development of quantum mechanics changed the view of molecular structures). The 3-dimensional geometry is an important factor for determine the properties (reactivity, toxicology, color, etc.) of a molecule.
A drawing program for chemists is not hard to image. When it comes to free software, we are so
lucky that we have more than one. GNOME users can use the molecular drawing program from the GNOME Chem
istry Utils project. It is called GChemPaint. As GChemPaint can only load a rather small number of file formats, you really learn to use OpenBabel rather quickly. It is an easy program to work with, and it is possible to save your drawing in most used image formats (both bitmap and vector formats). You can then easily insert your drawing
in your favorite word processing software prior to publication (take publication rather broad: everything from a high-school report to a paper in Nature).
As already said, drawing programs for chemists are not hard to imagine. Other projects in this area include titles likes bkchem, chemtool, easychem, xdrawchem, jchempaint, molsKetch (probably stalled).
Free chemistry software - Introduction
The year of 2011 has been declared the International Year of Chemistry by UNESCO (United Nations Educational, Scientific and Cultural Organization) and IUPAC (International Union of Pure and Applied Chemistry). The purpose of devoting a full year to chemistry is to spread the notion that chemistry is important for our daily life.
In this series of blog post I have examine the state of free software in chemistry. This first post is an outline of the usage of computers in chemistry.
It is hard to imagine modern life without the discoveries and developments done by chemists and chemical engineers over the last two or three centuries. Plastic, gasoline, and pharmaceuticals are products from the chemical industry. And forensic scientists use many chemical analysis in order to provide evidence for police investigations all over the world. But chemistry is more that an applied science. It also give us an insight to how our world works. In the recent decade, the modern cuisine has changed. For example, the cheif Heston Blumenthal has been using chemistry to create new dishes (this branch of chemistry is called molecular gastronomy).
As you can see, chemistry is a broad science and engineering discipline. Modern chemistry is divided into a number of branches. Traditionally, an academic education of chemistry consists of courses in general, organic, inorganic, physical and even analytical chemistry. Chemistry is a wet science, and as a student you spend a lot of time in laboratories. Amongst chemists, it is still discussed whether chemistry is a descriptive science (classification of observations) or an exact science (explaining observations).
Chemistry interfaces most other sciences, including physics, mathematics, statistics and biology. Quantum chemistry applies quantum mechanics to calculate properties of chemical substances. But as you might imagine, the three-body problem is a serious show-stopper for a chemist as very few molecules have only three nucleus and electrons.
Computers are heavily used in chemistry. One example is to perform quantum chemistry calculation as finding an analytical solution for a many-body problem is impossible. A rough break-down of the usage of computers in chemistry consists of three major areas. Firstly, you have the end-user applications used by every chemist. The applications are domain-specific applications - the domain is as broad as chemistry. The second area is chemoinformatics. It is a fairly young area (a decade or two only). Chemoinformatics applies techniques from informatics to transform chemical data to knowledge and thereby improving the decision making process. The usage of specialized databases and search algorithms is an integral part of chemoinformatics. Any non-trivial chemical compound can be represented in a number of ways. Even a small molecule like styrene can be named in different ways depending on how you look at it. Chemoinformaticians have introduced a string representation for all chemical compounds called the simplified molecular input line entry specification (SMILES). The SMILES code for styrene is C=CC1=CC=CC=C1. Image to find all compounds in your database with a certain substructure. You cannot use a regular expression or an SQL query. As molecules can be regarded as graphs (atoms are connected by chemical bonds), searching in chemical databases is a variant of find subgraphs. This is the core of chemoinformatics.
The third area where computers are used in chemistry is to perform calculations and it is often refered to as computational chemistry. It is an old area - calculation and simulation of properties of chemical compounds and reaction have been carried out as long as computers have been available to scientists. The calculations either use a classical-mechanical approach or a quantum mechanical approach. In the first approach, electrons is neglected and a force-field between the atoms are applied. This is possible to simulate large molecules using this approach. But if you need to predict the energy levels, thermodynamical properties, and charge distribution of a molecule, you have to use a quantum mechanical approach. This involves solving the time-independent Schrödinger equation (or at least an approximation to the equation called the Born-Oppenheimer approximation).
It is important to understand that most chemists are not educated as programmers. On the other hand, using computational techniques can save chemical industries huge fortunes. Today, most pharmaceutical companies have specialized departments for performing chemical calculations and supporting an informatics infrastructure. These departments are small in terms of man-power compared to the company as a whole. As the market is small and the potential benefits huge (time-to-market and saving expensive laboratory time), vendors often ask for very high license-fees. Vendors like Schrödinger, Wavefunction and Gaussian, and OpenEye offer software packages for chemists. Sadly, free software is a minor player in chemistry but you can find free chemistry software for most needs. |
9efde7a1f3d8f38e |
Time crystals would be a perpetuum mobile
One of the widely shared recent articles at was
Time crystals—how scientists created a new state of matter
What's going on?
Already at this point, you may see a misconception that leads to Wilczek's impossible proposals. There's a very general implicit problem in his usage of the word "material" or "object" for something that has some properties at different moments of time. Sorry but a "material" or an "object" is fully described by some information at a single moment of time, e.g. \(t=0\). If you need to talk about the values of observables at many or all values of time \(t\), then you are not talking about a "material" or "object" but rather a process. The misguided analogies between "crystals" and hypothetical "time crystals" may be said to result from this confusion mixing objects and processes. By the way, there were lots of the very same confusion in the literature about S-branes (Strominger or spacelike branes).
But let us look at the problem from a slightly different, although basically equivalent, angle.
More conceptually, Wilczek's time crystal is defined as an object that has the property that in its ground state (the state of lowest energy), an observable that we may call \(x(t)\) is a non-constant or periodic function of time. Something keeps on spinning indefinitely.
There exists a name for a hypothetical object that is oscillating indefinitely. The name is perpetuum mobile, or a perpetual motion machine of the first kind. In his original paper, Wilczek is aware of this point and acknowledges that his gadget therefore perilously close to fitting the definition of a perpetual motion machine.
Well, he's far too modest here. His gadget wouldn't be just close to a perpetuum mobile; it would be one. Wilczek just rebranded the perpetual motion machines just like cold fusion was rebranded as "low energy nuclear reactions" and creationism was rebranded as "the intelligent design". The main difference between the millions of "inventors" of perpetuum mobile in the past and Frank Wilczek is, we are led to believe, that Frank Wilczek is really smart and a Nobel prize winner and so on, so unlike the numerous losers and/or crackpots before him, he actually succeeded.
Design by a predecessor of Wilczek's.
The new papers don't show anything of the sort, I am confident although I haven't read them in their entirety. They just present some atomic physics systems that respond periodically when they're stimulated by some periodic pulses of laser light or something like that. What a shock that sustained, periodic external influences lead to sustained, periodic responses. As far as I can see, these "insights" are absolutely trivial and absolutely don't justify the statement that Wilczek's claims were shown true. I don't have enough motivation to read these papers because I find it obvious that they're just papers by clueless experimenters who observe something, they don't understand what they're actually seeing, and they say that it agrees with some theorist's wrong paper.
Shortly after Wilczek published his 2012 paper, exchanges between Wilczek and Patrick Bruno, a critic who indeed said that Wilczek's objects are impossible because they're perpetual motion machines, began. I guess that this 2013 paper with a no-go theorem was the last salvo by Bruno in his battles. Watanabe and Oshikawa added another no-go-theorem in their 2014 paper.
Instead of discussing any specifics of the experiments that just ignore all these results, let me say what I think is the deep theoretical misconception that led Wilczek to say all these things.
He clearly thinks that the spatial translations and temporal translations are analogous – after all, space and time are linked by the Lorentz symmetry in special relativity and they are naturally unified to spacetime translations – and because it's possible to spontaneously break the spatial translations (by creating crystals), it must be possible to do the same with temporal translations (by creating time crystals), so the only remaining task is to decide how to do it nicely.
But this "complete democracy" between space and time is wrong for a simple reason.
The reason is that things in the spacetime evolve and the observables (e.g. fields) aren't quite independent in every spacetime point. At most, you may determine the initial conditions e.g. at a spacelike hypersurface \(t=0\). Once you determine the fields (and their derivatives or canonical momenta) at \(t=0\), their values in the whole spacetime are determined by the field equations of motion. The surface where you can pick the initial conditions is referred to as the Cauchy surface and even in general relativity where many things are flexible, there exist very good reasons why this surface should better be space-like and contain no timelike vectors in it.
In quantum field theory, the reason why timelike Cauchy slices would be no good is simple: the field equations guarantee that timelike-separated fields almost always have nonzero commutators. So you simply can't determine these values independently because of the uncertainty principle!
Because the Cauchy surface is spacelike and not timelike, the "complete democracy" between space and time is broken. The equations of motion and commutators etc. are still Lorentz-covariant but the required spacelike signature of the Cauchy surfaces implies that you have much less freedom to determine how things depend on time than the freedom to decide how things depend on the spatial coordinates. And that's probably the key point that Wilczek is overlooking.
So if you choose the most general object that may exist in the spacetime, you may always determine it by some information at a Cauchy surface which is morally equivalent to the three-dimensional spacelike \(t=0\) hypersurface. And whether or not its evolution in time will be constant or non-constant, periodic or aperiodic, and damped or not damped, is completely determined by the dynamical laws of your physical theory. You simply cannot prescribe these things.
That differs from the case of ordinary crystals where you have the freedom to distribute the atoms to the lattice sites in the three-dimensional space. But you simply don't have the freedom to dictate whether some peaks reappear periodically after every period \(\Delta t\). The question what happens is dictated by the dynamical laws of physics.
Another pre-Wilczek model of a classical time crystal. Such concepts remind me of the big-government leftists. It's spectacularly clear that the more redistribution or more moving parts you add, the more energy (or money) the process will cost, waste, or demand for the machinery to run. But they always ignore or understate some energy cost, driven by the unchangeable belief that the perpetuum mobile or the big government is a great idea
OK, do the dynamical laws of physics allow the ground state of an object to indefinitely oscillate, to have an observable \(x(t)\) that is non-constant, like \(x_0\cos\omega t\)? Let's use the elementary rules of quantum mechanics to say something about this question. Well, it's easy. If the state \(\ket\psi\) is a ground state, it really means that it is an eigenstate of the energy operator\[
H\ket\psi = E_0 \ket\psi
\] where \(E_0\) is the lowest eigenvalue in the whole spectrum. But we don't even need to know that it's the lowest one – although this was the statement that Bruno – focusing on particular systems proposed by Wilczek – was proving incorrect. I only need that the state is an eigenstate. As undergraduate students learn in the first lectures of quantum mechanics – when they are taught about the time-independent and time-dependent Schrödinger equation – the evolution of the energy eigenstate in time is unavoidably stationary,\[
\ket{\psi(t)} = \exp(Ht / i\hbar) \ket{\psi(0)}.
\] Only the overall phase of the state is changing with time. That phase has no physical consequences (at an isolated moment of time) and it cancels in all the expectation values etc. which are therefore constant:\[
\bra{\psi(t)} x(t) \ket{\psi(t)} = {\rm const}.
\] So the ground states are simply stationary and no observable that may be measured in them may oscillate. Period. That's it. There are no quantum time crystals.
If some expectation value in a state depends on time, the state must unavoidably be a superposition of many energy eigenstates corresponding to different eigenvalues of the energy. You may split the state to pieces and pick the lowest-energy eigenstate contribution in it. And this true ground state will be stationary.
Even though the relativistic equations respect the Lorentz symmetry which is some kind of a "perfect" symmetry between the space and time, it's still true that relativity doesn't question the qualitative difference between objects that are timelike and objects that are spacelike. Indeed, whether e.g. a spacetime interval is timelike or spacelike is a question that all inertial observers will agree upon – the invariant squared length of the interval is either positive or negative and its value is Lorentz-invariant, a reason why Einstein was tempted to use the term Invariantentheorie for the theory that we know as the theory of relativity.
So world lines of true objects in consistent theories have to be timelike (or at most null) and not spacelike. This asymmetric treatment of the two signs is not in any conflict with the Lorentz symmetry because the Lorentz transformations do preserve the qualitative (timelike vs spacelike) character of intervals. For the same reason, one may consistently choose the initial conditions at spacelike Cauchy surfaces, but not surfaces of a mixed signature. As I mentioned, this difference boils down to the fact that fields' commutators vanish at spacelike separation but not timelike separation, so one can only determine the fields independently at spacelike hypersurfaces. Similarly, one can observe or build systems that spontaneously and permanently break the translational symmetry in space but not those that spontaneously and permanently break the translational symmetry in time.
Unless I am wrong, Wilczek's reasoning is probably rooted in "overinterpreting" the Lorentz symmetry in a certain way. Alternatively, we may say that people with common sense know that a perpetuum mobile is impossible – and within quantum mechanics, this fact is just demonstrated using a different formalism. It could of course be conceivable that quantum mechanics changes things so radically that it could allow the perpetual motion machines – after all, it allows the quantum tunneling and many other things that are impossible classically. But in the case of the perpetual motion machines, it just isn't the case.
Spinning nuclei
Under a 2012 blog post about the electron's electric quadrupole moment – which has to be zero (like the tensor properties of all particles with \(j=0\) or \(j=1/2\)) by the Wigner-Eckart theorem – someone asked how it's possible that people often say that uranium-238 is cigar-shaped i.e. has some "quadrupole moment" even though it has \(j=0\). He has also mentioned two \(j=1/2\) nuclei that are sometimes hinted to have a non-spherical shape, too.
It's confusing but if you really had just the state with \(j=0\) and nothing else, the Wigner-Eckart theorem – a general group-theoretical consequence of the addition of the angular momentum in quantum mechanics – would require all (traceless symmetric) tensors to be zero. That includes the ordinary and electric quadrupole moments. The spin \(j=0\) or \(j=1/2\) mean that the particles doesn't even carry enough spin-related information to remember the (sign-independent) axis along which it could be elongated or shrunk.
What's new about the nuclei is that they're composite which means that they have lots of excited states (describing various types of relative motion between the protons and neutrons – or quarks and gluons). In particular, there are states that look like the extra "orbital motion" added to the nucleus' spin. So aside from the \(j=0\) ground state, there is a \(j=2\) and \(j=4\) and \(j=6\) excited state of uranium-238. The dependence of the energy on the angular momentum goes approximately like \(a+bJ^2\) which allows you to extract something like a "moment of inertia" from \(b\). But this \(b\) isn't quite identified with the expectation value of any tensor in the state with \(j=0\) itself – that expectation value simply has to be zero.
Also, the states of uranium-238 with \(j=2,4,6,\dots\) are "excited" which means that they won't survive forever. These excited ("spinning") states of nuclei will emit a photon (gamma-ray) and drop to a lower value of \(j\) very quickly – they will reach the true ground state with \(j=0\) almost instantaneously. The energy and rates of transition of these jumps may also be used to deduce some nonzero values of the "quadrupole moments" (where the quotation marks indicate that you must be careful about the definition of the object because it's a generalization that doesn't necessarily coincide with other meanings of the phrase) – especially if it is true that the dominant emission is some electric quadrupole radiation. But if the transition results from some quadrupole radiation, the transition is determined from the matrix elements such as\[
\bra{\text{U-238}, j=0} Q_{ij} \ket{\text{U-238}, j=2}
\] which may be nonzero but they related two states of the nucleus. The matrix element above isn't an expectation value, it isn't a property of one state only, especially not the ground state.
So Wilczek's perpetuum mobile doesn't work for the nuclei, either. If the nuclei are spinning in a way that has some "classical component" – in the sense that some expectation value of an observable would be time-dependent – then they are superpositions of many energy eigenstates and the higher ones will collapse to the lower ones (e.g. by the emission of gamma quanta). At the end, you are left with the true ground state that simply has to be stationary.
For this reason, the temporal translational symmetry cannot be spontaneously broken, at least not in the sense envisioned by Wilczek. Note that the previous paragraph talks about the collapse to a lower state. This description works because the spectrum of energy is bounded from below. That's how it differs from the spatial counterpart, the momentum which is unbounded – both signs of any component of the momentum are equally good. You could view this "boundedness from below" as another example of the "qualitative differences" between spacelike and timelike entities in relativity. The combination \(k_\mu p^\mu\) of components of the energy-momentum operator \(p^\mu\) has a spectrum that is bounded from below exactly if \(k_\mu\) is timelike i.e. if \(k^\mu k_\mu\leq 0\), assuming the timelike (mostly minus) signature.
This "discrimination against" the timelike operators – and timelike crystals – is totally compatible with the Lorentz symmetry because the Lorentz symmetry only allows to transform spacetime intervals (or components of vectors or tensors, or slices, or other entities) of the same qualitative type to each other.
Another topic: LIGO and axions
Adrian Cho at Science discusses an April 2016 paper by Arvanitaki, Dimopoulos, et al. that just appeared in PRD.
When advanced LIGO sees thousands of black hole mergers in coming years, they say, it could also see signs of axionic (dark matter) waves created with the help of the black hole horizons – assuming that the axion mass is a picoelectronvolt, plus minus (or times over?) two orders of magnitude. Some smart folks say that it's a more exciting new thing that could be observed by LIGO than all the previous "possible future discoveries".
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1c6b72c543caab30 | viernes, 7 de mayo de 2010
• is a consequence of quantum theory that affects virtually all physical systems
• arises from unavoidable interaction of these systems with their natural environment
• explains why macroscopic systems seem to possess their familiar classical properties
No additional classical concepts are required for a consistent quantum description
• explains why certain microscopic objects ("particles") seem to be localized in space
There are no particles
• explains why microscopic systems are usually found in their energy eigenstates (and therefore seem to jump between them)
There are no quantum jumps
• thus explains why there appeared to be contradictory levels of description in physics (classical and quantum)
There is but ONE basic framework for all physical theories: quantum theory
• explains also how the Schrödinger equation of general relativity (the Wheeler-DeWitt equation) may describe the appearance of time in spite of being time-less
There is no time at a fundamental level
• is a direct consequence of the Schrödinger equation, but has nonetheless been essentially overlooked during the first 50 years of quantum theory
Decoherence is the theory of universal entanglement. Generically, it does not describe a distortion of the system by the environment, but rather a disturbance (change of state) of the environment by the system. This may nonetheless affect the system itself because of the fundamental quantum aspect of kinematical nonlocality.
The process of decoherence is based on an arrow of time in the form of a special (ultimately cosmological) initial condition.
Decoherence can not explain quantum probabilities without (a) introducing a novel definition of observer systems in quantum mechanical terms (this is usually done tacitly in classical terms), and (b) postulating the required probability measure (according to the Hilbert space norm).
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a22ed626387bee70 | partial differential equation
partial differential equation
In mathematics, an equation that contains partial derivatives, expressing a process of change that depends on more than one independent variable. It can be read as a statement about how a process evolves without specifying the formula defining the process. Given the initial state of the process (such as its size at time zero) and a description of how it is changing (i.e., the partial differential equation), its defining formula can be found by various methods, most based on integration. Important partial differential equations include the heat equation, the wave equation, and Laplace's equation, which are central to mathematical physics.
Learn more about partial differential equation with a free trial on
In mathematics, partial differential equations (PDE) are a type of differential equation, i.e., a relation involving an unknown function (or functions) of several independent variables and its (resp. their) partial derivatives with respect to those variables. Partial differential equations are used to formulate, and thus aid the solution of, problems involving functions of several variables; such as the propagation of sound or heat, electrostatics, electrodynamics, fluid flow, elasticity. Interestingly, seemingly distinct physical phenomena may have identical mathematical formulations, and thus be governed by the same underlying dynamic.
A relatively simple partial differential equation is
frac{partial}{partial x}u(x,y)=0, .
This relation implies that the values u(x,y) are independent of x. Hence the general solution of this equation is
u(x,y) = f(y),,
where f is an arbitrary function of y. The analogous ordinary differential equation is
which has the solution
u(x) = c,,
where c is any constant value (independent of x). These two examples illustrate that general solutions of ordinary differential equations involve arbitrary constants, but solutions of partial differential equations involve arbitrary functions. A solution of a partial differential equation is generally not unique; additional conditions must generally be specified on the boundary of the region where the solution is defined. For instance, in the simple example above, the function f(y) can be determined if u is specified on the line x=0.
Existence and uniqueness
Although the issue of the existence and uniqueness of solutions of ordinary differential equations has a very satisfactory answer with the Picard-Lindelöf theorem, that is far from the case for partial differential equations. There is a general theorem (the Cauchy-Kovalevskaya theorem) that states that the Cauchy problem for any partial differential equation that is analytic in the unknown function and its derivatives have a unique analytic solution. Although this result might appear to settle the existence and uniqueness of solutions, there are examples of linear partial differential equations whose coefficients have derivatives of all orders (which are nevertheless not analytic) but which have no solutions at all: see Lewy (1957). Even if the solution of a partial differential equation exists and is unique, it may nevertheless have undesirable properties. The mathematical study of these questions is usually in the more powerful context of weak solutions.
An example of pathological behavior is the sequence of Cauchy problems (depending upon n) for the Laplace equation
frac{part^2 u}{partial x^2} + frac{part^2 u}{partial y^2}=0,,
with initial conditions
u(x,0) = 0, ,
frac{partial u}{partial y}(x,0) = frac{sin n x}{n},,
where n is an integer. The derivative of u with respect to y approaches 0 uniformly in x as n increases, but the solution is
u(x,y) = frac{(sinh ny)(sin nx)}{n^2}.,
This solution approaches infinity if nx is not an integer multiple of π for any non-zero value of y. The Cauchy problem for the Laplace equation is called ill-posed or not well posed, since the solution does not depend continuously upon the data of the problem. Such ill-posed problems are not usually satisfactory for physical applications.
In PDEs, it is common to denote partial derivatives using subscripts. That is:
u_x = {partial u over partial x}
u_{xy} = {part^2 u over partial y, partial x} = {partial over partial y } left({partial u over partial x}right).
Especially in (mathematical) physics, one often prefers use of del (which in cartesian coordinates is written nabla=(part_x,part_y,part_z), ) for spatial derivatives and a dot dot u, for time derivatives, e.g. to write the wave equation (see below) as
ddot u=c^2triangle u.,(math notation)
ddot u=c^2nabla^2u.,(physics notation)
Heat equation in one space dimension
The equation for conduction of heat in one dimension for a homogeneous body has the form
u_t = alpha u_{xx} ,
where u(t,x) is temperature, and α is a positive constant that describes the rate of diffusion. The Cauchy problem for this equation consists in specifying u(0,x)= f(x), where f(x) is an arbitrary function.
General solutions of the heat equation can be found by the method of separation of variables. Some examples appear in the heat equation article. They are examples of Fourier series for periodic f and Fourier transforms for non-periodic f. Using the Fourier transform, a general solution of the heat equation has the form
u(t,x) = frac{1}{sqrt{2pi}} int_{-infty}^{infty} F(xi) e^{-alpha xi^2 t} e^{i xi x} dxi, ,
where F is an arbitrary function. In order to satisfy the initial condition, F is given by the Fourier transform of f, that is
F(xi) = frac{1}{sqrt{2pi}} int_{-infty}^{infty} f(x) e^{-i xi x}, dx. ,
If f represents a very small but intense source of heat, then the preceding integral can be approximated by the delta distribution, multiplied by the strength of the source. For a source whose strength is normalized to 1, the result is
F(xi) = frac{1}{sqrt{2pi}}, ,
and the resulting solution of the heat equation is
u(t,x) = frac{1}{2pi} int_{-infty}^{infty}e^{-alpha xi^2 t} e^{i xi x} dxi. ,
This is a Gaussian integral. It may be evaluated to obtain
u(t,x) = frac{1}{2sqrt{pi alpha t}} expleft(-frac{x^2}{4 alpha t} right). ,
This result corresponds to a normal probability density for x with mean 0 and variance 2αt. The heat equation and similar diffusion equations are useful tools to study random phenomena.
Wave equation in one spatial dimension
The wave equation is an equation for an unknown function u(t, x) of the form
u_{tt} = c^2 u_{xx}. ,
Here u might describe the displacement of a stretched string from equilibrium, or the difference in air pressure in a tube, or the magnitude of an electromagnetic field in a tube, and c is a number that corresponds to the velocity of the wave. The Cauchy problem for this equation consists in prescribing the initial displacement and velocity of a string or other medium:
u(0,x) = f(x), ,
u_t(0,x) = g(x), ,
where f and g are arbitrary given functions. The solution of this problem is given by d'Alembert's formula:
u(t,x) = frac{1}{2} left[f(x-ct) + f(x+ct)right] + frac{1}{2c}int_{x-ct}^{x+ct} g(y), dy. ,
This formula implies that the solution at (t,x) depends only upon the data on the segment of the initial line that is cut out by the characteristic curves
x - ct = hbox{constant,} quad x + ct = hbox{constant}, ,
that are drawn backwards from that point. These curves correspond to signals that propagate with velocity c forward and backward. Conversely, the influence of the data at any given point on the initial line propagates with the finite velocity c: there is no effect outside a triangle through that point whose sides are characteristic curves. This behavior is very different from the solution for the heat equation, where the effect of a point source appears (with small amplitude) instantaneously at every point in space. The solution given above is also valid if t is negative, and the explicit formula shows that the solution depends smoothly upon the data: both the forward and backward Cauchy problems for the wave equation are well-posed.
Spherical waves
Spherical waves are waves whose amplitude depends only upon the radial distance r from a central point source. For such waves, the three-dimensional wave equation takes the form
u_{tt} = c^2 left[u_{rr} + frac{2}{r} u_r right]. ,
This is equivalent to
(ru)_{tt} = c^2 left[(ru)_{rr} right],,
and hence the quantity ru satisfies the one-dimensional wave equation. Therefore a general solution for spherical waves has the form
u(t,r) = frac{1}{r} left[F(r-ct) + G(r+ct) right],,
where F and G are completely arbitrary functions. Radiation from an antenna corresponds to the case where G is identically zero. Thus the wave form transmitted from an antenna has no distortion in time: the only distorting factor is 1/r. This feature of undistorted propagation of waves is not present if there are two spatial dimensions.
Laplace equation in two dimensions
The Laplace equation for an unknown function of two variables φ has the form
varphi_{xx} + varphi_{yy} = 0.
Solutions of Laplace's equation are called harmonic functions.
Connection with holomorphic functions
Solutions of the Laplace equation in two dimensions are intimately connected with analytic functions of a complex variable (a.k.a. holomorphic functions): the real and imaginary parts of any analytic function are conjugate harmonic functions: they both satisfy the Laplace equation, and their gradients are orthogonal. If f=u+iv, then the Cauchy-Riemann equations state that
u_x = v_y, quad v_x = -u_y,,
and it follows that
u_{xx} + u_{yy} = 0, quad v_{xx} + v_{yy}=0. ,
Conversely, given any harmonic function in two dimensions, it is the real part of an analytic function, at least locally. Details are given in Laplace equation.
A typical boundary value problem
A typical problem for Laplace's equation is to find a solution that satisfies arbitrary values on the boundary of a domain. For example, we may seek a harmonic function that takes on the values u(θ) on a circle of radius one. The solution was given by Poisson:
varphi(r,theta) = frac{1}{2pi} int_0^{2pi} frac{1-r^2}{1 +r^2 -2rcos (theta -theta')} u(theta')dtheta'.,
Petrovsky (1967, p. 248) shows how this formula can be obtained by summing a Fourier series for φ. If r<1, the derivatives of φ may be computed by differentiating under the integral sign, and one can verify that φ is analytic, even if u is continuous but not necessarily differentiable. This behavior is typical for solutions of elliptic partial differential equations: the solutions may be much more smooth than the boundary data. This is in contrast to solutions of the wave equation, and more general hyperbolic partial differential equations, which typically have no more derivatives than the data.
Euler-Tricomi equation
The Euler-Tricomi equation is used in the investigation of transonic flow.
u_{xx} , =xu_{yy}.
Advection equation
The advection equation describes the transport of a conserved scalar psi in a velocity field {bold u}=(u,v,w). It is:
psi_t+(upsi)_x+(vpsi)_y+(wpsi)_z , =0.
If the velocity field is solenoidal (that is, nablacdot{bold u}=0), then the equation may be simplified to
psi_t+upsi_x+vpsi_y+wpsi_z , =0.
The one dimensional steady flow advection equation psi_t+u.psi_x=0 (where u is constant) is commonly referred to as the pigpen problem. If u is not constant and equal to psi the equation is referred to as Burgers' equation.
Ginzburg-Landau equation
The Ginzburg-Landau equation is used in modelling superconductivity. It is
iu_t+pu_{xx} +q|u|^2u , =igamma u where p,qinmathbb{C} and gammainmathbb{R} are constants and i is the imaginary unit.
The Dym equation
The Dym equation is named for Harry Dym and occurs in the study of solitons. It is
u_t , = u^3u_{xxx}.
Initial-boundary value problems
Many problems of Mathematical Physics are formulated as initial-boundary value problems.
Vibrating string
If the string is stretched between two points where x=0 and x=L and u denotes the amplitude of the displacement of the string, then u satisfies the one-dimensional wave equation in the region where 0<x<L and t is unlimited. Since the string is tied down at the ends, u must also satisfy the boundary conditions
u(t,0)=0, quad u(t,L)=0, ,
as well as the initial conditions
u(0,x)=f(x), quad u_t(0,x)=g(x). ,
The method of separation of variables for the wave equation
u_{tt} = c^2 u_{xx}, ,
leads to solutions of the form
u(t,x) = T(t) X(x),,
T + k^2 c^2 T=0, quad X + k^2 X=0,,
where the constant k must be determined. The boundary conditions then imply that X is a multiple of sin kx, and k must have the form
k= frac{npi}{L}, ,
where n is an integer. Each term in the sum corresponds to a mode of vibration of the string. The mode with n=1 is called the fundamental mode, and the frequencies of the other modes are all multiples of this frequency. They form the overtone series of the string, and they are the basis for musical acoustics. The initial conditions may then be satisfied by representing f and g as infinite sums of these modes. Wind instruments typically correspond to vibrations of an air column with one end open and one end closed. The corresponding boundary conditions are
X(0) =0, quad X'(L) = 0.,
The method of separation of variables can also be applied in this case, and it leads to a series of odd overtones.
The general problem of this type is solved in Sturm-Liouville theory.
Vibrating membrane
If a membrane is stretched over a curve C that forms the boundary of a domain D in the plane, its vibrations are governed by the wave equation
frac{1}{c^2} u_{tt} = u_{xx} + u_{yy}, ,
if t>0 and (x,y) is in D. The boundary condition is u(t,x,y) = 0 if (x,y) is on C . The method of separation of variables leads to the form
u(t,x,y) = T(t) v(x,y),,
which in turn must satisfy
frac{1}{c^2}T +k^2 T=0, ,
v_{xx} + v_{yy} + k^2 v =0.,
The latter equation is called the Helmholtz Equation. The constant k must be determined in order to allow a non-trivial v to satisfy the boundary condition on C. Such values of k2 are called the eigenvalues of the Laplacian in D, and the associated solutions are the eigenfunctions of the Laplacian in D. The Sturm-Liouville theory may be extended to this elliptic eigenvalue problem (Jost, 2002).
There are no generally applicable methods to solve non-linear PDEs. Still, existence and uniqueness results (such as the Cauchy-Kovalevskaya theorem) are often possible, as are proofs of important qualitative and quantitative properties of solutions (getting these results is a major part of analysis). Computational solution to the nonlinear PDEs, the Split-step method, exist for specific equations like nonlinear Schrödinger equation.
Nevertheless, some techniques can be used for several types of equations. The h-principle is the most powerful method to solve underdetermined equations. The Riquier-Janet theory is an effective method for obtaining information about many analytic overdetermined systems.
The method of characteristics (Similarity Transformation method) can be used in some very special cases to solve partial differential equations.
In some cases, a PDE can be solved via perturbation analysis in which the solution is considered to be a correction to an equation with a known solution. Alternatives are numerical analysis techniques from simple finite difference schemes to the more mature multigrid and finite element methods. Many interesting problems in science and engineering are solved in this way using computers, sometimes high performance supercomputers.
Other examples
The Schrödinger equation is a PDE at the heart of non-relativistic quantum mechanics. In the WKB approximation it is the Hamilton-Jacobi equation.
Except for the Dym equation and the Ginzburg-Landau equation, the above equations are linear in the sense that they can be written in the form Au = f for a given linear operator A and a given function f. Other important non-linear equations include the Navier-Stokes equations describing the flow of fluids, and Einstein's field equations of general relativity.
Also see the list of non-linear partial differential equations.
Some linear, second-order partial differential equations can be classified as parabolic, hyperbolic or elliptic. Others such as the Euler-Tricomi equation have different types in different regions. The classification provides a guide to appropriate initial and boundary conditions, and to smoothness of the solutions.
Equations of first order
Equations of second order
Assuming u_{xy}=u_{yx}, the general second-order PDE in two independent variables has the form
Au_{xx} + Bu_{xy} + Cu_{yy} + cdots = 0,
where the coefficients A, B, C etc. may depend upon x and y. This form is analogous to the equation for a conic section:
Ax^2 + Bxy + Cy^2 + cdots = 0.
Just as one classifies conic sections into parabolic, hyperbolic, and elliptic based on the discriminant B^2 - 4AC, the same can be done for a second-order PDE at a given point.
1. B^2 - 4AC , < 0 : solutions of elliptic PDEs are as smooth as the coefficients allow, within the interior of the region where the equation and solutions are defined. For example, solutions of Laplace's equation are analytic within the domain where they are defined, but solutions may assume boundary values that are not smooth. The motion of a fluid at subsonic speeds can be approximated with elliptic PDEs, and the Euler-Tricomi equation is elliptic where x<0.
2. B^2 - 4AC = 0, : equations that are parabolic at every point can be transformed into a form analogous to the heat equation by a change of independent variables. Solutions smooth out as the transformed time variable increases. The Euler-Tricomi equation has parabolic type on the line where x=0.
3. B^2 - 4AC , > 0 : hyperbolic equations retain any discontinuities of functions or derivatives in the initial data. An example is the wave equation. The motion of a fluid at supersonic speeds can be approximated with hyperbolic PDEs, and the Euler-Tricomi equation is hyperbolic where x>0.
If there are n independent variables x1, x2 , ..., xn, a general linear partial differential equation of second order has the form
L u =sum_{i=1}^nsum_{j=1}^n a_{i,j} frac{part^2 u}{partial x_i partial x_j} quad hbox{ plus lower order terms} =0. ,
The classification depends upon the signature of the eigenvalues of the coefficient matrix.
1. Elliptic: The eigenvalues are all positive or all negative.
2. Parabolic : The eigenvalues are all positive or all negative, save one which is zero.
3. Hyperbolic: There is only one negative eigenvalue and all the rest are positive, or there is only one positive eigenvalue and all the rest are negative.
4. Ultrahyperbolic: There is more than one positive eigenvalue and more than one negative eigenvalue, and there are no zero eigenvalues. There is only limited theory for ultrahyperbolic equations (Courant and Hilbert, 1962).
Systems of first-order equations and characteristic surfaces
The classification of partial differential equations can be extended to systems of first-order equations, where the unknown u is now a vector with m components, and the coefficient matrices A_nu are m by m matrices for nu=1, dots,n. The partial differential equation takes the form
Lu = sum_{nu=1}^{n} A_nu frac{partial u}{partial x_nu} + B=0, ,
where the coefficient matrices Aν and the vector B may depend upon x and u. If a hypersurface S is given in the implicit form
varphi(x_1, x_2, ldots, x_n)=0, ,
where φ has a non-zero gradient, then S is a characteristic surface for the operator L at a given point if the characteristic form vanishes:
Qleft(frac{partvarphi}{partial x_1}, ldots,frac{partvarphi}{partial x_n}right) =detleft[sum_{nu=1}^nA_nu frac{partial varphi}{partial x_nu}right]=0.,
The geometric interpretation of this condition is as follows: if data for u are prescribed on the surface S, then it may be possible to determine the normal derivative of u on S from the differential equation. If the data on S and the differential equation determine the normal derivative of u on S, then S is non-characteristic. If the data on S and the differential equation do not determine the normal derivative of u on S, then the surface is characteristic, and the differential equation restricts the data on S: the differential equation is internal to S.
1. A first-order system Lu=0 is elliptic if no surface is characteristic for L: the values of u on S and the differential equation always determine the normal derivative of u on S.
2. A first-order system is hyperbolic at a point if there is a space-like surface S with normal ξ at that point. This means that, given any non-trivial vector η orthogonal to ξ, and a scalar multiplier λ, the equation
Q(lambda xi + eta) =0, ,
has m real roots λ1, λ2, ..., λm. The system is strictly hyperbolic if these roots are always distinct. The geometrical interpretation of this condition is as follows: the characteristic form Q(ζ)=0 defines a cone (the normal cone) with homogeneous coordinates ζ. In the hyperbolic case, this cone has m sheets, and the axis ζ = λ ξ runs inside these sheets: it does not intersect any of them. But when displaced from the origin by η, this axis intersects every sheet. In the elliptic case, the normal cone has no real sheets.
Equations of mixed type
If a PDE has coefficients which are not constant, it is possible that it will not belong to any of these categories but rather be of mixed type. A simple but important example is the Euler-Tricomi equation
u_{xx} , = xu_{yy}
which is called elliptic-hyperbolic because it is elliptic in the region x < 0, hyperbolic in the region x > 0, and degenerate parabolic on the line x = 0.
Methods to solve PDEs
Separation of variables
The method of separation of variables will yield particular solutions of a linear PDE on very simple domains such as rectangles that may satisfy initial or boundary conditions.
Change of variable
Often a PDE can be reduced to a simpler form with a known solution by a suitable change of variables. For example the Black–Scholes PDE
frac{partial V}{partial t} + frac{1}{2}sigma^2 S^2frac{partial^2 V}{partial S^2} + rSfrac{partial V}{partial S} - rV = 0
is reducible to the Heat equation
frac{partial u}{partial tau} = frac{partial^2 u}{partial x^2}
by the change of variables (for complete details see Solution of the Black Scholes Equation):
V(S,t) = K v(x,tau),
x = ln(S/K),
tau = frac{1}{2} sigma^2 (T - t)
v(x,tau)=exp(-alpha x-betatau) u(x,tau).,
Method of characteristics
Superposition principle
Because any superposition of solutions of a linear PDE is again a solution, the particular solutions may then be combined to obtain more general solutions.
Fourier series
If the domain is finite or periodic, an infinite sum of solutions such as a Fourier series is appropriate, but an integral of solutions such as a Fourier integral is generally required for infinite domains. The solution for a point source for the heat equation given above is an example for use of a Fourier integral.
• R. Courant and D. Hilbert, Methods of Mathematical Physics, vol II. Wiley-Interscience, New York, 1962.
• L.C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998. ISBN 0-8218-0772-2
• F. John, Partial Differential Equations, Springer-Verlag, 1982.
• J. Jost, Partial Differential Equations, Springer-Verlag, New York, 2002.
• Hans Lewy (1957) An example of a smooth linear partial differential equation without solution. Annals of Mathematics, 2nd Series, 66(1),155-158.
• I.G. Petrovskii, Partial Differential Equations, W. B. Saunders Co., Philadelphia, 1967.
• A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002. ISBN 1-58488-299-9
• A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, Chapman & Hall/CRC Press, Boca Raton, 2004. ISBN 1-58488-355-3
• A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, Handbook of First Order Partial Differential Equations, Taylor & Francis, London, 2002. ISBN 0-415-27267-X
• D. Zwillinger, Handbook of Differential Equations (3rd edition), Academic Press, Boston, 1997.
• Y. Pinchover and J. Rubinstein, An Introduction to Partial Differential Equations, Cambridge University Press, Cambridge, 2005. ISBN 978-0-521-84886-2
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This article is about the scalar physical quantity. For an overview of and topical guide to energy, see Outline of energy. For other uses, see Energy (disambiguation).
"Energetic" redirects here. For other uses, see Energetic (disambiguation).
Energy transformation; In a typical lightning strike, 500 megajoules of electric potential energy is converted into the same amount of energy in other forms, most notably light energy, sound energy and thermal energy.
In physics, energy is a property of objects, transferable among them via fundamental interactions, which can be converted into different forms but not created or destroyed. The joule is the SI unit of energy, based on the amount transferred to an object by the mechanical work of moving it 1 metre against a force of 1 newton.[1]
Work and heat are two categories of processes or mechanisms that can transfer a given amount of energy. The second law of thermodynamics limits the amount of work that can be performed by energy that is obtained via a heating process—some energy is always lost as waste heat. The maximum amount that can go into work is called the available energy. Systems such as machines and living things often require available energy, not just any energy. Mechanical and other forms of energy can be transformed in the other direction into thermal energy without such limitations.
There are many forms of energy, but all these types must meet certain conditions such as being convertible to other kinds of energy, obeying conservation of energy, and causing a proportional change in mass in objects that possess it. Common energy forms include the kinetic energy of a moving object, the radiant energy carried by light and other electromagnetic radiation, the potential energy stored by virtue of the position of an object in a force field such as a gravitational, electric or magnetic field, and the thermal energy comprising the microscopic kinetic and potential energies of the disordered motions of the particles making up matter. Some specific forms of potential energy include elastic energy due to the stretching or deformation of solid objects and chemical energy such as is released when a fuel burns. Any object that has mass when stationary, such as a piece of ordinary matter, is said to have rest mass, or an equivalent amount of energy whose form is called rest energy, though this isn't immediately apparent in everyday phenomena described by classical physics.
According to mass–energy equivalence, all forms of energy (not just rest energy) exhibit mass. For example, adding 25 kilowatt-hours (90 megajoules) of energy to an object in the form of heat (or any other form) increases its mass by 1 microgram; if you had a sensitive enough mass balance or scale, this mass increase could be measured. Our Sun transforms nuclear potential energy to other forms of energy; its total mass does not decrease due to that in itself (since it still contains the same total energy even if in different forms), but its mass does decrease when the energy escapes out to its surroundings, largely as radiant energy.
Although any energy in any single form can be transformed into another form, the law of conservation of energy states that the total energy of a system can only change if energy is transferred into or out of the system. This means that it is impossible to create or destroy energy. The total energy of a system can be calculated by adding up all forms of energy in the system. Examples of energy transfer and transformation include generating or making use of electric energy, performing chemical reactions, or lifting an object. Lifting against gravity performs work on the object and stores gravitational potential energy; if it falls, gravity does work on the object which transforms the potential energy to the kinetic energy associated with its speed.
More broadly, living organisms require available energy to stay alive; humans get such energy from food along with the oxygen needed to metabolize it. Civilisation requires a supply of energy to function; energy resources such as fossil fuels are a vital topic in economics and politics. Earth's climate and ecosystem are driven by the radiant energy Earth receives from the sun (as well as the geothermal energy contained within the earth), and are sensitive to changes in the amount received. The word "energy" is also used outside of physics in many ways, which can lead to ambiguity and inconsistency. The vernacular terminology is not consistent with technical terminology. For example, while energy is always conserved (in the sense that the total energy does not change despite energy transformations), energy can be converted into a form, e.g., thermal energy, that cannot be utilized to perform work. When one talks about "conserving energy by driving less", one talks about conserving fossil fuels and preventing useful energy from being lost as heat. This usage of "conserve" differs from that of the law of conservation of energy.[2]
Main article: Forms of energy
Thermal energy is energy of microscopic constituents of matter, which may include both kinetic and potential energy.
The total energy of a system can be subdivided and classified in various ways. For example, classical mechanics distinguishes between kinetic energy, which is determined by an object's movement through space, and potential energy, which is a function of the position of an object within a field. It may also be convenient to distinguish gravitational energy, electric energy, thermal energy, several types of nuclear energy (which utilize potentials from the nuclear force and the weak force), electric energy (from the electric field), and magnetic energy (from the magnetic field), among others. Many of these classifications overlap; for instance, thermal energy usually consists partly of kinetic and partly of potential energy. Some types of energy are a varying mix of both potential and kinetic energy. An example is mechanical energy which is the sum of (usually macroscopic) kinetic and potential energy in a system. Elastic energy in materials is also dependent upon electrical potential energy (among atoms and molecules), as is chemical energy, which is stored and released from a reservoir of electrical potential energy between electrons, and the molecules or atomic nuclei that attract them.[need quotation to verify].The list is also not necessarily complete. Whenever physical scientists discover that a certain phenomenon appears to violate the law of energy conservation, new forms are typically added that account for the discrepancy.
Heat and work are special cases in that they are not properties of systems, but are instead properties of processes that transfer energy. In general we cannot measure how much heat or work are present in an object, but rather only how much energy is transferred among objects in certain ways during the occurrence of a given process. Heat and work are measured as positive or negative depending on which side of the transfer we view them from.
Potential energies are often measured as positive or negative depending on whether they are greater or less than the energy of a specified base state or configuration such as two interacting bodies being infinitely far apart. Wave energies (such as radiant or sound energy), kinetic energy, and rest energy are each greater than or equal to zero because they are measured in comparison to a base state of zero energy: "no wave", "no motion", and "no inertia", respectively.
The distinctions between different kinds of energy is not always clear-cut. As Richard Feynman points out:
These notions of potential and kinetic energy depend on a notion of length scale. For example, one can speak of macroscopic potential and kinetic energy, which do not include thermal potential and kinetic energy. Also what is called chemical potential energy is a macroscopic notion, and closer examination shows that it is really the sum of the potential and kinetic energy on the atomic and subatomic scale. Similar remarks apply to nuclear "potential" energy and most other forms of energy. This dependence on length scale is non-problematic if the various length scales are decoupled, as is often the case ... but confusion can arise when different length scales are coupled, for instance when friction converts macroscopic work into microscopic thermal energy.
Some examples of different kinds of energy:
Forms of energy
Type of energy Description
Kinetic (≥0), that of the motion of a body
Potential A category comprising many forms in this list
Mechanical The sum of (usually macroscopic) kinetic and potential energies
Mechanical wave (≥0), a form of mechanical energy propagated by a material's oscillations
Chemical that contained in molecules
Electric that from electric fields
Magnetic that from magnetic fields
Radiant (≥0), that of electromagnetic radiation including light
Nuclear that of binding nucleons to form the atomic nucleus
Ionization that of binding an electron to its atom or molecule
Elastic that of deformation of a material (or its container) exhibiting a restorative force
Gravitational that from gravitational fields
Intrinsic, the rest energy (≥0) that equivalent to an object's rest mass
Thermal A microscopic, disordered equivalent of mechanical energy
Heat an amount of thermal energy being transferred (in a given process) in the direction of decreasing temperature
Mechanical work an amount of energy being transferred in a given process due to displacement in the direction of an applied force
Thomas Young – the first to use the term "energy" in the modern sense.
The word energy derives from the Ancient Greek: ἐνέργεια energeia “activity, operation”,[3] which possibly appears for the first time in the work of Aristotle in the 4th century BC. In contrast to the modern definition, energeia was a qualitative philosophical concept, broad enough to include ideas such as happiness and pleasure.
In the late 17th century, Gottfried Leibniz proposed the idea of the Latin: vis viva, or living force, which defined as the product of the mass of an object and its velocity squared; he believed that total vis viva was conserved. To account for slowing due to friction, Leibniz theorized that thermal energy consisted of the random motion of the constituent parts of matter, a view shared by Isaac Newton, although it would be more than a century until this was generally accepted. The modern analog of this property, kinetic energy, differs from vis via only by a factor of two.
In 1807, Thomas Young was possibly the first to use the term "energy" instead of vis viva, in its modern sense.[4] Gustave-Gaspard Coriolis described "kinetic energy" in 1829 in its modern sense, and in 1853, William Rankine coined the term "potential energy". The law of conservation of energy, was also first postulated in the early 19th century, and applies to any isolated system. It was argued for some years whether heat was a physical substance, dubbed the caloric, or merely a physical quantity, such as momentum. In 1845 James Prescott Joule discovered the link between mechanical work and the generation of heat.
These developments led to the theory of conservation of energy, formalized largely by William Thomson (Lord Kelvin) as the field of thermodynamics. Thermodynamics aided the rapid development of explanations of chemical processes by Rudolf Clausius, Josiah Willard Gibbs, and Walther Nernst. It also led to a mathematical formulation of the concept of entropy by Clausius and to the introduction of laws of radiant energy by Jožef Stefan. According to Noether's theorem, the conservation of energy is a consequence of the fact that the laws of physics do not change over time.[5] Thus, since 1918, theorists have understood that the law of conservation of energy is the direct mathematical consequence of the translational symmetry of the quantity conjugate to energy, namely time.
Measurement and units
A schematic diagram of a Calorimeter - An instrument used by physicists to measure energy. In this example it is X-Rays.
Main article: Units of energy
Energy, like mass, is a scalar physical quantity. The joule is the International System of Units (SI) unit of measurement for energy. It is a derived unit of energy, work, or amount of heat. It is equal to the energy expended (or work done) in applying a force of one newton through a distance of one metre. However energy is also expressed in many other units such as ergs, calories, British Thermal Units, kilowatt-hours and kilocalories for instance. There is always a conversion factor for these to the SI unit; for instance; one kWh is equivalent to 3.6 million joules.[6]
The SI unit of power (energy per unit time) is the watt, which is simply a joule per second. Thus, a joule is a watt-second, so 3600 joules equal a watt-hour. The CGS energy unit is the erg, and the imperial and US customary unit is the foot pound. Other energy units such as the electron volt, food calorie or thermodynamic kcal (based on the temperature change of water in a heating process), and BTU are used in specific areas of science and commerce and have unit conversion factors relating them to the joule.
Because energy is defined as the ability to do work on objects, there is no absolute measure of energy. Only the transition of a system from one state into another can be defined and thus energy is measured in relative terms. The choice of a baseline or zero point is often arbitrary and can be made in whatever way is most convenient for a problem. For example in the case of measuring the energy deposited by X-rays as shown in the accompanying diagram, conventionally the technique most often employed is calorimetry. This is a thermodynamic technique that relies on the measurement of temperature using a thermometer or of intensity of radiation using a bolometer.
Energy density is a term used for the amount of useful energy stored in a given system or region of space per unit volume. For fuels, the energy per unit volume is sometimes a useful parameter. In a few applications, comparing, for example, the effectiveness of hydrogen fuel to gasoline it turns out that hydrogen has a higher specific energy than does gasoline, but, even in liquid form, a much lower energy density.
Scientific use
Classical mechanics
In classical mechanics, energy is a conceptually and mathematically useful property, as it is a conserved quantity. Several formulations of mechanics have been developed using energy as a core concept.
Work, a form of energy, is force times distance.
W = \int_C \mathbf{F} \cdot \mathrm{d} \mathbf{s}
This says that the work (W) is equal to the line integral of the force F along a path C; for details see the mechanical work article. Work and thus energy is frame dependent. For example, consider a ball being hit by a bat. In the center-of-mass reference frame, the bat does no work on the ball. But, in the reference frame of the person swinging the bat, considerable work is done on the ball.
The total energy of a system is sometimes called the Hamiltonian, after William Rowan Hamilton. The classical equations of motion can be written in terms of the Hamiltonian, even for highly complex or abstract systems. These classical equations have remarkably direct analogs in nonrelativistic quantum mechanics.[7]
Another energy-related concept is called the Lagrangian, after Joseph-Louis Lagrange. This is even more fundamental than the Hamiltonian, and can be used to derive the equations of motion. It was invented in the context of classical mechanics, but is generally useful in modern physics. The Lagrangian is defined as the kinetic energy minus the potential energy. Usually, the Lagrange formalism is mathematically more convenient than the Hamiltonian for non-conservative systems (such as systems with friction).
Noether's theorem (1918) states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. Noether's theorem has become a fundamental tool of modern theoretical physics and the calculus of variations. A generalisation of the seminal formulations on constants of motion in Lagrangian and Hamiltonian mechanics (1788 and 1833, respectively), it does not apply to systems that cannot be modeled with a Lagrangian; for example, dissipative systems with continuous symmetries need not have a corresponding conservation law.
In the context of chemistry, energy is an attribute of a substance as a consequence of its atomic, molecular or aggregate structure. Since a chemical transformation is accompanied by a change in one or more of these kinds of structure, it is invariably accompanied by an increase or decrease of energy of the substances involved. Some energy is transferred between the surroundings and the reactants of the reaction in the form of heat or light; thus the products of a reaction may have more or less energy than the reactants. A reaction is said to be exergonic if the final state is lower on the energy scale than the initial state; in the case of endergonic reactions the situation is the reverse. Chemical reactions are invariably not possible unless the reactants surmount an energy barrier known as the activation energy. The speed of a chemical reaction (at given temperature T) is related to the activation energy E, by the Boltzmann's population factor eE/kT – that is the probability of molecule to have energy greater than or equal to E at the given temperature T. This exponential dependence of a reaction rate on temperature is known as the Arrhenius equation.The activation energy necessary for a chemical reaction can be in the form of thermal energy.
Main articles: Bioenergetics and Food energy
Basic overview of energy and human life.
In biology, energy is an attribute of all biological systems from the biosphere to the smallest living organism. Within an organism it is responsible for growth and development of a biological cell or an organelle of a biological organism. Energy is thus often said to be stored by cells in the structures of molecules of substances such as carbohydrates (including sugars), lipids, and proteins, which release energy when reacted with oxygen in respiration. In human terms, the human equivalent (H-e) (Human energy conversion) indicates, for a given amount of energy expenditure, the relative quantity of energy needed for human metabolism, assuming an average human energy expenditure of 12,500kJ per day and a basal metabolic rate of 80 watts. For example, if our bodies run (on average) at 80 watts, then a light bulb running at 100 watts is running at 1.25 human equivalents (100 ÷ 80) i.e. 1.25 H-e. For a difficult task of only a few seconds' duration, a person can put out thousands of watts, many times the 746 watts in one official horsepower. For tasks lasting a few minutes, a fit human can generate perhaps 1,000 watts. For an activity that must be sustained for an hour, output drops to around 300; for an activity kept up all day, 150 watts is about the maximum.[8] The human equivalent assists understanding of energy flows in physical and biological systems by expressing energy units in human terms: it provides a "feel" for the use of a given amount of energy[9]
Sunlight is also captured by plants as chemical potential energy in photosynthesis, when carbon dioxide and water (two low-energy compounds) are converted into the high-energy compounds carbohydrates, lipids, and proteins. Plants also release oxygen during photosynthesis, which is utilized by living organisms as an electron acceptor, to release the energy of carbohydrates, lipids, and proteins. Release of the energy stored during photosynthesis as heat or light may be triggered suddenly by a spark, in a forest fire, or it may be made available more slowly for animal or human metabolism, when these molecules are ingested, and catabolism is triggered by enzyme action.
Any living organism relies on an external source of energy—radiation from the Sun in the case of green plants; chemical energy in some form in the case of animals—to be able to grow and reproduce. The daily 1500–2000 Calories (6–8 MJ) recommended for a human adult are taken as a combination of oxygen and food molecules, the latter mostly carbohydrates and fats, of which glucose (C6H12O6) and stearin (C57H110O6) are convenient examples. The food molecules are oxidised to carbon dioxide and water in the mitochondria
C6H12O6 + 6O2 → 6CO2 + 6H2O
C57H110O6 + 81.5O2 → 57CO2 + 55H2O
and some of the energy is used to convert ADP into ATP
ADP + HPO42− → ATP + H2O
The rest of the chemical energy in the carbohydrate or fat is converted into heat: the ATP is used as a sort of "energy currency", and some of the chemical energy it contains when split and reacted with water, is used for other metabolism (at each stage of a metabolic pathway, some chemical energy is converted into heat). Only a tiny fraction of the original chemical energy is used for work:[10]
gain in kinetic energy of a sprinter during a 100 m race: 4 kJ
gain in gravitational potential energy of a 150 kg weight lifted through 2 metres: 3kJ
Daily food intake of a normal adult: 6–8 MJ
It would appear that living organisms are remarkably inefficient (in the physical sense) in their use of the energy they receive (chemical energy or radiation), and it is true that most real machines manage higher efficiencies. In growing organisms the energy that is converted to heat serves a vital purpose, as it allows the organism tissue to be highly ordered with regard to the molecules it is built from. The second law of thermodynamics states that energy (and matter) tends to become more evenly spread out across the universe: to concentrate energy (or matter) in one specific place, it is necessary to spread out a greater amount of energy (as heat) across the remainder of the universe ("the surroundings").[11] Simpler organisms can achieve higher energy efficiencies than more complex ones, but the complex organisms can occupy ecological niches that are not available to their simpler brethren. The conversion of a portion of the chemical energy to heat at each step in a metabolic pathway is the physical reason behind the pyramid of biomass observed in ecology: to take just the first step in the food chain, of the estimated 124.7 Pg/a of carbon that is fixed by photosynthesis, 64.3 Pg/a (52%) are used for the metabolism of green plants,[12] i.e. reconverted into carbon dioxide and heat.
Earth sciences
In geology, continental drift, mountain ranges, volcanoes, and earthquakes are phenomena that can be explained in terms of energy transformations in the Earth's interior.,[13] while meteorological phenomena like wind, rain, hail, snow, lightning, tornadoes and hurricanes, are all a result of energy transformations brought about by solar energy on the atmosphere of the planet Earth.
Sunlight may be stored as gravitational potential energy after it strikes the Earth, as (for example) water evaporates from oceans and is deposited upon mountains (where, after being released at a hydroelectric dam, it can be used to drive turbines or generators to produce electricity). Sunlight also drives many weather phenomena, save those generated by volcanic events. An example of a solar-mediated weather event is a hurricane, which occurs when large unstable areas of warm ocean, heated over months, give up some of their thermal energy suddenly to power a few days of violent air movement.
In a slower process, radioactive decay of atoms in the core of the Earth releases heat. This thermal energy drives plate tectonics and may lift mountains, via orogenesis. This slow lifting represents a kind of gravitational potential energy storage of the thermal energy, which may be later released to active kinetic energy in landslides, after a triggering event. Earthquakes also release stored elastic potential energy in rocks, a store that has been produced ultimately from the same radioactive heat sources. Thus, according to present understanding, familiar events such as landslides and earthquakes release energy that has been stored as potential energy in the Earth's gravitational field or elastic strain (mechanical potential energy) in rocks. Prior to this, they represent release of energy that has been stored in heavy atoms since the collapse of long-destroyed supernova stars created these atoms.
In cosmology and astronomy the phenomena of stars, nova, supernova, quasars and gamma ray bursts are the universe's highest-output energy transformations of matter. All stellar phenomena (including solar activity) are driven by various kinds of energy transformations. Energy in such transformations is either from gravitational collapse of matter (usually molecular hydrogen) into various classes of astronomical objects (stars, black holes, etc.), or from nuclear fusion (of lighter elements, primarily hydrogen). The nuclear fusion of hydrogen in the Sun also releases another store of potential energy which was created at the time of the Big Bang. At that time, according to theory, space expanded and the universe cooled too rapidly for hydrogen to completely fuse into heavier elements. This meant that hydrogen represents a store of potential energy that can be released by fusion. Such a fusion process is triggered by heat and pressure generated from gravitational collapse of hydrogen clouds when they produce stars, and some of the fusion energy is then transformed into sunlight.
Quantum mechanics
Main article: Energy operator
In quantum mechanics, energy is defined in terms of the energy operator as a time derivative of the wave function. The Schrödinger equation equates the energy operator to the full energy of a particle or a system. In results can be considered as a definition of measurement of energy in quantum mechanics. The Schrödinger equation describes the space- and time-dependence of slow changing (non-relativistic) wave function of quantum systems. The solution of this equation for bound system is discrete (a set of permitted states, each characterized by an energy level) which results in the concept of quanta. In the solution of the Schrödinger equation for any oscillator (vibrator) and for electromagnetic waves in a vacuum, the resulting energy states are related to the frequency by Planck's relation: E = h\nu (where h is the Planck's constant and \nu the frequency). In the case of electromagnetic wave these energy states are called quanta of light or photons.
When calculating kinetic energy (work to accelerate a mass from zero speed to some finite speed) relativistically - using Lorentz transformations instead of Newtonian mechanics, Einstein discovered an unexpected by-product of these calculations to be an energy term which does not vanish at zero speed. He called it rest mass energy - energy which every mass must possess even when being at rest. The amount of energy is directly proportional to the mass of body:
E = m c^2 ,
m is the mass,
c is the speed of light in vacuum,
E is the rest mass energy.
For example, consider electronpositron annihilation, in which the rest mass of individual particles is destroyed, but the inertia equivalent of the system of the two particles (its invariant mass) remains (since all energy is associated with mass), and this inertia and invariant mass is carried off by photons which individually are massless, but as a system retain their mass. This is a reversible process - the inverse process is called pair creation - in which the rest mass of particles is created from energy of two (or more) annihilating photons. In this system the matter (electrons and positrons) is destroyed and changed to non-matter energy (the photons). However, the total system mass and energy do not change during this interaction.
In general relativity, the stress–energy tensor serves as the source term for the gravitational field, in rough analogy to the way mass serves as the source term in the non-relativistic Newtonian approximation.[14]
It is not uncommon to hear that energy is "equivalent" to mass. It would be more accurate to state that every energy has an inertia and gravity equivalent, and because mass is a form of energy, then mass too has inertia and gravity associated with it.
In classical physics, energy is a scalar quantity, the canonical conjugate to time. In special relativity energy is also a scalar (although not a Lorentz scalar but a time component of the energy–momentum 4-vector).[14] In other words, energy is invariant with respect to rotations of space, but not invariant with respect to rotations of space-time (= boosts).
Main article: Energy transformation
Energy may be transformed between different forms at various efficiencies. Items that transform between these forms are called transducers. Examples of transducers include a battery, from chemical energy to electric energy; a dam: gravitational potential energy to kinetic energy of moving water (and the blades of a turbine) and ultimately to electric energy through an electric generator.
There are strict limits to how efficiently energy can be converted into other forms of energy via work, and heat as described by Carnot's theorem and the second law of thermodynamics. These limits are especially evident when an engine is used to perform work. However, some energy transformations can be quite efficient. The direction of transformations in energy (what kind of energy is transformed to what other kind) is often determined by entropy (equal energy spread among all available degrees of freedom) considerations. In practice all energy transformations are permitted on a small scale, but certain larger transformations are not permitted because it is statistically unlikely that energy or matter will randomly move into more concentrated forms or smaller spaces.
Energy transformations in the universe over time are characterized by various kinds of potential energy that has been available since the Big Bang, later being "released" (transformed to more active types of energy such as kinetic or radiant energy), when a triggering mechanism is available. Familiar examples of such processes include nuclear decay, in which energy is released that was originally "stored" in heavy isotopes (such as uranium and thorium), by nucleosynthesis, a process ultimately using the gravitational potential energy released from the gravitational collapse of supernovae, to store energy in the creation of these heavy elements before they were incorporated into the solar system and the Earth. This energy is triggered and released in nuclear fission bombs or in civil nuclear power generation. Similarly, in the case of a chemical explosion, chemical potential energy is transformed to kinetic energy and thermal energy in a very short time. Yet another example is that of a pendulum. At its highest points the kinetic energy is zero and the gravitational potential energy is at maximum. At its lowest point the kinetic energy is at maximum and is equal to the decrease of potential energy. If one (unrealistically) assumes that there is no friction or other losses, the conversion of energy between these processes would be perfect, and the pendulum would continue swinging forever.
Energy is also transferred from potential energy (E_p) to kinetic energy (E_k) and then back to potential energy constantly. This is referred to as conservation of energy. In this closed system, energy cannot be created or destroyed; therefore, the initial energy and the final energy will be equal to each other. This can be demonstrated by the following:
E_{pi} + E_{ki} = E_{pF} + E_{kF}
The equation can then be simplified further since E_p = mgh (mass times acceleration due to gravity times the height) and E_k = \frac{1}{2} mv^2 (half mass times velocity squared). Then the total amount of energy can be found by adding E_p + E_k = E_{total}.
Conservation of energy and mass in transformation
Energy gives rise to weight when it is trapped in a system with zero momentum, where it can be weighed. It is also equivalent to mass, and this mass is always associated with it. Mass is also equivalent to a certain amount of energy, and likewise always appears associated with it, as described in mass-energy equivalence. The formula E = mc², derived by Albert Einstein (1905) quantifies the relationship between rest-mass and rest-energy within the concept of special relativity. In different theoretical frameworks, similar formulas were derived by J. J. Thomson (1881), Henri Poincaré (1900), Friedrich Hasenöhrl (1904) and others (see Mass-energy equivalence#History for further information).
Matter may be converted to energy (and vice versa), but mass cannot ever be destroyed; rather, mass/energy equivalence remains a constant for both the matter and the energy, during any process when they are converted into each other. However, since c^2 is extremely large relative to ordinary human scales, the conversion of ordinary amount of matter (for example, 1 kg) to other forms of energy (such as heat, light, and other radiation) can liberate tremendous amounts of energy (~9\times 10^{16} joules = 21 megatons of TNT), as can be seen in nuclear reactors and nuclear weapons. Conversely, the mass equivalent of a unit of energy is minuscule, which is why a loss of energy (loss of mass) from most systems is difficult to measure by weight, unless the energy loss is very large. Examples of energy transformation into matter (i.e., kinetic energy into particles with rest mass) are found in high-energy nuclear physics.
Reversible and non-reversible transformations
Thermodynamics divides energy transformation into two kinds: reversible processes and irreversible processes. An irreversible process is one in which energy is dissipated (spread) into empty energy states available in a volume, from which it cannot be recovered into more concentrated forms (fewer quantum states), without degradation of even more energy. A reversible process is one in which this sort of dissipation does not happen. For example, conversion of energy from one type of potential field to another, is reversible, as in the pendulum system described above. In processes where heat is generated, quantum states of lower energy, present as possible excitations in fields between atoms, act as a reservoir for part of the energy, from which it cannot be recovered, in order to be converted with 100% efficiency into other forms of energy. In this case, the energy must partly stay as heat, and cannot be completely recovered as usable energy, except at the price of an increase in some other kind of heat-like increase in disorder in quantum states, in the universe (such as an expansion of matter, or a randomisation in a crystal).
As the universe evolves in time, more and more of its energy becomes trapped in irreversible states (i.e., as heat or other kinds of increases in disorder). This has been referred to as the inevitable thermodynamic heat death of the universe. In this heat death the energy of the universe does not change, but the fraction of energy which is available to do work through a heat engine, or be transformed to other usable forms of energy (through the use of generators attached to heat engines), grows less and less.
Conservation of energy
According to conservation of energy, energy can neither be created (produced) nor destroyed by itself. It can only be transformed. The total inflow of energy into a system must equal the total outflow of energy from the system, plus the change in the energy contained within the system. Energy is subject to a strict global conservation law; that is, whenever one measures (or calculates) the total energy of a system of particles whose interactions do not depend explicitly on time, it is found that the total energy of the system always remains constant.[15]
Richard Feynman said during a 1961 lecture:[16]
There is a fact, or if you wish, a law, governing all natural phenomena that are known to date. There is no known exception to this law—it is exact so far as we know. The law is called the conservation of energy. It states that there is a certain quantity, which we call energy, that does not change in manifold changes which nature undergoes. That is a most abstract idea, because it is a mathematical principle; it says that there is a numerical quantity which does not change when something happens. It is not a description of a mechanism, or anything concrete; it is just a strange fact that we can calculate some number and when we finish watching nature go through her tricks and calculate the number again, it is the same.
Most kinds of energy (with gravitational energy being a notable exception)[17] are subject to strict local conservation laws as well. In this case, energy can only be exchanged between adjacent regions of space, and all observers agree as to the volumetric density of energy in any given space. There is also a global law of conservation of energy, stating that the total energy of the universe cannot change; this is a corollary of the local law, but not vice versa.[2][16]
This law is a fundamental principle of physics. As shown rigorously by Noether's theorem, the conservation of energy is a mathematical consequence of translational symmetry of time,[18] a property of most phenomena below the cosmic scale that makes them independent of their locations on the time coordinate. Put differently, yesterday, today, and tomorrow are physically indistinguishable. This is because energy is the quantity which is canonical conjugate to time. This mathematical entanglement of energy and time also results in the uncertainty principle - it is impossible to define the exact amount of energy during any definite time interval. The uncertainty principle should not be confused with energy conservation - rather it provides mathematical limits to which energy can in principle be defined and measured.
Each of the basic forces of nature is associated with a different type of potential energy, and all types of potential energy (like all other types of energy) appears as system mass, whenever present. For example, a compressed spring will be slightly more massive than before it was compressed. Likewise, whenever energy is transferred between systems by any mechanism, an associated mass is transferred with it.
In quantum mechanics energy is expressed using the Hamiltonian operator. On any time scales, the uncertainty in the energy is by
\Delta E \Delta t \ge \frac { \hbar } {2 }
which is similar in form to the Heisenberg Uncertainty Principle (but not really mathematically equivalent thereto, since H and t are not dynamically conjugate variables, neither in classical nor in quantum mechanics).
In particle physics, this inequality permits a qualitative understanding of virtual particles which carry momentum, exchange by which and with real particles, is responsible for the creation of all known fundamental forces (more accurately known as fundamental interactions). Virtual photons (which are simply lowest quantum mechanical energy state of photons) are also responsible for electrostatic interaction between electric charges (which results in Coulomb law), for spontaneous radiative decay of exited atomic and nuclear states, for the Casimir force, for van der Waals bond forces and some other observable phenomena.
Transfer between systems
Main article: Energy transfer
Closed systems
Energy transfer usually refers to movements of energy between systems which are closed to transfers of matter. The portion of the energy which is transferred by conservative forces over a distance is measured as the work the source system does on the receiving system. The portion of the energy which does not do work doing during the transfer is called heat.[19] Energy can be transferred between systems in a variety of ways. Examples include the transmission of electromagnetic energy via photons, physical collisions which transfer kinetic energy,[20] and the conductive transfer of thermal energy.
Energy is strictly conserved and is also locally conserved wherever it can be defined. Mathematically, the process of energy transfer is described by the first law of thermodynamics:
\Delta{}E = W + Q
where E is the amount of energy transferred, W represents the work done on the system, and Q represents the heat flow into the system.[21] As a simplification, the heat term, Q, is sometimes ignored, especially when the thermal efficiency of the transfer is high.
\Delta{}E = W
This simplified equation is the one used to define the joule, for example.
Open systems
There are other ways in which an open system can gain or lose energy. In chemical systems, energy can be added to a system by means of adding substances with different chemical potentials, which potentials are then extracted (both of these process are illustrated by fueling an auto, a system which gains in energy thereby, without addition of either work or heat). These terms may be added to the above equation, or they can generally be subsumed into a quantity called "energy addition term E" which refers to any type of energy carried over the surface of a control volume or system volume. Examples may be seen above, and many others can be imagined (for example, the kinetic energy of a stream of particles entering a system, or energy from a laser beam adds to system energy, without either being either work-done or heat-added, in the classic senses).
\Delta{}E = W + Q + E
Where E in this general equation represents other additional advected energy terms not covered by work done on a system, or heat added to it.
Internal energy
Internal energy is the sum of all microscopic forms of energy of a system. It is the energy needed to create the system. It is related to the potential energy, e.g., molecular structure, crystal structure, and other geometric aspects, as well as the motion of the particles, in form of kinetic energy. Thermodynamics is chiefly concerned with changes in internal energy and not its absolute value, which is impossible to determine with thermodynamics alone.[22]
First law of thermodynamics
The first law of thermodynamics asserts that energy (but not necessarily thermodynamic free energy) is always conserved[23] and that heat flow is a form of energy transfer. For homogeneous systems, with a well-defined temperature and pressure, a commonly used corollary of the first law is that, for a system subject only to pressure forces and heat transfer (e.g., a cylinder-full of gas), the differential change in the internal energy of the system (with a gain in energy signified by a positive quantity) is given as
\mathrm{d}E = T\mathrm{d}S - P\mathrm{d}V\,,
where the first term on the right is the heat transferred into the system, expressed in terms of temperature T and entropy S (in which entropy increases and the change dS is positive when the system is heated), and the last term on the right hand side is identified as work done on the system, where pressure is P and volume V (the negative sign results since compression of the system requires work to be done on it and so the volume change, dV, is negative when work is done on the system).
This equation is highly specific, ignoring all chemical, electrical, nuclear, and gravitational forces, effects such as advection of any form of energy other than heat and pV-work. The general formulation of the first law (i.e., conservation of energy) is valid even in situations in which the system is not homogeneous. For these cases the change in internal energy of a closed system is expressed in a general form by
\mathrm{d}E=\delta Q+\delta W
where \delta Q is the heat supplied to the system and \delta W is the work applied to the system.
Equipartition of energy
The energy of a mechanical harmonic oscillator (a mass on a spring) is alternatively kinetic and potential. At two points in the oscillation cycle it is entirely kinetic, and alternatively at two other points it is entirely potential. Over the whole cycle, or over many cycles, net energy is thus equally split between kinetic and potential. This is called equipartition principle; total energy of a system with many degrees of freedom is equally split among all available degrees of freedom.
This principle is vitally important to understanding the behaviour of a quantity closely related to energy, called entropy. Entropy is a measure of evenness of a distribution of energy between parts of a system. When an isolated system is given more degrees of freedom (i.e., given new available energy states that are the same as existing states), then total energy spreads over all available degrees equally without distinction between "new" and "old" degrees. This mathematical result is called the second law of thermodynamics.
See also
Notes and references
1. ^ Energy units are usually defined in terms of the work they can do. However, because work is an indirect measurement of energy, (One example of the difficulties involved: if you use the first law of thermodynamics to define energy as the work an object can do, you must perform a perfectly reversible process, which is impossible in a finite time.) many experts emphasize understanding how energy behaves, specifically the conservation of energy, rather than trying to explain what energy "is". "The Feynman Lectures on Physics Vol I.". Retrieved 3 Apr 2014.
2. ^ a b The Laws of Thermodynamics including careful definitions of energy, free energy, et cetera.
3. ^ Harper, Douglas. "Energy". Online Etymology Dictionary. Retrieved May 1, 2007.
4. ^ Smith, Crosbie (1998). The Science of Energy – a Cultural History of Energy Physics in Victorian Britain. The University of Chicago Press. ISBN 0-226-76420-6.
5. ^ Lofts, G; O'Keeffe D; et al. (2004). "11 — Mechanical Interactions". Jacaranda Physics 1 (2 ed.). Milton, Queensland, Australia: John Willey & Sons Australia Ltd. p. 286. ISBN 0-7016-3777-3.
6. ^ Ristinen, Robert A., and Kraushaar, Jack J. Energy and the Environment. New York: John Wiley & Sons, Inc., 2006.
7. ^ The Hamiltonian MIT OpenCourseWare website 18.013A Chapter 16.3 Accessed February 2007
8. ^ "Retrieved on May-29-09". Retrieved 2010-12-12.
9. ^ Bicycle calculator - speed, weight, wattage etc. [1].
10. ^ These examples are solely for illustration, as it is not the energy available for work which limits the performance of the athlete but the power output of the sprinter and the force of the weightlifter. A worker stacking shelves in a supermarket does more work (in the physical sense) than either of the athletes, but does it more slowly.
11. ^ Crystals are another example of highly ordered systems that exist in nature: in this case too, the order is associated with the transfer of a large amount of heat (known as the lattice energy) to the surroundings.
12. ^ Ito, Akihito; Oikawa, Takehisa (2004). "Global Mapping of Terrestrial Primary Productivity and Light-Use Efficiency with a Process-Based Model." in Shiyomi, M. et al. (Eds.) Global Environmental Change in the Ocean and on Land. pp. 343–58.
13. ^ "Earth's Energy Budget". Retrieved 2010-12-12.
14. ^ a b Misner, Thorne, Wheeler (1973). Gravitation. San Francisco: W. H. Freeman. ISBN 0-7167-0344-0.
15. ^ Berkeley Physics Course Volume 1. Charles Kittel, Walter D Knight and Malvin A Ruderman
16. ^ a b Feynman, Richard (1964). The Feynman Lectures on Physics; Volume 1. U.S.A: Addison Wesley. ISBN 0-201-02115-3.
17. ^ "E. Noether's Discovery of the Deep Connection Between Symmetries and Conservation Laws". 1918-07-16. Retrieved 2010-12-12.
18. ^ "Time Invariance". Retrieved 2010-12-12.
19. ^ Although heat is "wasted" energy for a specific energy transfer,(see: waste heat) it can often be harnessed to do useful work in subsequent interactions. However, the maximum energy that can be "recycled" from such recovery processes is limited by the second law of thermodynamics.
20. ^ The mechanism for most macroscopic physical collisions is actually electromagnetic, but it is very common to simplify the interaction by ignoring the mechanism of collision and just calculate the beginning and end result.
21. ^ The signs in this equation follow the IUPAC convention.
22. ^ I. Klotz, R. Rosenberg, Chemical Thermodynamics - Basic Concepts and Methods, 7th ed., Wiley (2008), p.39
23. ^ Kittel and Kroemer (1980). Thermal Physics. New York: W. H. Freeman. ISBN 0-7167-1088-9.
Further reading
• Alekseev, G. N. (1986). Energy and Entropy. Moscow: Mir Publishers.
• Crowell, Benjamin (2011) [2003]. Light and Matter. Fullerton, California: Light and Matter.
• Ross, John S. (23 April 2002). "Work, Power, Kinetic Energy". Project PHYSNET. Michigan State University.
• Smil, Vaclav (2008). Energy in nature and society: general energetics of complex systems. Cambridge, USA: MIT Press. ISBN 0-262-19565-8.
• Walding, Richard, Rapkins, Greg, Rossiter, Glenn (1999-11-01). New Century Senior Physics. Melbourne, Australia: Oxford University Press. ISBN 0-19-551084-4.
External links |
5f0347f93cec651d | Notes to Action at a Distance in Quantum Mechanics
1. Intuitively, the spin-component of a particle in a certain direction can be thought of as its intrinsic angular momentum along that direction. But, as we shall see in section 5, the nature of spin properties depends on the interpretation of quantum mechanics. In any case, the exact nature of this quantity will not be essential for what follows in sections 1-4. The important thing is that in various quantum states the properties of distant physical systems may be curiously correlated.
2. Recall Bell's (1981) example of Bertlmann's socks. ‘Dr. Bertlmann likes to wear two socks of different colours. Which colour he will have on a given foot on a given day is quite unpredictable. But when you see that the first sock is pink you can already be sure that the second sock will not be pink. Observation of the first, and experience of Bertlmann, gives immediate information about the second.’
3. Two comments:
(i) In some Bell-type models of the EPR/B experiment, it is assumed that in addition to the pair's state and the settings of the measurement apparatuses, there are other factors that may be relevant for the probabilities of measurement outcomes. In particular, in his presentation of stochastic, local models of the EPR/B experiment Bell (1971, p. 37) assumes that the setting of the apparatuses need not specify their entire relevant states. The outcomes may also be influenced by some other aspects of the apparatus microstates, which may be different for the same settings (see also Jarrett 1984). More generally, in addition to the state of the L- (R-) particle and the setting of the L- (R-) measurement apparatus, there may be some other (local) physical quantities that are relevant for the probability of the L- (R-) measurement outcome. That is, letting α and β denote all the relevant local physical quantities (other than the settings) that are relevant for the probability of the L- and the R-outcome respectively, in such models the single and joint probabilities of outcomes will be: Pλ l α(xl), Pλ r β(xl) and Pλ lr α β(xl & yr). We shall refer to this type of models in section 7. But, for the sake of simplicity, in the rest of this entry we shall focus on the simpler models above.
(ii) There are two different approaches to modeling the probabilities in Bell-type models of the EPR/B experiment: The many-spaces and the big-space approaches (see Butterfield 1989, 1992a). In the many-space approach, which we use in this review, each triple of pair's state, L- and R-setting labels a different probability space of measurement outcomes. For example, letting l and l′ be different L-apparatus settings, the probability Pλ lr(xl & yr) belongs to one probability space, whereas the probability Pλ l′r(xl′ & yr) belongs to another. By contrast, in the big-space approach, all the probabilities of a Bell-type model belong to one big probability space. In this approach the probabilities of outcomes are expressed in terms of conditional probabilities. For example, the probabilities P(xl & yr / λ & l & r) and P(xl′ & yr / λ & l′ & r) correspond to Pλ l r(xl & yr) and Pλ l′ r(xl′ & yr), respectively. (Note that in contrast to the above notation, in the literature probabilities of spin-measurement outcomes in the big-space approach are frequently expressed as conditional probabilities of non-specific spin-outcomes, i.e. the non-specific outcomes ‘up’ or ‘down’, given certain settings: P(x & y / λ & l & r), where ‘x’ and ‘y’ denote non-specific outcomes.) Mathematically, the two approaches can easily be related to each other. In particular, one can construct a big-probability space in which the conditional probabilities of outcomes, given a pair's state and apparatus settings, are equal to the corresponding unconditional probabilities in the many-spaces approaches: P(xl & yr / λ & l & r) = Pλ l r(xl & yr), P(xl′ & yr / λ & l′ & r) = Pλ l′ r(xl′ & yr), etc. But, conceptually the two approaches are different. First, in contrast to the many-spaces approach, in the big-space approach it is presupposed that settings always have definite probabilities (Butterfield 1989, p. 118, 1992a, section 2). Secondly, some of the probabilities of the big-space approach, e.g. P(xl / λ & r), have no correspondence in the many-spaces approach. Third, as we shall see in the next section, factorizability can be analyzed into two conditions: parameter independence and outcome independence. Berkovitz (2002) argues that the meaning of parameter independence need not be the same in the two different approaches. That is, in some circumstances the parameter independence of the big-space approach expresses different properties than the parameter independence of the many-spaces approach. Indeed, in these circumstances the parameter independence of the many-spaces approach fails, whereas the parameter independence of the big-space approach holds. For arguments for the superiority of the many-spaces approach over the big-space approach, see Butterfield (1989, p. 118), and Berkovitz (2002, section 4.2).
4. Or when the range of the values of λ is discrete,
Pψ l r(xl & yr) = ∑λ Pλ l r(xl & yr) · ρψ l r(λ),
Pψ l (xl) = ∑λ Pλ l (xl) · ρψ l(λ), and
Pψ r(yr) = ∑λ Pλ r(yr) · ρψ r(λ).
5. For a dissenting view, see Fine (1981, 1982a), Cartwright (1989) and Chang and Cartwright (1993). We shall discuss this view in section 9.
6. While the fullest analysis of factorizability is due to Jarrett, precursors are Suppes and Zanotti (1976) and van Fraassen (1982).
7. See van Fraassen (1982) and Jarrett (1984, 1989).
8. As we shall see below, in the literature the term ‘interpretation’ is frequently used to refer to alternative quantum theories. The question of whether this use is justified and the criteria for distinguishing between an interpretation of orthodox quantum mechanics and an alternative quantum theory will be insubstantial for the considerations below.
9. For a history of the notion of action at a distance, see Hesse (1969).
10. In reality, the position of different particles will be different: |upi>pi (|downi>pi). But this is immaterial for the analysis below.
11. This is a variant of the so-called ‘tails problem’ (see the entry on collapse theories, section 12, and Albert 1992, chapter 5).
12. For a recent interesting discussion of Newton's view of action at a distance, see Henry (1994), and references therein.
13. For the Clarke-Leibniz correspondence, see Alexander (1956).
14. Of course, here ‘field’ is not intended to mean a field in the sense of quantum field theory.
15. For discussions of this version of Bohm's theory, see for example Dürr, Goldstein and Zanghì (1992a), Albert (1992) and Cushing (1994).
16. The wave function propagates according to the Schrödinger equation in the ‘configuration’ space of the particles, which for an N-particle system is a 3N-dimensional space, coordinatized by the 3N position coordinates of the particles. For more details, see the entry on Bohmian mechanics.
17. Here follows a more technical account of the above experiment according to the minimal Bohm theory. Let the wave function, i.e. the state of the guiding field, before any measurement occurs be:
ψ = 1/√2f1(z1) f2(z2) ( |z-up>1 |z-down>2 − |z-down>1 |z-up>2),
where f1(z1) and f2(z2) are non-overlapping Gaussian wavepackets; z1 and z2 are respectively the positions of particle 1 and particle 2 along the z-direction; and |z-up> and |z-down> are z-spin eigenstates (i.e. z-spin ‘up’ and z-spin ‘down,’ respectively). Suppose that we perform a z-spin measurement on particle 1, and switch off the R-apparatus. Then (suppressing for simplicity's sake the free time evolution of the two wavepackets as they move towards their respective Stern-Gerlach devices and the states of the Stern-Gerlach devices), the state of the guiding field during the L-measurement will be:
1/√2 f2(z2) ( f1(z1 + g1T) |z-up>1 |z-down>2f1(z1g1T) |z-down>1 |z-up>2)
where g1 is the coupling constant for the spin measurement on particle 1 (coupling the position and the spin degrees of freedoms that are related to the guidance of particle 1); and T is the duration of the measurement. Since the guiding field of the particle pair factorizes into f2(z2) and ( f1(z1 + g1T) |z-up>1 |z-down>2f1(z1g1T) |z-down>1 |z-up>2), it follows from the guiding equation that particle 2's velocity along the z-axis does not depend on particle 1's position.
18. See Bohm, Schiller and Tiomno (1955), Dewdney, Holland and Kyprianidis (1987), Bohm and Hiley (1993, chapter 10), and Holland (1993).
19. While in the minimal and the non-minimal Bohm theories, the wave function is interpreted as a field, Dürr, Goldstein and Zanghì (1997, section 12) propose that the wave function should be interpreted as a parameter of a physical law. This, they argue, may explain why there is no action of configurations of particles on wave functions.
20. For discussions of the above experiment in the non-minimal theory, see Dewdney, Holland and Kyprianidis (1987), Bohm and Hiley (1993, section 10.6), and Holland (1993, section 11.3).
21. For discussions of the prospects of relativistic modal interpretations, see Dickson and Clifton (1998), Arntzenius (1998), Myrvold (2002a), Earman and Ruetsche (2005) and Berkovitz and Hemmo (2005, 2006a,b). We shall discuss this issue at the end of this section and in section 10.2.
22. If some of the ci are degenerate, the Schmidt biorthogonal decomposition is not unique, and the properties assigned by the above rule are projections onto multi-dimensional subspaces.
23. Note the difference between |ψ9> and the singlet state |ψ3>. In |ψ9>, the coefficients of the two branches of the superposition are unequal. And while EPR/B-like experiments can be prepared with both the state |ψ3> and the state |ψ9>, the difference between these states is significant for the above interpretation. For unlike |ψ3>, |ψ9> has a unique factorization. Accordingly, the L- and the R-particle each have definite spin properties in the state |ψ9> but not in |ψ3>.
24. In fact, as we shall see later in this section (in the discussion of ‘property composition’), this claim needs some qualification.
25. In the original modal interpretations, the question of the relation between the dynamics of properties of systems and the dynamics of the properties of their subsystems has been largely overlooked. For a discussion of this issue, see Vermaas (1997, 1999), Berkovitz and Hemmo (2005, 2006a,b).
26. Similarly to any other physical object, the brain of a human observer has many different sets of relational properties, i.e. sets of properties that are related to different systems. Brain properties that are defined relative to different systems are generally different. Thus, the question arises as to which of these different brain properties are correlated to our beliefs about the properties of physical systems that figure in our experience. For a discussion of this question, see Berkovitz and Hemmo (2005, section 6, 2006b, section 6).
27. The challenge is to explicate the nature of such holistic properties and to relate them to our experience.
28. That is, the property of a system is given by the spectral decomposition of its so-called ‘reduced state’ (a statistical operator obtained by a partial tracing). For example, the reduced state of the L-particle in the state |ψ9> is obtained by a partial tracing of |ψ9> over the Hilbert space of the R-particle.
29. The above formulation of screening off is motivated by the fact that we work in the framework of the many-spaces approach to the probabilities of outcomes in Bell-type models of the EPR/B experiment. In the literature, the formulation of screening off is slightly different:
P(x/y & CC(x,y)) = P(x / CC(x,y)) P(y / CC(x,y)) ≠ 0
P(y/x & CC(x,y)) = P(y / CC(x,y)) P(x / CC(x,y)) ≠ 0
30. In his celebrated theorem, Bell did not mention Reichenbach's principle or FactorUCP. But it is reasonable to assume that he had in mind some similar principles.
31. There are some obvious candidates for superluminal signaling. First, the potentials in the Schrödinger equation are Newtonian. Therefore, if one is allowed to vary the potential somewhere, this will be felt instantaneously throughout space. But, in the context of this entry such superluminal signaling is less interesting because it will be due to Newtonian effects rather than quantum effects. Second, wave functions can spread instantaneously: If you have a particle confined to a box (so that its wave function is zero outside the box) and open the box, the wave function will instantaneously be non-zero everywhere, and superluminal signaling will be possible. It is noteworthy, however, that the preparation of such state requires the existence of an infinite potential barrier—a state that is impossible. In any case, in what follows we shall focus on the question of whether the non-locality in the EPR/B experiment, as depicted by various interpretations of quantum mechanics, can be exploited to give rise to superluminal signaling.
32. Note that according to this suggestion, the statistical predictions of Bohm's theory slightly deviate from the statistical predictions of orthodox quantum mechanics.
33. It is noteworthy, however, that while separability does not imply OI, the prospects of separable models that violate OI are dim. To see why, let us consider Maudlin's (1994, p. 98) criticism of Howard's claim that OI follows from spatiotemporal separability. Maudlin invites us to consider the following model for the EPR/B experiment. Suppose that each particle had some means of superluminal communication, which may be realized by a tachyon. Suppose also that each of the particles carries the same instructions: If it arrives at a measurement apparatus without having received a message from its partner, and the measurement apparatus is in a state of being ready to measure z-spin, its state of not having definite z-spin evolves with equal chance to either the state of having z-spin ‘up’ or the state of having z-spin ‘down.’ It then communicates to its partner the setting of its measurement apparatus and the outcome of the measurement. If it receives a message from its partner, its state is modified accordingly, so that the new chance of spin outcomes agrees with the predictions of orthodox quantum mechanics. Such a model will involve a violation of OI, but by construction it is separable: The particles and the tachyons have separable states at all times, and the joint state of any two systems is just the product of their individual states. Yet, the model will be separable in the intended sense only if the above set of communication instructions could be encoded into the qualitative, intrinsic properties of each of the particles, and each of the particles could keep an open line of communication with its partner and no other particles. But a little reflection on the grave difficulties involved with that task suggests that the physical feasibility and plausibility of any such separable model of the EPR/B experiment will be highly questionable.
34. Friedman (1983, sections 4.6-4.7) holds that special relativity per se does not prohibit superluminal signaling, but that such signaling will lead to paradoxes of time travel. Maudlin (1994, pp. 112-116) argues that superluminal signaling need not imply such paradoxes, as the conditions for them are much more complex than merely the existence of superluminal signaling.
35. As Maudlin (1996, pp. 292-293) notes, it is not clear that a general criterion for identifying a structure of spacetime as intrinsic could be found.
Copyright © 2007 by
Joseph Berkovitz <>
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4a8065ed69778522 | Viewpoint: Reconnecting to superfluid turbulence
Joseph J. Niemela, The Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, 34014 Trieste, Italy
Published October 6, 2008 | Physics 1, 26 (2008) | DOI: 10.1103/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.
Velocity Statistics Distinguish Quantum Turbulence from Classical Turbulence
M. S. Paoletti, Michael E. Fisher, K. R. Sreenivasan, and D. P. Lathrop
Published October 6, 2008 | PDF (free)
+Enlarge image Figure 1
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.
M. S. Paoletti et al.
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.
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].
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).
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: Joseph J. Niemela
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.
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4.2 Quantum models
4.2.1 Bose-Einstein condensates
We have seen that one of the main aims of research in analogue models of gravity is the possibility of simulating semiclassical gravity phenomena, such as the Hawking radiation effect or cosmological particle production. In this sense systems characterised by a high degree of quantum coherence, very cold temperatures, and low speeds of sound offer the best test field. Hence it is not surprising that in recent years Bose-Einstein condensates (BECs) have become the subject of extensive study as possible analogue models of general relativity [136Jump To The Next Citation Point137Jump To The Next Citation Point16Jump To The Next Citation Point19Jump To The Next Citation Point18Jump To The Next Citation Point115Jump To The Next Citation Point114Jump To The Next Citation Point].
Let us start by very briefly reviewing the derivation of the acoustic metric for a BEC system, and show that the equations for the phonons of the condensate closely mimic the dynamics of a scalar field in a curved spacetime. In the dilute gas approximation, one can describe a Bose gas through a quantum field Y satisfying
( 2 ) ih @-Y = - h-- \~/ 2 + V (x) + k(a) Y &#8224;Y Y. (183) @t 2m ext
Here k parameterises the strength of the interactions between the different bosons in the gas. It can be re-expressed in terms of the scattering length as
4pah2 k(a) = ------. (184) m
As usual, the quantum field can be separated into a macroscopic (classical) condensate and a fluctuation: Y = y + f, with <Y > = y. Then, by adopting the self-consistent mean field approximation (see for example [153])
f&#8224;ff -~ 2&lt;f &#8224;f &gt; f + &lt;ff &gt; f&#8224;, (185)
one can arrive at the set of coupled equations:
( ) @-- h2-- 2 ih @ty(t, x) = - 2m \~/ + Vext(x) + k nc y(t,x) * +k {2~ny(t, x) + ~my (t,x)}; (186) ( ) @ h2 2 ih --f(t, x) = - --- \~/ + Vext(x) + k 2nT f(t,x) @t 2m +k mT f&#8224;(t,x). (187)
2 2 nc =_ |y(t,x) | ; mc =_ y (t,x); (188) n~ =_ &lt;f&#8224;f &gt;; m~ =_ &lt;f f &gt;; (189) nT = nc + ~n; mT = mc + ~m. (190)
The equation for the classical wave function of the condensate is closed only when the back-reaction effect due to the fluctuations are neglected. (This back-reaction is hiding in the parameters ~m and ~n.) This is the approximation contemplated by the Gross-Pitaevskii equation. In general one will have to solve both equations simultaneously. Adopting the Madelung representation for the wave function of the condensate
V~ -------- y(t,x) = nc(t,x) exp[- ih(t,x)/h], (191)
and defining an irrotational “velocity field” by v =_ \~/ h/m, the Gross-Pitaevskii equation can be rewritten as a continuity equation plus an Euler equation:
@-- @tnc + \~/ .(ncv) = 0, (192) ( 2 2 2 V~ --) m @-v + \~/ mv---+ Vext(t,x) + knc - h-- \~/ - V~ -nc = 0. (193) @t 2 2m nc
These equations are completely equivalent to those of an irrotational and inviscid fluid apart from the existence of the so-called quantum potential
V ~ --- V ~ --- Vquantum = -h2 \~/ 2 nc/(2m nc), (194)
which has the dimensions of an energy. Note that
[ 2 2 V~ --] [ 2 ] nc \~/ iVquantum =_ nc \~/ i --h- \~/ V~ -nc = \~/ j - -h--nc \~/ i\ ~/ j ln nc , (195) 2m nc 4m
which justifies the introduction of the so-called quantum stress tensor
quantum h2 sij = - ----nc \~/ i\ ~/ j lnnc. (196) 4m
This tensor has the dimensions of pressure, and may be viewed as an intrinsically quantum anisotropic pressure contributing to the Euler equation. If we write the mass density of the Madelung fluid as r = m nc, and use the fact that the flow is irrotational then the Euler equation takes the form
[ ] [ ] [ ] @-- Vext(t,x)- kr2- quantum r @tv + (v . \~/ )v + r \~/ m + \~/ 2m2 + \~/ .s = 0. (197)
Note that the term Vext/m has the dimensions of specific enthalpy, while 2 kr /(2m) represents a bulk pressure. When the gradients in the density of the condensate are small one can neglect the quantum stress term leading to the standard hydrodynamic approximation. Because the flow is irrotational, the Euler equation is often more conveniently written in Hamilton-Jacobi form:
( V ~ --) @ [ \~/ h]2 h2 \~/ 2 nc m ---h + ------+ Vext(t,x) + knc - ----- V~ --- = 0. (198) @t 2m 2m nc
Apart from the wave function of the condensate itself, we also have to account for the (typically small) quantum perturbations of the system (187View Equation). These quantum perturbations can be described in several different ways, here we are interested in the “quantum acoustic representation”
( 1 V~ n-- ) f(t,x) = e-ih/h - V~ ---n1 - i ---c-h1 , (199) 2 nc h
where n ,h 1 1 are real quantum fields. By using this representation Equation (187View Equation) can be rewritten as
1 ( ) @tn1 + -- \~/ . n1 \~/ h + nc \~/ h1 = 0, (200) m 1- h2-- @th1 + m \~/ h . \~/ h1 + k(a) n1- 2m D2n1 = 0. (201)
Here D2 represents a second-order differential operator obtained from linearizing the quantum potential. Explicitly:
D n =_ - 1n- 3/2 [ \~/ 2(n+1/2)] n + 1n- 1/2 \~/ 2(n -1/2n ). (202) 2 1 2 c c 1 2 c c 1
The equations we have just written can be obtained easily by linearizing the Gross-Pitaevskii equation around a classical solution: nc --> nc + n1, f --> f + f1. It is important to realise that in those equations the back-reaction of the quantum fluctuations on the background solution has been assumed negligible. We also see in Equations (200View Equation, 201View Equation), that time variations of V ext and time variations of the scattering length a appear to act in very different ways. Whereas the external potential only influences the background Equation (198View Equation) (and hence the acoustic metric in the analogue description), the scattering length directly influences both the perturbation and background equations. From the previous equations for the linearised perturbations it is possible to derive a wave equation for h1 (or alternatively, for n1). All we need is to substitute in Equation (200View Equation) the n1 obtained from Equation (201View Equation). This leads to a PDE that is second-order in time derivatives but infinite order in space derivatives - to simplify things we can construct the symmetric 4 × 4 matrix
|_ _| f00 ... f0j ................... f mn(t,x) =_ . . (203) |_ fi0 .. fij _|
(Greek indices run from 0-3, while Roman indices run from 1-3.) Then, introducing (3+1)-dimensional space-time coordinates
xm =_ (t; xi) (204)
the wave equation for h1 is easily rewritten as
mn @m(f @nh1) = 0. (205)
Where the f mn are differential operators acting on space only:
[ h2 ]-1 f00 = - k(a) - ----D2 (206) [ 2m ] 0j h2 -1 \~/ jh0 f = - k(a) - ----D2 ----- (207) [ 2m m] i0 \~/ ih0 h2 -1 f = - -m--- k(a)- 2m--D2 (208) [ ]1- ij nc-dij- \~/ ih0 h2-- \~/ jh0 f = m - m k(a) - 2m D2 m . (209)
Now, if we make a spectral decomposition of the field h1 we can see that for wavelengths larger than h/mcs (this corresponds to the “healing length”, as we will explain below), the terms coming from the linearization of the quantum potential (the D 2) can be neglected in the previous expressions, in which case the mn f can be approximated by numbers, instead of differential operators. (This is the heart of the acoustic approximation.) Then, by identifying
V~ --- mn mn - g g = f , (210)
the equation for the field h1 becomes that of a (massless minimally coupled) quantum scalar field over a curved background
1 ( V~ --- mn ) Dh1 =_ V~ ---g @m - g g @n h1 = 0, (211)
with an effective metric of the form
| _ _| - {cs(a,nc)2- v2}... - vj n gmn(t,x) =_ -----c----- ............ ........ . (212) m cs(a,nc) |_ - vi .. dij _|
Here the magnitude cs(nc,a) represents the speed of the phonons in the medium:
k(a) nc cs(a,nc)2 = -------. (213) m
With this effective metric now in hand, the analogy is fully established, and one is now in a position to start asking more specific physics questions.
4.2.2 BEC models in the eikonal approximation
It is interesting to consider the case in which the above “hydrodynamical” approximation for BECs does not hold. In order to explore a regime where the contribution of the quantum potential cannot be neglected we can use the so called eikonal approximation, a high-momentum approximation where the phase fluctuation h1 is itself treated as a slowly-varying amplitude times a rapidly varying phase. This phase will be taken to be the same for both n1 and h1 fluctuations. In fact, if one discards the unphysical possibility that the respective phases differ by a time varying quantity, any time-constant difference can be safely reabsorbed in the definition of the (complex) amplitudes. Specifically, we shall write
h1(t,x) = Re {Ah exp(- if)}, (214) n1(t,x) = Re {Ar exp(- if)}. (215)
As a consequence of our starting assumptions, gradients of the amplitude, and gradients of the background fields, are systematically ignored relative to gradients of f. (Warning: What we are doing here is not quite a “standard” eikonal approximation, in the sense that it is not applied directly on the fluctuations of the field y(t,x) but separately on their amplitudes and phases r1 and f1.) We adopt the notation
@f w = --; ki = \~/ if. (216) @t
Then the operator D2 can be approximated as
1 -3/2 +1/2 1 -1/2 -1/2 D2 n1 =_ ---nc [D(n c )] n1 +--nc D(n c n1) (217) 2 2 ~~ + 1-n-c1 [Dn1] (218) 2 1- -1 2 = - 2 nc k n1. (219)
A similar result holds for D2 acting on h1. That is, under the eikonal approximation we effectively replace the operator D2 by the function
1- -1 2 D2 --&gt; - 2 nc k . (220)
For the matrix mn f this effectively results in the replacement
[ ]-1 00 h2-k2- f -- &gt; - k(a) + 4m n (221) [ c]-1 0j h2-k2- \~/ jh0 f -- &gt; - k(a) + 4m nc m (222) i [ 2 2]1- i0 \~/ -h0 -h--k- f -- &gt; - m k(a) + 4m nc (223) ij i [ 2 2]-1 j fij-- &gt; nc-d--- \~/ -h0 k(a) + -h-k-- \~/ -h0. (224) m m 4m nc m
(As desired, this has the net effect of making mn f a matrix of numbers, not operators.) The physical wave equation (205View Equation) now becomes a nonlinear dispersion relation
f00 w2 + (f 0i + fi0) w ki + fij ki kj = 0. (225)
After substituting the approximate D2 into this dispersion relation and rearranging, we see (remember: 2 2 ij k = ||k|| = d ki kj)
[ ] 2 i nck2 h2 2 i 2 - w + 2 v0 wki +----- k(a) + -----k - (v0 ki) = 0. (226) m 4mnc
That is:
( )2 nck2 [ h2 ] w - vi0 ki = ----- k(a) + -----k2 . (227) m 4mnc
Introducing the speed of sound cs this takes the form:
V~ -------(-------)2- i 2 2 -h-- 2 w = v0 ki± csk + 2m k . (228)
At this stage some observations are in order:
1. It is interesting to recognise that the dispersion relation (228View Equation) is exactly in agreement with that found in 1947 by Bogoliubov [36] (reprinted in [310]; see also [223]) for the collective excitations of a homogeneous Bose gas in the limit T --> 0 (almost complete condensation). In his derivation Bogoliubov applied a diagonalization procedure for the Hamiltonian describing the system of bosons.
2. It is easy to see that (228View Equation) actually interpolates between two different regimes depending on the value of the wavelength c = 2p/ ||k|| with respect to the “acoustic Compton wavelength” cc = h/(mcs). (Remember that cs is the speed of sound; this is not a standard particle physics Compton wavelength.) In particular, if we assume v0 = 0 (no background velocity), then for large wavelengths c » cc one gets a standard phonon dispersion relation w ~~ c||k||. For wavelengths c « cc the quasi-particle energy tends to the kinetic energy of an individual gas particle and in fact 2 2 w ~~ h k /(2m).
We would also like to highlight that in relative terms, the approximation by which one neglects the quartic terms in the dispersion relation gets worse as one moves closer to a horizon where v0 = - cs. The non-dimensional parameter that provides this information is defined by
V~ ------- -c2c 2 d =_ --1-+-4c2---1- -~ ----1------cc-. (229) (1 - v0/cs) (1 - v0/cs)8c2
As we will discuss in Section 5.1.3, this is the reason why sonic horizons in a BEC can exhibit different features from those in standard general relativity.
3. The dispersion relation (228View Equation) exhibits a contribution due to the background flow i v 0 ki, plus a quartic dispersion at high momenta. The group velocity is
( ) 2 h2--2 i @w-- i ----c-+--2m2k------ i vg = @ki = v0 ± V~ 2 2 (h--- 2)2 k . (230) ck + 2m k
Dispersion relations of this type (but in most cases with the sign of the quartic term reversed) have been used by Corley and Jacobson in analysing the issue of trans-Planckian modes in the Hawking radiation from general relativistic black holes [185Jump To The Next Citation Point186Jump To The Next Citation Point88Jump To The Next Citation Point]. The existence of modified dispersion relations (MDR), that is, dispersion relations that break Lorentz invariance, can be taken as a manifestation of new physics showing up at high energies/short wavelengths. In their analysis, the group velocity reverses its sign for large momenta. (Unruh’s analysis of this problem used a slightly different toy model in which the dispersion relation saturated at high momentum [377Jump To The Next Citation Point].) In our case, however, the group velocity grows without bound allowing high-momentum modes to escape from behind the horizon. Thus the acoustic horizon is not absolute in these models, but is instead frequency dependent, a phenomenon that is common once non-trivial dispersions are included.
Indeed, with hindsight the fact that the group velocity goes to infinity for large k was pre-ordained: After all, we started from the generalised nonlinear Schrödinger equation, and we know what its characteristic curves are. Like the diffusion equation the characteristic curves of the Schrödinger equation (linear or nonlinear) move at infinite speed. If we then approximate this generalised nonlinear Schrödinger equation in any manner, for instance by linearization, we cannot change the characteristic curves: For any well behaved approximation technique, at high frequency and momentum we should recover the characteristic curves of the system we started with. However, what we certainly do see in this analysis is a suitably large region of momentum space for which the concept of the effective metric both makes sense, and leads to finite propagation speed for medium-frequency oscillations.
This type of superluminal dispersion relation has also been analysed by Corley and Jacobson [90Jump To The Next Citation Point]. They found that this escape of modes from behind the horizon often leads to self-amplified instabilities in systems possessing both an inner horizon as well as an outer horizon, possibly causing them to disappear in an explosion of phonons. This is also in partial agreement with the stability analysis performed by Garay et al. [136Jump To The Next Citation Point137Jump To The Next Citation Point] using the whole Bogoliubov equations. Let us however leave further discussion regarding these developments to the Section 5.1.3 on horizon stability.
4.2.3 The Heliocentric universe
Helium is one of the most fascinating elements provided by nature. Its structural richness confers on helium a paradigmatic character regarding the emergence of many and varied macroscopic properties from the microscopic world (see [418Jump To The Next Citation Point] and references therein). Here, we are interested in the emergence of effective geometries in helium, and their potential use in testing aspects of semiclassical gravity.
Helium four, a bosonic system, becomes superfluid at low temperatures (2.17 K at vapour pressure). This superfluid behaviour is associated with the condensation in the vacuum state of a macroscopically large number of atoms. A superfluid is automatically an irrotational and inviscid fluid, so in particular one can apply to it the ideas worked out in Section 2. The propagation of classical acoustic waves (scalar waves) over a background fluid flow can be described in terms of an effective Lorentzian geometry: the acoustic geometry. However, in this system one can naturally go considerably further, into the quantum domain. For long wavelengths, the quasiparticles in this system are quantum phonons. One can separate the classical behaviour of a background flow (the effective geometry) from the behaviour of the quantum phonons over this background. In this way one can reproduce, in laboratory settings, different aspects of quantum field theory over curved backgrounds. The speed of sound in the superfluid phase is typically of the order of cm/sec. Therefore, at least in principle, it should not be too difficult to establish configurations with supersonic flows and their associated ergoregions.
Helium three, the fermionic isotope of helium, in contrast becomes superfluid at very much lower temperatures (below 2.5 milli-K). The reason behind this rather different behaviour is the pairing of fermions to form effective bosons (Cooper pairing), which are then able to condense. In the so-called 3He - A phase, the structure of the fermionic vacuum is such that it possesses two Fermi points, instead of the more typical Fermi surface. In an equilibrium configuration one can choose the two Fermi points to be located at {px = 0,py = 0,pz = ± pF } (in this way, the z-axis signals the direction of the angular momentum of the pairs). Close to either Fermi point the spectrum of quasiparticles becomes equivalent to that of Weyl fermions. From the point of view of the laboratory, the system is not isotropic, it is axisymmetric. There is a speed for the propagation of quasiparticles along the z-axis, c|| -~ cm/sec, and a different speed, -5 c _L -~ 10 c||, for propagation perpendicular to the symmetry axis. However, from an internal observer’s point of view this anisotropy is not “real”, but can be made to disappear by an appropriate rescaling of the coordinates. Therefore, in the equilibrium case, we are reproducing the behaviour of Weyl fermions over Minkowski spacetime. Additionally, the vacuum can suffer collective excitations. These collective excitations will be experienced by the Weyl quasiparticles as the introduction of an effective electromagnetic field and a curved Lorentzian geometry. The control of the form of this geometry provides the sought for gravitational analogy.
Apart from the standard way to provide a curved geometry based on producing non-trivial flows, there is also the possibility of creating topologically non-trivial configurations with a built-in non-trivial geometry. For example, it is possible to create a domain-wall configuration [200Jump To The Next Citation Point199Jump To The Next Citation Point] (the wall contains the z-axis) such that the transverse velocity c _L acquires a profile in the perpendicular direction (say along the x-axis) with c _L passing through zero at the wall (see Figure 8View Image). This particular arrangement could be used to reproduce a black hole-white hole configuration only if the soliton is set up to move with a certain velocity along the x-axis. This configuration has the advantage than it is dynamically stable, for topological reasons, even when some supersonic regions are created.
View Image
Figure 8: Domain wall configuration in 3He.
A third way in which superfluid Helium can be used to create analogues of gravitational configurations is the study of surface waves (or ripplons) on the interface between two different phases of 3 He [415Jump To The Next Citation Point417Jump To The Next Citation Point]. In particular, if we have a thin layer of 3He - A in contact with another thin layer of 3He - B, the oscillations of the contact surface “see” an effective metric of the form [415Jump To The Next Citation Point417Jump To The Next Citation Point]
[ ( ) ] ds2 = ------1------- - 1 - W 2- aAaBU 2 dt2- 2W .dx dt + dx .dx , (231) (1- aAaBU 2)
W =_ aAvA + aBvB; U =_ vA - vB; (232)
----hBrA------ ----hArB------ aA =_ h r + h r ; aB =_ h r + h r . (233) A B B A A B B A
(All of this provided that we are looking at wavelengths larger than the layer thickness, khA « 1 and k hB « 1.)
View Image
Figure 9: Ripplons in the interface between two sliding superfluids.
The advantage of using surface waves instead of bulk waves in superfluids is that one could create horizons without reaching supersonic speeds in the bulk fluid. This could alleviate the appearance of dynamical instabilities in the system, that in this case are controlled by the strength of the interaction of the ripplons with bulk degrees of freedom [415Jump To The Next Citation Point417].
4.2.4 Slow light
The geometrical interpretation of the motion of light in dielectric media leads naturally to conjecture that the use of flowing dielectrics might be useful for simulating general relativity metrics with ergoregions and black holes. Unfortunately, these types of geometry require flow speeds comparable to the group velocity of the light. Since typical refractive indexes in non-dispersive media are quite close to unity, it is then clear that it is practically impossible to use them to simulate such general relativistic phenomena. However recent technological advances have radically changed this state of affairs. In particular the achievement of controlled slowdown of light, down to velocities of a few meters per second (or even down to complete rest) [38320452211309374348], has opened a whole new set of possibilities regarding the simulation of curved-space metrics via flowing dielectrics.
But how can light be slowed down to these “snail-like” velocities? The key effect used to achieve this takes the name of Electromagnetically Induced Transparency (EIT). A laser beam is coupled to the excited levels of some atom and used to strongly modify its optical properties. In particular one generally chooses an atom with two long-lived metastable (or stable) states, plus a higher energy state that has some decay channels into these two lower states. The coupling of the excited states induced by the laser light can affect the transition from a lower energy state to the higher one, and hence the capability of the atom to absorb light with the required transition energy. The system can then be driven into a state where the transitions between each of the lower energy states and the higher energy state exactly cancel out, due to quantum interference, at some specific resonant frequency. In this way the higher-energy level has null averaged occupation number. This state is hence called a “dark state”. EIT is characterised by a transparency window, centred around the resonance frequency, where the medium is both almost transparent and extremely dispersive (strong dependence on frequency of the refractive index). This in turn implies that the group velocity of any light probe would be characterised by very low real group velocities (with almost vanishing imaginary part) in proximity to the resonant frequency.
Let us review the most common setup envisaged for this kind of analogue model. A more detailed analysis can be found in [232Jump To The Next Citation Point]. One can start by considering a medium in which an EIT window is opened via some control laser beam which is oriented perpendicular to the direction of the flow. One then illuminates this medium, now along the flow direction, with some probe light (which is hence perpendicular to the control beam). This probe beam is usually chosen to be weak with respect to the control beam, so that it does not modify the optical properties of the medium. In the case in which the optical properties of the medium do not vary significantly over several wavelengths of the probe light, one can neglect the polarization and can hence describe the propagation of the latter with a simple scalar dispersion relation [235Jump To The Next Citation Point124]
w2 k2 - ---[1 + x(w)] , (234) c2
where x is the susceptibility of the medium, related to the refractive index n via the simple relation n = V~ 1-+-x-.
It is easy to see that in this case the group and phase velocities differ
vg = @w- = --------c--------; vph = w- = V~ -c----. (235) @k V~ 1-+-x-+ -w-@x- k 1 + x 2n @w
So even for small refractive indexes one can get very low group velocities, due to the large dispersion in the transparency window, and in spite of the fact that the phase velocity remains very near to c. (The phase velocity is exactly c at the resonance frequency w0). In an ideal EIT regime the probe light experiences a vanishing susceptibility x near the the critical frequency w0, this allows us to express the susceptibility near the critical frequency via the expansion
2a- [ 3] x(w) = w0 (w - w0) + O (w - w0) , (236)
where a is sometimes called the “group refractive index”. The parameter a depends on the dipole moments for the transition from the metastable states to the high energy one, and most importantly depends on the ratio between the probe-light energy per photon, hw0, and the control-light energy per atom [232]. This might appear paradoxical because it seems to suggest that for a dimmer control light the probe light would be further slowed down. However this is just an artificial feature due to the extension of the EIT regime beyond its range of applicability. In particular in order to be effective the EIT requires the control beam energy to dominate all processes and hence it cannot be dimmed at will.
At resonance we have
vg = @w---&gt; --c--- ~~ -c; vph = w---&gt; c. (237) @k 1 + a a k
We can now generalise the above discussion to the case in which our highly dispersive medium flows with a characteristic velocity profile u(x, t). In order to find the dispersion relation of the probe light in this case we just need to transform the dispersion relation (234View Equation) from the comoving frame of the medium to the laboratory frame. Let us consider for simplicity a monochromatic probe light (more realistically a pulse with a very narrow range of frequencies w near w0). The motion of the dielectric medium creates a local Doppler shift of the frequency
w --&gt; g (w0 - u .k) , (238)
where g is the usual relativistic factor. Given that k2 - w2/c2 is a Lorentz invariant, it is then easy to see that this Doppler detuning affects the dispersion relation (234View Equation) only via the susceptibility dependent term. Given further that in any realistic case one would deal with non-relativistic fluid velocities u « c we can then perform an expansion of the dispersion relation up to second order in u/c. Expressing the susceptibility via (236View Equation) we can then rewrite the dispersion relation in the form [235]
gmnkmkn = 0, (239)
( ) k = w0,- k , (240) n c
and (most of the relevant articles adopt the signature (+ - - - ), as we also do for this particular section)
[ | ] -1 +-au2/c2|-------auT-/c2------- gmn = 2 | T 2 . (241) au/c -I3×3 + 4au ox u /c
The inverse of this tensor will be the covariant effective metric experienced by the probe light, whose rays would then be null geodesics of the line element 2 m n ds = gmndx dx. In this sense the probe light will propagate as in a curved background. Explicitly one finds the covariant metric to be
[ | ] A | BuT gmn = ---|----------------T- , (242) Bu |-I3× 3 + Cu ox u
1- 4au2/c2 A = ------2-------2--2-----2--4--4; (243) 1 + (a - 3a)u /c - 4a u /c B = --------------1---------------; (244) 1 + (a2 - 3a)u2/c2 - 4a2u4/c4 1 - (4/a + 4u2/c2) C = ------------------------------. (245) 1 + (a2 - 3a)u2/c2 - 4a2u4/c4
Several comments are in order concerning the metric (242View Equation). First of all it is clear that although more complicated than an acoustic metric it will be still possible to cast it into the Arnowitt-Deser-Misner-like form [392]
| [- [c2 - gabua ub ]|[ueff]i ] gmn = ----eff-------eff--eff-|------ , (246) [ueff]j |[geff]ij
where the effective speed ueff is proportional to the fluid flow speed u and the three space effective metric geff is (rather differently from the acoustic case) non-trivial.
In any case, the existence of this ADM form already tells us that an ergoregion will always appear once the norm of the effective velocity exceeds the effective speed of light (which for slow light is approximately c/a where a can be extremely large due to the huge dispersion in the transparency window around the resonance frequency w0). However a trapped surface (and hence an optical black hole) will form only if the inward normal component of the effective flow velocity exceeds the group velocity of light. In the slow light setup so far considered such a velocity turns out to be u = c/(2V ~ a).
The realization that ergoregions and event horizons can be simulated via slow light may lead one to the (erroneous) conclusion that this is an optimal system for simulating particle creation by gravitational fields. However, as pointed out by Unruh in [284Jump To The Next Citation Point379Jump To The Next Citation Point], such a conclusion would turn out to be over-enthusiastic. In order to obtain particle creation an inescapable requirement is to have so-called “mode mixing”, that is, mixing between the positive and negative frequency modes of the incoming and outgoing states. This is tantamount to saying that there must be regions where the frequency of the quanta as seen by a stationary observer at infinity (laboratory frame) becomes negative beyond the ergosphere at g00 = 0.
In a flowing medium this can in principle occur thanks to the tilting of the dispersion relation due to the Doppler effect caused by the velocity of the flow Equation (238View Equation), but this also tells us that the condition w0 - u .k < 0 can be satisfied only if the velocity of the medium exceeds |w0/k | which is the phase velocity of the probe light, not its group velocity. Since the phase velocity in the slow light setup we are considering is very close to c, the physical speed of light in vacuum, not very much hope is left for realizing analogue particle creation in this particular laboratory setting.
However it was also noticed by Unruh and Schützhold [379Jump To The Next Citation Point] that a different setup for slow light might deal with this and other issues (see [379Jump To The Next Citation Point] for a detailed summary). In the setup suggested by these authors there are two strong background counter-propagating control beams illuminating the atoms. The field describing the beat fluctuations of this electromagnetic background can be shown to satisfy, once the dielectric medium is in motion, the same wave equation as that on a curved background. In this particular situation the phase velocity and the group velocity are approximately the same, and both can be made small, so that the previously discussed obstruction to mode mixing is removed. So in this new setup it is concretely possible to simulate classical particle creation such as, e.g., super-radiance in the presence of ergoregions.
Nonetheless the same authors showed that this does not open the possibility for a simulation of quantum particle production (e.g., Hawking radiation). This is because that effect also requires the commutation relations of the field to generate the appropriate zero-point energy fluctuations (the vacuum structure) according to the Heisenberg uncertainty principle. This is not the case for the effective field describing the beat fluctuations of the system we have just described, which is equivalent to saying that it does not have a proper vacuum state (i.e., analogue to any physical field). Hence one has to conclude that any simulation of quantum particle production is precluded.
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ac73a2c7f8c8bb7d | Superfluid vacuum theory
From Wikipedia, the free encyclopedia
(Redirected from Logarithmic BEC vacuum)
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Superfluid vacuum theory (SVT), sometimes known as the BEC vacuum theory, is an approach in theoretical physics and quantum mechanics where the fundamental physical vacuum (non-removable background) is viewed as superfluid or as a Bose–Einstein condensate (BEC).
The microscopic structure of this physical vacuum is currently unknown and is a subject of intensive studies in SVT. An ultimate goal of this approach is to develop scientific models that unify quantum mechanics (describing three of the four known fundamental interactions) with gravity, making SVT a candidate for the theory of quantum gravity and describing all known interactions in the Universe, at both microscopic and astronomic scales, as different manifestations of the same entity, superfluid vacuum.
The concept of a luminiferous aether as a medium sustaining electromagnetic waves was discarded after the advent of the special theory of relativity. The aether, as conceived in classical physics leads to several contradictions; in particular, aether having a definite velocity at each space-time point will exhibit a preferred direction. This conflicts with the relativistic requirement that all directions within a light cone are equivalent. However, as early as in 1951 P.A.M. Dirac published two papers where he pointed out that we should take into account quantum fluctuations in the flow of the aether.[1][2] His arguments involve the application of the uncertainty principle to the velocity of aether at any space-time point, implying that the velocity will not be a well-defined quantity. In fact, it will be distributed over various possible values. At best, one could represent the aether by a wave function representing the perfect vacuum state for which all aether velocities are equally probable. These works can be regarded as the birth point of the theory.
Inspired by the Dirac ideas, K. P. Sinha, C. Sivaram and E. C. G. Sudarshan published in 1975 a series of papers that suggested a new model for the aether according to which it is a superfluid state of fermion and anti-fermion pairs, describable by a macroscopic wave function.[3][4][5] They noted that particle-like small fluctuations of superfluid background obey the Lorentz symmetry, even if the superfluid itself is non-relativistic. Nevertheless, they decided to treat the superfluid as the relativistic matter - by putting it into the stress–energy tensor of the Einstein field equations. This did not allow them to describe the relativistic gravity as a small fluctuation of the superfluid vacuum, as subsequent authors have noted.
As an alternative to the better known string theories, a very different theory by Friedwardt Winterberg proposes instead, that the vacuum is a kind of superfluid plasma compound of positive and negative Planck masses, called a Planck mass plasma.[6][7][citation needed]
Since then, several theories have been proposed within the SVT framework. They differ in how the structure and properties of the background superfluid must look like. In absence of observational data which would rule out some of them, these theories are being pursued independently.
Relation to other concepts and theories[edit]
Lorentz and Galilean symmetries[edit]
According to the approach, the background superfluid is assumed to be essentially non-relativistic whereas the Lorentz symmetry is not an exact symmetry of Nature but rather the approximate description valid only for small fluctuations. An observer who resides inside such vacuum and is capable of creating or measuring the small fluctuations would observe them as relativistic objects - unless their energy and momentum are sufficiently high to make the Lorentz-breaking corrections detectable.[8] If the energies and momenta are below the excitation threshold then the superfluid background behaves like the ideal fluid, therefore, the Michelson–Morley-type experiments would observe no drag force from such aether.[1][2]
Further, in the theory of relativity the Galilean symmetry (pertinent to our macroscopic non-relativistic world) arises as the approximate one - when particles' velocities are small compared to speed of light in vacuum. In SVT one does not need to go through Lorentz symmetry to obtain the Galilean one - the dispersion relations of most non-relativistic superfluids are known to obey the non-relativistic behavior at large momenta.[9][10][11]
To summarize, the fluctuations of vacuum superfluid behave like relativistic objects at "small"[nb 1] momenta (a.k.a. the "phononic limit")
E^2 \propto |\vec p|^2
and like non-relativistic ones
E \propto |\vec p|^2
at large momenta. The yet unknown nontrivial physics is believed to be located somewhere between these two regimes.
Relativistic quantum field theory[edit]
In the relativistic quantum field theory the physical vacuum is also assumed to be some sort of non-trivial medium to which one can associate certain energy. This is because the concept of absolutely empty space (or "mathematical vacuum") contradicts to the postulates of quantum mechanics. According to QFT, even in absence of real particles the background is always filled by pairs of creating and annihilating virtual particles. However, a direct attempt to describe such medium leads to the so-called ultraviolet divergences. In some QFT models, such as quantum electrodynamics, these problems can be "solved" using the renormalization technique, namely, replacing the diverging physical values by their experimentally measured values. In other theories, such as the quantum general relativity, this trick does not work, and reliable perturbation theory cannot be constructed.
According to SVT, this is because in the high-energy ("ultraviolet") regime the Lorentz symmetry starts failing so dependent theories cannot be regarded valid for all scales of energies and momenta. Correspondingly, while the Lorentz-symmetric quantum field models are obviously a good approximation below the vacuum-energy threshold, in its close vicinity the relativistic description becomes more and more "effective" and less and less natural since one will need to adjust the expressions for the covariant field-theoretical actions by hand.
Curved space-time[edit]
According to general relativity, the gravitational interaction is described in terms of space-time curvature using the mathematical formalism of Riemannian geometry. This was supported by numerous experiments and observations in the regime of low energies. However, the attempts to quantize general relativity led to various severe problems, therefore, the microscopic structure of gravity is still ill-defined. There may be a fundamental reason for this—the degrees of freedom of general relativity are based on may be only approximate and effective. The question of whether general relativity is an effective theory has been raised for a long time.[12]
According to SVT, the curved space-time arises as the small-amplitude collective excitation mode of the non-relativistic background condensate.[8][13] The mathematical description of this is similar to fluid-gravity analogy which is being used also in the analog gravity models.[14] Thus, relativistic gravity is essentially a long-wavelength theory of the collective modes whose amplitude is small compared to the background one. Outside this requirement the curved-space description of gravity in terms of the Riemannian geometry becomes incomplete or ill-defined.
Cosmological constant[edit]
The notion of the cosmological constant makes sense in a relativistic theory only, therefore, within the SVT framework this constant can refer at most to the energy of small fluctuations of the vacuum above a background value but not to the energy of vacuum itself.[15] Thus, in SVT this constant does not have any fundamental physical meaning and the related problems, such as the vacuum catastrophe, simply do not occur in first place.
Gravitational waves and gravitons[edit]
According to general relativity, the conventional gravitational wave is:
1. the small fluctuation of curved spacetime which
2. has been separated from its source and propagates independently.
Theory of superfluid vacuum brings into question that the relativistic object possessing both of these properties may exist in Nature.[13] Indeed, according to the approach, the curved spacetime itself is the small collective excitation of the superfluid background, therefore, the property (1) means that the graviton would be in fact the "small fluctuation of the small fluctuation" which does not look like a physically robust concept (as if somebody tried to introduce small fluctuations inside a phonon, for instance). As a result, it may be not just a coincidence that in general relativity the gravitational field alone has no well-defined stress–energy tensor, only the pseudotensor one.[16] Therefore, the property (2) cannot be completely justified in a theory with exact Lorentz symmetry which the general relativity is. Though, SVT does not a priori forbid an existence of the non-localized wave-like excitations of the superfluid background which might be responsible for the astrophysical phenomena which are currently being attributed to gravitational waves, such as the Hulse–Taylor binary. However, such excitations cannot be correctly described within the framework of a fully relativistic theory.
Mass generation and Higgs boson[edit]
The Higgs boson is the spin-0 particle which has been introduced in the electroweak theory to give the mass to the weak bosons. The origin of mass of the Higgs boson itself is not explained by the electroweak theory. Instead, this mass introduced as a free parameter by means of the Higgs potential which thus makes it yet another free parameter of the Standard Model.[17] Within a framework of the Standard Model (or its extensions) the theoretical estimates of this parameter's value are possible only indirectly and results differ from each other significantly.[18] Thus, the usage of the Higgs boson (or any other elementary particle with predefined mass) alone is not the most fundamental solution of the mass generation problem but only its reformulation ad infinitum. Another known issue of the Glashow–Weinberg–Salam model is the wrong sign of mass term in the (unbroken) Higgs sector for energies above the symmetry-breaking scale.[nb 2]
While SVT does not explicitly forbid the existence of the electroweak Higgs particle, it has its own idea of the fundamental mass generation mechanism - elementary particles acquire mass due to the interaction with the vacuum condensate, similarly to the gap generation mechanism in superconductors or superfluids.[13][19] Although this idea is not entirely new, one could recall the relativistic Coleman-Weinberg approach,[20] SVT gives the meaning to the symmetry-breaking relativistic scalar field as describing small fluctuations of background superfluid which can be interpreted as an elementary particle only under certain conditions.[21] In general, one allows two scenarios to happen:
• Higgs boson exists: in this case SVT provides the mass generation mechanism which underlies the electroweak one and explains the origin of mass of the Higgs boson itself;
• Higgs boson does not exist: then the weak bosons acquire mass by directly interacting with the vacuum condensate.
Thus, the Higgs boson, even if it exists, would be a by-product of the fundamental mass generation phenomenon rather than its cause.[21]
Also, some versions of SVT favor a wave equation based on the logarithmic potential rather than on the quartic one. The former potential has not only the Mexican-hat shape, necessary for the spontaneous symmetry breaking, but also some other features which make it more suitable for the vacuum's description.
Logarithmic BEC vacuum theory[edit]
In this model the physical vacuum is conjectured to be strongly-correlated 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 whereas relativistic elementary particles can be described by the particle-like modes in the limit of low energies and momenta.[19] The essential difference of this theory from others is that in the logarithmic superfluid the maximal velocity of fluctuations is constant in the leading (classical) order. This allows to fully recover the relativity postulates in the "phononic" (linearized) limit.[13]
The proposed theory has many observational consequences. They are based on the fact that at high energies and momenta the behavior of the particle-like modes eventually becomes distinct from the relativistic one - they can reach the speed of light limit at finite energy.[22] Among other predicted effects is the superluminal propagation and vacuum Cherenkov radiation.[23]
Theory advocates the mass generation mechanism which is supposed to replace or alter the electroweak Higgs one. It was shown that masses of elementary particles can arise as a result of interaction with the superfluid vacuum, similarly to the gap generation mechanism in superconductors.[13][19] For instance, the photon propagating in the average interstellar vacuum acquires a tiny mass which is estimated to be about 10−35 electronvolt. One can also derive an effective potential for the Higgs sector which is different from the one used in the Glashow–Weinberg–Salam model, yet it yields the mass generation and it is free of the imaginary-mass problem[nb 2] appearing in the conventional Higgs potential.[21]
See also[edit]
1. ^ The term "small" refers here to the linearized limit, in practice the values of these momenta may not be small at all.
2. ^ a b If one expands the Higgs potential then the coefficient at the quadratic term appears to be negative. This coefficient has a physical meaning of squared mass of a scalar particle.
1. ^ a b Dirac, P. A. M. (24 November 1951). "Is there an Æther?". Letters to Nature (Nature) 168 (4282): 906–907. Bibcode:1951Natur.168..906D. doi:10.1038/168906a0. Retrieved 16 October 2012.
2. ^ a b Dirac, P. A. M. (26 April 1952). "Is there an Æther?". Nature 169 (4304): 702–702. Bibcode:1952Natur.169..702D. doi:10.1038/169702b0.
3. ^ K. P. Sinha, C. Sivaram, E. C. G. Sudarshan, Found. Phys. 6, 65 (1976).
4. ^ K. P. Sinha, C. Sivaram, E. C. G. Sudarshan, Found. Phys. 6, 717 (1976).
5. ^ K. P. Sinha and E. C. G. Sudarshan, Found. Phys. 8, 823 (1978).
6. ^ Winterberg, Friedwardt (1988). "Substratum Approach to a Unified Theory of Elementary Particles". Z.f. Naturforsch.-Physical Sciences. 43a.
7. ^ Winterberg, Friedwardt (2003). "Planck Mass Plasma Vacuum Conjecture". Z. Naturforsch 58a: 231–267.
8. ^ a b G. E. Volovik, The Universe in a helium droplet, Int. Ser. Monogr. Phys. 117 (2003) 1-507.
9. ^ N. N. Bogoliubov, Izv. Acad. Nauk USSR 11, 77 (1947).
10. ^ N.N. Bogoliubov, J. Phys. 11, 23 (1947)
11. ^ V. L. Ginzburg, L. D. Landau, Zh. Eksp. Teor. Fiz. 20, 1064 (1950).
12. ^ A. D. Sakharov, Sov. Phys. Dokl. 12, 1040 (1968). This paper was reprinted in Gen. Rel. Grav. 32, 365 (2000) and commented in: M. Visser, Mod. Phys. Lett. A 17, 977 (2002).
13. ^ a b c d e K. G. Zloshchastiev, Spontaneous symmetry breaking and mass generation as built-in phenomena in logarithmic nonlinear quantum theory, Acta Phys. Polon. B 42 (2011) 261-292 ArXiv:0912.4139.
14. ^ M. Novello, M. Visser, G. Volovik, Artificial Black Holes, World Scientific, River Edge, USA, 2002, p391.
15. ^ G.E. Volovik, Int. J. Mod. Phys. D15, 1987 (2006) ArXiv: gr-qc/0604062.
16. ^ L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields, (1951), Pergamon Press, chapter 11.96.
17. ^ V. A. Bednyakov, N. D. Giokaris and A. V. Bednyakov, Phys. Part. Nucl. 39 (2008) 13-36 ArXiv:hep-ph/0703280.
18. ^ B. Schrempp and M. Wimmer, Prog. Part. Nucl. Phys. 37 (1996) 1-90 ArXiv:hep-ph/9606386.
19. ^ a b c A. V. Avdeenkov and K. G. Zloshchastiev, Quantum Bose liquids with logarithmic nonlinearity: Self-sustainability and emergence of spatial extent, J. Phys. B: At. Mol. Opt. Phys. 44 (2011) 195303. ArXiv:1108.0847.
20. ^ S. R. Coleman and E. J. Weinberg, Phys. Rev. D7, 1888 (1973).
21. ^ a b c V. Dzhunushaliev and K.G. Zloshchastiev (2013). "Singularity-free model of electric charge in physical vacuum: Non-zero spatial extent and mass generation". Cent. Eur. J. Phys. 11 (3): 325–335. arXiv:1204.6380. Bibcode:2013CEJPh..11..325D. doi:10.2478/s11534-012-0159-z.
22. ^ K. G. Zloshchastiev, Logarithmic nonlinearity in theories of quantum gravity: Origin of time and observational consequences, Grav. Cosmol. 16 (2010) 288-297 ArXiv:0906.4282.
23. ^ K. G. Zloshchastiev, Vacuum Cherenkov effect in logarithmic nonlinear quantum theory, Phys. Lett. A 375 (2011) 2305–2308 ArXiv:1003.0657. |
9a7d69f5f513f53c | Take the 2-minute tour ×
In my notes, I have the Time Independent Schrodinger equation for a free particle $$\frac{\partial^2 \psi}{\partial x^2}+\frac{p^2}{\hbar^2}\psi=0\tag1$$
The solution to this is given, in my notes, as $$\Large \psi(x)=C e^\left(\frac{ipx}{\hbar}\right)\tag2$$
Now, since (1) is a second order homogeneous equation with constant coefficients, given the coefficients we have, we get a pair of complex roots:$$r_{1,2}=\pm \frac{ip}{\hbar}\tag3$$
Thus, the most general solution looks something like:$$\psi(x)=c_1 \cos \left(\frac{px}{\hbar}\right)+c_2 \sin \left(\frac{px}{\hbar}\right)\tag4$$
However, instead of writing the solution as a cosine plus a sin, the professor seems to have taken a special case of the general solution (with $c_1=1$ and $c_2=i$) and converted the resulting $$\psi(x)=\cos \left(\frac{px}{\hbar}\right)+ i\sin \left(\frac{px}{\hbar}\right)\tag5$$ into exponential form, using $$e^{i\theta}=\cos \theta + i\sin \theta \tag6$$ to get (2).
The main question I have concerning this is: shouldn't we be going after real solutions, and ignoring the complex ones for this particular situation? According to my understanding $\Psi(x,t)$ is complex but $\psi(x)$ should be real. Thanks in advance.
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The wavefunction needn't and shouldn't be real. – Mew Feb 8 '13 at 10:38
There are cases where you can get away with a real wavefunction, but the complex case is more general and fundamental. The free particle Hamiltonian $\hat{H}$ commutes with reflection $x\rightarrow -x$,$p\rightarrow -p$, so states with momenta $\pm p$ are both solutions. In equation (2) they have chosen the solution which is an eigenvalue of the momentum operator $\hat{p}$ with a plus sign $+$. The other sign is also a solution, representing a wave going in the opposite direction. Your real solution contains both left moving and right moving waves. – Michael Brown Feb 8 '13 at 11:05
If you look at the particle current $\vec{j}\propto \psi^\star \nabla \psi - \psi \nabla \psi^\star$ you'll see that real wavefunctions correspond to states where there is no net current, so you can only really expect them to turn up when you have bound states. If there is nothing to reflect a particle back the way it came then it is free to move off to infinity and the current can't vanish, so the wavefunction can't be real. – Michael Brown Feb 8 '13 at 11:10
Related: The book of Griffiths, Intro to QM, Problem 2.1b, p.24; and this Phys.SE post. – Qmechanic Feb 8 '13 at 15:54
1 Answer 1
There is no need for the solution $\psi(x)$ to be real. What must be real is the probability density that is "carried" by $\psi(x)$. In some loose and imprecise intuitive way, you may think about a TV image carried by electromagnetic waves. The signal that travels is not itself the image, but it carries it, and you can recover the image by decoding the signal properly.
Somewhat similarly, the complex wave function that is found by solving Schrödinger equation carries the information of "where the particle is likely to be", but in an indirect manner. The information on the probability density $P(x)$ of finding the particle is recovered from $\psi(x)$ simply by multiplying it times its complex conjugate:
$$\psi(x)^*\psi(x) = P(x)$$
that gives a real function as a result. Note that it is a density: what you compute eventually is the probability of finding the particle between $x=a$ and $x=b$ as $\int_{a}^{b} P(x) dx$
As you know, when you multiply a complex number(/function) times its complex conjugate, the information on the phase is lost:
$$\rho e^{i \theta}\rho e^{-i \theta}=\rho^{2}$$
For that reason, in some places one can (not quite correctly) read that the phase has no physical meaning (see footnote), and then you may wonder "if I eventually get real numbers, why did not they invent a theory that directly handles real functions?".
The answer is that, among other reasons, complex wave functions make life interesting because, since the Schrödinger equation is linear, the superposition principle holds for its solutions. Wave functions add, and it is in that addition where the relative phases play the most important role.
The archetypical case happens in the double slit experiment. If $\psi_{1}$ and $\psi_{2}$ are the wave functions that represent the particle coming from the hole number $1$ and $2$ respectively, the final wave function is $$\psi_{1}+\psi_{2}$$ and thus the probability density of finding the particle after it has crossed the screen with two holes is found from $$P_{1+2}= (\psi_{1}+\psi_{2})^{*}(\psi_{1}+\psi_{2}) $$
That is, you shall first add the wave functions representing the individual holes to have the combined complex wave function, and then compute the probability density. In that addition, the phase informations carried by $\psi_{1}$ and $\psi_{2}$ play the most important role, since they give rise to interference patterns.
Comment: Feynman is quoted to have said "One of the miseries of life is that everybody names things a little bit wrong, and so it makes everything a little harder to understand in the world than it would be if it were named differently." It is quite similar here. Every book says that the phase of the wave function has no physical meaning. That is not 100% correct, as you see.
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3af10ff4bac37c1c | Schrödinger equation
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In quantum mechanics, the Schrödinger equation is a partial differential equation that describes how the quantum state of a physical system changes with time. It was formulated in late 1925, and published in 1926, by the Austrian physicist Erwin Schrödinger.[1]
In classical mechanics, the equation of motion is Newton's second law, (F = ma), used to mathematically predict what the system will do at any time after the initial conditions of the system. In quantum mechanics, the analogue of Newton's law is Schrödinger's equation for a quantum system (usually atoms, molecules, and subatomic particles whether free, bound, or localized). It is not a simple algebraic equation, but in general a linear partial differential equation, describing the time-evolution of the system's wave function (also called a "state function").[2]:1–2
The concept of a wavefunction is a fundamental postulate of quantum mechanics. Schrödinger's equation is also often presented as a separate postulate, but some authors[3]:Chapter 3 assert it can be derived from symmetry principles. Generally, "derivations" of the Schrödinger equation demonstrate its mathematical plausibility for describing wave–particle duality.
In the standard interpretation of quantum mechanics, the wave function is the most complete description that can be given of a physical system. Solutions to Schrödinger's equation describe not only molecular, atomic, and subatomic systems, but also macroscopic systems, possibly even the whole universe.[4]:292ff The Schrödinger equation, in its most general form, is consistent with both classical mechanics and special relativity, but the original formulation by Schrödinger himself was non-relativistic.
The Schrödinger equation is not the only way to make predictions in quantum mechanics - other formulations can be used, such as Werner Heisenberg's matrix mechanics, and Richard Feynman's path integral formulation.
Time-dependent equation[edit]
The form of the Schrödinger equation depends on the physical situation (see below for special cases). The most general form is the time-dependent Schrödinger equation, which gives a description of a system evolving with time:[5]:143
A wave function that satisfies the non-relativistic Schrödinger equation with V = 0. In other words, this corresponds to a particle traveling freely through empty space. The real part of the wave function is plotted here.
Time-dependent Schrödinger equation (general)
where i is the imaginary unit, ħ is the Planck constant divided by 2π, the symbol ∂/∂t indicates a partial derivative with respect to time t, Ψ (the Greek letter Psi) is the wave function of the quantum system, and Ĥ is the Hamiltonian operator (which characterizes the total energy of any given wave function and takes different forms depending on the situation).
Each of these three rows is a wave function which satisfies the time-dependent Schrödinger equation for a harmonic oscillator. Left: The real part (blue) and imaginary part (red) of the wave function. Right: The probability distribution of finding the particle with this wave function at a given position. The top two rows are examples of stationary states, which correspond to standing waves. The bottom row is an example of a state which is not a stationary state. The right column illustrates why stationary states are called "stationary".
The most famous example is the non-relativistic Schrödinger equation for a single particle moving in an electric field (but not a magnetic field; see the Pauli equation):[6]
Time-dependent Schrödinger equation
(single non-relativistic particle)
i\hbar\frac{\partial}{\partial t} \Psi(\mathbf{r},t) = \left [ \frac{-\hbar^2}{2\mu}\nabla^2 + V(\mathbf{r},t)\right ] \Psi(\mathbf{r},t)
where μ is the particle's "reduced mass", V is its potential energy, 2 is the Laplacian (a differential operator), and Ψ is the wave function (more precisely, in this context, it is called the "position-space wave function"). In plain language, it means "total energy equals kinetic energy plus potential energy", but the terms take unfamiliar forms for reasons explained below.
Given the particular differential operators involved, this is a linear partial differential equation. It is also a diffusion equation, but unlike the heat equation, this one is also a wave equation given the imaginary unit present in the transient term.
The term "Schrödinger equation" can refer to both the general equation (first box above), or the specific nonrelativistic version (second box above and variations thereof). The general equation is indeed quite general, used throughout quantum mechanics, for everything from the Dirac equation to quantum field theory, by plugging in various complicated expressions for the Hamiltonian. The specific nonrelativistic version is a simplified approximation to reality, which is quite accurate in many situations, but very inaccurate in others (see relativistic quantum mechanics and relativistic quantum field theory).
To apply the Schrödinger equation, the Hamiltonian operator is set up for the system, accounting for the kinetic and potential energy of the particles constituting the system, then inserted into the Schrödinger equation. The resulting partial differential equation is solved for the wave function, which contains information about the system.
Time-independent equation[edit]
The time-independent Schrödinger equation predicts that wave functions can form standing waves, called stationary states (also called "orbitals", as in atomic orbitals or molecular orbitals). These states are important in their own right, and if the stationary states are classified and understood, then it becomes easier to solve the time-dependent Schrödinger equation for any state. The time-independent Schrödinger equation is the equation describing stationary states. (It is only used when the Hamiltonian itself is not dependent on time. However, even in this case the total wave function still has a time dependency.)
Time-independent Schrödinger equation (general)
E\Psi=\hat H \Psi
In words, the equation states:
When the Hamiltonian operator acts on a certain wave function Ψ, and the result is proportional to the same wave function Ψ, then Ψ is a stationary state, and the proportionality constant, E, is the energy of the state Ψ.
The time-independent Schrödinger equation is discussed further below. In linear algebra terminology, this equation is an eigenvalue equation.
As before, the most famous manifestation is the non-relativistic Schrödinger equation for a single particle moving in an electric field (but not a magnetic field):
Time-independent Schrödinger equation (single non-relativistic particle)
E \Psi(\mathbf{r}) = \left[ \frac{-\hbar^2}{2\mu}\nabla^2 + V(\mathbf{r}) \right] \Psi(\mathbf{r})
with definitions as above.
The Schrödinger equation and its solutions introduced a breakthrough in thinking about physics. Schrödinger's equation was the first of its type, and solutions led to consequences that were very unusual and unexpected for the time.
Total, kinetic, and potential energy[edit]
The overall form of the equation is not unusual or unexpected as it uses the principle of the conservation of energy. The terms of the nonrelativistic Schrödinger equation can 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.
The Schrödinger equation predicts that if certain properties of a system are measured, the result may be quantized, meaning that only specific discrete values can occur. One example is energy quantization: the energy of an electron in an atom is always one of the quantized energy levels, a fact discovered via atomic spectroscopy. (Energy quantization is discussed below.) Another example is quantization of angular momentum. This was an assumption in the earlier Bohr model of the atom, but it is a prediction of the Schrödinger equation.
Another result of the Schrödinger equation is that not every measurement gives a quantized result in quantum mechanics. For example, position, momentum, time, and (in some situations) energy can have any value across a continuous range.[7]:165–167
Measurement and uncertainty[edit]
In classical mechanics, a particle has, at every moment, an exact position and an exact momentum. These values change deterministically as the particle moves according to Newton's laws. In quantum mechanics, particles do not have exactly determined properties, and when they are measured, the result is randomly drawn from a probability distribution. The Schrödinger equation predicts what the probability distributions are, but fundamentally cannot predict the exact result of each measurement.
The Heisenberg uncertainty principle is the statement of the inherent measurement uncertainty in quantum mechanics. It states that the more precisely a particle's position is known, the less precisely its momentum is known, and vice versa.
The Schrödinger equation describes the (deterministic) evolution of the wave function of a particle. However, even if the wave function is known exactly, the result of a specific measurement on the wave function is uncertain.
Quantum tunneling[edit]
Main article: Quantum tunneling
Quantum tunneling through a barrier. A particle coming from the left does not have enough energy to climb the barrier. However, it can sometimes "tunnel" to the other side.
In classical physics, when a ball is rolled slowly up a large hill, it will come to a stop and roll back, because it doesn't have enough energy to get over the top of the hill to the other side. However, the Schrödinger equation predicts that there is a small probability that the ball will get to the other side of the hill, even if it has too little energy to reach the top. This is called quantum tunneling. It is related to the distribution of energy: Although the ball's assumed position seems to be on one side of the hill, there is a chance of finding it on the other side.
Particles as waves[edit]
A double slit experiment showing the accumulation of electrons on a screen as time passes.
The nonrelativistic Schrödinger equation is a type of partial differential equation called a wave equation. Therefore it is often said particles can exhibit behavior usually attributed to waves. In most modern interpretations this description is reversed – the quantum state, i.e. wave, is the only genuine physical reality, and under the appropriate conditions it can show features of particle-like behavior.
Two-slit diffraction is a famous example of the strange behaviors that waves regularly display, that are not intuitively associated with particles. The overlapping waves from the two slits cancel each other out in some locations, and reinforce each other in other locations, causing a complex pattern to emerge. Intuitively, one would not expect this pattern from firing a single particle at the slits, because the particle should pass through one slit or the other, not a complex overlap of both.
However, since the Schrödinger equation is a wave equation, a single particle fired through a double-slit does show this same pattern (figure on right). Note: The experiment must be repeated many times for the complex pattern to emerge. The appearance of the pattern proves that each electron passes through both slits simultaneously.[8][9][10] Although this is counterintuitive, the prediction is correct; in particular, electron diffraction and neutron diffraction are well understood and widely used in science and engineering.
Related to diffraction, particles also display superposition and interference.
The superposition property allows the particle to be in a quantum superposition of two or more states with different classical properties at the same time. For example, a particle can have several different energies at the same time, and can be in several different locations at the same time. In the above example, a particle can pass through two slits at the same time. This superposition is still a single quantum state, as shown by the interference effects, even though that conflicts with classical intuition.
Interpretation of the wave function[edit]
The Schrödinger equation provides a way to calculate the wave function of a system and how it changes dynamically in time. However, the Schrödinger equation does not directly say what, exactly, the wave function is. Interpretations of quantum mechanics address questions such as what the relation is between the wave function, the underlying reality, and the results of experimental measurements.
An important aspect is the relationship between the Schrödinger equation and wavefunction collapse. In the oldest Copenhagen interpretation, particles follow the Schrödinger equation except during wavefunction collapse, during which they behave entirely differently. The advent of quantum decoherence theory allowed alternative approaches (such as the Everett many-worlds interpretation and consistent histories), wherein the Schrödinger equation is always satisfied, and wavefunction collapse should be explained as a consequence of the Schrödinger equation.
Historical background and development[edit]
Erwin Schrödinger
Following Max Planck's quantization of light (see black body radiation), Albert Einstein interpreted Planck's quanta to be photons, particles of light, and proposed that the energy of a photon is proportional to its frequency, one of the first signs of wave–particle duality. Since energy and momentum are related in the same way as frequency and wavenumber in special relativity, it followed that the momentum p of a photon is inversely proportional to its wavelength λ, or proportional to its wavenumber k.
p = \frac{h}{\lambda} = \hbar k
where h is Planck's constant. Louis de Broglie hypothesized that this is true for all particles, even particles which have mass such as electrons. He showed that, assuming that the matter waves propagate along with their particle counterparts, electrons form standing waves, meaning that only certain discrete rotational frequencies about the nucleus of an atom are allowed.[11] These quantized orbits correspond to discrete energy levels, and de Broglie reproduced the Bohr model formula for the energy levels. The Bohr model was based on the assumed quantization of angular momentum L according to:
L = n{h \over 2\pi} = n\hbar.
According to de Broglie the electron is described by a wave and a whole number of wavelengths must fit along the circumference of the electron's orbit:
n \lambda = 2 \pi r.\,
This approach essentially confined the electron wave in one dimension, along a circular orbit of radius r.
In 1921, prior to de Broglie, Arthur C. Lunn at the University of Chicago had used the same argument based on the completion of the relativistic energy–momentum 4-vector to derive what we now call the de Broglie relation[12] Unlike de Broglie, Lunn went on to formulate the differential equation now known as the Schrödinger equation, and solve for its energy eigenvalues for the hydrogen atom. Unfortunately the paper was rejected by the Physical Review, as recounted by Kamen.[13]
Following up on de Broglie's ideas, physicist Peter Debye made an offhand comment that if particles behaved as waves, they should satisfy some sort of wave equation. Inspired by Debye's remark, Schrödinger decided to find a proper 3-dimensional wave equation for the electron. He was guided by William R. Hamilton's analogy between mechanics and optics, encoded in the observation that the zero-wavelength limit of optics resembles a mechanical system — the trajectories of light rays become sharp tracks that obey Fermat's principle, an analog of the principle of least action.[14] A modern version of his reasoning is reproduced below. The equation he found is:[15]
However, by that time, Arnold Sommerfeld had refined the Bohr model with relativistic corrections.[16][17] Schrödinger used the relativistic energy momentum relation to find what is now known as the Klein–Gordon equation in a Coulomb potential (in natural units):
\left(E + {e^2\over r} \right)^2 \psi(x) = - \nabla^2\psi(x) + m^2 \psi(x).
He found the standing waves of this relativistic equation, but the relativistic corrections disagreed with Sommerfeld's formula. Discouraged, he put away his calculations and secluded himself in an isolated mountain cabin in December 1925.[18]
While at the cabin, Schrödinger decided that his earlier non-relativistic calculations were novel enough to publish, and decided to leave off the problem of relativistic corrections for the future. Despite the difficulties in solving the differential equation for hydrogen (he had sought help from his friend the mathematician Hermann Weyl[19]:3) Schrödinger showed that his non-relativistic version of the wave equation produced the correct spectral energies of hydrogen in a paper published in 1926.[19]:1[20] In the equation, Schrödinger computed the hydrogen spectral series by treating a hydrogen atom's electron as a wave Ψ(x, t), moving in a potential well V, created by the proton. This computation accurately reproduced the energy levels of the Bohr model. In a paper, Schrödinger himself explained this equation as follows:
This 1926 paper was enthusiastically endorsed by Einstein, who saw the matter-waves as an intuitive depiction of nature, as opposed to Heisenberg's matrix mechanics, which he considered overly formal.[22]
The Schrödinger equation details the behavior of Ψ but says nothing of its nature. Schrödinger tried to interpret it as a charge density in his fourth paper, but he was unsuccessful.[23]:219 In 1926, just a few days after Schrödinger's fourth and final paper was published, Max Born successfully interpreted Ψ as the probability amplitude, whose absolute square is equal to probability density.[23]:220 Schrödinger, though, always opposed a statistical or probabilistic approach, with its associated discontinuities—much like Einstein, who believed that quantum mechanics was a statistical approximation to an underlying deterministic theory— and never reconciled with the Copenhagen interpretation.[24]
Louis de Broglie in his later years proposed a real valued wave function connected to the complex wave function by a proportionality constant and developed the De Broglie–Bohm theory.
The wave equation for particles[edit]
The Schrödinger equation is a wave equation, since the solutions are functions which describe wave-like motions. Wave equations in physics can normally be derived from other physical laws – the wave equation for mechanical vibrations on strings and in matter can be derived from Newton's laws – where the wave function represents the displacement of matter, and electromagnetic waves from Maxwell's equations, where the wave functions are electric and magnetic fields. The basis for Schrödinger's equation, on the other hand, is the energy of the system and a separate postulate of quantum mechanics: the wave function is a description of the system.[25] The Schrödinger equation is therefore a new concept in itself; as Feynman put it:
Spherical harmonics are to the Schrödinger equation what the math of Henri Poincaré was to Einstein's theory of relativity: foundational.
The foundation of the equation is structured to be a linear differential equation based on classical energy conservation, and consistent with the De Broglie relations. The solution is the wave function ψ, which contains all the information that can be known about the system. In the Copenhagen interpretation, the modulus of ψ is related to the probability the particles are in some spatial configuration at some instant of time. Solving the equation for ψ can be used to predict how the particles will behave under the influence of the specified potential and with each other.
The Schrödinger equation was developed principally from the De Broglie hypothesis, a wave equation that would describe particles,[27] and can be constructed as shown informally in the following sections.[28] For a more rigorous description of Schrödinger's equation, see also.[29]
Consistency with energy conservation[edit]
The total energy E of a particle is the sum of kinetic energy T and potential energy V, this sum is also the frequent expression for the Hamiltonian H in classical mechanics:
E = T + V =H \,\!
Explicitly, for a particle in one dimension with position x, mass m and momentum p, and potential energy V which generally varies with position and time t:
E = \frac{p^2}{2m}+V(x,t)=H.
For three dimensions, the position vector r and momentum vector p must be used:
E = \frac{\mathbf{p}\cdot\mathbf{p}}{2m}+V(\mathbf{r},t)=H
This formalism can be extended to any fixed number of particles: the total energy of the system is then the total kinetic energies of the particles, plus the total potential energy, again the Hamiltonian. However, there can be interactions between the particles (an N-body problem), so the potential energy V can change as the spatial configuration of particles changes, and possibly with time. The potential energy, in general, is not the sum of the separate potential energies for each particle, it is a function of all the spatial positions of the particles. Explicitly:
E=\sum_{n=1}^N \frac{\mathbf{p}_n\cdot\mathbf{p}_n}{2m_n} + V(\mathbf{r}_1,\mathbf{r}_2\cdots\mathbf{r}_N,t) = H \,\!
The simplest wavefunction is a plane wave of the form:
\Psi(\mathbf{r},t) = A e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)} \,\!
where the A is the amplitude, k the wavevector, and ω the angular frequency, of the plane wave. In general, physical situations are not purely described by plane waves, so for generality the superposition principle is required; any wave can be made by superposition of sinusoidal plane waves. So if the equation is linear, a linear combination of plane waves is also an allowed solution. Hence a necessary and separate requirement is that the Schrödinger equation is a linear differential equation.
For discrete k the sum is a superposition of plane waves:
\Psi(\mathbf{r},t) = \sum_{n=1}^\infty A_n e^{i(\mathbf{k}_n\cdot\mathbf{r}-\omega_n t)} \,\!
for some real amplitude coefficients An, and for continuous k the sum becomes an integral, the Fourier transform of a momentum space wavefunction:[30]
\Psi(\mathbf{r},t) = \frac{1}{(\sqrt{2\pi})^3}\int\Phi(\mathbf{k})e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}d^3\mathbf{k} \,\!
where d3k = dkxdkydkz is the differential volume element in k-space, and the integrals are taken over all k-space. The momentum wavefunction Φ(k) arises in the integrand since the position and momentum space wavefunctions are Fourier transforms of each other.
Consistency with the De Broglie relations[edit]
Diagrammatic summary of the quantities related to the wavefunction, as used in De broglie's hypothesis and development of the Schrödinger equation.[27]
Einstein's light quanta hypothesis (1905) states that the energy E of a photon is proportional to the frequency ν (or angular frequency, ω = 2πν) of the corresponding quantum wavepacket of light:
E = h\nu = \hbar \omega \,\!
Likewise De Broglie's hypothesis (1924) states that any particle can be associated with a wave, and that the momentum p of the particle is inversely proportional to the wavelength λ of such a wave (or proportional to the wavenumber, k = 2π/λ), in one dimension, by:
p = \frac{h}{\lambda} = \hbar k\;,
while in three dimensions, wavelength λ is related to the magnitude of the wavevector k:
\mathbf{p} = \hbar \mathbf{k}\,,\quad |\mathbf{k}| = \frac{2\pi}{\lambda} \,.
The Planck–Einstein and de Broglie relations illuminate the deep connections between energy with time, and space with momentum, and express wave–particle duality. In practice, natural units comprising ħ = 1 are used, as the De Broglie equations reduce to identities: allowing momentum, wavenumber, energy and frequency to be used interchangeably, to prevent duplication of quantities, and reduce the number of dimensions of related quantities. For familiarity SI units are still used in this article.
Schrödinger's insight,[citation needed] late in 1925, was to express the phase of a plane wave as a complex phase factor using these relations:
\Psi = Ae^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)} = Ae^{i(\mathbf{p}\cdot\mathbf{r}-Et)/\hbar}
and to realize that the first order partial derivatives were:
with respect to space:
\nabla\Psi = \dfrac{i}{\hbar}\mathbf{p}Ae^{i(\mathbf{p}\cdot\mathbf{r}-Et)/\hbar} = \dfrac{i}{\hbar}\mathbf{p}\Psi
with respect to time:
\dfrac{\partial \Psi}{\partial t} = -\dfrac{i E}{\hbar} Ae^{i(\mathbf{p}\cdot\mathbf{r}-Et)/\hbar} = -\dfrac{i E}{\hbar} \Psi
Another postulate of quantum mechanics is that all observables are represented by linear Hermitian operators which act on the wavefunction, and the eigenvalues of the operator are the values the observable takes. The previous derivatives are consistent with the energy operator, corresponding to the time derivative,
where E are the energy eigenvalues, and the momentum operator, corresponding to the spatial derivatives (the gradient ),
\hat{\mathbf{p}} \Psi = -i\hbar\nabla \Psi = \mathbf{p} \Psi
where p is a vector of the momentum eigenvalues. In the above, the "hats" ( ^ ) indicate these observables are operators, not simply ordinary numbers or vectors. The energy and momentum operators are differential operators, while the potential energy function V is just a multiplicative factor.
Substituting the energy and momentum operators into the classical energy conservation equation obtains the operator:
E= \dfrac{\mathbf{p}\cdot\mathbf{p}}{2m}+V \quad \rightarrow \quad \hat{E} = \dfrac{\hat{\mathbf{p}}\cdot\hat{\mathbf{p}}}{2m} + V
so in terms of derivatives with respect to time and space, acting this operator on the wavefunction Ψ immediately led Schrödinger to his equation:[citation needed]
Wave–particle duality can be assessed from these equations as follows. The kinetic energy T is related to the square of momentum p. As the particle's momentum increases, the kinetic energy increases more rapidly, but since the wavenumber |k| increases the wavelength λ decreases. In terms of ordinary scalar and vector quantities (not operators):
\mathbf{p}\cdot\mathbf{p} \propto \mathbf{k}\cdot\mathbf{k} \propto T \propto \dfrac{1}{\lambda^2}
The kinetic energy is also proportional to the second spatial derivatives, so it is also proportional to the magnitude of the curvature of the wave, in terms of operators:
\hat{T} \Psi = \frac{-\hbar^2}{2m}\nabla\cdot\nabla \Psi \, \propto \, \nabla^2 \Psi \,.
As the curvature increases, the amplitude of the wave alternates between positive and negative more rapidly, and also shortens the wavelength. So the inverse relation between momentum and wavelength is consistent with the energy the particle has, and so the energy of the particle has a connection to a wave, all in the same mathematical formulation.[27]
Wave and particle motion[edit]
Increasing levels of wavepacket localization, meaning the particle has a more localized position.
In the limit ħ → 0, the particle's position and momentum become known exactly. This is equivalent to the classical particle.
Schrödinger required that a wave packet solution near position r with wavevector near k will move along the trajectory determined by classical mechanics for times short enough for the spread in k (and hence in velocity) not to substantially increase the spread in r. Since, for a given spread in k, the spread in velocity is proportional to Planck's constant ħ, it is sometimes said that in the limit as ħ approaches zero, the equations of classical mechanics are restored from quantum mechanics.[31] Great care is required in how that limit is taken, and in what cases.
The limiting short-wavelength is equivalent to ħ tending to zero because this is limiting case of increasing the wave packet localization to the definite position of the particle (see images right). Using the Heisenberg uncertainty principle for position and momentum, the products of uncertainty in position and momentum become zero as ħ → 0:
\sigma(x) \sigma(p_x) \geqslant \frac{\hbar}{2} \quad \rightarrow \quad \sigma(x) \sigma(p_x) \geqslant 0 \,\!
where σ denotes the (root mean square) measurement uncertainty in x and px (and similarly for the y and z directions) which implies the position and momentum can only be known to arbitrary precision in this limit.
The Schrödinger equation in its general form
i\hbar \frac{\partial}{\partial t} \Psi\left(\mathbf{r},t\right) = \hat{H} \Psi\left(\mathbf{r},t\right) \,\!
is closely related to the Hamilton–Jacobi equation (HJE)
\frac{\partial}{\partial t} S(q_i,t) = H\left(q_i,\frac{\partial S}{\partial q_i},t \right) \,\!
where S is action and H is the Hamiltonian function (not operator). Here the generalized coordinates qi for i = 1, 2, 3 (used in the context of the HJE) can be set to the position in Cartesian coordinates as r = (q1, q2, q3) = (x, y, z).[31]
\Psi = \sqrt{\rho(\mathbf{r},t)} e^{iS(\mathbf{r},t)/\hbar}\,\!
where ρ is the probability density, into the Schrödinger equation and then taking the limit ħ → 0 in the resulting equation, yields the Hamilton–Jacobi equation.
The implications are:
• The motion of a particle, described by a (short-wavelength) wave packet solution to the Schrödinger equation, is also described by the Hamilton–Jacobi equation of motion.
• The Schrödinger equation includes the wavefunction, so its wave packet solution implies the position of a (quantum) particle is fuzzily spread out in wave fronts. On the contrary, the Hamilton–Jacobi equation applies to a (classical) particle of definite position and momentum, instead the position and momentum at all times (the trajectory) are deterministic and can be simultaneously known.
Non-relativistic quantum mechanics[edit]
The quantum mechanics of particles without accounting for the effects of special relativity, for example particles propagating at speeds much less than light, is known as non-relativistic quantum mechanics. Following are several forms of Schrödinger's equation in this context for different situations: time independence and dependence, one and three spatial dimensions, and one and N particles.
In actuality, the particles constituting the system do not have the numerical labels used in theory. The language of mathematics forces us to label the positions of particles one way or another, otherwise there would be confusion between symbols representing which variables are for which particle.[29]
Time independent[edit]
If the Hamiltonian is not an explicit function of time, the equation is separable into a product of spatial and temporal parts. In general, the wavefunction takes the form:
\Psi(\text{space coords},t)=\psi(\text{space coords})\tau(t)\,.
where ψ(space coords) is a function of all the spatial coordinate(s) of the particle(s) constituting the system only, and τ(t) is a function of time only.
Substituting for ψ into the Schrödinger equation for the relevant number of particles in the relevant number of dimensions, solving by separation of variables implies the general solution of the time-dependent equation has the form:[15]
\Psi(\text{space coords},t) = \psi(\text{space coords}) e^{-i{E t/\hbar}} \,.
Since the time dependent phase factor is always the same, only the spatial part needs to be solved for in time independent problems. Additionally, the energy operator Ê = /t can always be replaced by the energy eigenvalue E, thus the time independent Schrödinger equation is an eigenvalue equation for the Hamiltonian operator:[5]:143ff
\hat{H} \psi = E \psi
This is true for any number of particles in any number of dimensions (in a time independent potential). This case describes the standing wave solutions of the time-dependent equation, which are the states with definite energy (instead of a probability distribution of different energies). In physics, these standing waves are called "stationary states" or "energy eigenstates"; in chemistry they are called "atomic orbitals" or "molecular orbitals". Superpositions of energy eigenstates change their properties according to the relative phases between the energy levels.
The energy eigenvalues from this equation form a discrete spectrum of values, so mathematically energy must be quantized. More specifically, the energy eigenstates form a basis – any wavefunction may be written as a sum over the discrete energy states or an integral over continuous energy states, or more generally as an integral over a measure. This is the spectral theorem in mathematics, and in a finite state space it is just a statement of the completeness of the eigenvectors of a Hermitian matrix.
One-dimensional examples[edit]
For a particle in one dimension, the Hamiltonian is:
\hat{H} = \frac{\hat{p}^2}{2m} + V(x) \,, \quad \hat{p} = -i\hbar \frac{d}{d x}
and substituting this into the general Schrödinger equation gives:
-\frac{\hbar^2}{2m}\frac{d^2}{d x^2}\psi(x) + V(x)\psi(x) = E\psi(x)
This is the only case the Schrödinger equation is an ordinary differential equation, rather than a partial differential equation. The general solutions are always of the form:
\Psi(x,t)=\psi(x) e^{-iEt/\hbar} \, .
For N particles in one dimension, the Hamiltonian is:
\hat{H} = \sum_{n=1}^{N}\frac{\hat{p}_n^2}{2m_n} + V(x_1,x_2,\cdots x_N) \,,\quad \hat{p}_n = -i\hbar \frac{\partial}{\partial x_n}
where the position of particle n is xn. The corresponding Schrödinger equation is:
-\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\frac{\partial^2}{\partial x_n^2}\psi(x_1,x_2,\cdots x_N) + V(x_1,x_2,\cdots x_N)\psi(x_1,x_2,\cdots x_N) = E\psi(x_1,x_2,\cdots x_N) \, .
so the general solutions have the form:
\Psi(x_1,x_2,\cdots x_N,t) = e^{-iEt/\hbar}\psi(x_1,x_2\cdots x_N)
For non-interacting distinguishable particles,[32] the potential of the system only influences each particle separately, so the total potential energy is the sum of potential energies for each particle:
V(x_1,x_2,\cdots x_N) = \sum_{n=1}^N V(x_n) \, .
and the wavefunction can be written as a product of the wavefunctions for each particle:
\Psi(x_1,x_2,\cdots x_N,t) = e^{-i{E t/\hbar}}\prod_{n=1}^N\psi(x_n) \, ,
For non-interacting identical particles, the potential is still a sum, but wavefunction is a bit more complicated - it is a sum over the permutations of products of the separate wavefunctions to account for particle exchange. In general for interacting particles, the above decompositions are not possible.
Free particle[edit]
For no potential, V = 0, so the particle is free and the equation reads:[5]:151ff
- E \psi = \frac{\hbar^2}{2m}{d^2 \psi \over d x^2}\,
which has oscillatory solutions for E > 0 (the Cn are arbitrary constants):
\psi_E(x) = C_1 e^{i\sqrt{2mE/\hbar^2}\,x} + C_2 e^{-i\sqrt{2mE/\hbar^2}\,x}\,
and exponential solutions for E < 0
\psi_{-|E|}(x) = C_1 e^{\sqrt{2m|E|/\hbar^2}\,x} + C_2 e^{-\sqrt{2m|E|/\hbar^2}\,x}.\,
The exponentially growing solutions have an infinite norm, and are not physical. They are not allowed in a finite volume with periodic or fixed boundary conditions.
See also free particle and wavepacket for more discussion on the free particle.
Constant potential[edit]
Animation of a de Broglie wave incident on a barrier.
For a constant potential, V = V0, the solution is oscillatory for E > V0 and exponential for E < V0, corresponding to energies that are allowed or disallowed in classical mechanics. Oscillatory solutions have a classically allowed energy and correspond to actual classical motions, while the exponential solutions have a disallowed energy and describe a small amount of quantum bleeding into the classically disallowed region, due to quantum tunneling. If the potential V0 grows to infinity, the motion is classically confined to a finite region. Viewed far enough away, every solution is reduced an exponential; the condition that the exponential is decreasing restricts the energy levels to a discrete set, called the allowed energies.[30]
Harmonic oscillator[edit]
A harmonic oscillator in classical mechanics (A–B) and quantum mechanics (C–H). In (A–B), a ball, attached to a spring, oscillates back and forth. (C–H) are six solutions to the Schrödinger Equation for this situation. The horizontal axis is position, the vertical axis is the real part (blue) or imaginary part (red) of the wavefunction. Stationary states, or energy eigenstates, which are solutions to the time-independent Schrödinger Equation, are shown in C,D,E,F, but not G or H.
The Schrödinger equation for this situation is
E\psi = -\frac{\hbar^2}{2m}\frac{d^2}{d x^2}\psi + \frac{1}{2}m\omega^2x^2\psi
It is a notable quantum system to solve for; since the solutions are exact (but complicated – in terms of Hermite polynomials), and it can describe or at least approximate a wide variety of other systems, including vibrating atoms, molecules,[33] and atoms or ions in lattices,[34] and approximating other potentials near equilibrium points. It is also the basis of perturbation methods in quantum mechanics.
There is a family of solutions – in the position basis they are
where n = 0,1,2,..., and the functions Hn are the Hermite polynomials.
Three-dimensional examples[edit]
The extension from one dimension to three dimensions is straightforward, all position and momentum operators are replaced by their three-dimensional expressions and the partial derivative with respect to space is replaced by the gradient operator.
The Hamiltonian for one particle in three dimensions is:
\hat{H} = \frac{\hat{\mathbf{p}}\cdot\hat{\mathbf{p}}}{2m} + V(\mathbf{r}) \,, \quad \hat{\mathbf{p}} = -i\hbar \nabla
generating the equation:
-\frac{\hbar^2}{2m}\nabla^2\psi(\mathbf{r}) + V(\mathbf{r})\psi(\mathbf{r}) = E\psi(\mathbf{r})
with stationary state solutions of the form:
\Psi(\mathbf{r},t) = \psi(\mathbf{r}) e^{-iEt/\hbar}
where the position of the particle is r. Two useful coordinate systems for solving the Schrödinger equation are Cartesian coordinates so that r = (x, y, z) and spherical polar coordinates so that r = (r, θ, φ), although other orthogonal coordinates are useful for solving the equation for systems with certain geometric symmetries.
For N particles in three dimensions, the Hamiltonian is:
\hat{H} = \sum_{n=1}^{N}\frac{\hat{\mathbf{p}}_n\cdot\hat{\mathbf{p}}_n}{2m_n} + V(\mathbf{r}_1,\mathbf{r}_2,\cdots\mathbf{r}_N) \,,\quad \hat{\mathbf{p}}_n = -i\hbar \nabla_n
where the position of particle n is rn and the gradient operators are partial derivatives with respect to the particle's position coordinates. In Cartesian coordinates, for particle n, the position vector is rn = (xn, yn, zn) while the gradient and Laplacian operator are respectively:
\nabla_n = \mathbf{e}_x \frac{\partial}{\partial x_n} + \mathbf{e}_y\frac{\partial}{\partial y_n} + \mathbf{e}_z\frac{\partial}{\partial z_n}\,,\quad \nabla_n^2 = \nabla_n\cdot\nabla_n = \frac{\partial^2}{{\partial x_n}^2} + \frac{\partial^2}{{\partial y_n}^2} + \frac{\partial^2}{{\partial z_n}^2}
The Schrödinger equation is:
-\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\nabla_n^2\Psi(\mathbf{r}_1,\mathbf{r}_2,\cdots\mathbf{r}_N) + V(\mathbf{r}_1,\mathbf{r}_2,\cdots\mathbf{r}_N)\Psi(\mathbf{r}_1,\mathbf{r}_2,\cdots\mathbf{r}_N) = E\Psi(\mathbf{r}_1,\mathbf{r}_2,\cdots\mathbf{r}_N)
with stationary state solutions:
\Psi(\mathbf{r}_1,\mathbf{r}_2\cdots \mathbf{r}_N,t) = e^{-iEt/\hbar}\psi(\mathbf{r}_1,\mathbf{r}_2\cdots \mathbf{r}_N)
Again, for non-interacting distinguishable particles the potential is the sum of particle potentials
V(\mathbf{r}_1,\mathbf{r}_2,\cdots \mathbf{r}_N) = \sum_{n=1}^N V(\mathbf{r}_n)
and the wavefunction is a product of the particle wavefuntions
\Psi(\mathbf{r}_1,\mathbf{r}_2\cdots \mathbf{r}_N,t) = e^{-i{E t/\hbar}}\prod_{n=1}^N\psi(\mathbf{r}_n) \, .
For non-interacting identical particles, the potential is a sum but the wavefunction is a sum over permutations of products. The previous two equations do not apply to interacting particles.
Following are examples where exact solutions are known. See the main articles for further details.
Hydrogen atom[edit]
This form of the Schrödinger equation can be applied to the hydrogen atom:[25][27]
E \psi = -\frac{\hbar^2}{2\mu}\nabla^2\psi - \frac{e^2}{4\pi\varepsilon_0 r}\psi
where e is the electron charge, r is the position of the electron (r = |r| is the magnitude of the position), the potential term is due to the Coulomb interaction, wherein ε0 is the electric constant (permittivity of free space) and
\mu = \frac{m_em_p}{m_e+m_p}
is the 2-body reduced mass of the hydrogen nucleus (just a proton) of mass mp and the electron of mass me. The negative sign arises in the potential term since the proton and electron are oppositely charged. The reduced mass in place of the electron mass is used since the electron and proton together orbit each other about a common centre of mass, and constitute a two-body problem to solve. The motion of the electron is of principle interest here, so the equivalent one-body problem is the motion of the electron using the reduced mass.
The wavefunction for hydrogen is a function of the electron's coordinates, and in fact can be separated into functions of each coordinate.[35] Usually this is done in spherical polar coordinates:
\psi(r,\theta,\phi) = R(r)Y_\ell^m(\theta, \phi) = R(r)\Theta(\theta)\Phi(\phi)
where R are radial functions and \scriptstyle Y_{\ell}^{m}(\theta, \phi ) \, are spherical harmonics of degree and order m. This is the only atom for which the Schrödinger equation has been solved for exactly. Multi-electron atoms require approximative methods. The family of solutions are:[36]
\psi_{n\ell m}(r,\theta,\phi) = \sqrt {{\left ( \frac{2}{n a_0} \right )}^3\frac{(n-\ell-1)!}{2n[(n+\ell)!]} } e^{- r/na_0} \left(\frac{2r}{na_0}\right)^{\ell} L_{n-\ell-1}^{2\ell+1}\left(\frac{2r}{na_0}\right) \cdot Y_{\ell}^{m}(\theta, \phi )
n & = 1,2,3, \dots \\
\ell & = 0,1,2, \dots, n-1 \\
m & = -\ell,\dots,\ell \\
NB: generalized Laguerre polynomials are defined differently by different authors—see main article on them and the hydrogen atom.
Two-electron atoms or ions[edit]
The equation for any two-electron system, such as the neutral helium atom (He, Z = 2), the negative hydrogen ion (H, Z = 1), or the positive lithium ion (Li+, Z = 3) is:[28]
E\psi = -\hbar^2\left[\frac{1}{2\mu}\left(\nabla_1^2 +\nabla_2^2 \right) + \frac{1}{M}\nabla_1\cdot\nabla_2\right] \psi + \frac{e^2}{4\pi\varepsilon_0}\left[ \frac{1}{r_{12}} -Z\left( \frac{1}{r_1}+\frac{1}{r_2} \right) \right] \psi
where r1 is the position of one electron (r1 = |r1| is its magnitude), r2 is the position of the other electron (r2 = |r2| is the magnitude), r12 = |r12| is the magnitude of the separation between them given by
|\mathbf{r}_{12}| = |\mathbf{r}_2 - \mathbf{r}_1 | \,\!
μ is again the two-body reduced mass of an electron with respect to the nucleus of mass M, so this time
\mu = \frac{m_e M}{m_e+M} \,\!
and Z is the atomic number for the element (not a quantum number).
The cross-term of two laplacians
is known as the mass polarization term, which arises due to the motion of atomic nuclei. The wavefunction is a function of the two electron's positions:
\psi = \psi(\mathbf{r}_1,\mathbf{r}_2).
There is no closed form solution for this equation.
Time dependent[edit]
This is the equation of motion for the quantum state. In the most general form, it is written:[5]:143ff
and the solution, the wavefunction, is a function of all the particle coordinates of the system and time. Following are specific cases.
For one particle in one dimension, the Hamiltonian
\hat{H} = \frac{\hat{p}^2}{2m} + V(x,t) \,,\quad \hat{p} = -i\hbar \frac{\partial}{\partial x}
generates the equation:
i\hbar\frac{\partial}{\partial t}\Psi(x,t) = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}\Psi(x,t) + V(x,t)\Psi(x,t)
For N particles in one dimension, the Hamiltonian is:
\hat{H} = \sum_{n=1}^{N}\frac{\hat{p}_n^2}{2m_n} + V(x_1,x_2,\cdots x_N,t) \,,\quad \hat{p}_n = -i\hbar \frac{\partial}{\partial x_n}
where the position of particle n is xn, generating the equation:
i\hbar\frac{\partial}{\partial t}\Psi(x_1,x_2\cdots x_N,t) = -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\frac{\partial^2}{\partial x_n^2}\Psi(x_1,x_2\cdots x_N,t) + V(x_1,x_2\cdots x_N,t)\Psi(x_1,x_2\cdots x_N,t) \, .
For one particle in three dimensions, the Hamiltonian is:
generating the equation:
For N particles in three dimensions, the Hamiltonian is:
where the position of particle n is rn, generating the equation:[5]:141
i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r}_1,\mathbf{r}_2,\cdots\mathbf{r}_N,t) = -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\nabla_n^2\Psi(\mathbf{r}_1,\mathbf{r}_2,\cdots\mathbf{r}_N,t) + V(\mathbf{r}_1,\mathbf{r}_2,\cdots\mathbf{r}_N,t)\Psi(\mathbf{r}_1,\mathbf{r}_2,\cdots\mathbf{r}_N,t)
This last equation is in a very high dimension, so the solutions are not easy to visualize.
Solution methods[edit]
General techniques:
The Schrödinger equation has the following properties: some are useful, but there are shortcomings. Ultimately, these properties arise from the Hamiltonian used, and solutions to the equation.
In the development above, the Schrödinger equation was made to be linear for generality, though this has other implications. If two wave functions ψ1 and ψ2 are solutions, then so is any linear combination of the two:
\displaystyle \psi = a\psi_1 + b \psi_2
where a and b are any complex numbers (the sum can be extended for any number of wavefunctions). This property allows superpositions of quantum states to be solutions of the Schrödinger equation. Even more generally, it holds that a general solution to the Schrödinger equation can be found by taking a weighted sum over all single state solutions achievable. For example, consider a wave function Ψ(x, t) such that the wave function is a product of two functions: one time independent, and one time dependent. If states of definite energy found using the time independent Shrödinger equation are given by ψE(x) with amplitude An and time dependent phase factor is given by
e^{{-iE_n t}/\hbar},
then a valid general solution is
\displaystyle \Psi(x,t) = \sum\limits_{n} A_n \psi_{E_n}(x) e^{{-iE_n t}/\hbar}.
Additionally, the ability to scale solutions allows one to solve for a wave function without normalizing it first. If one has a set of normalized solutions ψn, then
\displaystyle \Psi = \sum\limits_{n} A_n \psi_n
can be normalized by ensuring that
\displaystyle \sum\limits_{n}|A_n|^2 = 1.
This is much more convenient than having to verify that
\displaystyle \int\limits_{-\infty}^{\infty}|\Psi(x)|^2\,dx = \int\limits_{-\infty}^{\infty}\Psi(x)\Psi^{*}(x)\,dx = 1.
Real energy eigenstates[edit]
For the time-independent equation, an additional feature of linearity follows: if two wave functions ψ1 and ψ2 are solutions to the time-independent equation with the same energy E, then so is any linear combination:
\hat H (a\psi_1 + b \psi_2 ) = a \hat H \psi_1 + b \hat H \psi_2 = E (a \psi_1 + b\psi_2).
Two different solutions with the same energy are called degenerate.[30]
In an arbitrary potential, if a wave function ψ solves the time-independent equation, so does its complex conjugate, denoted ψ*. By taking linear combinations, the real and imaginary parts of ψ are each solutions. If there is no degeneracy they can only differ by a factor.
In the time-dependent equation, complex conjugate waves move in opposite directions. If Ψ(x, t) is one solution, then so is Ψ(x, –t). The symmetry of complex conjugation is called time-reversal symmetry.
Space and time derivatives[edit]
Continuity of the wavefunction and its first spatial derivative (in the x direction, y and z coordinates not shown), at some time t.
The Schrödinger equation is first order in time and second in space, which describes the time evolution of a quantum state (meaning it determines the future amplitude from the present).
Explicitly for one particle in 3-dimensional Cartesian coordinates – the equation is
i\hbar{\partial \Psi \over \partial t} = - {\hbar^2\over 2m} \left ( {\partial^2 \Psi \over \partial x^2} + {\partial^2 \Psi \over \partial y^2} + {\partial^2 \Psi \over \partial z^2} \right ) + V(x,y,z,t)\Psi.\,\!
The first time partial derivative implies the initial value (at t = 0) of the wavefunction
\Psi(x,y,z,0) \,\!
is an arbitrary constant. Likewise – the second order derivatives with respect to space implies the wavefunction and its first order spatial derivatives
& \Psi(x_b,y_b,z_b,t) \\
& \frac{\partial}{\partial x}\Psi(x_b,y_b,z_b,t) \quad \frac{\partial}{\partial y}\Psi(x_b,y_b,z_b,t) \quad \frac{\partial}{\partial z}\Psi(x_b,y_b,z_b,t)
\end{align} \,\!
are all arbitrary constants at a given set of points, where xb, yb, zb are a set of points describing boundary b (derivatives are evaluated at the boundaries). Typically there are one or two boundaries, such as the step potential and particle in a box respectively.
As the first order derivatives are arbitrary, the wavefunction can be a continuously differentiable function of space, since at any boundary the gradient of the wavefunction can be matched.
On the contrary, wave equations in physics are usually second order in time, notable are the family of classical wave equations and the quantum Klein–Gordon equation.
Local conservation of probability[edit]
The Schrödinger equation is consistent with probability conservation. Multiplying the Schrödinger equation on the right by the complex conjugate wavefunction, and multiplying the wavefunction to the left of the complex conjugate of the Schrödinger equation, and subtracting, gives the continuity equation for probability:[37]
{ \partial \over \partial t} \rho\left(\mathbf{r},t\right) + \nabla \cdot \mathbf{j} = 0,
is the probability density (probability per unit volume, * denotes complex conjugate), and
\mathbf{j} = {1 \over 2m} \left( \Psi^*\hat{\mathbf{p}}\Psi - \Psi\hat{\mathbf{p}}\Psi^* \right)\,\!
is the probability current (flow per unit area).
Hence predictions from the Schrödinger equation do not violate probability conservation.
Positive energy[edit]
If the potential is bounded from below, meaning there is a minimum value of potential energy, the eigenfunctions of the Schrödinger equation have energy which is also bounded from below. This can be seen most easily by using the variational principle, as follows. (See also below).
For any linear operator  bounded from below, the eigenvector with the smallest eigenvalue is the vector ψ that minimizes the quantity
\langle \psi |\hat{A}|\psi \rangle
over all ψ which are normalized.[37] In this way, the smallest eigenvalue is expressed through the variational principle. For the Schrödinger Hamiltonian Ĥ bounded from below, the smallest eigenvalue is called the ground state energy. That energy is the minimum value of
\langle \psi|\hat{H}|\psi\rangle = \int \psi^*(\mathbf{r}) \left[ - \frac{\hbar^2}{2m} \nabla^2\psi(\mathbf{r}) + V(\mathbf{r})\psi(\mathbf{r})\right] d^3\mathbf{r} = \int \left[ \frac{\hbar^2}{2m}|\nabla\psi|^2 + V(\mathbf{r}) |\psi|^2 \right] d^3\mathbf{r} = \langle \hat{H}\rangle
(using integration by parts). Due to the complex modulus of ψ squared (which is positive definite), the right hand side always greater than the lowest value of V(x). In particular, the ground state energy is positive when V(x) is everywhere positive.
For potentials which are bounded below and are not infinite over a region, there is a ground state which minimizes the integral above. This lowest energy wavefunction is real and positive definite – meaning the wavefunction can increase and decrease, but is positive for all positions. It physically cannot be negative: if it were, smoothing out the bends at the sign change (to minimize the wavefunction) rapidly reduces the gradient contribution to the integral and hence the kinetic energy, while the potential energy changes linearly and less quickly. The kinetic and potential energy are both changing at different rates, so the total energy is not constant, which can't happen (conservation). The solutions are consistent with Schrödinger equation if this wavefunction is positive definite.
The lack of sign changes also shows that the ground state is nondegenerate, since if there were two ground states with common energy E, not proportional to each other, there would be a linear combination of the two that would also be a ground state resulting in a zero solution.
Analytic continuation to diffusion[edit]
The above properties (positive definiteness of energy) allow the analytic continuation of the Schrödinger equation to be identified as a stochastic process. This can be interpreted as the Huygens–Fresnel principle applied to De Broglie waves; the spreading wavefronts are diffusive probability amplitudes.[37]
For a free particle (not subject to a potential) in a random walk, substituting τ = it into the time-dependent Schrödinger equation gives:[38]
{\partial \over \partial \tau} X(\mathbf{r},\tau) = \frac{\hbar}{2m} \nabla ^2 X(\mathbf{r},\tau) \, , \quad X(\mathbf{r},\tau) = \Psi(\mathbf{r},\tau/i)
which has the same form as the diffusion equation, with diffusion coefficient ħ/2m.
Relativistic quantum mechanics[edit]
Relativistic quantum mechanics is obtained where quantum mechanics and special relativity simultaneously apply. In general, one wishes to build relativistic wave equations from the relativistic energy–momentum relation
E^2 = (pc)^2 + (m_0c^2)^2 \, ,
instead of classical energy equations. The Klein–Gordon equation and the Dirac equation are two such equations. The Klein–Gordon equation was the first such equation to be obtained, even before the non-relativistic one, and applies to massive spinless particles. The Dirac equation arose from taking the "square root" of the Klein–Gordon equation by factorizing the entire relativistic wave operator into a product of two operators – one of these is the operator for the entire Dirac equation.
The general form of the Schrödinger equation remains true in relativity, but the Hamiltonian is less obvious. For example, the Dirac Hamiltonian for a particle of mass m and electric charge q in an electromagnetic field (described by the electromagnetic potentials φ and A) is:
\hat{H}_{\text{Dirac}}= \gamma^0 \left[c \boldsymbol{\gamma}\cdot\left(\hat{\mathbf{p}} - q \mathbf{A}\right) + mc^2 + \gamma^0q \phi \right]\,,
in which the γ = (γ1, γ2, γ3) and γ0 are the Dirac gamma matrices related to the spin of the particle. The Dirac equation is true for all spin-1/2 particles, and the solutions to the equation are 4-component spinor fields with two components corresponding to the particle and the other two for the antiparticle.
For the Klein–Gordon equation, the general form of the Schrödinger equation is inconvenient to use, and in practice the Hamiltonian is not expressed in an analogous way to the Dirac Hamiltonian. The equations for relativistic quantum fields can be obtained in other ways, such as starting from a Lagrangian density and using the Euler-Lagrange equations for fields, or use the representation theory of the Lorentz group in which certain representations can be used to fix the equation for a free particle of given spin (and mass).
In general, the Hamiltonian to be substituted in the general Schrödinger equation is not just a function of the position and momentum operators (and possibly time), but also of spin matrices. Also, the solutions to a relativistic wave equation, for a massive particle of spin s, are complex-valued 2(2s + 1)-component spinor fields.
Quantum field theory[edit]
The general equation is also valid and used in quantum field theory, both in relativistic and non-relativistic situations. However, the solution ψ is no longer interpreted as a "wave", but should be interpreted as an operator acting on states existing in a Fock space.
See also[edit]
1. ^ Schrödinger, E. (1926). "An Undulatory Theory of the Mechanics of Atoms and Molecules" (PDF). Physical Review 28 (6): 1049–1070. Bibcode:1926PhRv...28.1049S. doi:10.1103/PhysRev.28.1049. Archived from the original (PDF) on 17 December 2008.
3. ^ Ballentine, Leslie (1998), Quantum Mechanics: A Modern Development, World Scientific Publishing Co., ISBN 9810241054
5. ^ a b c d e Shankar, R. (1994). Principles of Quantum Mechanics (2nd ed.). Kluwer Academic/Plenum Publishers. ISBN 978-0-306-44790-7.
6. ^
7. ^ Nouredine Zettili (17 February 2009). Quantum Mechanics: Concepts and Applications. John Wiley & Sons. ISBN 978-0-470-02678-6.
8. ^ Donati, O; Missiroli, G F; Pozzi, G (1973). "An Experiment on Electron Interference". American Journal of Physics 41: 639–644. doi:10.1119/1.1987321.
9. ^ Brian Greene, The Elegant Universe, p. 110
10. ^ Feynman Lectures on Physics (Vol. 3), R. Feynman, R.B. Leighton, M. Sands, Addison-Wesley, 1965, ISBN 0-201-02118-8
11. ^ de Broglie, L. (1925). "Recherches sur la théorie des quanta" [On the Theory of Quanta] (PDF). Annales de Physique 10 (3): 22–128. Translated version at the Wayback Machine (archived May 9, 2009).
12. ^ Weissman, M.B.; V. V. Iliev; I. Gutman (2008). "A pioneer remembered: biographical notes about Arthur Constant Lunn". Communications in Mathematical and in Computer Chemistry 59 (3): 687–708.
13. ^ Kamen, Martin D. (1985). Radiant Science, Dark Politics. Berkeley and Los Angeles, CA: University of California Press. pp. 29–32. ISBN 0-520-04929-2.
14. ^ Schrodinger, E. (1984). Collected papers. Friedrich Vieweg und Sohn. ISBN 3-7001-0573-8. See introduction to first 1926 paper.
15. ^ a b Encyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC publishers, 1991, (Verlagsgesellschaft) 3-527-26954-1, (VHC Inc.) ISBN 0-89573-752-3
16. ^ Sommerfeld, A. (1919). Atombau und Spektrallinien. Braunschweig: Friedrich Vieweg und Sohn. ISBN 3-87144-484-7.
17. ^ For an English source, see Haar, T. "The Old Quantum Theory".
18. ^ Rhodes, R. (1986). Making of the Atomic Bomb. Touchstone. ISBN 0-671-44133-7.
19. ^ a b Erwin Schrödinger (1982). Collected Papers on Wave Mechanics: Third Edition. American Mathematical Soc. ISBN 978-0-8218-3524-1.
20. ^ Schrödinger, E. (1926). "Quantisierung als Eigenwertproblem; von Erwin Schrödinger". Annalen der Physik 384: 361–377. doi:10.1002/andp.19263840404.
21. ^ Erwin Schrödinger, "The Present situation in Quantum Mechanics," p. 9 of 22. The English version was translated by John D. Trimmer. The translation first appeared first in Proceedings of the American Philosophical Society, 124, 323–38. It later appeared as Section I.11 of Part I of Quantum Theory and Measurement by J.A. Wheeler and W.H. Zurek, eds., Princeton University Press, New Jersey 1983).
22. ^ Einstein, A.; et. al. "Letters on Wave Mechanics: Schrodinger–Planck–Einstein–Lorentz".
23. ^ a b c Moore, W.J. (1992). Schrödinger: Life and Thought. Cambridge University Press. ISBN 0-521-43767-9.
24. ^ It is clear that even in his last year of life, as shown in a letter to Max Born, that Schrödinger never accepted the Copenhagen interpretation.[23]:220
25. ^ a b Molecular Quantum Mechanics Parts I and II: An Introduction to Quantum Chemistry (Volume 1), P.W. Atkins, Oxford University Press, 1977, ISBN 0-19-855129-0
26. ^ The New Quantum Universe, T.Hey, P.Walters, Cambridge University Press, 2009, ISBN 978-0-521-56457-1
27. ^ a b c d Quanta: A handbook of concepts, P.W. Atkins, Oxford University Press, 1974, ISBN 0-19-855493-1
28. ^ a b Physics of Atoms and Molecules, B.H. Bransden, C.J.Joachain, Longman, 1983, ISBN 0-582-44401-2
29. ^ a b Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (2nd Edition), R. Resnick, R. Eisberg, John Wiley & Sons, 1985, ISBN 978-0-471-87373-0
30. ^ a b c Quantum Mechanics Demystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN(10) 0 07 145546 9
31. ^ a b Analytical Mechanics, L.N. Hand, J.D. Finch, Cambridge University Press, 2008, ISBN 978-0-521-57572-0
32. ^ N. Zettili. Quantum Mechanics: Concepts and Applications (2nd ed.). p. 458. ISBN 978-0-470-02679-3.
33. ^ Physical chemistry, P.W. Atkins, Oxford University Press, 1978, ISBN 0-19-855148-7
37. ^ a b c Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, ISBN 978-0-13-146100-0
38. ^
External links[edit] |
06af11a2f9d6204c | Sunday, June 28, 2015
Thinking outside the quantum box
Doctor: Don't tell me you're lost too.
Shardovan: No, but as you guessed, Doctor, we people of Castrovalva are too much part of this thing you call the occlusion.
Doctor: But you do see it, the spatial anomaly.
Shardovan: With my eyes, no — but, in my philosophy.
— Doctor Who, Castrovalva.
I've made no particular secret, on this blog, that I'm looking (in an adventuresome sort of way) for alternatives to quantum theory. So far, though, I've mostly gone about it rather indirectly, fishing around the edges of the theory for possible angles of attack without ever engaging the theory on its home turf. In this post I'm going to shave things just a bit closer — fishing still, but doing so within line-of-sight of the NO FISHING sign. I'm also going to explain why I'm being so indirect, which bears on what sort of fish I think most likely here.
To remind, in previous posts I've mentioned two reasons for looking for an alternative to quantum theory. Both reasons are indirect, considering quantum theory in the larger context of other theories of physics. First, I reasoned that when a succession of theories are getting successively more complicated, this suggests some wrong assumption may be shared by all of them (here). Later I observed that quantum physics and relativity are philosophically disparate from each other (here), a disparity that has been an important motivator for TOE (Theory of Everything) physicists for decades.
The earlier post looked at a few very minor bits of math, just enough to derive Bell's Inequality, but my goal was only to point out that a certain broad strategy could, in a sense, sidestep the nondeterminism and nonlocality of quantum theory. I made no pretense of assembling a full-blown replacement for standard quantum theory based on the strategy (though some researchers are attempting to do so, I believe, under the banner of the transactional interpretation). In the later post I was even less concrete, with no equations at all.
The quantum meme
How to fish
Why to fish
Hygiene again
The structure of quantum math
The structure of reality
The quantum meme
Why fish for alternatives away from the heart of the quantum math? Aside, that is, from the fact that any answers to be found in the heart of the math already have, presumably, plenty of eyeballs looking there for them. If the answer is to be found there after all, there's no lasting harm to the field in someone looking elsewhere; indeed, those who looked elsewhere can cheerfully write off their investment knowing they played their part in covering the bases — if it was at least reasonable to cover those bases. But going into that investigation, one wants to choose an elsewhere that's a plausible place to look.
Supposing quantum theory can be successfully challenged, I suggest it's quite plausible the successful challenge might not be found by direct assault (even though eventual confrontation would presumably occur, if it were really successful). Consider Thomas Kuhn's account of how science progresses. In normal science, researchers work within a paradigm, focusing their energies on problems within the paradigm's framework and thereby making, hopefully, rapid progress on those problems because they're not distracting themselves with broader questions. Eventually, he says, this focused investigation within the paradigm highlights shortcomings of the paradigm so they become impossible to ignore, researchers have a crisis of confidence in the paradigm, and after a period of distress to those within the field, a new paradigm emerges, through the process he calls a scientific revolution. I've advocated a biological interpretation of this, in which sciences are a variety of memetic organisms, and scientific revolution is the organisms' reproductive process. But if this is so, then scientific paradigms are being selected by Darwinian evolution. What are they being selected for?
Well, the success of science hinges on paradigms being selected for how effectively they allow us to understand reality. Science is a force to be reckoned with because our paradigms have evolved to be very good at helping us understand reality. That's why the scientific species has evolved mechanisms that promote empirical testing: in the long run, if you promote empirical testing and pass that trait on to your descendants, your descendants will be more effective, and therefore thrive. So far so good.
In theory, one could imagine that eventually a paradigm would come along so consistent with physical reality, and with such explanatory power, that it would never break down and need replacing. In theory. However, there's another scenario where a paradigm could get very difficult to break down. Suppose a paradigm offers the only available way to reason about a class of situations; and within that class of situations are some "chinks in the armor", that is, some considerations whose study could lead to a breakdown of the paradigm; but the only way to apply the paradigm is to frame things in a way that prevents the practitioner from thinking of the chinks-in-the-armor. The paradigm would thus protect itself from empirical attack, not by being more explanatory, but by selectively preventing empirical questions from being asked.
What characteristics might we expect such a paradigm to have, and would they be heritable? Advanced math that appears unavoidable would seem a likely part of such a complex. If learning the subject requires indoctrination in the advanced math, then whatever that math is doing to limit your thinking will be reliably done to everyone in the field; and if any replacement paradigm can only be developed by someone who's undergone the indoctrination, that will favor passing on the trait to descendant paradigms. General relativity and quantum theory both seem to exhibit some degree of this characteristic. But while advanced math may be an enabler, it might not be enough in itself. A more directly effective measure, likely to be enabled by a suitable base of advanced math, might be to make it explicitly impossible to ask any question without first framing the question in the form prescribed by the paradigm — as quantum theory does.
This suggests to me that the mathematical details of quantum theory may be a sort of tarpit, that pulls you in and prevents you from leaving. I'm therefore trying to look at things from lots of different perspectives in the general area without ever getting quite so close as to be pulled in. Eventually I'll have to move further and further in; but the more outside ideas I've tied lines to before then, the better I'll be able to pull myself out again.
How to fish
What I'm hoping to get out of this fishing expedition is new ideas, new ways of thinking about the problem. That's ideas, plural. It's not likely the first new idea one comes up with will be the key to unlocking all the mysteries of the universe. It's not even likely that just one new idea would ever do it. One might need a lot of new ideas, many of which wouldn't actually be part of a solution — but the whole collection of them, including all the ones not finally used, helps to get a sense of the overall landscape of possibilities, which may help in turning up yet more new ideas inspired from earlier ones, and indeed may make it easier to recognize when one actually does strike on some combination of ideas that produce a useful theory.
Hence my remark, in an aside in an earlier post, that I'm okay with absurd as long as it's different and shakes up my thinking.
Case in point. In the early 1500s, there was this highly arrogant and abrasive iconoclastic fellow who styled himself Philippus Aureolus Theophrastus Bombastus von Hohenheim; ostensibly our word "bombastic" comes from his name. He rejected the prevailing medical paradigm of his day, which was based on ancient texts, and asserted his superiority to the then-highly-respected ancient physician Celsus by calling himself "Paracelsus", which is the name you've probably heard of him under. He also shook up alchemical theory; but I mention him here for his medical ideas. Having rejected the prevailing paradigm, he was rather in the market for alternatives. He advocated observing nature, an idea that really began to take off after he shook things up. He advocated keeping wounds clean instead of applying cow dung to them, which seems a good idea. He proposed that disease is caused by some external agent getting into the body, rather than by an imbalance of humours, which sounds rather clever of him. But I'm particularly interested that he also, grasping for alternatives to the prevailing paradigm, borrowed from folk medicine the principle of like affects like. Admittedly, you couldn't do much worse than some of the prevailing practices of the day. But I'm fascinated by his latching on to like-effects-like, because it demonstrates how bits of replicative material may be pulled in from almost anywhere when trying to form a new paradigm. Having seen that, it figured later into my ideas on memetic organisms.
It also, along the way, flags out the existence of a really radically different way of picturing the structure of reality. Like-affects-like is a wildly different way of thinking, and therefore ought to be a great limbering-up exercise.
In fact, like-affects-like is, I gather, the principle underlying the anthropological phenomenon of magic — sympathetic magic, it's called. I somewhat recall an anthropologist expounding at length (alas, I wish I could remember where) that anthropologically this can be understood as the principle underlying all magic. So I got to thinking, what sort of mathematical framework might one use for this sort of thing? I haven't resolved a specific answer for the math framework, yet; but I've tried to at least set my thoughts in order.
What I'm interested in here is the mathematical and thus scientific utility of the like-affects-like principle, not its manifestation in the anthropological phenomenon of magic (as Richard Cavendish observed, "The religious impulse is to worship, the scientific to explain, the magical to dominate and command"). Yet the term "like affects like" is both awkward and vague; so I use the term sympathy for discussing it from a mathematical or scientific perspective.
How might a rigorous model of this work, structurally? Taking a stab at it, one might have objects, each capable of taking on characteristics with a potentially complex structure, and patterns which can arise in the characteristics of the objects. Interactions between the objects occur when the objects share a pattern. The characteristics of objects might be dispensed with entirely, retaining only the patterns, provided one specifies the structure of the range of possible patterns (perhaps a lattice of patterns?). There may be a notion of degrees of similarity of patterns, giving rise to varying degrees of interaction. This raises the question of whether one ought to treat similar patterns as sharing some sort of higher-level pattern and themselves interacting sympathetically. More radically, one might ask whether an object is merely an intersection of patterns, in which case one might aspire to — in some sense — dispense with the objects entirely, and have only a sort of web of patterns. Evidently, the whole thing hinges on figuring out what patterns are and how they relate to each other, then setting up interactions on that basis.
I distinguish between three types of sympathy:
• Pseudo-sympathy (type 0). The phenomenon can be understood without recourse to the sympathetic principle, but it may be convenient to use sympathy as a way of modeling it.
• Weak sympathy (type 1). The phenomenon may in theory arise from a non-sympathetic reality, but in practice there's no way to understand it without recourse to sympathy.
• Strong sympathy (type 2). The phenomenon cannot, even in theory, arise from a non-sympathetic reality.
All of which gives, at least, a lower bound on how far outside the box one might think. One doesn't have to apply the sympathetic principle in a theory, in order to benefit from the reminder to keep one's thinking limber.
(It is, btw, entirely possible to imagine a metric space of patterns, in which degree of similarity between patterns becomes distance between patterns, and one slides back into a geometrical model after all. To whatever extent the merit of the sympathetic model is in its different way of thinking, to that extent one ought to avoid setting up a metric space of patterns, as such.)
Why to fish
Asking questions is, broadly speaking, good. A line comes to mind from James Gleick's biography of Feynman (quoted favorably by Freeman Dyson): "He believed in the primacy of doubt, not as a blemish upon our ability to know but as the essence of knowing." Nevertheless, one does have to pick and choose which questions are worth spending most effort on; as mentioned above, the narrow focus of normal scientific research enables its often-rapid progress. I've been grounding my questions about quantum mechanics in observations about the character of the theory in relation to other theories of physics.
By contrast, one could choose to ground one's questions in reasoning about what sort of features reality can plausibly have. Einstein did this when maintaining that the quantum theory was an incomplete theory of the physical world — that it was missing some piece of reality. An example he cited is the Schrödinger's cat thought-experiment: Until observed, a quantum system can exist in a superposition of states. So, set up an experiment in which a quantum event is magnified into a macroscopic event — through a detector, the outcome of the quantum event causes a device to either kill or not kill a cat. Put the whole experimental apparatus, including the cat, in a box and close it so the outcome cannot be observed. Until you open the box, the cat is in a superposition of states, both alive and dead. Einstein reasoned that since the quantum theory alone would lead to this conclusion, there must be something more to reality that would disallow this superposition of cat.
The trouble with using this sort of reasoning to justify a line of research is, all it takes to undermine the justification is to say there's no reason reality can't be that strange.
Hence my preference for motivations based on the character of the theory, rather than the plausibility of the reality it depicts. My reasoning is still subjective — which is fine, since I'm motivating asking a question, not accepting an answer — but at least the reasoning is then not based on intuition about the nature of reality. Intuition specifically about physical reality could be right, of course, but has gotten a bad reputation — as part of the necessary process by which the quantum paradigm has secured its ecological niche — so it's better in this case to base intuition on some other criterion.
Hygiene again
To make sure I'm fully girded for battle — this is rough stuff, one can't be too well armed for it — I want to revisit some ideas I collected in earlier blog posts, and squeeze just a bit more out of them than I did before.
My previous thought relating explicitly to Theories of Everything was that, drawing an analogy with vau-calculi, spacetime geometry should perhaps be viewed not as a playing field on which all action occurs, but rather as a hygiene condition on the interactions that make up the universe. This analogy can be refined further. The role of variables in vau-calculi is coordinating causal connections between distant parts of the term. There are four kinds of variables, but unboundedly many actual variables of each kind; and α-renaming keeps these actual variables from bleeding into each other. A particular variable, though we may think of it as a very simple thing — a syntactic atom, in fact — is perhaps better understood as a distributed, complex-structured entity woven throughout the fabric of a branch of the term's syntax tree, humming with the dynamically maintained hygiene condition that keeps it separate from other variables. It may impinge on a large part of the α-renaming infrastructure, but most of its complex distributed structure is separate from the hygiene condition. The information content of the term is largely made up of these complex, distributed entities, with various local syntactic details decorating the syntax tree and regulating the rewriting actions that shape the evolution of the term. Various rewriting actions cause propagation across one (or perhaps more than one) of these distributed entities — and it doesn't actually matter how many rewriting steps are involved in this propagation, as for example even the substitution operations could be handled by gradually distributing information across a branch of the syntax tree via some sort of "sinking" structure, mirror to the binding structures that "rise" through the tree.
Projecting some of this, cautiously, through the analogy to physics, we find ourselves envisioning a structure of reality in which spacetime is a hygiene condition on interwoven, sprawling complex entities that impinge on spacetime but are not "inside" it; whose distinctness from each other is maintained by the hygiene condition; and whose evolution we expect to describe by actions in a dimension orthogonal to spacetime. The last part of which is interestingly suggestive of my other previous post on physics, where I noted, with mathematical details sufficient to make the point, that while quantum physics is evidently nondeterministic and nonlocal as judged relative to the time dimension, one can recover determinism and locality relative to an orthogonal dimension of "meta-time" across which spacetime evolves.
One might well ask why this hygiene condition in physics should take the form of a spacetime geometry that, at least at an intermediate scale, approximates a Euclidean geometry of three space and one time dimension. I have a thought on this, drawing from another of my irons in the fire; enough, perhaps, to move thinking forward on the question. This 3+1 dimension structure is apparently that of quaternions. And quaternions are, so at least I suspect (I've been working on a blog post exploring this point), the essence of rotation. So perhaps we should think of our hygiene condition as some sort of rotational constraint, and the structure of spacetime follows from that.
I also touched on Theories of Everything in a recent post while exploring the notion that nature is neither discrete nor continuous but something between (here). If there is a balance going on between discrete and continuous facets of physical worldview, apparently the introduction of discrete elementary particles is not, in itself, enough discreteness to counterbalance the continuous feature provided by the wave functions of these particles, and the additional feature of wave-function collapse or the like is needed to even things out. One might ask whether the additional discreteness associated with wave-function collapse could be obviated by backing off somewhat on the continuous side. The uncertainty principle already suggests that the classical view of particles in continuous spacetime — which underlies the continuous wave function (more about that below) — is an over-specification; the need for additional balancing discreteness might be another consequence of the same over-specification.
Interestingly, variables in λ-like calculi are also over-specified: that's why there's a need for α-renaming in the first place, because the particular name chosen for a variable is arbitrary as long as it maintains its identity relative to other variables in the term. And α-renaming is the hygiene device analogized to geometry in physics. Raising the prospect that to eliminate this over-specification might also eliminate the analogy, or make it much harder to pin down. There is, of course, Curry's combinatorial calculus which has no variables at all; personally I find Church's variable-using approach easier to read. Tracing that through the analogy, one might conjecture the possibility of constructing a Theory of Everything that didn't need the awkward additional discreteness, by eliminating the distributed entities whose separateness from each other is maintained by the geometrical hygiene condition, thus eliminating the geometry itself in the process. Following the analogy, one would expect this alternative description of physical reality to be harder to understand than conventional physics. Frankly I have no trouble believing that a physics without geometry would be harder to understand.
The idea that quantum theory as a model of reality might suffer from having had too much put into it, does offer a curious counterpoint to Einstein's suggestion that quantum theory is missing some essential piece of reality.
The structure of quantum math
The structure of the math of quantum theory is actually pretty simple... if you stand back far enough. Start with a physical system. This is a small piece of reality that we are choosing to study. Classically, it's a finite set of elementary things described by a set of parameters. Hamilton (yes, that's the same guy who discovered quaternions) proposed to describe the whole behavior of such a system by a single function, since called a Hamiltonian function, which acts on the parameters describing the instantaneous state of the system together with parameters describing the abstract momentum of each state parameter (essentially, how the parameters change with respect to time). So the Hamiltonian is basically an embodiment of the whole classical dynamics of the system, treated as a lump rather than being broken into separate descriptions of the individual parts of the system. Since quantum theory doesn't "do" separate parts, instead expecting everything to affect everything else, it figures the Hamiltonian approach would be particularly compatible with the quantum worldview. Nevertheless, in the classical case it's still possible to consider the parts separately. For a system with a bunch of parts, the number of parameters to the Hamiltonian will be quite large (typically, at least six times the number of parts — three coordinates for position and three for momentum of each part).
Now, the quantum state of the system is described by a vector over a complex Hilbert space of, typically, infinite dimension. Wait, what? Yes, that's an infinite number of complex numbers. In fact, it might be an uncountably infinite number of complex numbers. Before you completely freak out over this, it's only fair to point out that if you have a real-valued field over three-dimensional space, that's an uncountably infinite number of real numbers (the number of locations in three-space being uncountably infinite). Still, the very fact that you're putting this thing in a Hilbert space, which is to say you're not asking for any particular kind of simple structure relating the different quantities, such as a three-dimensional Euclidean continuum, is kind of alarming. Rather than a smooth geometric structure, this is a deliberately disorganized mess, and honestly I don't think it's unfair to wish there were some more coherent reality "underneath" that gives rise to this infinite structure. Indeed, one might suspect this is a major motive for wanting a hidden variable theory — not wishing for determinism, or wishing for locality, but just wishing for a simpler model of what's going on. David Bohm's hidden variable theory, although it did show one could recover determinism with actual classical particles "underneath", did so without simplifying the mathematics — the mathematical structure of the quantum state was still there, just given a makeover as a potential field. In my earlier account of this bit of history, I noted that Einstein, seeing Bohm's theory, remarked, "This is not at all what I had in mind." I implied that Einstein didn't like Bohm's theory because it was nonlocal; but one might also object that Bohm's theory doesn't offer a simpler underlying reality, rather a more complicated one.
The elements of the vector over Hilbert space are observable classical states of the system; so this vector is indexed by, essentially, the sets of possible inputs to the Hamiltonian. One can see how, step by step, we've ended up with a staggering level of complexity in our description, which we cope with by (ironically) not looking at it. By which I mean, we represent this vast amorphous expanse of information by a single letter (such as ψ), to be manipulated as if it were a single entity using operations that perform some regimented, impersonal operation on all its components that doesn't in general require it to have any overall shape. I don't by any means deride such treatments, which recover some order out of the chaos; but it's certainly not reassuring to realize how much lack of structure is hidden beneath such neat-looking formulae as the Schrödinger equation. And the amorphism beneath the elegant equations also makes it hard to imagine an alternative when looking at the specifics of the math (as suspected based on biological assessment of the evolution of physics).
The quantum situation gets its structure, and its dynamics, from the Hamiltonian, that single creature embodying the whole of the rules of classical behavior for the system. The Schrödinger equation (or whatever alternative plays its role) governs the evolution of the quantum state vector over time, and contains within it a differential operator based on the classical Hamiltonian function.
iℏ Ψ
= Ĥ Ψ .
One really wants to stop and admire this equation. It's a linear partial differential equation, which is wonderful; nonlinearity is what gives rise to chaos in the technical sense, and one would certainly rather deal with a linear system. Unfortunately, the equation only describes the evolution of the system so long as it remains a purely quantum system; the moment you open the box to see whether the cat is dead, this wave function collapses into observation of one of the classical states indexing the quantum state vector, with (to paint in broad strokes) the amplitudes of the complex numbers in the vector determining the probability distribution of observed classical states.
It also satisfies James Clerk Maxwell's General Maxim of Physical Science, which says (as recounted by Ludwik Silberstein) that when we take the derivatives of our system with respect to time, we should end up with expressions that do not themselves explicitly involve time. When this is so, the system is "complete", or, "undisturbed". (The idea here is that if the rules governing the system change over time, it's because the system is being affected by some other factor that is varying over time.)
The equation is, indeed, highly seductive. Although I'm frankly on guard against it, yet here I am, being drawn into making remarks on its properties. Back to the question of structure. This equation effectively segregates the mathematical description of the system into a classical part that drives the dynamics (the Hamiltonian), and a quantum part that smears everything together (the quantum state vector). The wave function Ψ, described by the equation, is the adapter used to plug these two disparate elements together. The moment you start contemplating the equation, this manner of segregating the description starts to seem inevitable. So, having observed these basic elements of the quantum math, let us step back again before we get stuck.
The key structural feature of the quantum description, in contrast to classical physics, is that the parts can't be considered separately. This classical separability produced the sense of simplicity that, I speculated above, could be an ulterior motive for hidden variable theories. The term for this is superposition of states, i.e., a quantum state that could collapse into any of multiple classical states, and therefore must contain all of those classical states in its description.
A different view of this is offered by so-called quantum logic. The idea here (notably embraced by physicist David Finkelstein, who I've mentioned in an earlier post because he was lead author of some papers in the 1960s on quaternion quantum theory) is that quantum theory is a logic of propositions about the physical world, differing fundamentally from classical propositional logic because of the existence of superposition as a propositional principle. There's a counterargument that this isn't really a "logic", because it doesn't describe reasoning as such, just the behavior of classical observations when applied as a filter to quantum systems; and indeed one can see that something of the sort is happening in the Schrödinger equation, above — but that would be pulling us back into the detailed math. Quantum logic, whatever it doesn't apply to, does apply to observational propositions under the regime of quantum mechanics, while remaining gratifyingly abstracted from the detailed quantum math.
Formally, in classical logic we have the distributive law
P and (Q or R) = (P and Q) or (P and R) ;
but in quantum logic, (Q or R) is superpositional in nature, saying that we can eliminate options that are neither, yet allowing more than the union of situations where one holds and situations where the other holds; and this causes the distributive law to fail. If we know P, and we know that either Q or R (but we may be fundamentally unable to determine which), this is not the same as knowing that either both P and Q, or both P and R. We aren't allowed to refactor our proposition so as to treat Q separately from R, without changing the nature of our knowledge.
[note: I've fixed the distributive law, above, which I botched and didn't even notice till, thankfully, a reader pointed it out to me. Doh!]
One can see in this broadly the reason why, when we shift from classical physics to quantum physics, we lose our ability to consider the underlying system as made up of elementary things. In considering each classical elementary thing, we summed up the influences on that thing from each of the other elementary things, and this sum was a small tidy set of parameters describing that one thing alone. The essence of quantum logic is that we can no longer refactor the system in order to take this sum; the one elementary thing we want to consider now has a unique relationship with each of the other elementary things in the system.
Put that way, it seems that the one elementary thing we want to consider would actually have a close personal relationship with each other elementary thing in the universe. A very large Rolodex indeed. One might object that most of those elementary things in the universe are not part of the system we are considering — but what if that's what we're doing wrong? Sometimes, a whole can be naturally decomposable into parts in one way, but when you try to decompose it into parts in a different way you end up with a complicated mess because all of your "parts" are interacting with each other. I suggested, back in my first blog post on physics, that there might be some wrong assumption shared by both classical and quantum physics; well, the idea that the universe is made up of elementary particles (or quanta, whatever you prefer to call them) is something shared by both theories. The quantum math (Schrödinger equation again, above) has this classical decomposition built into its structure, pushing us to perceive the subsequent quantum weirdness as intrinsic to reality, or perhaps intrinsic to our observation of reality — but what if it's rather intrinsic to that particular way of slicing off a piece of the universe for consideration?
The quantum folks have been insisting for years that quantum reality seems strange only because we're imposing our intuitions from the macroscopic world onto the quantum-scale world where it doesn't apply. Okay... Our notion that the universe is made up of individual things is certainly based on our macroscopic experience. What if it breaks down sooner than we thought — what if, instead of pushing the idea of individual things down to a smaller and smaller scale until they sizzle apart into a complex Hilbert space, we should instead have concluded that individual things are something of an illusion even at macroscopic scales?
The structure of reality
One likely objection is that no matter how you split up reality, you'd still have to observe it classically and the usual machinery of quantum mechanics would apply just the same. There are at least a couple of ways — two come to mind atm — for some differently shaped 'slice' of reality to elude the quantum machinery.
• The alternative slice might not be something directly observable.
Here an extreme example comes in handy (as hoped). Recall the sympathetic hypothesis, above. A pattern would not be subject to direct observation, any more than a Platonic ideal like "table" or "triangle" would be. (Actually, it seems possible a pattern would be a Platonic ideal.)
This is also reminiscent of the analogy with vau-calculus. I noted above that much of the substance of a calculus term is made up of variables, where by a variable I meant the entire dynamically interacting web delineated by a variable binding construct and all its matching variable instances. A variable in this sense isn't, so to speak, observable; one can observe a particular instance of a variable, but a variable instance is just an atom, and not particularly interesting.
• The alternative slice might be something quantum math can't practically cope with. Quantum math is very difficult to apply in practice; some simple systems can be solved, but others are intractable. (It's fashionable in some circles to assume more powerful computers will solve all math problems. I'm reminded of a quote attributed to Eugene Wigner, commenting on a large quantum calculation: "It is nice to know that the computer understands the problem. But I would like to understand it, too.") It's not inconceivable that phenomena deviating from quantum predictions are "hiding in plain sight". My own instinct is that if this were so, they probably wouldn't be just on the edge of what we can cope with mathematically, but well outside that perimeter.
This raises the possibility that quantum mechanics might be an idealized approximation, holding asymptotically in a degenerate case — in somewhat the same way that Newtonian mechanics holds approximately for macroscopic problems that don't involve very high velocities.
We have several reasons, by this point, to suspect that whatever it is we're contemplating adding to our model of reality, it's nonlocal (that is, nonlocal relative to the time dimension, as is quantum theory). On one hand, bluntly, classical physics has had its chance and not worked out; we're already conjecturing that insisting on a classical approach is what got us into the hole we're trying to get out of. On the other hand, under the analogy we're exploring with vau-calculus, we've already noted that most of the term syntax is occupied by distributed variables — which are, in a deep sense, fundamentally nonlocal. The idea of spacetime as a hygiene condition rather than a base medium seems, on the face of it, to call for some sort of nonlocality; in fact, saying reality has a substantial component that doesn't follow the contours of spacetime is evidently equivalent to saying it's nonlocal. Put that way, saying that reality can be usefully sliced in a way that defies the division into elementary particles/things is also another way of saying it's nonlocal, since when we speak of dividing reality into elementary "things", we mean, things partitioned away from each other by spacetime. So what we have here is several different views of the same sort of conjectured property of reality. Keeping in mind, multiple views of a single structure is a common and fruitful phenomenon in mathematics.
I'm inclined to doubt this nonlocality would be of the sort already present in quantum theory. Quantum nonlocality might be somehow a degenerate case of a more general principle; but, again bluntly, quantum theory too has had its chance. Moreover, it seems we may be looking for something that operates on macroscopic scales, and quantum nonlocality (entanglement) tends to break down (decohere) at these scales. This suggests the prospect of some form of robust nonlocality, in contrast to the more fragile quantum effects.
So, at this point I've got in my toolkit of ideas (not including sympathy, which seems atm quite beyond the pale, limited to the admittedly useful role of devil's advocate):
• a physical structure substantially not contained within spacetime.
• space emergent as a hygiene condition, perhaps rotation-related.
• robust nonlocality, with quantum nonlocality perhaps as an asymptotic degenerate case.
• some non-spacetime dimension over which one can recover abstract determinism/locality.
• decomposition of reality into coherent "finite slices" in some way other than into elementary things in spacetime.
• slices may be either non-observable or out of practical quantum scope.
• the structural role of the space hygiene condition may be to keep slices distinct from each other.
• conceivably an alternative decomposition of reality may allow some over-specified elements in classical descriptions to be dropped entirely from the theory, at unknown price to descriptive clarity.
I can't make up my mind if this is appallingly vague, or consolidating nicely. Perhaps both. At any rate, the next phase of this operation would seem likely to shift further along the scale toward identifying concrete structures that meet the broad criteria. In that regard, it is probably worth remarking that current paradigm physics already decomposes reality into nonlocal slices (though not in the sense suggested here): the types of elementary particles. The slices aren't in the spirit of the "finite" condition, as there are only (atm) seventeen of them for the whole of reality; and they may, perhaps, be too closely tied to spacetime geometry — but they are, in themselves, certainly nonlocal.
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4e9f126300cae654 | fredag 9 maj 2014
Why Insist on Quantum Mechanics Based on Magic and Contradiction?
The ground state of Helium is postulated to be $1s^2$ with two overlaying electrons with opposite spin and identical spherically symmetric spatial wave-functions in the first shell, which is not the ground state because its energy is too large. This is the starting point for the Schrödinger equation for many-electron atoms.
Here is a further motivation why it may be of interest to consider wave-functions for an atom with $N$ electrons as a sum of $N$ functions $\psi_1(x)$,…,$\psi_N(x)$, all depending on a common three-dimensional space coordinate $x$ (plus time) as suggested in a previous post:
• $\psi (x)=\psi_1(x)+\psi_2(x)+…+\psi_N(x)$.
We recall that Schrödinger's equation for the Hydrogen atom as the basis of quantum mechanics, takes the form:
• $ih\frac{\partial\psi}{\partial t}=-\frac{h^2}{2m}\Delta\psi +V\psi$ for all $x$ and $t>0$,
with kernel potential $V(x)=-\frac{1}{\vert x\vert}$, $x$ a three-dimensional space coordinate, $t>0$ time, $h$ Planck's constant, $m$ the mass of an electron and corresponding one-electron wave-function $\psi (x,t)$ as solution. This equation is magically pulled out of a hat from the relation
• $E =\frac{p^2}{2m} + V(x)$
expressing conservation of energy $E$ of a body of mass $m$ with position $x(t)$ moving in a potential $V(x)$ with momentum $p=m\frac{dx}{dt}$, by the following formal substitutions:
• $E\rightarrow ih\frac{\partial}{\partial t}$,
• $p\rightarrow\frac{h}{i}\nabla$,
followed by formal multiplication by $\psi$. Energy conservation for the Hydrogen atom then takes the form:
• $E=K(t)+P(t)$ for all $t>0$, where
• $K(t) =\frac{h^2}{2m}\int\vert\nabla\psi (x,t)\vert^2\, dx$ is the kinetic energy,
• $P(t)=\int \frac{\vert\psi (x, t)\vert^2}{\vert x\vert}dx$ is the potential energy
of the electron, under the normalization
• $\int\vert\psi (x,t)\vert^2\, dx=1$.
So far so good: The different energy levels $E$ of time-periodic solutions to Schrödinger's equation give the observed spectrum of the Hydrogen atom with corresponding wave-functions describing the distribution of the electron around the kernel. We see that the Laplace term gives rise to the kinetic energy as an effect of gradient regularization.
But consider now the accepted standard text-book generalization of Schrödinger's equation to an atom with $N$ electrons:
• $ih\frac{\partial\psi}{\partial t}=-\sum_{j=1}^N(\frac{h^2}{2m}\Delta_j -\frac{N}{\vert x_j\vert})\psi + \sum_{k < j}\frac{1}{\vert x_j-x_k\vert}\psi$,
where $\psi (x_1,…,x_N,t)$ depends on $N$ three-dimensional space coordinates $x_1,…, x_N$ and time $t$, and $\Delta_j$ is the Laplace operator with respect to coordinate $x_j$, under the normalization
• $\int\vert\psi\vert^2\, dx_1….dx_N=1$.
We see the appearance of the one-electron operators
with corresponding one-electron kinetic energies:
• $K_j(t) =\frac{h^2}{2m}\int\vert\nabla_j\psi\vert^2\, dx_1…dx_N$,
and electron-electron repulsion expressed by the coupling potential
• $\sum_{k < j}\frac{1}{\vert x_j-x_k\vert}$.
We see that in this model each electron $j$ is equipped with its own three-dimensional space with coordinate $x_j$ and its own kinetic energy $K_j$, with interaction between the electrons only through the coupling potential.
The electron individuality and high dimensionality of the wave function $\psi (x_1,…x_N)$ is reduced by restriction to wave functions as products $\psi_1(x_1)…\psi (x_N)$ built from three-dimensional wave functions $\psi_1,…,\psi_N$ combined with symmetry or antisymmetry under permutations of the coordinates $x_1,…,x_N$, which eliminates all individuality of the electrons.
Extreme electron individuality is thus countered by permutations removing all individuality, but the individual one-electron kinetic energies $K_j$ are kept as if each electron keeps its individuality. This is strange.
To see the result, recall that the ground state of minimal energy of Helium with two electrons is supposed to be given by a symmetric wave function $\psi (x_1,x_2)$
• $\psi (x_1,x_2)=\phi (x_1)\phi (x_2)$,
where $\phi (x_1)\sim \exp(-2\vert x_1\vert )$ is spherically symmetric, the same for both electrons. The two electrons of the ground state of Helium are thus supposed to have identical spherically symmetric distributions denoted $1s^2$, see the periodic table above. The trouble is now that this configuration has energy (in Hartree units) $- 2.75$ while the observed energy is $-2.903$.
The true ground state is thus different from $1s^2$ and to handle this situation, while insisting that ground state still is $1s^2$ as in the table above, a so-called corrective perturbation is made introducing a dependence of $\psi (x_1,x_2)$ on $\vert x_1-x_2\vert$ in a Rayleigh-Ritz minimization procedure. This way a better correspondence with observation is reached, because separation of the electrons is now possible: If one electron is on one side of the kernel then the other electron is on the other side. But the standard message is contradictory:
• The ground state configuration for Helium is $1s^2$, which however is not the ground state because its energy is too large ($-2.75$ instead of $-2.903$).
• Smaller energy can be obtained by a perturbation computation but the corresponding electron configuration is hidden to readers of physics books, because the ground state is still postulated to be $1s^2$.
If we minimize energy over wave functions of product form
• $\psi (x_1,x_2)=\psi_1(x_1)\psi_2(x_2)$,
without asking for symmetry, we find that the minimum is achieved with spherically symmetric $\psi_1=\psi_2$, with too large energy as just noted. However, if we instead compute the kinetic energy based on the sum with common space space coordinate $x$
• $\psi_1(x) +\psi_2(x)$
as suggested in the previous post, then separation of the electrons is advantageous allowing discontinuous electron distributions (joining smoothly) without cost of kinetic energy and better correspondence with observation is achieved.
• The standard attribution of individual kinetic energy appears to make the individual electron distributions "too stiff" and thus favors overlaying electrons rather than separated electrons, requiring Pauli's exclusion principle to prevent overlaying of more than two electrons.
• If kinetic energy is instead computed from the sum of individual electron distributions, electron "stiffness" is reduced and separation favored.
• Since the standard individual one-electron attribution of kinetic energy is ad hoc, there is little reason to insist that kinetic energy must be computed this way, in particular when it leads to an incorrect ground state already for Helium.
• Attributing kinetic energy to a sum of electron wave-functions allows discontinuous electron distributions joining smoothly without cost of kinetic energy. Electron individuality is here kept as individual distribution in space, while kinetic energy is collectively computed from the assembly. This would be the way to handle individuality in a collective macroscopic setting and there is no reason why this would not be operational also for microscopics.
• Since the stated ground state as $1s^2$ for Helium is incorrect, there is no reason to believe that any of the other ground states listed in the standard periodic table is correct.
• If so, then the claim that the standard Schrödinger's equation explains the periodic table has little reason.
PS1 The standard argument is that the standard multi-d Schrödinger equation must be correct since there is no case known for which the multi-d wave-function solution does not agree exactly with what is observed! But this is not a correct argument, because (i) the multi-d Schrödinger equation cannot be solved, (ii) even if the wave-function could be determined its physical meaning is unclear and so comparison with reality is impossible. The standard argument is to turn (i) and (ii) from scientific disaster into monumental success by claiming that since the wave-function is impossible to determine, there is no way to prove that it is not correct.
Realizing that arguing this way does not follow basic scientific principle may open to searching for different forms of Schrödinger's equation, as non-linear systems of equations in three space dimensions instead of linear multi-d scalar equations, which are computable and have physical meaning, as suggested.
PS2 The standard way to handle that the standard linear multi-d Schrödinger equation is uncomputable is using Density Functional Theory (DFT) awarded the 1998 Nobel Prize in Chemistry, as a non-linear 3d scalar system in electron density. DFT results from averaging in the standard linear multi-d Schrödinger equation producing exchange correlation potentials which are impossible to determine. If the standard multi-d linear Schrödinger equation is questionable, then so is DFT.
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24f0c1c872490dc5 | Monday, August 29, 2016
Interpreted programming languages
Last night I drifted off while reading a Lisp book.
xkcd 224.
It's finally registered on me that much of the past half century of misunderstandings and confusions about the semantics of Lisp, of quotation, and, yes, of fexprs, can be accounted for by failure to recognize there is a theoretical difference between an interpreted programming language and a compiled programming language. If we fail to take this difference into account, our mathematical technology for studying compiled programming languages will fail when applied to interpreted languages, leading to loss of coherence in language designs and a tendency to blame the language rather than the theory.
Technically, a compiler translates the program into an executable form to be run thereafter, while an interpreter figures out what the program says to do and just does it immediately. Compilation allows higher-performance execution, because the compiler takes care of reasoning about source-code before execution, usually including how to optimize the translation for whatever resources are prioritized (time, space, peripherals). It's easy to suppose this is all there is to it; what the computer does is an obvious focus for attention. One might then suppose that interpretation is a sort of cut-rate alternative to compilation, a quick-and-dirty way to implement a language if you don't care about performance. I think this misses some crucial point about interpretation, some insight to be found not in the computer, but in the mind of the programmer. I don't understand that crucial insight clearly enough — yet — to focus a whole blog post on it; but meanwhile, there's this theoretical distinction between the two strategies which also isn't to be found in the computer's-eye view, and which I do understand enough about to focus this blog post on.
It's not safe to say the language is the same regardless of which way it's processed, because the language design and the processing strategy aren't really independent. In principle a given language might be processed either way; but the two strategies provide different conceptual frameworks for thinking about the language, lending themselves to different language design choices, with different purposes — for which different mathematical properties are of interest and different mathematical techniques are effective. This is a situation where formal treatment has to be considered in the context of intent. (I previously blogged about another case where this happens, formal grammars and term-rewriting calculi, which are both term-rewriting systems but have nearly diametrically opposite purposes; over thar.)
I was set onto the topic of this post recently by some questions I was asked about Robert Muller's M-Lisp. My dissertation mentions Muller's work only lightly, because Muller's work and mine are so far apart. However, because they seem to start from the same place yet lead in such different directions, one naturally wants to know why, and I've struck on a way to make sense of it: starting from the ink-blot of Lisp, Muller and I both looked to find a nearby design with greater elegance — and we ended up with vastly different languages because Muller's search space was shaped by a conceptual framework of compilation while mine was shaped by a conceptual framework of interpretation. I will point out, below, where his path and mine part company, and remark briefly on how this divergence affected his destination.
The mathematical technology involved here, I looked at from a lower-level perspective in an earlier post. It turns out, from a higher-level perspective, that the technology itself can be used for both kinds of languages, but certain details in the way it is usually applied only work with compiled languages and, when applied to interpreted languages, result in the trivialization of theory noted by Wand's classic paper, "The Theory of Fexprs is Trivial".
Universal program
Half a century's worth of misunderstandings and confusions
I'll explore this through a toy programming language which I'll then modify, starting with something moderately similar to what McCarthy originally described for Lisp (before it got unexpectedly implemented).
This is a compiled programming language, without imperative features, similar to λ-calculus, for manipulating data structures that are nested trees of atomic symbols. The syntax of this language has two kinds of expressions: S-expressions (S for Symbolic), which don't specify any computation but merely specify constant data structures — trees of atomic symbols — and M-expressions (M for Meta), which specify computations that manipulate these data structures.
S-expressions, the constants of the language, take five forms:
S ::= s | () | #t | #f | (S . S) .
That is, an S-expression is either a symbol, an empty list, true, false, or a pair (whose elements are called, following Lisp tradition, its car and cdr). A symbol name is a sequence of one or more upper-case letters. (There should be no need, given the light usage in this post, for any elaborate convention to clarify the difference between a symbol name and a nonterminal such as S.)
I'll assume the usual shorthand for lists, (S ...) ≡ (S . (...)), so for example (FOO BAR QUUX) ≡ (FOO . (BAR . (QUUX . ()))); but I won't complicate the formal grammar with this shorthand since it doesn't impact the abstract syntax.
The forms of M-expressions start out looking exactly like λ-calculus, then add on several other compound forms and, of course, S-expressions which are constants:
M ::= x | [λx.M] | [M M] | [if M M M] | [car M] | [cdr M] | [cons M M] | [eq? M M] | S .
The first form is a variable, represented here by nonterminal x. A variable name will be a sequence of one or more lower-case letters. Upper- versus lower-case letters is how McCarthy distinguished between symbols and variables in his original description of Lisp.
Variables, unlike symbols, are not constants; rather, variables are part of the computational infrastructure of the language, and any variable might stand for an arbitrary computation M.
The second form constructs a unary function, via λ; the third applies a unary function to a single argument. if expects its first argument to be boolean, and returns its second if the first is true, third if the first is false. car and cdr expect their argument to be a pair and extract its first and second element respectively. cons constructs a pair. eq? expects its two arguments to be S-expressions, and returns #t if they're identical, #f if they're different.
Compound unary functions, constructed by λ, are almost first-class. They can be returned as the values of expressions, and they can be assigned to variables; but as the grammar is set out, they cannot appear as elements of data structures. A pair expression is built up from two S-expressions, and a compound unary function is not an S-expression. McCarthy's original description of Lisp defines S-expressions this way; his stated purpose was only to manipulate trees of symbols. Trained as I am in the much later Lisp culture of first-class functions and minimal constraints, it felt unnatural to follow this narrower definition of S-expressions; but in the end I had to define it so. I'm trying to reproduce the essential factors that caused Lisp to come out the way it did, and, strange to tell, everything might have come out differently if not for the curtailed definition of S-expressions.
(One might ask, couldn't we indirectly construct a pair with a function in it using cons? A pair with a function in it would thus be a term that can't be represented directly by a source expression. This point likely doesn't matter directly to how things turned out; but fwiw, I suspect McCarthy didn't have in mind to allow that, no. It's entirely possible he also hadn't really thought about it yet at the preliminary stage of design where this detail had its impact on the future of Lisp. It's the sort of design issue one often discovers by playing around with a prototype — and in this case, playing around with a prototype is how things got out of hand; more on that later.)
Somehow we want to specify the meanings of programs in our programming language. Over the decades, a number of techniques for formal PL semantics have been entertained. One of the first things tried was to set up a term-rewriting machine modeled roughly on λ-calculus, that would perform small finite steps until it reached a final state; that was Peter Landin's notion, when Christopher Strachey set him onto the problem in the 1960s. It took some years to get the kinks out of that approach, and meanwhile other techniques were tried — such as denotational semantics, meta-circular evaluation — but setting up a term-rewriting calculus has been quite a popular technique since the major technical obstacles to it were overcome. Of the three major computational models tied together in the 1930s by the Church-Turing thesis, two of them were based on term-rewriting: Turing machines, which were the convincing model, the one that lent additional credibility to the others when they were all proven equivalent; and λ-calculus, which had mathematical elegance that the nuts-and-bolts Turing-machine model lacked. The modern "small-step" rewriting approach to semantics (as opposed to "big-step", where one deduces how to go in a single step from start to finish) does a credible job of combining the strengths of Turing-machines and λ-calculus; I've a preference for it myself, and it's the strategy I'll use here. I described the technique in somewhat more depth in my previous post on this material.
Small-step semantics applies easily to this toy language because every intermediate computational state of the system is naturally represented by a source-code expression. That is, there is no obvious need to go beyond the source-code grammar we've already written. Some features that have been omitted from the toy language would make it more difficult to limit all computational states to source expressions; generally these would be "stateful" features, such as input/output or a mutable store. Landin used a rewriting system whose terms (computational states) were not source-code expressions. One might ask whether there are any language features that would make it impossible to limit computational states to source expressions, and the answer is essentially yes, there are — features related not to statefulness, but to interpretation. For now, though, we'll assume that all terms are source expressions.
We can define the semantics of the language in reasonably few rewriting schemata.
[[λx.M1] M2] → M1[x ← M2]
[if #t M1 M2] → M1
[if #f M1 M2] → M2
[car (S1 . S2)] → S1
[cdr (S1 . S2)] → S2
[cons S1 S2] → (S1 . S2)
[eq? S S] → #t
[eq? S1 S2] → #f if the Sk are different.
Rewriting relation → is the compatible closure of these schemata; that is, for any context C and terms Mk, if M1 → M2 then C[M1] → C[M2]. Relation → is also Church-Rosser: although a given term M may be rewritable in more than one way, any resulting difference can always be eliminated by later rewriting. That is, the reflexive transitive closure →* has the diamond property: if M1 →* M2 and M1 →* M3, then there exists M4 such that M2 →* M4 and M3 →* M4.
Formal equality of terms, =, is the symmetric closure of →* (thus, the reflexive symmetric transitive compatible closure of the schemata, which is to say, the least congruence containing the schemata).
Another important relation is operational equivalence, ≅. Two terms are operationally equivalent just if replacing either by the other in any possible context preserves the observable result of the computation. M1 ≅ M2 iff for every context C and S-expression S, C[M1] ↦* S iff C[M2] ↦* S. (Fwiw, relation ↦ is what the computation actually does, versus → which is anything the rewriting calculus could do; → is compatible Church-Rosser and therefore nice to work with mathematically, but ↦ is deterministic and therefore we can be sure it does what we meant it to. Another way of putting it is that → has the mathematical character of λ-calculus while ↦ has the practical character of Turing-machines.)
Operational equivalence is exactly what must be guaranteed in order for an optimizing compiler to safely perform a local source-to-source transformation: as long as the two terms are operationally equivalent, the compiler can replace one with the other in any context. The rewriting calculus is operationally sound if formal equality implies operational equivalence; then the rewriting calculus can supply proofs of operational equivalence for use in optimizing compilation.
Before moving on, two other points of interest about operational equivalence:
• Operational equivalence of S-expressions is trivial; that is, S1 and S2 are operationally equivalent only if they are identical. This follows immediately by plugging the trivial context into the definition of operational equivalence, C[Sk] ≡ Sk. Thus, in every non-trivial operational equivalence M1 ≅ M2, at least one of the Mk is not an S-expression.
• All terms in the calculus — all M-expressions — are source expressions; but if there were any terms in the calculus that were not source expressions, they would be irrelevant to a source-to-source optimizer; however, if these non-source terms could be usefully understood as terms in an intermediate language used by the compiler, an optimizer might still be able to make use of them and their formal equalities.
Universal program
McCarthy's Lisp language was still in its infancy when the project took an uncontrollable turn in a radically different direction than McCarthy had envisioned going with it. Here's what happened.
A standard exercise in theory of computation is to construct a universal Turing machine, which can take as input an encoding of an arbitrary Turing machine T and an input w to T, and simulate what T would do given input w. This is an extremely tedious exercise; the input to a Turing machine looks nothing like the control device of a Turing machine, so the encoding is highly intrusive, and the control device of the universal machine is something of an unholy mess. McCarthy set out to lend his new programming language mathematical street cred by showing that not only could it simulate itself with a universal program, but the encoding would be much more lucid and the logic simpler in contrast to the unholy mess of the universal Turing machine.
The first step of this plan was to describe an encoding of programs in the form of data structures that could be used as input to a program. That is to say, an encoding of M-expressions as S-expressions. Much of this is a very straightforward homomorphism, recursively mapping the non-data M-expression structure onto corresponding S-expressions; for our toy language, encoding φ would have
• φ(x) = symbol s formed by changing the letters of its name from lower-case to upper-case. Thus, φ(foo) = FOO.
• φ([λx.M]) = (LAMBDA φ(x) φ(M)).
• φ([M1 M2]) = (φ(M1) φ(M2)).
• φ([if M1 M2 M3]) = (IF φ(M1) φ(M2) φ(M3)).
• φ([car M]) = (CAR φ(M)).
• φ([cdr M]) = (CDR φ(M)).
• φ([cons M1 M2]) = (CONS φ(M1) φ(M2)).
• φ([eq? M1 M2]) = (EQ φ(M1) φ(M2)).
(This encoding ignores the small detail that certain symbol names used in the encoding — LAMBDA, IF, CAR, CDR, CONS, EQ — must not also be used as variable names, if the encoding is to behave correctly. McCarthy seems not to have been fussed about this detail, and nor should we be.)
For a proper encoding, though, S-expressions have to be encoded in a way that unambiguously distinguishes them from the encodings of other M-expressions. McCarthy's solution was
• φ(S) = (QUOTE S).
Now, in some ways this is quite a good solution. It has the virtue of simplicity, cutting the Gordian knot. It preserves the readability of the encoded S-expression, which supports McCarthy's desire for a lucid encoding. The main objection one could raise is that it isn't homomorphic; that is, φ((S1 . S2)) is not built up from φ(S1) and φ(S2).
As McCarthy later recounted, they had expected to have plenty of time to refine the language design before it could be implemented. (The FORTRAN compiler, after all, had been a massive undertaking.) Meanwhile, to experiment with the language they began hand-implementing particular functions. The flaw in this plan was that, because McCarthy had been so successful in demonstrating a universal Lisp function eval with simple logic, it wasn't difficult to hand-implement eval; and, because he had been so successful in making the encoding lucid, this instantly produced a highly usable Lisp interpreter. The sudden implementation precipitated a user community and substantial commitment to specifics of what had been a preliminary language design.
All this might have turned out differently if the preliminary design had allowed first-class functions to be elements in pairs. A function has to be encoded, homomorphically, which would require a homomorphic encoding of pairs, perhaps φ((M1 . M2)) = (CONS φ(M1) φ(M2)); once we allow arbitrary M-expressions within the pair syntax, (M1 . M2), that syntax itself becomes a pair constructor and there's really no need for a separate cons operator in the M-language; then CONS can encode the one constructor. One might then reasonably restrict QUOTE to base cases; more, self-encode () #t and #f, leaving only the case of encoding symbols, and rename QUOTE to SYMBOL. The encoding would then be fully homomorphic — but the encodings of large constant data structures would become unreadable. For example, fairly readable constant structure
would encode through φ as
That didn't happen, though.
The homomorphic, non-QUOTE encoding would naturally tend to produce a universal function with no practical potential for meta-programming. In theory, one could use the non-homomorphic QUOTE encoding and still not offer any native meta-programming power. However, the QUOTE-based encoding means there are data structures lying around that look exactly like working executable code except that they happen to be QUOTEd. In practice, the psychology of the notation makes it rather inevitable that various facilities in the language would allow a blurring of the line between data and code. Lisp, I was told when first taught the language in the 1980s, treats code as data. Sic: I was not told Lisp treats data as code, but that it treats code as data.
In other words, Lisp had accidentally become an interpreted language; a profoundly different beast than the compiled language McCarthy had set out to create, and one whose character naturally suggests a whole different set of features that would not have occurred to someone designing a compiled language in 1960. There were, of course, some blunders along the way (dynamic scope is an especially famous one, and I would rate the abandonment of fexprs in favor of macros as another of similar magnitude); but in retrospect I see all that as part of exploring a whole new design space of interpreted features. Except that over the past three decades or so the Lisp community seems to have somewhat lost track of its interpreted roots; but, I'll get back to that.
Of interest:
• In S-expression Lisp, all source expressions are S-expressions. It is no less true now than before that an operational equivalence M1 ≅ M2 can only be nontrivial if at least one of the Mk is not an S-expression; but now, if the Mk are source expressions, we can be absolutely certain that they are both S-expressions. So if the operational equivalence relation is restricted to source expressions, it's trivial. This isn't disastrous; it just means that, in order to have nontrivial theory, we are going to have to have some terms that are not source expressions (as Landin did, though for a different reason); and if we choose to compile the language, we won't be allowed to call our optimizations "local source-to-source" (any given optimization could be one or the other, but not both at once).
• This is the fork in the road where Muller and I went our separate ways. Muller's M-Lisp, taking the compiled-language view, supposes that S-expressions are encoded homomorphically, resulting in a baseline language with no native meta-programming power. He then considers how to add some meta-programming power to the resulting language. However, practical meta-programming requires the programmer be able to write lucid code that can also be manipulated as data; and the homomorphic encoding isn't lucid. So instead, meta-programming in Muller's extended M-Lisp uses general M-expressions directly (rather than their encodings). If an M-expression turns out to be wanted as data, it then gets belatedly encoded — with the drawback that the M-expression can't be rewritten until the rewriting schemata can tell it won't be needed as data. This causes difficulties with operational equivalence of general M-expressions; in effect, as the burden of meta-programming is shifted from S-expressions to general M-expressions, it carries along with it the operational-equivalence difficulties that had been limited to S-expressions.
McCarthy hadn't finished the details of M-expressions, so S-expression Lisp wasn't altogether an encoding of anything; it was itself, leaving its implementors rather free to invent it as they went along. Blurring the boundary between quoted data and unquoted code provided meta-programming facilities that hadn't been available in compiled languages (essentially, I suggest, a sort of flexibility we enjoy in natural languages). In addition to QUOTE itself (which has a rather fraught history entangled with first-class functions and dynamic scope, cf. §3.3.2 of my dissertation), from very early on the language had fexprs, which are like LAMBDA-constructed functions except that they treat their operands as data — as if the operands had been QUOTEd — rather than evaluating them as code (which may later be done, if desired, explicitly by the fexpr using EVAL). In 1963, macros were added — not the mere template-substitution macros found in various other languages, but macros that treat their operands as data and perform an arbitrary computation to generate a data structure as output, which is then interpreted as code at the point of call.
But how exactly do we specify the meanings of programs in this interpreted S-expression language? We could resort to the meta-circular evaluator technique; this is a pretty natural strategy since an evaluator is exactly what we have as our primary definition of the language. That approach, though, is difficult to work with mathematically, and in particular doesn't lend itself to proofs of operational equivalence. If we try to construct a rewriting system the way we did before, we immediately run into the glaring practical problem that the same representation is used for executable code, which we want to have nontrivial theory, and passive data which necessarily has perfectly trivial theory. That is, as noted earlier, all source expressions are S-expressions and operational equivalence of S-expressions is trivial. It's possible to elaborate in vivid detail the theoretical train wreck that results from naive application of the usual rewriting semantics strategy to Lisp with quotation (or, worse, fexprs); but this seems to be mainly of interest if one is trying to prove that something can't be done. I'm interested in what can be done.
If what you want is nontrivial theory, that could in principle be used to guide optimizations, this is not difficult to arrange (once you know how; cf. my past discussion of profundity index). As mentioned, all nontrivial operational equivalences must have at least one of the two terms not a source expression (S-expression), therefore we need some terms that aren't source expressions; and our particular difficulty here is having no way to mark a source expression unmistakably as code; so, introduce a primitive context that says "evaluate this source expression". The new context only helps with operational equivalence if it's immune to QUOTE, and no source expression is immune to QUOTE, so that's yet another way to see that the new context must form a term that isn't a source expression.
Whereas the syntax of the compiled M-language had two kinds of terms — constant S-expressions and computational M-expressions — the syntax of the interpreted S-language will have three kinds of terms. There are, again, the "constant" terms, the S-expressions, which are now exactly the source expressions. There are the "computational" terms, which are needed for the actual work of computation; these are collectively shaped something like λ-calculus. We expect a big part of the equational strength of the rewriting calculus to reside in these computational terms, roughly the same equational strength as λ-calculus itself, and therefore of course those terms have to be entirely separate from the source expressions which can't have nontrivial equational theory. And then there are the "interpretation" terms, the ones that orchestrate the gradual conversion of source expressions into computational expressions. The code-marking terms are of this sort. The rewriting rules involving these "interpretation" terms will amount to an algorithm for interpreting source code.
This neat division of terms into three groups won't really be as crisp as I've just made it sound. Interpretation is by nature a gradual process whose coordination seeps into other parts of the grammar. Some non-interpretation terms will carry along environment information, in order to make it available for later use. This blurring of boundaries is perhaps another part of the flexibility that (I'll again suggest) makes interpreted languages more similar to natural languages.
I'll use nonterminal T for arbitrary terms. Here are the interpretation forms.
T ::= [eval T T] | ⟨wrap T⟩ | e .
Form [eval T1 T2] is a term that stands for evaluating a term T1 in an environment T2. This immediately allows us to distinguish between statements such as
S1 ≅ S2
[eval S1 e0] ≅ [eval S2 e0]
∀e, [eval S1 e] ≅ [eval S2 e] .
The first proposition is the excessively strong statement that S-expressions Sk are operationally equivalent — interchangeable in any context — which can only be true if the Sk are identically the same S-expression. The second proposition says that evaluating S1 in environment e0 is operationally equivalent to evaluating S2 in environment e0 — that is, for all contexts C and all S-expressions S, C[[eval S1 e0]] ↦* S iff C[[eval S2 e0]] ↦* S. The third proposition says that evaluating S1 in any environment e is operationally equivalent to evaluating S2 in the same e — which is what we would ordinarily have meant, in a compiled language, if we said that two executable code (as opposed to data) expressions Sk were operationally equivalent.
The second form, ⟨wrap T⟩, is a wrapper placed around a function T, that induces evaluation of the operand passed to the function. If T is used without such a wrapper (and presuming T isn't already a wrapped function), it acts directly on its unevaluated operand — that is, T is a fexpr.
The third form, e, is simply an environment. An environment is a series of symbol-value bindings, ⟪sk←Tk⟫; there's no need to go into gory detail here (though I did say more in a previous post).
The computational forms are, as mentioned, similar to λ-calculus with some environments carried along.
T ::= x | [combine T T T] | ⟨λx.T⟩ | ⟨εx.T⟩ .
Here we have a variable, a combination, and two kinds of function. Form ⟨λx.T⟩ is a function that substitutes its operand for x in its body T. Variant ⟨εx.T⟩ substitutes its dynamic environment for x in its body T.
Form [combine T1 T2 T3] is a term that stands for combining a function T1 with an operand T2 in a dynamic environment T3. The dynamic environment is the set of bindings in force at the point where the function is called; as opposed to the static environment, the set of bindings in force at the point where the function is constructed. Static environments are built into the bodies of functions by the function constructor, so they don't show up in the grammar. For example, [eval (LAMBDA X FOO) e0] would evaluate to a function with static environment e0, of the form ⟨wrap ⟨λx.[eval FOO ⟪...⟫]⟩⟩ with the contents of e0 embedded somewhere in the ⟪...⟫.
Putting it all together,
T ::= x | [combine T T T] | ⟨λx.T⟩ | ⟨εx.T⟩ | [eval T T] | ⟨wrap T⟩ | e | S .
The rewriting schemata naturally fall into two groups, those for internal computation and those for source-code interpretation. (There are of course no schemata associated with the third group of syntactic forms, the syntactic forms for passive data, because passive.) The computation schemata closely resemble λ-calculus, except with the second form of function used to capture the dynamic environment (which fexprs sometimes need).
[combine ⟨λx.T1⟩ T2 T3] → T1[x ← T2]
[combine ⟨εx.T1⟩ T2 T3] → [combine T1[x ← T3] T2 T3] .
The interpretation schemata look very much like the dispatching logic of a Lisp interpreter.
[eval d T] → d
if d is an empty list, boolean, λ-function, ε-function, or environment
[eval s e] → lookup(s,e) if symbol s is bound in environment e
[eval (T1 T2) T3] → [combine T1 T2 T3]
[combine ⟨wrap T1⟩ T2 T3] → [combine T1 [eval T2 T3] T3] .
(There would also be some atomic constants representing primitive first-class functions and reserved operators such as if, and schemata specifying what they do.)
Half a century's worth of misunderstandings and confusions
As I remarked earlier, Lisp as we know it might not have happened — at least, not when and where it did — if McCarthy had thought to allow first-class functions to occur in pairs. The thing is, though, I don't think it's all that much of an "accident". He didn't think to allow first-class functions to occur in pairs, and perhaps the reason we're likely to think to allow them today is that our thinking has been shaped by decades of the free-wheeling attitude fostered by the language that Lisp became because he didn't think to then. The actual sequence of events seems less unlikely than one might first suppose.
Researchers trying to set up semantics for Lisp have been led astray, persistently over the decades, by the fact that the primary Lisp constructor of first-class functions is called LAMBDA. Its behavior is not that of calculus λ, exactly because it's entangled with the process of interpreting Lisp source code. This becomes apparent when contemplating rewriting calculi for Lisp of the sort I've constructed above (and have discussed before on this blog): When you evaluate a LAMBDA expression you get a wrapped function, one that explicitly evaluates its operand and then passes the result to a computational function; that is, passes the result to a fexpr. Scan that: ordinary Lisp functions do not correspond directly to calculus λ-abstractions, but fexprs do correspond directly to calculus λ-abstractions. In its relation to Lisp, λ-calculus is a formal calculus of fexprs.
Much consternation has also been devoted to the perceived theoretical difficulty presented by Lisp's quotation operator (and presented in more extreme form by fexprs), because it presents a particular context that can distinguish any two S-expressions placed into it: (QUOTE S1) and (QUOTE S2) are observably distinct whenever the Sk are distinct from each other. Yet, this observation only makes sense in a compiled programming language. Back in the day, it would have been an unremarkable observation that Lisp only has syntax for data structures, no syntax at all for control. Two syntactically distinct Lisp source expressions are operationally non-equivalent even without any quotation or fexpr context, because they don't represent programs at all; they're just passive data structures. The context that makes a source expression code rather than data is patently not in the source; it's in what program you send the source expression to. Conventional small-step operational semantics presumes the decision to compile, along with a trivial interpretive mapping between source expression and internal computational terms (so the interpretive mapping doesn't have to appear explicitly in the rewriting schemata). It is true that, without any such constructs as quotation or fexprs, there would be no reason to treat the language as interpreted rather than compiled; but once you've crossed that Rubicon, the particular constructs like quotation or fexprs are mere fragments of, and can be distractions from, the main theoretical challenge of defining the semantics of an interpreted language.
The evolution of Lisp features has itself been a long process of learning how best to realize the flexibility offered by interpreted language. Fexprs were envisioned just about from the very beginning — 1960 — but were sabotaged by dynamic scope, a misfeature that resulted from early confusion over how to handle symbol bindings in an interpreter. Macros were introduced in 1963, and unlike fexprs they lend themselves to preprocessing at compile time if one chooses to use a compiler; but macros really ought to be much less mathematically elegant... except that in the presence of dynamic scope, fexprs are virulently unstable. Then there was the matter of first-class functions; that's an area where Lisp ought to have had a huge advantage; but first-class functions don't really come into their own without static scope (The Art of the Interpreter noted this) — and first-class functions also present a difficulty for compilers (which is why procedures in ALGOL were second-class). The upshot was that after Lisp 1.5, when Lisp splintered into multiple dialects, first-class functions went into eclipse until they reemerged in the mid-1970s when Scheme introduced static scope into the language. Fexprs held on longer but, ironically, were finally rejected by the Lisp community in 1980 — just a little ahead of the mainstream adoption of Scheme's static-scope innovation. So for the next twenty years and more, Lisp had static scope and first-class functions, but macros and no fexprs. Meanwhile, EVAL — key workhorse of meta-linguistic flexibility — was expunged from the new generation of mainstream Lisps and has had great difficulty getting back in.
The latter half of Lisp's history has been colored by a long-term trend in programming language design as a whole. I've alluded to this several times above. I have no specific sources to suggest for this; it's visible in the broad sweep of what languages were created, what research was done, and I've sensed it though my interactions with the Lisp community over the past thirty years. When I learned Lisp in the mid-1980s, it was from the Revised Maclisp Manual, Saturday Evening Edition (which I can see a few feet away on my bookshelf as I write this, proof that manuals can be well-written). Maclisp was a product of the mentality of the 1970s. Scheme too was a product of that mentality. And what comes through to me now, looking back, isn't the differences between those languages (different though they are), but that those people knew, gut deep, that Lisp is an interpreted language — philosophically, regardless of the technical details of the language processing software. The classic paper I cited above for the relationship between first-class functions and static scope was one of the "lambda" papers associated with the development of Scheme: "The Art of the Interpreter". The classic textbook — the Wizard Book — that emerged from the Scheme design is Structure and Interpretation of Computer Programs.
But then things changed. Compilation had sometimes intruded into Lisp design, yes (with unfortunate results, as I've mentioned), but the intrusion became more systematic. Amongst Scheme's other achievements it had provided improved compilation techniques, a positive development but which also encouraged greater focus on the challenges of compilation. We refined our mathematical technology for language semantics of compiled languages, we devised complex type systems for use with compiled languages, more and more we designed our languages to fit these technologies — and as Lisp didn't fit, more and more we tweaked our Lisp dialects to try to make them fit. Of course some of the indigenous features of Lisp couldn't fit, because the mathematical tools were fundamentally incompatible with them (no pun intended). And somewhere along the line, somehow, we forgot, perhaps not entirely but enough, that Lisp is interpreted. Second-class syntax has lately been treated more and more as if it were a primary part of the language, rather than a distraction from the core design. Whatever merits such languages have, wholeheartedly embracing the interpretation design stance is not among them.
I'm a believer in trying more rather than less. I don't begrudge anyone their opportunity to follow the design path that speaks to them; but not all those paths speak to me. Second-class syntax doesn't speak to me, nor recasting Lisp into a compiled language. I'm interested in compiling Lisp, but want the language design to direct those efforts rather than the other way around. To me, the potential of interpretation beckons; the exciting things we've already found on that path suggest to me there's more to find further along, and the only way to know is to follow the path and see. To do that, it seems to me we have to recognize that it is a distinct path, the distinct philosophy of interpretation; and, in company with that, we need to hone our mathematical technology for interpreted languages.
These are your father's parentheses
Elegant weapons
for a more... civilized age.
xkcd 297.
Saturday, June 11, 2016
The co-hygiene principle
The mathematics is not there till we put it there.
Sir Arthur Eddington, The Philosophy of Physical Science, 1938.
Investigating possible connections between seemingly unrelated branches of science and mathematics can be very cool. Independent of whether the connections actually pan out. It can be mind-bending either way — I'm a big fan of mind-bending, as a practical cure for rigid thinking — and you can get all sorts of off-beat insights into odd corners that get illuminated along the way. The more unlikely the connection, the more likely potential for mind bending; and also the more likely potential for pay-off if somehow it does pan out after all.
Two hazards you need to avoid, with this sort of thing: don't overplay the chances it'll pan out — and don't underplay the chances it'll pan out. Overplay and you'll sound like a crackpot and, worse, you might turn yourself into one. Relish the mind bending, take advantage of it to keep your thinking limber, and don't get upset when you're not finding something that might not be there. And at the same time, if you're after something really unlikely, say with only one chance in a million it'll pan out, and you don't leave yourself open to the possibility it will, you might just draw that one chance in a million and miss it, which would be just awful. So treat the universe as if it has a sense of humor, and be prepared to roll with the punchlines.
Okay, the particular connection I'm chasing is an observed analogy between variable substitution in rewriting calculi and fundamental forces in physics. If you know enough about those two subjects to say that makes no sense, that's what I thought too when I first noticed the analogy. It kept bothering me, though, because it hooks into something on the physics side that's already notoriously anomalous — gravity. The general thought here is that when two seemingly disparate systems share some observed common property, there may be some sort of mathematical structure that can be used to describe both of them and gives rise to the observed common property; and a mathematical modeling structure that explains why gravity is so peculiar in physics is an interesting prospect. So I set out to understand the analogy better by testing its limits, elaborating it until it broke down. Except, the analogy has yet to cooperate by breaking down, even though I've now featured it on this blog twice (1, 2).
So, building on the earlier explorations, in this post I tackle the problem from the other end, and try to devise a type of descriptive mathematical model that would give rise to the pattern observed in the analogy.
This sort of pursuit, as I go about it, is a game of endurance; again and again I'll lay out all the puzzle pieces I've got, look at them together, and try to accumulate a few more insights to add to the collection. Then gather up the pieces and save them away for a while, and come back to the problem later when I'm fresh on it again. Only this time I've kind-of succeeded in reaching my immediate goal. The resulting post, laying out the pieces and accumulating insights, is therefore both an explanation of where the result comes from and a record of the process by which I got there. There are lots of speculations within it shooting off in directions that aren't where I ended up. I pointedly left the stray speculations in place. Some of those tangents might turn out to be valuable after all; and taking them out would create a deceptive appearance of things flowing inevitably to a conclusion when, in the event, I couldn't tell whether I was going anywhere specific until I knew I'd arrived.
Naturally, for finding a particular answer — here, a mathematical structure that can give rise to the observed pattern — the reward is more questions.
Noether's Theorem
Determinism and rewriting
Nondeterminism and the cosmic footprint
Massive interconnection
Epilog: hygiene
Noether's Theorem
Noether's theorem (pedantically, Noether's first theorem) says that each differentiable invariant in the action of a system gives rise to a conservation law. This is a particularly celebrated result in mathematical physics; it's explicitly about how properties of a system are implied by the mathematical structure of its description; and invariants — the current fad name for them in physics is "symmetries" — are close kin to both hygiene and geometry, which relate to each other through the analogy I'm pursuing; so Noether's theorem has a powerful claim on my attention.
The action of a system always used to seem very mysterious to me, until I figured out it's one of those deep concepts that, despite its depth, is also quite shallow. It comes from Lagrangian mechanics, a mathematical formulation of classical mechanics alternative to the Newtonian mechanics formulation. This sort of thing is ubiquitous in mathematics, alternative formulations that are provably equivalent to each other but make various problems much easier or harder to solve.
Newtonian mechanics seeks to describe the trajectory of a thing in terms of its position, velocity, mass, and the forces acting on it. This approach has some intuitive advantages but is sometimes beastly difficult to solve for practical problems. The Lagrangian formulation is sometimes much easier to solve. Broadly, the time evolution of the system follows a trajectory through abstract state-space, and a function called the Lagrangian of the system maps each state into a quantity that... er... well, its units are those of energy. For each possible trajectory of the system through state-space, the path integral of the Lagrangian is the action. The principle of least action says that starting from a given state, the system will evolve along the trajectory that minimizes the action. Solving for the behavior of the system is then a matter of finding the trajectory whose action is smallest. (How do you solve for the trajectory with least action? Well, think of the trajectories as abstract values subject to variation, and imagine taking the "derivative" of the action over these variations. The least action will be a local minimum, where this derivative is zero. There's a whole mathematical technology for solving problems of just that form, called the "calculus of variations".)
The Lagrangian formulation tends to be good for systems with conserved quantities; one might prefer the Newtonian approach for, say, a block sliding on a surface with friction acting on it. And this Lagrangian affinity for conservative systems is where Noether's theorem comes in: if there's a differentiable symmetry of the action — no surprise it has to be differentiable, seeing how central integrals and derivatives are to all this — the symmetry manifests itself in the system behavior as a conservation law.
And what, you may ask, is this magical Lagrangian function, whose properties studied through the calculus of variations reveal the underlying conservation laws of nature? Some deeper layer of reality, the secret structure that underlies all? Not exactly. The Lagrangian function is whatever works: some function that causes the principle of least action to correctly predict the behavior of the system. In quantum field theory — so I've heard, having so far never actually grappled with QFT myself — the Lagrangian approach works for some fields but there is no Lagrangian for others. (Yes, Lagrangians are one of those mathematical devices from classical physics that treats systems in such an abstract, holistic way that it's applicable to quantum mechanics. As usual for such devices, its history involves Sir William Rowan Hamilton, who keeps turning up on this blog.)
This is an important point: the Lagrangian is whatever function makes the least-action principle work right. It's not "really there", except in exactly the sense that if you can devise a Lagrangian for a given system, you can then use it via the action integral and the calculus of variations to describe the behavior of the system. Once you have a Lagrangian function that does in fact produce the system behavior you want it to, you can learn things about that behavior from mathematical exploration of the Lagrangian. Such as Noether's theorem. When you find there is, or isn't, a certain differentiable symmetry in the action, that tells you something about what is or isn't conserved in the behavior of the system, and that result really may be of great interest; just don't lose sight of the fact that you started with the behavior of the system and constructed a suitable Lagrangian from which you are now deducing things about what the behavior does and doesn't conserve.
In 1543, Copernicus's heliocentric magnum opus De revolutionibus orbium coelestium was published with an unsigned preface by Lutheran theologian Andreas Osiander saying, more or less, that of course it'd be absurd to suggest the Earth actually goes around the Sun but it's a very handy fiction for the mathematics. Uhuh. It's unnecessary to ask whether our mathematical models are "true"; we don't need them to be true, just useful. When Francis Bacon remarked that what is most useful in practice is most correct in theory, he had a point — at least, for practical purposes.
The rewriting-calculus side of the analogy has a structural backstory from at least the early 1960s (some of which I've described in an earlier post, though with a different emphasis). Christopher Strachey hired Peter Landin as an assistant, and encouraged him to do side work exploring formal foundations for programming languages. Landin focused on tying program semantics to λ-calculus; but this approach suffered from several mismatches between the behavioral properties of programming languages versus λ-calculus, and in 1975 Gordon Plotkin published a solution for one of these mismatches, in one of the all-time classic papers in computer science, "Call-by-name, call-by-value and the λ-calculus" (pdf). Plotkin defined a slight variant of λ-calculus, by altering the conditions for the β-rule so that the calculus became call-by-value (the way most programming languages behaved while ordinary λ-calculus did not), and proved that the resulting λv-calculus was fully Church-Rosser ("just as well-behaved" as ordinary λ-calculus). He further set up an operational semantics — a rewriting system that ignored mathematical well-behavedness in favor of obviously describing the correct behavior of the programming language — and proved a set of correspondence theorems between the operational semantics and λv-calculus.
[In the preceding paragraph I perhaps should have mentioned compatibility, the other crucial element of rewriting well-behavedness; which you might think I'd have thought to mention since it's a big deal in my own work, though less flashy and more taken-for-granted than Church-Rosser-ness.]
Then in the 1980s, Matthias Felleisen applied Plotkin's approach to some of the most notoriously "unmathematical" behaviors of programs: side-effects in both data (mutable variables) and control (goto and its ilk). Like Plotkin, he set up an operational semantics and a calculus, and proved correspondence theorems between them, and well-behavedness for the calculus. He introduced the major structural innovation of treating a side-effect as an explicit syntactic construct that could move upward within its term. This upward movement would be a fundamentally different kind of rewrite from the function-application — the β-rule — of λ-calculus; abstractly, a side-effect is represented by a context 𝓢, which moves upward past some particular context C and, in the process, modifies C to leave in its wake some other context C': C[𝓢[T]] → 𝓢[C'[T]] . A side-effect is thus viewed as something that starts in a subterm and expands outward to affect more and more of the term until, potentially, it affects the whole term — if it's allowed to expand that far. Of course, a side-effect might never expand that far if it's trapped inside a context that it can't escape from; notably, no side-effect can escape from context λ.[ ] , which is to say, no side-effect can escape from inside the body of a function that hasn't been called.
This is where I started tracking the game, and developing my own odd notions. There seemed to me to be two significant drawbacks to Felleisen's approach, in its original published form. For one thing, the transformation of context C to C', as 𝓢 moved across it, could be quite extensive; Felleisen himself aptly called these transformations "bubbling up"; as an illustration of how messy things could get, here are the rules for a continuation-capture construct C expanding out of the operator or operand of a function call:
V(CT) → C(λx1.(T(λx2.(A(x1(Vx2)))))) for unused xk.
The other drawback to the approach was that as published at the time, it didn't actually provide the full measure of well-behavedness from Plotkin's treatment of call-by-value. One way or another, a constraint had to be relaxed somewhere. What does the side-effect construct 𝓢 do once it's finished moving upward? The earliest published solution was to wait till 𝓢 reaches the top of the term, and then get rid of it by a whole-term rewriting rule; that works, but the whole-term rewriting rule is explicitly not well-behaved: calculus well-behavedness requires that any rewriting on a whole term can also be done to a subterm, and here we've deliberately introduced a rewriting rule that can't be applied to subterms. So we've weakened the calculus well-behavedness. Another solution is to let 𝓢 reach the top of the term, then let it settle into some sort of normal form, and relax the semantics–calculus correspondence theorems to allow for equivalent normal forms. So the correspondence is weaker or, at least, more complicated. A third solution is to introduce an explicit context-marker — in both the calculus and the operational semantics — delimiting the possible extent of the side-effect. So you've got full well-behavedness but for a different language than you started out with. (Felleisen's exploration of this alternative is part of the prehistory of delimited continuations, but that's another story.)
[In a galling flub, I'd written in the preceding paragraph Church-Rosser-ness instead of well-behavedness; fixed now.]
It occurred to me that a single further innovation should be able to eliminate both of these drawbacks. If each side-effect were delimited by a context-marker that can move upward in the term, just as the side-effect itself can, then the delimiter would restore full Church-Rosser-ness without altering the language behavior; but, in general, the meanings of the delimited side-effects depend on the placement of the delimiter, so to preserve the meaning of the term, moving the delimiter may require some systematic alteration to the matching side-effect markers. To support this, let the delimiter be a variable-binding construct, with free occurrences of the variable in the side-effect markers. The act of moving the delimiter would then involve a sort of substitution function that propagates needed information to matching side-effect markers. What with one thing and another, my academic pursuits dragged me away from this line of thought for years, but then in the 2000s I found myself developing an operational semantics and calculus as part of my dissertation, in order to demonstrate that fexprs really are well-behaved (though I should have anticipated that some people, having been taught otherwise, would refuse to believe it even with proof). So I seized the opportunity to also elaborate my binding-delimiters approach to things that — unlike fexprs — really are side-effects.
This second innovation rather flew in the face of a tradition going back about seven or eight decades, to the invention of λ-calculus. Alonzo Church was evidently quite concerned about what variables mean; he maintained that a proposition with free variables in it doesn't have a clear meaning, and he wanted to have just one variable-binding construct, λ, whose β-rule defines the practical meanings of all variables. This tradition of having just one kind of variable, one binding construct, and one kind of variable-substitution (β-substitution) has had a powerful grip on researchers' imaginations for generations, to the point where even when other binding constructs are introduced they likely still have most of the look-and-feel of λ. My side-effect-ful variable binders are distinctly un-λ-like, with rewriting rules, and substitution functions, bearing no strong resemblance to the β-rule. Freedom from the β mold had the gratifying effect of allowing much simpler rewriting rules for moving upward through a term, without the major perturbations suggested by the term "bubbling up"; but, unsurprisingly, the logistics of a wild profusion of new classes of variables were not easy to work out. Much elegant mathematics surrounding λ-calculus rests squarely on the known simple properties of its particular take on variable substitution. The chapter of my dissertation that grapples with the generalized notion of substitution (Chapter 13, "Substitutive Reduction Systems", for anyone keeping score) has imho appallingly complicated foundations, although the high-level theorems at least are satisfyingly powerful. One thing that did work out neatly was enforcement of variable hygiene, which in ordinary λ-calculus is handled by α-renaming. In order to apply any nontrivial term-rewriting rule without disaster, you have to first make sure there aren't some two variables using the same name whose distinction from each other would be lost during the rewrite. It doesn't matter, really, what sort of variables are directly involved in the rewrite rule: an unhygienic rewrite could mess up variables that aren't even mentioned by the rule. Fortunately, it's possible to define a master α-renaming function that recurses through the term renaming variables to maintain hygiene, and whenever you add a new sort of variable to the system, just extend the master function with particular cases for that new sort of variable. Each rewriting rule can then invoke the master function, and everything works smoothly.
I ended up with four classes of variables. "Ordinary" variables, of the sort supported by λ-calculus, I found were actually wanted only for a specific (and not even technically necessary) purpose: to support partial evaluation. You could build the whole calculus without them and everything would work right, but the equational theory would be very weak. (I blogged on this point in detail here.) A second class of variable supported continuations; in effect, the side-effect marker was a "throw" and the binding construct was a "catch". Mutable state was more complicated, involving two classes of variables, one for assignments and one for lookups. The variables for assignment were actually environment identities; each assignment side-effect would then specify a value, a symbol, and a free variable identifying the environment. The variables for lookup stood for individual environment-symbol queries; looking up a symbol in an environment would generate queries for that environment and each of its ancestor environments. The putative result of the lookup would be a leaf subterm with free variable occurrences for all the queries involved, waiting to assimilate the query results, while the queries themselves would rise through the term in search of matching assignments. Whenever a query found a matching assignment, it would self-annihilate while using substitution to report the result to all waiting free variable occurrences.
Does all this detail matter to the analogy with physics? Well, that's the question, isn't it. There's a lot there, a great deal of fodder to chew on when considering how an analogy with something else might have a structural basis.
Amongst the four classes of variables, partial-evaluation variables have a peculiar sort of... symmetry. If you constructed a vau-calculus with, say, only continuation variables, you'd still have two different substitution functions — one to announce that a delimiting "catch" has been moved upward, and one for α-renaming. If you constructed a vau-calculus with only mutable-state variables, you'd have, well, a bunch of substitution functions, but in particular all the substitutions used to enact rewriting operations would be separate from α-renaming. β-substitution, though, is commensurate with α-renaming; once you've got β-substitution of partial-evaluation variables, you can use it to α-rename them as well, which is why ordinary λ-calculus has, apparently at least, only one substitution function.
Qualitatively, partial-evaluation variables seem more integrated into the fabric of the calculus, in contrast to the other classes of variables.
All of which put me powerfully in mind of physics because it's a familiar observation that gravity seems qualitatively more integrated into the fabric of spacetime, in contrast to the other fundamental forces (xkcd). General relativity portrays gravity as the shape of spacetime, whereas the other forces merely propagate through spacetime, and a popular strategy for aspiring TOEs (Theories Of Everything) is to integrate the other fundamental forces into the geometry as well — although, looking at the analogy, perhaps that popular strategy isn't such a good idea after all. Consider: The analogy isn't just between partial-evaluation variables and gravity. It's between the contrast of partial-evaluation variables against the other classes of variables, and the contrast of gravity against the other fundamental forces. All the classes of variables, and all the fundamental forces, are to some extent involved. I've already suggested that Felleisen's treatment of side-effects was both weakened and complicated by its too-close structural imitation of λ, whereas a less λ-like treatment of side-effects can be both stronger and simpler; so, depending on how much structure carries through the analogy, perhaps trying to treat the other fundamental forces too much like gravity should be expected to weaken and complicate a TOE.
Projecting through the analogy suggests alternative ways to structure theories of physics, which imho is worthwhile independent of whether the analogy is deep or shallow; as I've remarked before, I actively look for disparate ways of thinking as a broad base for basic research. The machinery of calculus variable hygiene, with which partial-evaluation variables have a special affinity, is only one facet of term structure; and projecting this through to fundamental physics, where gravity has a special affinity with geometry, this suggests that geometry itself might usefully be thought of, not as the venue where physics takes place, but merely as part of the rules by which the game is played. Likewise, the different kinds of variables differ from each other by the kinds of structural transformations they involve; and projecting that through the analogy, one might try to think of the fundamental forces as differing from each other not (primarily) by some arbitrary rules of combination and propagation, but by being different kinds of structural manipulations of reality. Then, if there is some depth to the analogy, one might wonder if some of the particular technical contrasts between different classes of variables might be related to particular technical contrasts between different fundamental forces — which, frankly, I can't imagine deciphering until and unless one first sets the analogy on a solid technical basis.
I've speculated several times on this blog on the role of non-locality in physics. Bell's Theorem says that the statistical distribution of quantum predictions cannot be explained by any local, deterministic theory of physics if, by 'local and deterministic', you mean 'evolving forward in time in a local and deterministic way'; but it's quite possible to generate this same statistical distribution of spacetime predictions using a theory that evolves locally and deterministically in a fifth dimension orthogonal to spacetime. Which strikes a familiar chord through the analogy with calculus variables, because non-locality is, qualitatively at least, the defining characteristic of what we mean by "side-effects", and the machinery of α-renaming maintains hygiene for these operations exactly by going off and doing some term rearranging on the side (as if in a separate dimension of rewriting that we usually don't bother to track). Indeed, thought of this way, a "variable" seems to be an inherently distributed entity, spread over a region of the term — called its scope — rather than located at a specific point. A variable instance might appear to have a specific location, but only because we look at a concrete syntactic term; naturally we have to have a particular concrete term in order to write it down, but somehow this doesn't seem to do justice to the reality of the hygiene machinery. One could think of an equivalence class of terms under α-renaming, but imho even that is a bit passive. The reality of a variable, I've lately come to think, is a dynamic distributed entity weaving through the term, made up of the binding construct (such as a λ), all the free instances within its scope, and the living connections that tie all those parts together; I imagine if you put your hand on any part of that structure you could feel it humming with vitality.
To give a slightly less hand-wavy description of my earlier post on Bell's Theorem — since it is the most concrete example we have to inform our view of the analogy on the physics side:
Bell looked at a refinement of the experiment from the EPR paradox. A device emits two particles with entangled spin, which shoot off in opposite directions, and their spins are measured by oriented detectors at some distance from the emitter. The original objection of Einstein Podolsky and Rosen was that the two measurements are correlated with each other, but because of the distance between the two detectors, there's no way for information about either measurement to get to where the other measurement takes place without "spooky action at a distance". Bell refined this objection by noting that the correlation of spin measurements depends on the angle θ between the detectors. If you suppose that the orientations of the detectors at measurement is not known at the time and place where the particles are emitted, and that the outcomes of the measurements are determined by some sort of information — "hidden variable" — propagating from the emission event at no more than the speed of light, then there are limits (called Bell's Inequality) on how the correlation can be distributed as a function of θ, no matter what the probability distribution of the hidden variable. The distribution predicted by quantum mechanics violates Bell's Inequality; so if the actual probability distribution of outcomes from the experiment matches the quantum mechanical prediction, we're living in a world that can't be explained by a local hidden-variable theory.
My point was that this whole line of reasoning supposes the state of the world evolves forward in time. If it doesn't, then we have to rethink what we even mean by "locality", and I did so. Suppose our entire four-dimensional reality is generated by evolving over a fifth dimension, which we might as well call "metatime". "Locality" in this model means that information about the state of one part of spacetime takes a certain interval of metatime to propagate a certain distance to another part of spacetime. Instead of trying to arrange the probability distribution of a hidden variable at the emission event so that it will propagate through time to produce the desired probability distribution of measurements — which doesn't work unless quantum mechanics is seriously wrong about this simple system — we can start with some simple, uniform probability distribution of possible versions of the entire history of the experiment, and by suitably arranging the rules by which spacetime evolves, we can arrange that eventually spacetime will settle into a stable state where the probability distribution is just what quantum mechanics predicts. In essence, it works like this: let the history of the experiment be random (we don't need nondeterminism here; this is just a statement of uniformly unknown initial conditions), and suppose that the apparent spacetime "causation" between the emission and the measurements causes the two measurements to be compared to each other. Based on θ, let some hidden variable decide whether this version of history is stable; and if it isn't stable, just scramble up a new one (we can always do that by pulling it out of the uniform distribution of the hidden variable, without having to posit fundamental nondeterminism). By choosing the rule for how the hidden variable interacts with θ, you can cause the eventual stable history of the experiment to exhibit any probability distribution you choose.
That immense power is something to keep a cautious eye on: not only can this technique produce the probability distribution predicted by quantum mechanics, it can produce any other probability distribution as well. So, if the general structure of this mathematical theory determines something about the structure of the physical reality it depicts, what it determines is apparently not, in any very straightforward fashion, that probability distribution.
The side of the analogy we have prior detailed structural knowledge about is the vau-calculus side. Whatever useful insights we may hope to extract from the metatime approach to Bell's Theorem, it's very sketchy compared to vau-calculus. So if we want to work out a structural pattern that applies to both sides of the analogy, it's plausible to start building from the side we know about, questioning and generalizing as we go along. To start with,
• Suppose we have a complex system, made up of interconnected parts, evolving by some sort of transformative steps according to some simple rules.
Okay, freeze frame. Why should the system be made up of parts? Well, in physics it's (almost) always the parts we're interested in. We ourselves are, apparently, parts of reality, and we interact with parts of reality. Could we treat the whole as a unit and then somehow temporarily pull parts out of it when we need to talk about them? Maybe, but the form with parts is still the one we're primarily interested in. And what about "transformative steps"; do we want discrete steps rather than continuous equations? Actually, yes, that is my reading of the situation; not only does fundamental physics appear to be shot through with discreteness (I expanded on this point a while back), but the particular treatment I used for my metatime proof-of-concept (above) used an open-ended sequence of discrete trials to generate the requisite probability distribution. If a more thoroughly continuous treatment is really wanted, one might try to recover continuity by taking a limit a la calculus.
• Suppose we separate the transformation rules into two groups, which we call bookkeeping rules and operational rules; and suppose we have a set of exclusive criteria on system configurations, call these hygiene conditions, which must be satisfied before any operational rule can be applied.
Freeze again. At first glance, this looks pretty good. From any unhygienic configuration, we can't move forward operationally until we've done bookkeeping to ensure hygiene. Both calculus rewriting and the metatime proof-of-concept seemingly conform to this pattern; but the two cases differ profoundly in how their underlying hygiene (supposing that's what it is, in the physics case) affects the form of the modeled system, and we'll need to consider the difference carefully if we mean to build our speculations on a sound footing.
Determinism and rewriting
Hygiene in rewriting is all about preserving properties of a term (to wit, variable instance–binding correspondences), whereas our proof-of-concept metatime transformations don't appear to be about perfectly preserving something but rather about shaping probability distributions. One might ask whether it's possible to set up the internals of our metatime model so that the probability distribution is a consequence, or symptom, of conserving something behind the scenes. Is the seemingly nondeterministic outcome of our quantum observation in a supposedly small quantum system here actually dictated by the need to maintain some cosmic balance that can't be directly observed because it's distributed over a ridiculously large number of entities (such as the number of electrons in the universe)? That could lead to some bracing questions about how to usefully incorporate such a notion into a mathematical theory.
As an alternative, one might decide that the probability distribution in the metatime model should not be a consequence of absolutely preserving a condition. There are two philosophically disparate sorts of models involving probabilities: either the probability comes from our lack of knowledge (the hidden-variable hypothesis), and in the underlying model the universe is computing an inevitable outcome; or the probability is in the foundations (God playing dice, in the Einsteinian phrase), and in the underlying model the universe is exploring the range of possible outcomes. I discussed this same distinction, in another form, in an earlier post, where it emerged as the defining philosophical distinction between a calculus and a grammar (here). In those terms, if our physics model is fundamentally deterministic then it's a calculus and by implication has that structural affinity with the vau-calculi on the other side of the analogy; but if our physics model is fundamentally nondeterministic then it's a grammar, and our analogy has to try to bridge that philosophical gap. Based on past experience, though, I'm highly skeptical of bridging the gap; if the analogy can be set on a concrete technical basis, the TOE on the physics side seems to me likely to be foundationally deterministic.
The foundationally deterministic approach to probability is to start with a probabilistic distribution of deterministic initial states, and evolve them all forward to produce a probabilistic distribution of deterministic final states. Does the traditional vau-calculus side of our analogy, where we have so much detail to start with, have anything to say about this? In the most prosaic sense, ones suspects not; probability distributions don't traditionally figure into deterministic computation semantics, where this approach would mean considering fuzzy sets of terms. There may be some insight lurking, though, in the origins of calculus hygiene.
When Alonzo Church's 1932/3 formal logic turned out to be inconsistent, he tried to back off and find some subset of it that was provably consistent. Here consistent meant that not all propositions are equivalent to each other, and the subset of the logic that he and his student J. Barkley Rosser proved consistent in this sense was what we now call λ-calculus. The way they did it was to show that if any term T1 can be reduced in the calculus in two different ways, as T2 and T3, then there must be some T4 that both of them can be reduced to. Since logical equivalence of terms is defined as the smallest congruence generated by the rewriting relation of the calculus, from the Church-Rosser property it follows that if two terms are equivalent, there must be some term that they both can be reduced to; and therefore, two different irreducible terms cannot possibly be logically equivalent to each other.
Proving the Church-Rosser theorem for λ-calculus was not, originally, a simple matter. It took three decades before a simple proof began to circulate, and the theorem for variant calculi continues to be a challenge. And this is (in one view of the matter, at least) where hygiene comes into the picture. Church had three major rewriting rules in his system, later called the α, β, and γ rules. The α rule was the "bookkeeping" rule; it allowed renaming a bound variable as long as you don't lose its distinction from other variables in the process. The β rule is now understood as the single operational rule of λ-calculus, how to apply a function to an argument. The γ rule is mostly forgotten now; it was simply doing a β-step backward, and was later dropped in favor of starting with just α and β and then taking the congruent closure (reflexive, symmetric, transitive, and compatible). Ultimately the Church-Rosser theorem allows terms to be sorted into β-equivalence classes; but the terms in each class aren't generally thought of as "the same term", just "equivalent terms". α-equivalent terms, though, are much closer to each other, and for many purposes would actually be thought of as "the same term, just written differently". Recall my earlier description of a variable as a distributed entity, weaving through the term, made up of binding construct, instances, and living connections between them. If you have a big term, shot through with lots of those dynamic distributed entities, the interweaving could really be vastly complex. So factoring out the α-renaming is itself a vast simplification, which for a large term may dwarf what's left after factoring to complete the Church-Rosser proof. To see by just how much the bookkeeping might dwarf the remaining operational complexity, imagine scaling the term up to the sort of cosmological scope mentioned earlier — like the number of electrons in the universe.
It seems worth considering, that hygiene may be a natural consequence of a certain kind of factoring of a vastly interconnected system: you sequester almost all of the complexity into bookkeeping with terrifically simple rules applied on an inhumanly staggering scale, and comparatively nontrivial operational rules that never have to deal directly with the sheer scale of the system because that part of the complexity was factored into the bookkeeping. In that case, at some point we'll need to ask when and why that sort of factoring is possible. Maybe it isn't really possible for the cosmos, and a flaw in our physics is that we've been trying so hard to factor things this way; when we really dive into that question we'll be in deep waters.
It's now no longer clear, btw, that geometry corresponds quite directly to α-renaming. There was already some hint of that in the view of vau-calculus side-effects as "non-local", which tends to associate geometry with vau-calculus term structure rather than α-renaming as such. Seemingly, hygiene is then a sort of adjunct to the geometry, something that allows the geometry to coexist with the massive interconnection of the system.
But now, with massive interconnection resonating between the two sides of the analogy, it's definitely time to ask some of those bracing questions about incorporating cosmic connectivity into a mathematical theory of physics.
Nondeterminism and the cosmic footprint
We want to interpret our probability distribution as a footprint, and reconstruct from it the massively connected cosmic order that walked there. Moreover, we're conjecturing that the whole system is factorable into bookkeeping/hygiene on one hand(?), and operations that amount to what we'd ordinarily call "laws of physics" on the other; and we'd really like to deduce, from the way quantum mechanics works, something about the nature of the bookkeeping and the factorization.
Classically, if we have a small system that's acted on by a lot of stuff we don't know about specifically, we let all those influences sum to a potential field. One might think of this classical approach as a particular kind of cosmological factorization in which the vast cosmic interconnectedness is reduced to a field, so one can then otherwise ignore almost everything to model the behavior of the small system of interest using a small operational set of physical laws. We know the sort of cosmic order that reduces that way, it's the sort with classical locality (relative to time evolution); and the vaster part of the factorization — the rest of the cosmos, that reduced to a potential field — is of essentially the same kind as the small system. The question we're asking at this point is, what sort of cosmic order reduces such that its operational part is quantum mechanics, and what does its bookkeeping part look like? Looking at vau-calculus, with its α-equivalence and Church-Rosser β-equivalence, it seems fairly clear that hygiene is an asymmetric factorization: if the cosmos factors this way, the bookkeeping part wouldn't have to look at all like quantum mechanics. A further complication is that quantum mechanics may be an approximation only good when the system you're looking at is vastly smaller than the universe as a whole; indeed, this conjecture seems rather encouraged by what happens when we try to apply our modern physical theories to the cosmos as a whole: notably, dark matter. (The broad notion of asymmetric factorizations surrounding quantum mechanics brings to mind both QM's notorious asymmetry between observer and observed, and Einstein's suggestion that QM is missing some essential piece of reality.)
For this factorization to work out — for the cosmic system as a whole to be broadly "metaclassical" while factoring through the bookkeeping to either quantum mechanics or a very good approximation thereof — the factorization has to have some rather interesting properties. In a generic classical situation where one small thing is acted on by a truly vast population of other things, we tend to expect all those other things to average out (as typically happens with a classical potential field), so that the vastness of the population makes their combined influence more stable rather than less; and also, as our subsystem interacts with the outside influence and we thereby learn more about that influence, we become more able to allow for it and still further reduce any residual unpredictability of the system.
Considered more closely, though, the expectation that summing over a large population would tend to average out is based broadly on the paradigm of signed magnitudes on an unbounded scale that attenuate over distance. If you don't have attenuation, and your magnitudes are on a closed loop, such as angles of rotation, increasing the population just makes things more unpredictable. Interestingly, I'd already suggested in one of my earlier explorations of the hygiene analogy that the physics hygiene condition might be some sort of rotational constraint, for the — seemingly — unrelated reason that the primary geometry of physics has 3+1 dimension structure, which is apparently the structure of quaternions, and my current sense of quaternions is that they're the essence of rotation. Though when things converge like this, it can be very hard to distinguish between an accidental convergence and one that simply reassembles fragments of a deeper whole.
I'll have a thought on the other classical problem — losing unpredictability as we learn more about outside influences over time — after collecting some further insights on the structural dynamics of bookkeeping.
Massive interconnection
Given a small piece of the universe, which other parts of the universe does it interact with, and how?
In the classical decomposition, all interactions with other parts of the universe are based on relative position in the geometry — that is, locality. Interestingly, conventional quantum mechanics retains this manner of selecting interactions, embedding it deeply into the equational structure of its mathematics. Recall the Schrödinger equation,
iℏ Ψ
= Ĥ Ψ .
The element that shapes the time evolution of the system — the Hamiltonian function Ĥ — is essentially an embodiment of the classical expectation of the system behavior; which is to say, interaction according to the classical rules of locality. (I discussed the structure of the Schrödinger equation at length in an earlier post, yonder.) Viewing conventional QM this way, as starting with classical interactions and then tacking on quantum machinery to "fix" it, I'm put in mind of Ptolemaic epicycles, tacked on to the perfect-circles model of celestial motion to make it work. (I don't mean the comparison mockingly, just critically; Ptolemy's system worked pretty well, and Copernicus used epicycles too. Turns out there's a better way, though.)
How does interaction-between-parts play out in vau-calculus, our detailed example of hygiene at work?
The whole syntax of a calculus term is defined by two aspects: the variables — by which I mean those "distributed entities" woven through the term, each made of a binding, its bound instances, and the connections that hygiene maintains between them — and, well, everything else. Once you ignore the specific identities of all the variable instances, you've just got a syntax tree with each node labeled by a context-free syntax production; and the context-free syntax doesn't have very many rules. In λ-calculus there are only four syntax rules: a term is either a combination, or a λ-abstraction, or a variable, or a constant. Some treatments simplify this by using variables for the "constants", and it's down to only three syntax rules. The lone operational rule β,
((λx.T1)T2) → T1[x ← T2] ,
gives a purely local pattern in the syntax tree for determining when the operation can be applied: any time you have a parent node that's a combination whose left child is a λ-abstraction. Operational rules stay nearly this simple even in vau-calculi; the left-hand side of each operational rule specifies when it can be applied by a small pattern of adjacent nodes in the syntax tree, and mostly ignores variables (thanks to hygienic bookkeeping). The right-hand side is where operational substitution may be introduced. So evidently vau-calculus — like QM — is giving local proximity a central voice in determining when and how operations apply, with the distributed aspect (variables) coming into play in the operation's consequences (right-hand side).
Turning it around, though, if you look at a small subterm — analogous, presumably, to a small system studied in physics — what rules govern its non-local connections to other parts of the larger term? Let's suppose the term is larger than our subterm by a cosmically vast amount. The free variables in the subterm are the entry points by which non-local influences from the (vast) context can affect the subterm (via substitution functions). And there is no upper limit to how fraught those sorts of interconnections can get (which is, after all, what spurs advocates of side-effect-less programming). That complexity belongs not to the "laws of physics" (neither the operational nor even the bookkeeping rules), but to the configuration of the system. From classical physics, we're used to having very simple laws, with all the complexity of our problems coming from the particular configuration of the small system we're studying; and now we've had that turned on its head. From the perspective of the rules of the calculus, yes, we still have very simple rules of manipulation, and all the complexity is in the arrangement of the particular term we're working with; but from the perspective of the subterm of interest, the interconnections imposed by free variables look a lot like "laws of physics" themselves. If we hold our one subterm fixed and allow the larger term around it to vary probabilistically then, in general, we don't know what the rules are and have no upper bound on how complicated those rules might be. All we have are subtle limits on the shape of the possible influences by those rules, imposed roundaboutly by the nature of the bookkeeping-and-operational transformations.
One thing about the shape of these nonlocal influences: they don't work like the local part of operations. The substitutive part of an operation typically involves some mixture of erasing bound variable instances and copying fragments from elsewhere. The upshot is that it rearranges the nonlocal topology of the term, that is, rearranges the way the variables interweave — which is, of course, why the bookkeeping rules are needed, to maintain the integrity of the interweaving as it winds and twists. And this is why a cosmic system of this sort doesn't suffer from a gradual loss of unpredictability as the subsystem interacts with its neighbors in the nonlocal network and "learns" about them: each nonlocal interaction that affects it changes its nonlocal-network neighbors. As long as the overall system is cosmically vast compared to the the subsystem we're studying, in practice we won't run out of new network neighbors no matter how many nonlocal actions our subsystem undergoes.
This also gives us a specific reason to suspect that quantum mechanics, by relying on this assumption of an endless supply of fresh network neighbors, would break down when studying subsystems that aren't sufficiently small compared to the cosmos as a whole. Making QM (as I've speculated before), like Newtonian physics, an approximation that works very well in certain cases.
Here's what the reconstructed general mathematical model looks like so far:
• The system as a whole is made up of interconnected parts, evolving by transformative steps according to simple rules.
• The interconnections form two subsystems: local geometry, and nonlocal network topology.
• The transformation rules are of two kinds, bookkeeping and operational.
• Operational rules can only be applied to a system configuration satisfying certain hygiene conditions on its nonlocal network topology; bookkeeping, which only acts on nonlocal network topology, makes it possible to achieve the hygiene conditions.
• Operational rules are activated based on local criteria (given hygiene). When applied, operations can have both local and nonlocal effects, while the integrity of the nonlocal network topology is maintained across both kinds of effect via hygiene, hence bookkeeping.
To complete this picture, it seems, we want to consider what a small local system consists of, and how it relates to the whole. This is all the more critical since we're entertaining the possibility that quantum mechanics might be an approximation that only works for a small local system, so that understanding how a local system relates to the whole may be crucial to understanding how quantum mechanics can arise locally from a globally non-quantum TOE.
A local system consists of some "local state", stuff that isn't interconnection of either kind; and some interconnections of (potentially) both kinds that are entirely encompassed within the local system. For example, in vau-calculi — our only prior source for complete examples — we might have a subterm (λx.(yx)). Variables are nonlocal network topology, of course, but in this case variable x is entirely contained within the local system. The choice of variable name "x" is arbitrary, as long as it remains different from "y" (hygiene). But what about the choice of "y"? In calculus reasoning we would usually say that because y is free in this subterm, we can't touch it; but that's only true if we're interested in comparing it to specific contexts, or to other specific subterms. If we're only interested in how this subterm relates to the rest of the universe, and we have no idea what the rest of the universe is, then the free variable y really is just one end of a connection whose other end is completely unknown to us; and a different choice of free variable would tell us exactly as much, and as little, as this one. We would be just as well off with (λx.(zx)), or (λz.(wz)), or even (λy.(xy)) — as long as we maintain the hygienic distinction between the two variables. The local geometry that can occur just outside this subterm, in its surrounding context, is limited to certain specific forms (by the context-free grammar of the calculus); the nonlocal network topology is vastly less constrained.
The implication here is that all those terms just named are effectively equivalent to (λx.(yx)). One might be tempted to think of this as simply different ways of "writing" the same local system, as in physics we might describe the same local system using different choices of coordinate axes; but the choice of coordinate axes is about local geometry, not nonlocal network topology. Here we're starting with simple descriptions of local systems, and then taking the quotient under the equivalence induced by the bookkeeping operations. It's tempting to think of the pre-quotient simple descriptions as "classical" and analogize the quotient to "quantum", but in fact there is a second quotient to be taken. In the metatime proof-of-concept, the rewriting kept reshuffling the entire history of the experiment until it reached a steady state — the obvious analogy is to a calculus irreducible term, the final result of the operational rewrite relation of the calculus. And this, at last, is where Church-Rosser-ness comes in. Church-Rosser is what guarantees that the same irreducible form (if any) is reached no matter in what order operational rules are applied. It's the enabler for each individual state of the system to evolve deterministically. To emphasize this point: Church-Rosser-ness applies to an individual possible system state, thus belongs to the deterministic-foundations approach to probabilities. The probability distribution itself is made up of individual possibilities each one of which is subject to Church-Rosser-ness. (Church-Rosser-ness is also, btw, a property of the mathematical model: one doesn't ask "Why should these different paths of system state evolution all come back together to the same normal form?", because that's simply the kind of mathematical model one has chosen to explore.)
The question we're trying to get a handle on is why the nonlocal effects of some operational rules would appear to be especially compatible with the bookkeeping quotient of the local geometry.
In vau-calculi, the nonlocal operational effects (i.e., operational substitution functions) that do not integrate smoothly with bookkeeping (i.e., with α-renaming) are the ones that support side-effects; and the one nonlocal operational effect that does integrate smoothly with bookkeeping — β-substitution — supports partial evaluation and turns out to be optional, in the sense that the operational semantics of the system could be described without that kind of nonlocal effect and the mathematics would still be correct with merely a weaker equational theory.
This suggests that in physics, gravity could be understood without bringing nonlocal effects into it at all, though there may be some sort of internal mathematical advantage to bringing them in anyway; while the other forces may be thought of as, in some abstract sense, side-effects.
So, what exactly makes a side-effect-ful substitution side-effect-ful? Conversely, β-substitution is also a form of substitution; it engages the nonlocal network topology, reshuffling it by distributing copies of subterms, the sort of thing I speculated above may be needed to maintain the unpredictability aspect of quantum mechanics. So, what makes β-substitution not side-effect-ful in character? Beyond the very specific technicalities of β- and α-substitution; and just how much, or little, should we be abstracting away from those technicalities? I'm supposing we have to abstract away at least a bit, on the principle that physics isn't likely to be technically close to vau-calculi in its mathematical details.
Here's a stab at a sufficient condition:
• A nonlocal operational effect is side-effect-ful just if it perturbs the pre-existing local geometry.
The inverse property, called "purity" in a programming context (as in "pure function"), is that the nonlocal operational effect doesn't perturb the pre-existing local geometry. β-substitution is pure in this sense, as it replaces a free variable-instance with a subterm but doesn't affect anything local other than the variable-instance itself. Contrast this with the operational substitution for control variables; the key clauses (that is, nontrivial base cases) of the two substitutions are
x[x ← T] → T .
The control substitution alters the local-geometric distance between pre-existing structure T and whatever pre-existing immediate context surrounds the subterm acted on. Both substitutions have the — conjecturally important — property that they substantially rearrange the nonlocal network topology by injecting arbitrary new network connections (that is, new free variables). The introduction of new free variables is a major reason why vau-calculi need bookkeeping to maintain hygiene; although, interestingly, it's taken all this careful reasoning about bookkeeping to conclude that bookkeeping isn't actually necessary to the notion of purity/impurity (or side-effect-ful/non-side-effect-ful); apparently, bookkeeping is about perturbations of the nonlocal network topology, whereas purity/impurity is about perturbations of the local geometry. To emphasize the point, one might call this non-perturbation of local geometry co-hygiene — all the nonlocal operational effects must be hygienic, which might or might not require bookkeeping depending on internals of the mathematics, but only the β- and gravity nonlocal effects are co-hygienic.
Abstracting away from how we got to it, here's what we have:
• A complex system of parts, evolving through a Church-Rosser transformation step relation.
• Interconnections within a system state, partitioned into local (geometry) and nonlocal (network).
• Each transformation step is selected locally.
• The nonlocal effects of each transformation step rearrange — scramble — nonlocal connections at the locus where applied.
• Certain transformative operations have nonlocal effects that do not disrupt pre-existing local structure — that are co-hygienic — and thereby afford particularly elegant description.
What sort of elegance is involved in the description of a co-hygienic operation depends on the technical character of the mathematical model; for β-reduction, what we've observed is functional compatibility between β- and α-substitution, while for gravity we've observed the general-relativity integration between gravity and the geometry of spacetime.
So my proposed answer to the conundrum I've been pondering is that the affinity between gravity and geometry suggests a modeling strategy with a nonlocal network pseudo-randomly scrambled by locally selected operational transformations evolving toward a stable state of spacetime, in which the gravity operations are co-hygienic. A natural follow-on question is just what sort of mathematical machinery, if any, would cause this network-scrambling to approximate quantum mechanics.
On the side, I've got intimations here that quantum mechanics may be an approximation induced by the pseudo-random network scrambling when the system under study is practically infinitesimal compared to the cosmos as a whole, and perhaps that the network topology has a rotational aspect.
Meanwhile, an additional line of possible inquiry has opened up. All along I've been trying to figure out what the analogy says about physics; but now it seems one might study the semantics of a possibly-side-effect-ful program fragment by some method structurally akin to quantum mechanics. The sheer mathematical perversity of quantum mechanics makes me skeptical that this could be a practical approach to programming semantics; on the other hand, it might provide useful insights for the TOE mathematics.
Epilog: hygiene
So, what happened to hygiene? It was a major focus of attention through nearly the whole investigation, and then dropped out of the plot near the end.
At its height of prestige, when directly analogized to spacetime geometry (before that fell through), hygiene motivated the suggestion that geometry might be thought of not as the venue where physics happens but merely as part of its rules. That suggestion is still somewhat alive since the proposed solution treats geometry as abstractly just one of the two forms of interconnection in system state, though there's a likely asymmetry of representation between the two forms. There was also some speculation that understanding hygiene on the physics side could be central to making sense of the model; that, I'd now ease off on, but do note that in seeking a possible model for physics one ought to keep an eye out for a possible bookkeeping mechanism, and certainly resolving the nonlocal topology of the model would seem inseparable from resolving its bookkeeping. So hygiene isn't out of the picture, and may even play an important role; just not with top billing.
Would it be possible for the physics model to be unhygienic? In the abstract sense I've ended up with, lack of hygiene would apparently mean an operation whose local effect causes nonlocal perturbation. Whether or not dynamic scope qualifies would depend on whether dynamically scoped variables are considered nonlocal; but since we expect some nonlocal connectivity, and those variables couldn't be perturbed by local operations without losing most/all of their nonlocality, probably the physics model would have to be hygienic. My guess is that if a TOE actually tracked the nonlocal network (as opposed to, conjecturally, introducing a quantum "blurring" as an approximation for the cumulative effect of the network), the tracking would need something enough like calculus variables that some sort of bookkeeping would be called for. |
bd2ba152f5e5c18a | June 2, 2021
International team including University of Calgary researcher proves 'imaginary' numbers have real function in quantum world
Faculty of Science mathematician on team that showed complex numbers are necessary for distinguishing quantum states
Visualize putting two apples on a table. That number “2” is easy to recognize and understand, because it’s a real number clearly observable in the real world.
Now try putting the square root of negative one, or √(−1), of apple on a table. It’s impossible because this number has very different properties from a real number — nothing in the real world can represent it.
Such numbers are called “imaginary” numbers.
The term was originally coined in the 17th century by French mathematician René Descartes, who regarded such numbers as fictitious or useless.
However, imaginary numbers have actually turned out to be useful. Combining them with real numbers makes “complex” numbers, used in many fields of science and engineering because mathematical calculations can be done faster and in a more practical way using these numbers.
Yet in such general calculations, “complex numbers are not strictly necessary. They’re just a tool,” says Dr. Carlo Maria Scandolo, PhD, assistant professor of mathematical statistics in the Department of Mathematics and Statistics in the Faculty of Science.
In quantum mechanics, complex numbers are used in the equations that describe the behaviour of objects that can behave like particles under some conditions and like waves in others, says Scandolo, a member of UCalgary’s Institute for Quantum Science and Technology.
For example, the most famous equation of quantum mechanics, called the Schrödinger equation, contains the imaginary number i, which is also called the “imaginary unit.”
Carlo Maria Scandolo
Carlo Maria Scandolo.
Imaginary numbers have real consequences
Theoretical physicists have been puzzled for nearly a century by complex numbers in quantum theory. Do these numbers actually have physical consequences in quantum mechanics and are they are necessary to describe the quantum world?
Now, a team of researchers at the University of Science and Technology in China, the University of Warsaw in Poland, and Scandolo at the University of Calgary, has answered that question.
In a new study, the team has shown for the first time that complex numbers — including their imaginary component — carry real information about quantum states that can be observed and measured in the quantum world.
“They are not a mere mathematical artifact,” says study co-author Scandolo. “Complex numbers really do exist and have operational meaning.”
The team’s research increases the fundamental understanding of how the quantum world works and how scientists might better harness quantum resources to, for example, build powerful quantum computers and quantum networks.
Their research is published in two papers, one in Physical Review Letters and the other in the journal Physical Review A.
Study showed complex numbers are a 'quantum resource'
The research team used an approach known as resource theories to devise a theoretical description, or mathematical framework, for whether complex numbers are a “resource.” In quantum theory, a resource is a property that enables new actions that would otherwise be impossible.
For example, quantum entanglement is a resource because it allows actions such as teleportation, or the transfer of quantum bits of information between separate locations.
The team’s mathematical framework, for which Scandolo did part of the calculations and provided some ideas, suggested that complex numbers are indeed a quantum resource.
To test whether this was the case, the Chinese researchers set up a “game,” a linear optics experiment in which the gamemaster sent a pair of quantum-entangled photons (particles of light) to two researchers, “Alice” and “Bob,” who were in separate locations. Each received one of the entangled protons.
Their task was to identify which quantum states were prepared by the gamemaster. They could do local measurements on their own photon and then compare measurements, to calculate their probability of guessing the correct state.
Alice and Bob were able to identify some of the quantum states with 100-per-cent accuracy, but only if they were allowed to use complex numbers (including imaginary numbers) in their local measurements.
However, if they were constrained to using only real numbers in their measurements, “suddenly they had no clue about which state was prepared,” Scandolo notes.
“This experiment showed that we can associate a task for which only the presence of imaginary numbers allows us to do well,” he says.
Exploiting complex numbers as a quantum resource
Albert Einstein famously said that quantum mechanics should allow two objects to affect each other’s behaviour instantly across vast distances, which he called “spooky action at a distance.”
The research team’s study indicates that complex numbers limit the amount of non-local behaviour in some sense, because without these numbers scientists wouldn’t be able to discriminate between quantum states locally.
“Complex numbers are a resource because they allow us to do this local discrimination,” Scandolo says.
Gaining a better understanding of this resource would enable scientists to exploit complex numbers to their fullest in situations where using quantum information is advantageous.
The research team’s next step is to explore other situations involving local discrimination (of other real quantum objects, for example) in which complex numbers could be a quantum resource.
Scandolo began this line of research as a postdoctoral scholar at the University of Calgary. His postdoc work was supported by a Faculty of Science Grand Challenges Award and by funding from the Pacific Institute for the Mathematical Sciences.
The University of Calgary’s researchers continue to be leaders in quantum science. |
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Discrete & Continuous Dynamical Systems
June 2018 , Volume 38 , Issue 6
Select all articles
Ergodic theorems for nonconventional arrays and an extension of the Szemerédi theorem
Yuri Kifer
2018, 38(6): 2687-2716 doi: 10.3934/dcds.2018113 +[Abstract](3959) +[HTML](183) +[PDF](561.84KB)
The paper is primarily concerned with the asymptotic behavior as \begin{document}$N\to∞$\end{document} of averages of nonconventional arrays having the form \begin{document}${N^{ - 1}}\sum\limits_{n = 1}^N {\prod\limits_{j = 1}^\ell {{T^{{P_j}(n,N)}}} } {f_j}$\end{document} where \begin{document}$f_j$\end{document}'s are bounded measurable functions, \begin{document}$T$\end{document} is an invertible measure preserving transformation and \begin{document}$P_j$\end{document}'s are polynomials of \begin{document}$n$\end{document} and \begin{document}$N$\end{document} taking on integer values on integers. It turns out that when \begin{document}$T$\end{document} is weakly mixing and \begin{document}$P_j(n, N) = p_jn+q_jN$\end{document} are linear or, more generally, have the form \begin{document}$P_j(n, N) = P_j(n)+Q_j(N)$\end{document} for some integer valued polynomials \begin{document}$P_j$\end{document} and \begin{document}$Q_j$\end{document} then the above averages converge in \begin{document}$L^2$\end{document} but for general polynomials \begin{document}$P_j$\end{document} of both \begin{document}$n$\end{document} and \begin{document}$N$\end{document} the \begin{document}$L^2$\end{document} convergence can be ensured even in the "conventional" case \begin{document}$\ell = 1$\end{document} only when \begin{document}$T$\end{document} is strongly mixing while for \begin{document}$\ell>1$\end{document} strong \begin{document}$2\ell$\end{document}-mixing should be assumed. Studying also weakly mixing and compact extensions and relying on Furstenberg's structure theorem we derive an extension of Szemerédi's theorem saying that for any subset of integers \begin{document}$\Lambda $\end{document} with positive upper density there exists a subset \begin{document}${\cal N}_\Lambda $\end{document} of positive integers having uniformly bounded gaps such that for \begin{document}$N∈{\cal N}_\Lambda $\end{document} and at least \begin{document}$\varepsilon N, \, \varepsilon >0$\end{document} of \begin{document}$n$\end{document}'s all numbers \begin{document}$p_jn+q_jN, \, j = 1, ..., \ell, $\end{document} belong to \begin{document}$\Lambda $\end{document}. We obtain also a version of these results for several commuting transformations which yields a corresponding extension of the multidimensional Szemerédi theorem.
Partially hyperbolic sets with a dynamically minimal lamination
Luiz Felipe Nobili França
2018, 38(6): 2717-2729 doi: 10.3934/dcds.2018114 +[Abstract](3365) +[HTML](172) +[PDF](385.01KB)
We study partially hyperbolic sets of \begin{document}$C^1$\end{document}-diffeomorphisms. For these sets there are defined the strong stable and strong unstable laminations.A lamination is called dynamically minimal when the orbit of each leaf intersects the set densely.
We prove that partially hyperbolic sets having a dynamically minimal lamination have empty interior. We also study the Lebesgue measure and the spectral decomposition of these sets. These results can be applied to \begin{document}$C^1$\end{document}-generic/robustly transitive attractors with one-dimensional center bundle.
Asymptotic properties of various stochastic Cucker-Smale dynamics
Laure Pédèches
2018, 38(6): 2731-2762 doi: 10.3934/dcds.2018115 +[Abstract](3726) +[HTML](194) +[PDF](986.9KB)
Starting from the stochastic Cucker-Smale model introduced in [14], we look into its asymptotic behaviours for different kinds of interaction. First in term of ergodicity, when $t$ goes to infinity, seeking invariant probability measures and using Lyapunov functionals. Second, when the number $N$ of particles becomes large, leading to results about propagation of chaos.
Remarks on the critical coupling strength for the Cucker-Smale model with unit speed
Seung-Yeal Ha, Dongnam Ko and Yinglong Zhang
2018, 38(6): 2763-2793 doi: 10.3934/dcds.2018116 +[Abstract](4013) +[HTML](206) +[PDF](871.21KB)
We present a non-trivial lower bound for the critical coupling strength to the Cucker-Smale model with unit speed constraint and short-range communication weight from the viewpoint of a mono-cluster(global) flocking. For a long-range communication weight, the critical coupling strength is zero in the sense that the mono-cluster flocking emerges from any initial configurations for any positive coupling strengths, whereas for a short-range communication weight, a mono-cluster flocking can emerge from an initial configuration only for a sufficiently large coupling strength. Our main interest lies on the condition of non-flocking. We provide a positive lower bound for the critical coupling strength. We also present numerical simulations for the upper and lower bounds for the critical coupling strength depending on initial configurations and compare them with analytical results.
Synchronization of positive solutions for coupled Schrödinger equations
Chuangye Liu and Zhi-Qiang Wang
2018, 38(6): 2795-2808 doi: 10.3934/dcds.2018118 +[Abstract](4052) +[HTML](269) +[PDF](441.85KB)
In this paper, we analyze synchronized positive solutions for a coupled nonlinear Schrödinger equation
where \begin{document}$ 2< p<\frac{n}{n-2}, $\end{document} if \begin{document}$ n\ge 3$\end{document} and \begin{document}$ 2< p<+∞ $\end{document}, if \begin{document}$ n = 1, 2, $\end{document} and \begin{document}$μ_1, μ_2, β>0 $\end{document} are positive constants. Our goal is two fold. On one hand we study the question under what conditions the ground states are nontrivial synchronized positive solutions, giving precise conditions in terms of the size of the coupling constant. On the other hand, we examine the questions on whether all positive solutions are synchronized solutions. We have a complete answer for the case \begin{document}$ n = 1 $\end{document} by proving that positivity implies synchronization. The latter result enables us to obtain the exact number of positive solutions even though no uniqueness result holds in the case, and this is quite different from the case \begin{document}$ p = 2 $\end{document} for which uniqueness of positive solutions was known ([19]).
Ruelle's inequality in negative curvature
Felipe Riquelme
2018, 38(6): 2809-2825 doi: 10.3934/dcds.2018119 +[Abstract](3659) +[HTML](167) +[PDF](390.47KB)
In this paper we study different notions of entropy for measure-preserving dynamical systems defined on noncompact spaces. We see that some classical results for compact spaces remain partially valid in this setting. We define a new kind of entropy for dynamical systems defined on noncompact Riemannian manifolds, which satisfies similar properties to the classical ones. As an application, we prove Ruelle's inequality and Pesin's entropy formula for the geodesic flow in manifolds with pinched negative sectional curvature.
Introduction to tropical series and wave dynamic on them
Nikita Kalinin and Mikhail Shkolnikov
2018, 38(6): 2827-2849 doi: 10.3934/dcds.2018120 +[Abstract](3469) +[HTML](180) +[PDF](516.7KB)
The theory of tropical series, that we develop here, firstly appeared in the study of the growth of pluriharmonic functions. Motivated by waves in sandpile models we introduce a dynamic on the set of tropical series, and it is experimentally observed that this dynamic obeys a power law. So, this paper serves as a compilation of results we need for other articles and also introduces several objects interesting by themselves.
Reducibility of three dimensional skew symmetric system with Liouvillean basic frequencies
Dongfeng Zhang, Junxiang Xu and Xindong Xu
2018, 38(6): 2851-2877 doi: 10.3934/dcds.2018123 +[Abstract](3522) +[HTML](169) +[PDF](518.58KB)
In this paper we consider the system \begin{document}$\dot{x} = (A(\epsilon)+ \epsilon^{m} P(t;\epsilon)) x, x∈\mathbb{R}^{3}, $\end{document} where \begin{document}$\epsilon$\end{document} is a small parameter, \begin{document}$A, P$\end{document} are all \begin{document}$3×3$\end{document} skew symmetric matrices, \begin{document}$A$\end{document} is a constant matrix with eigenvalues \begin{document}$± i\bar{λ}(\epsilon)$\end{document} and 0, where \begin{document}$\bar{λ}(\epsilon) = λ+a_{m_{0}}\epsilon^{m_{0}} + O(\epsilon^{m_{0}+1}) (m_{0}< m),$\end{document} \begin{document}$a_{m_{0}}≠ 0,$\end{document} \begin{document}$P$\end{document} is a quasi-periodic matrix with basic frequencies \begin{document}$ω = (1,α)$\end{document} with \begin{document}$α$\end{document} being irrational. First, it is proved that for most of sufficiently small parameters, this system can be reduced to a rotation system. Furthermore, if the basic frequencies satisfy that \begin{document}$ 0≤β(α) < r,$\end{document} where \begin{document}$β(α)$\end{document} measures how Liouvillean \begin{document}$α$\end{document} is, \begin{document}$r$\end{document} is the initial analytic radius, it is proved that for most of sufficiently small parameters, this system can be reduced to constant system by means of a quasi-periodic change of variables.
Incompressible limit for the compressible flow of liquid crystals in $ L^p$ type critical Besov spaces
Qunyi Bie, Haibo Cui, Qiru Wang and Zheng-An Yao
2018, 38(6): 2879-2910 doi: 10.3934/dcds.2018124 +[Abstract](4326) +[HTML](181) +[PDF](629.27KB)
The present paper is devoted to the compressible nematic liquid crystal flow in the whole space \begin{document}$ \mathbb{R}^N\,(N≥ 2)$\end{document}. Here we concentrate on the incompressible limit in the \begin{document}$ L^p$\end{document} type critical Besov spaces setting. We first establish the existence of global solutions in the framework of \begin{document}$ L^p$\end{document} type critical spaces provided that the initial data are close to some equilibrium states. Based on the global existence, we then consider the incompressible limit problem in the ill prepared data case. We justify the low Mach number convergence to the incompressible flow of liquid crystals in proper function spaces. In addition, the accurate converge rates are obtained.
Stability of transonic jets with strong rarefaction waves for two-dimensional steady compressible Euler system
Min Ding and Hairong Yuan
2018, 38(6): 2911-2943 doi: 10.3934/dcds.2018125 +[Abstract](3575) +[HTML](174) +[PDF](582.8KB)
We study supersonic flow past a convex corner which is surrounded by quiescent gas. When the pressure of the upstream supersonic flow is larger than that of the quiescent gas, there appears a strong rarefaction wave to rarefy the supersonic gas. Meanwhile, a transonic characteristic discontinuity appears to separate the supersonic flow behind the rarefaction wave from the static gas. In this paper, we employ a wave front tracking method to establish structural stability of such a flow pattern under non-smooth perturbations of the upcoming supersonic flow. It is an initial-value/free-boundary problem for the two-dimensional steady non-isentropic compressible Euler system. The main ingredients are careful analysis of wave interactions and construction of suitable Glimm functional, to overcome the difficulty that the strong rarefaction wave has a large total variation.
Isolated singularities for elliptic equations with hardy operator and source nonlinearity
Huyuan Chen and Feng Zhou
2018, 38(6): 2945-2964 doi: 10.3934/dcds.2018126 +[Abstract](3722) +[HTML](179) +[PDF](470.64KB)
In this paper, we concern the isolated singular solutions for semi-linear elliptic equations involving Hardy-Leray potential
We classify the isolated singularities and obtain the existence and stability of positive solutions of (1). Our results are based on the study of nonhomogeneous Hardy problem in a new distributional sense.
Lozi-like maps
Michał Misiurewicz and Sonja Štimac
2018, 38(6): 2965-2985 doi: 10.3934/dcds.2018127 +[Abstract](3864) +[HTML](188) +[PDF](583.08KB)
We define a broad class of piecewise smooth plane homeomorphisms which have properties similar to the properties of Lozi maps, including the existence of a hyperbolic attractor. We call those maps Lozi-like. For those maps one can apply our previous results on kneading theory for Lozi maps. We show a strong numerical evidence that there exist Lozi-like maps that have kneading sequences different than those of Lozi maps.
Propagation of monostable traveling fronts in discrete periodic media with delay
Shi-Liang Wu and Cheng-Hsiung Hsu
2018, 38(6): 2987-3022 doi: 10.3934/dcds.2018128 +[Abstract](4193) +[HTML](251) +[PDF](606.44KB)
This paper is devoted to study the front propagation for a class of discrete periodic monostable equations with delay and nonlocal interaction. We first establish the existence of rightward and leftward spreading speeds and prove their coincidence with the minimal wave speeds of the pulsating traveling fronts in the right and left directions, respectively. The dependency of the speeds of propagation on the heterogeneity of the medium and the delay term is also investigated. We find that the periodicity of the medium increases the invasion speed, in comparison with a homogeneous medium; while the delay decreases the invasion speed. Further, we prove the uniqueness of all noncritical pulsating traveling fronts. Finally, we show that all noncritical pulsating traveling fronts are globally exponentially stable, as long as the initial perturbations around them are uniformly bounded in a weight space.
High energy solutions of the Choquard equation
Daomin Cao and Hang Li
2018, 38(6): 3023-3032 doi: 10.3934/dcds.2018129 +[Abstract](4029) +[HTML](180) +[PDF](369.22KB)
In this paper we are concerned with the existence of positive high energy solutions of the Choquard equation. Under certain assumptions, the ground state of Choquard equation does not exist. However, by global compactness analysis, we prove that there exists a positive high energy solution.
A singular cahn-hilliard-oono phase-field system with hereditary memory
Monica Conti, Stefania Gatti and Alain Miranville
2018, 38(6): 3033-3054 doi: 10.3934/dcds.2018132 +[Abstract](3625) +[HTML](197) +[PDF](447.78KB)
We consider a phase-field system modeling phase transition phenomena, where the Cahn-Hilliard-Oono equation for the order parameter is coupled with the Coleman-Gurtin heat law for the temperature. The former suitably describes both local and nonlocal (long-ranged) interactions in the material undergoing phase-separation, while the latter takes into account thermal memory effects. We study the well-posedness and longtime behavior of the corresponding dynamical system in the history space setting, for a class of physically relevant and singular potentials. Besides, we investigate the regularization properties of the solutions and, for sufficiently smooth data, we establish the strict separation property from the pure phases.
Interface stabilization of a parabolic-hyperbolic pde system with delay in the interaction
Gilbert Peralta and Karl Kunisch
2018, 38(6): 3055-3083 doi: 10.3934/dcds.2018133 +[Abstract](3780) +[HTML](229) +[PDF](523.25KB)
A coupled parabolic-hyperbolic system of partial differential equations modeling the interaction of a structure submerged in a fluid is studied. The system being considered incorporates delays in the interaction on the interface between the fluid and the solid. We study the stability properties of the interaction model under suitable assumptions between the competing strengths of the delays and the feedback controls.
Liouville theorems for periodic two-component shallow water systems
Qiaoyi Hu, Zhixin Wu and Yumei Sun
2018, 38(6): 3085-3097 doi: 10.3934/dcds.2018134 +[Abstract](3713) +[HTML](174) +[PDF](393.31KB)
We establish Liouville-type theorems for periodic two-component shallow water systems, including a two-component Camassa-Holm equation (2CH) and a two-component Degasperis-Procesi (2DP) equation. More presicely, we prove that the only global, strong, spatially periodic solutions to the equations, vanishing at some point \begin{document}$(t_0, x_0)$\end{document}, are the identically zero solutions. Also, we derive new local-in-space blow-up criteria for the dispersive 2CH and 2DP.
Exit time asymptotics for small noise stochastic delay differential equations
David Lipshutz
2018, 38(6): 3099-3138 doi: 10.3934/dcds.2018135 +[Abstract](4321) +[HTML](177) +[PDF](672.4KB)
Dynamical system models with delayed dynamics and small noise arise in a variety of applications in science and engineering. In many applications, stable equilibrium or periodic behavior is critical to a well functioning system. Sufficient conditions for the stability of equilibrium points or periodic orbits of certain deterministic dynamical systems with delayed dynamics are known and it is of interest to understand the sample path behavior of such systems under the addition of small noise. We consider a small noise stochastic delay differential equation (SDDE). We obtain asymptotic estimates, as the noise vanishes, on the time it takes a solution of the stochastic equation to exit a bounded domain that is attracted to a stable equilibrium point or periodic orbit of the corresponding deterministic equation. To obtain these asymptotics, we prove a sample path large deviation principle (LDP) for the SDDE that is uniform over initial conditions in bounded sets. The proof of the uniform sample path LDP uses a variational representation for exponential functionals of strong solutions of the SDDE. We anticipate that the overall approach may be useful in proving uniform sample path LDPs for other infinite-dimensional small noise stochastic equations.
Sign-changing multi-bump solutions for Kirchhoff-type equations in $\mathbb{R}^3$
Yinbin Deng and Wei Shuai
2018, 38(6): 3139-3168 doi: 10.3934/dcds.2018137 +[Abstract](4466) +[HTML](325) +[PDF](610.26KB)
We are interested in the existence of sign-changing multi-bump solutions for the following Kirchhoff equation
where \begin{document}$λ$\end{document}>0 is a parameter and the potential \begin{document}$V(x)$\end{document} is a nonnegative continuous function with a potential well \begin{document}$Ω: = int(V^{-1}(0))$\end{document} which possesses \begin{document}$k$\end{document} disjoint bounded components \begin{document}$Ω_1,Ω_2,···,Ω_k$\end{document}. Under some conditions imposed on \begin{document}$f(u)$\end{document}, multiple sign-changing multi-bump solutions are obtained. Moreover, the concentration behavior of these solutions as \begin{document}$λ→ +∞$\end{document} are also studied.
Normality and uniqueness of Lagrange multipliers
Karla L. Cortez and Javier F. Rosenblueth
2018, 38(6): 3169-3188 doi: 10.3934/dcds.2018138 +[Abstract](3999) +[HTML](219) +[PDF](151.64KB)
In this paper we study, for certain problems in the calculus of variations and optimal control, two different questions related to uniqueness of multipliers appearing in first order necessary conditions. One deals with conditions under which a given multiplier associated with an extremal of a fixed function is unique, a property which, in nonlinear programming, is known to be equivalent to the strict Mangasarian-Fromovitz constraint qualification. We show that, for isoperimetric problems in the calculus of variations, a similar characterization holds, but not in optimal control where the corresponding condition is only sufficient for the uniqueness of the multiplier. The other question is related to the set of multipliers associated with all functions for which a solution to the constrained problem is given. We prove that, for both types of problems, this set is a singleton if and only if a strong normality assumption holds.
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8a48c7f161452103 | Making Sense Of Quantum Mechanics: Geometric Quantum Theory
The theory of quantum mechanics lies at the basis of our modern understanding of how microscopic particles such as electrons and atomic nuclei behave. Despite this fact, controversy on what it actually says about the physical world has accompanied the further development of the theory and its various off-springs up to today.
Perhaps the main question in the debate on the interpretation of quantum mechanics has been determining the physical meaning of the “wave function,” the central mathematical object in the theory. In one of the simplest quantum mechanical models, the 1-particle Schrödinger theory, this wave function is a mathematical function assigning to each point in space and time a (complex) number. If one knows the initial value of the wave function at each point in space, then the wave function at some later time can in principle be determined by solving the Schrödinger equation.
How to use the wave function
In spite of the addressed controversy on its interpretation, quantum mechanics does give a universally-agreed-upon answer to this question: First, the wave function is used to compute the probability to find the particle in a given region of space. Second, the wave function determines the expectation values of so-called observables of the particle – as, for instance, its energy, its velocity and, of course, its position. That is, in order to apply the theory sensibly, one considers a multitude of repetitions of the same experiment.
To be more specific, think of the case of shooting an electron through an experimental apparatus and letting it hit a detector screen, as is done in the famous double-slit-experiment. As explained, one repeats the experiment under the same circumstances until enough data has been gathered. The theory itself is relevant when one wants to know the final statistics (where the electrons tend to hit on the screen) from the initial statistics (where the electrons started out) without having done the experiment.
More precisely, one measures the initial position and velocity of each electron in the ensemble (setting the clock anew each time the experiment starts over) and can determine an initial wave function from that data. Again, solving the Schrödinger equation for this initial wave function and apparatus yields the wave function at any later time. This can then be translated back into the pattern on the screen, which emerges when plotting all of the individual impacts on the screen (see figure). It is important to emphasize, however, that it is impossible to predict, where an individual particle will appear on the screen. Rather, the wave function predicts the final pattern or, better to say, the probability to find a particle in the ensemble within a certain region on the detector.
Figure: Results of a double-slit-experiment performed by Dr. Tonomura showing the build-up of an interference pattern of single electrons. Numbers of electrons are 11 (a), 200 (b), 6000 (c), 40000 (d), 140000 (e). This picture was licensed by Belsazar under the GNU Free Documentation License.
The Controversy
The above discussion suggests that quantum mechanics is a purely statistical theory, i.e. one only used to compute the probabilities of the whereabouts of particles and expectation values of the respective observables. Yet there are many physicists, who do not subscribe to this view, also known as the ensemble interpretation. In most interpretations of quantum mechanics, the wave function is given a meaning going beyond its purely statistical interpretation. The book by J. Baggot shall be recommended here for further reading.
For instance, in the Copenhagen interpretation, the wave function describes an actual wave propagating through space and time, which collapses into a pointlike particle once a measurement takes place. This is still the orthodox point of view, shared by the largest portion of physicists in the field, but we are far from reaching any kind of consensus.
The ensemble interpretation is minimalist in the sense that it makes the theory usable without asserting much on “the nature of reality.” Yet, taking it to its logical conclusions, as the author has done in a recent article, is in itself a challenge to the orthodoxy.
The Findings
In 1926, the physicist E. Madelung rewrote the Schrödinger equation in a manner that exposed how the theory is linked to Newtonian (continuum) mechanics, obtaining what is today known as the Madelung equations. In the 1950s David Bohm employed this to develop his own interpretation of quantum mechanics, which is still an active area of research.
If, however, the ensemble interpretation is applied to the Madelung equations, as the author has done in his research, the Madelung equations show that the reason for this unpredictability is the universal presence of a noise. This “Bohm noise” acts on each individual particle in the ensemble and it can be proven that it is only relevant for particles of sufficiently small mass – explaining why the relevant quantum effects are not noticeable on our everyday scales. The physical origin of the Bohm noise is still unknown, one possible explanation is the existence of a gravitational background radiation.
Further, this ansatz questions the established status of the uncertainty principle: When quantum mechanics is taken as a purely statistical theory, the mathematical relation corresponding to the “principle” is one of statistical nature, not one that refers to any individual particle. Therefore, this approach to the Schrödinger theory does not put an upper bound on the possible precision of simultaneous measurement of position and velocity of any individual particle, even though experimental limitations may do this in practice. This, in turn, allows for the physical existence of particle paths – a finding fitting well into the recent discovery of macroscopic quantum analogs.
Finally, it was shown that an ensemble interpretation of the Madelung equations allows for the experimental reconciliation of the Schrödinger theory with the established theory of probability. To be more specific, it has been proven that the fundamental change of the 1-particle Schrödinger theory to accord with the rules of standard probability theory yields the same predictions as standard quantum mechanics in the experimentally relevant cases, but the change also leads to different predictions in experimentally less accessible instances.
Since the Madelung equations are also more suitable to study quantum dynamics with geometric constraints (think e.g. of an electron restricted to move on the surface of a sphere) and this geometric-analytic approach of studying the Schrödinger theory was inspired by the theory of geometric quantization, the author has termed the adapted theory geometric quantum theory.
These findings are described in the article entitled The Madelung Picture as a Foundation of Geometric Quantum Theory, recently published in the journal Foundations of Physics. This work was conducted by Maik Reddiger while study at the Technische Universität Berlin.
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a9353937632c5c14 | @article{2716, abstract = {Multi-dimensional mean-payoff and energy games provide the mathematical foundation for the quantitative study of reactive systems, and play a central role in the emerging quantitative theory of verification and synthesis. In this work, we study the strategy synthesis problem for games with such multi-dimensional objectives along with a parity condition, a canonical way to express ω ω -regular conditions. While in general, the winning strategies in such games may require infinite memory, for synthesis the most relevant problem is the construction of a finite-memory winning strategy (if one exists). Our main contributions are as follows. First, we show a tight exponential bound (matching upper and lower bounds) on the memory required for finite-memory winning strategies in both multi-dimensional mean-payoff and energy games along with parity objectives. This significantly improves the triple exponential upper bound for multi energy games (without parity) that could be derived from results in literature for games on vector addition systems with states. Second, we present an optimal symbolic and incremental algorithm to compute a finite-memory winning strategy (if one exists) in such games. Finally, we give a complete characterization of when finite memory of strategies can be traded off for randomness. In particular, we show that for one-dimension mean-payoff parity games, randomized memoryless strategies are as powerful as their pure finite-memory counterparts.}, author = {Chatterjee, Krishnendu and Randour, Mickael and Raskin, Jean}, journal = {Acta Informatica}, number = {3-4}, pages = {129 -- 163}, publisher = {Springer}, title = {{Strategy synthesis for multi-dimensional quantitative objectives}}, doi = {10.1007/s00236-013-0182-6}, volume = {51}, year = {2014}, } @article{2852, abstract = {A robust combiner for hash functions takes two candidate implementations and constructs a hash function which is secure as long as at least one of the candidates is secure. So far, hash function combiners only aim at preserving a single property such as collision-resistance or pseudorandomness. However, when hash functions are used in protocols like TLS they are often required to provide several properties simultaneously. We therefore put forward the notion of robust multi-property combiners and elaborate on different definitions for such combiners. We then propose a combiner that provably preserves (target) collision-resistance, pseudorandomness, and being a secure message authentication code. This combiner satisfies the strongest notion we propose, which requires that the combined function satisfies every security property which is satisfied by at least one of the underlying hash function. If the underlying hash functions have output length n, the combiner has output length 2 n. This basically matches a known lower bound for black-box combiners for collision-resistance only, thus the other properties can be achieved without penalizing the length of the hash values. We then propose a combiner which also preserves the property of being indifferentiable from a random oracle, slightly increasing the output length to 2 n+ω(log n). Moreover, we show how to augment our constructions in order to make them also robust for the one-wayness property, but in this case require an a priory upper bound on the input length.}, author = {Fischlin, Marc and Lehmann, Anja and Pietrzak, Krzysztof Z}, journal = {Journal of Cryptology}, number = {3}, pages = {397 -- 428}, publisher = {Springer}, title = {{Robust multi-property combiners for hash functions}}, doi = {10.1007/s00145-013-9148-7}, volume = {27}, year = {2014}, } @inproceedings{2905, abstract = {Persistent homology is a recent grandchild of homology that has found use in science and engineering as well as in mathematics. This paper surveys the method as well as the applications, neglecting completeness in favor of highlighting ideas and directions.}, author = {Edelsbrunner, Herbert and Morozovy, Dmitriy}, location = {Kraków, Poland}, pages = {31 -- 50}, publisher = {European Mathematical Society Publishing House}, title = {{Persistent homology: Theory and practice}}, doi = {10.4171/120-1/3}, year = {2014}, } @inproceedings{8044, abstract = {Many questions concerning models in quantum mechanics require a detailed analysis of the spectrum of the corresponding Hamiltonian, a linear operator on a suitable Hilbert space. Of particular relevance for an understanding of the low-temperature properties of a system is the structure of the excitation spectrum, which is the part of the spectrum close to the spectral bottom. We present recent progress on this question for bosonic many-body quantum systems with weak two-body interactions. Such system are currently of great interest, due to their experimental realization in ultra-cold atomic gases. We investigate the accuracy of the Bogoliubov approximations, which predicts that the low-energy spectrum is made up of sums of elementary excitations, with linear dispersion law at low momentum. The latter property is crucial for the superfluid behavior the system.}, author = {Seiringer, Robert}, booktitle = {Proceeding of the International Congress of Mathematicans}, isbn = {9788961058063}, location = {Seoul, South Korea}, pages = {1175--1194}, publisher = {Kyung Moon SA}, title = {{Structure of the excitation spectrum for many-body quantum systems}}, volume = {3}, year = {2014}, } @inproceedings{1702, abstract = {In this paper we present INTERHORN, a solver for recursion-free Horn clauses. The main application domain of INTERHORN lies in solving interpolation problems arising in software verification. We show how a range of interpolation problems, including path, transition, nested, state/transition and well-founded interpolation can be handled directly by INTERHORN. By detailing these interpolation problems and their Horn clause representations, we hope to encourage the emergence of a common back-end interpolation interface useful for diverse verification tools.}, author = {Gupta, Ashutosh and Popeea, Corneliu and Rybalchenko, Andrey}, booktitle = {Electronic Proceedings in Theoretical Computer Science, EPTCS}, location = {Vienna, Austria}, pages = {31 -- 38}, publisher = {Open Publishing}, title = {{Generalised interpolation by solving recursion free-horn clauses}}, doi = {10.4204/EPTCS.169.5}, volume = {169}, year = {2014}, } @inproceedings{1708, abstract = {It has been long argued that, because of inherent ambiguity and noise, the brain needs to represent uncertainty in the form of probability distributions. The neural encoding of such distributions remains however highly controversial. Here we present a novel circuit model for representing multidimensional real-valued distributions using a spike based spatio-temporal code. Our model combines the computational advantages of the currently competing models for probabilistic codes and exhibits realistic neural responses along a variety of classic measures. Furthermore, the model highlights the challenges associated with interpreting neural activity in relation to behavioral uncertainty and points to alternative population-level approaches for the experimental validation of distributed representations.}, author = {Savin, Cristina and Denève, Sophie}, location = {Montreal, Canada}, number = {January}, pages = {2024 -- 2032}, publisher = {Neural Information Processing Systems}, title = {{Spatio-temporal representations of uncertainty in spiking neural networks}}, volume = {3}, year = {2014}, } @article{1733, abstract = {The classical (boolean) notion of refinement for behavioral interfaces of system components is the alternating refinement preorder. In this paper, we define a distance for interfaces, called interface simulation distance. It makes the alternating refinement preorder quantitative by, intuitively, tolerating errors (while counting them) in the alternating simulation game. We show that the interface simulation distance satisfies the triangle inequality, that the distance between two interfaces does not increase under parallel composition with a third interface, that the distance between two interfaces can be bounded from above and below by distances between abstractions of the two interfaces, and how to synthesize an interface from incompatible requirements. We illustrate the framework, and the properties of the distances under composition of interfaces, with two case studies.}, author = {Cerny, Pavol and Chmelik, Martin and Henzinger, Thomas A and Radhakrishna, Arjun}, journal = {Theoretical Computer Science}, number = {3}, pages = {348 -- 363}, publisher = {Elsevier}, title = {{Interface simulation distances}}, doi = {10.1016/j.tcs.2014.08.019}, volume = {560}, year = {2014}, } @inbook{1806, abstract = {The generation of asymmetry, at both cellular and tissue level, is one of the most essential capabilities of all eukaryotic organisms. It mediates basically all multicellular development ranging from embryogenesis and de novo organ formation till responses to various environmental stimuli. In plants, the awe-inspiring number of such processes is regulated by phytohormone auxin and its directional, cell-to-cell transport. The mediators of this transport, PIN auxin transporters, are asymmetrically localized at the plasma membrane, and this polar localization determines the directionality of intercellular auxin flow. Thus, auxin transport contributes crucially to the generation of local auxin gradients or maxima, which instruct given cell to change its developmental program. Here, we introduce and discuss the molecular components and cellular mechanisms regulating the generation and maintenance of cellular PIN polarity, as the general hallmarks of cell polarity in plants.}, author = {Baster, Pawel and Friml, Jiří}, booktitle = {Auxin and Its Role in Plant Development}, editor = {Zažímalová, Eva and Petrášek, Jan and Benková, Eva}, pages = {143 -- 170}, publisher = {Springer}, title = {{Auxin on the road navigated by cellular PIN polarity}}, doi = {10.1007/978-3-7091-1526-8_8}, year = {2014}, } @article{1816, abstract = {Watermarking techniques for vector graphics dislocate vertices in order to embed imperceptible, yet detectable, statistical features into the input data. The embedding process may result in a change of the topology of the input data, e.g., by introducing self-intersections, which is undesirable or even disastrous for many applications. In this paper we present a watermarking framework for two-dimensional vector graphics that employs conventional watermarking techniques but still provides the guarantee that the topology of the input data is preserved. The geometric part of this framework computes so-called maximum perturbation regions (MPR) of vertices. We propose two efficient algorithms to compute MPRs based on Voronoi diagrams and constrained triangulations. Furthermore, we present two algorithms to conditionally correct the watermarked data in order to increase the watermark embedding capacity and still guarantee topological correctness. While we focus on the watermarking of input formed by straight-line segments, one of our approaches can also be extended to circular arcs. We conclude the paper by demonstrating and analyzing the applicability of our framework in conjunction with two well-known watermarking techniques.}, author = {Huber, Stefan and Held, Martin and Meerwald, Peter and Kwitt, Roland}, journal = {International Journal of Computational Geometry and Applications}, number = {1}, pages = {61 -- 86}, publisher = {World Scientific Publishing}, title = {{Topology-preserving watermarking of vector graphics}}, doi = {10.1142/S0218195914500034}, volume = {24}, year = {2014}, } @article{1821, abstract = {We review recent progress towards a rigorous understanding of the Bogoliubov approximation for bosonic quantum many-body systems. We focus, in particular, on the excitation spectrum of a Bose gas in the mean-field (Hartree) limit. A list of open problems will be discussed at the end.}, author = {Seiringer, Robert}, journal = {Journal of Mathematical Physics}, number = {7}, publisher = {American Institute of Physics}, title = {{Bose gases, Bose-Einstein condensation, and the Bogoliubov approximation}}, doi = {10.1063/1.4881536}, volume = {55}, year = {2014}, } @article{1822, author = {Jakšić, Vojkan and Pillet, Claude and Seiringer, Robert}, journal = {Journal of Mathematical Physics}, number = {7}, publisher = {American Institute of Physics}, title = {{Introduction}}, doi = {10.1063/1.4884877}, volume = {55}, year = {2014}, } @inbook{1829, abstract = {Hitting and batting tasks, such as tennis forehands, ping-pong strokes, or baseball batting, depend on predictions where the ball can be intercepted and how it can properly be returned to the opponent. These predictions get more accurate over time, hence the behaviors need to be continuously modified. As a result, movement templates with a learned global shape need to be adapted during the execution so that the racket reaches a target position and velocity that will return the ball over to the other side of the net or court. It requires altering learned movements to hit a varying target with the necessary velocity at a specific instant in time. Such a task cannot be incorporated straightforwardly in most movement representations suitable for learning. For example, the standard formulation of the dynamical system based motor primitives (introduced by Ijspeert et al (2002b)) does not satisfy this property despite their flexibility which has allowed learning tasks ranging from locomotion to kendama. In order to fulfill this requirement, we reformulate the Ijspeert framework to incorporate the possibility of specifying a desired hitting point and a desired hitting velocity while maintaining all advantages of the original formulation.We show that the proposed movement template formulation works well in two scenarios, i.e., for hitting a ball on a string with a table tennis racket at a specified velocity and for returning balls launched by a ball gun successfully over the net using forehand movements.}, author = {Muelling, Katharina and Kroemer, Oliver and Lampert, Christoph and Schölkopf, Bernhard}, booktitle = {Learning Motor Skills}, editor = {Kober, Jens and Peters, Jan}, pages = {69 -- 82}, publisher = {Springer}, title = {{Movement templates for learning of hitting and batting}}, doi = {10.1007/978-3-319-03194-1_3}, volume = {97}, year = {2014}, } @article{1842, abstract = {We prove polynomial upper bounds of geometric Ramsey numbers of pathwidth-2 outerplanar triangulations in both convex and general cases. We also prove that the geometric Ramsey numbers of the ladder graph on 2n vertices are bounded by O(n3) and O(n10), in the convex and general case, respectively. We then apply similar methods to prove an (Formula presented.) upper bound on the Ramsey number of a path with n ordered vertices.}, author = {Cibulka, Josef and Gao, Pu and Krcál, Marek and Valla, Tomáš and Valtr, Pavel}, journal = {Discrete & Computational Geometry}, number = {1}, pages = {64 -- 79}, publisher = {Springer}, title = {{On the geometric ramsey number of outerplanar graphs}}, doi = {10.1007/s00454-014-9646-x}, volume = {53}, year = {2014}, } @article{1844, abstract = {Local protein interactions ("molecular context" effects) dictate amino acid replacements and can be described in terms of site-specific, energetic preferences for any different amino acid. It has been recently debated whether these preferences remain approximately constant during evolution or whether, due to coevolution of sites, they change strongly. Such research highlights an unresolved and fundamental issue with far-reaching implications for phylogenetic analysis and molecular evolution modeling. Here, we take advantage of the recent availability of phenotypically supported laboratory resurrections of Precambrian thioredoxins and β-lactamases to experimentally address the change of site-specific amino acid preferences over long geological timescales. Extensive mutational analyses support the notion that evolutionary adjustment to a new amino acid may occur, but to a large extent this is insufficient to erase the primitive preference for amino acid replacements. Generally, site-specific amino acid preferences appear to remain conserved throughout evolutionary history despite local sequence divergence. We show such preference conservation to be readily understandable in molecular terms and we provide crystallographic evidence for an intriguing structural-switch mechanism: Energetic preference for an ancestral amino acid in a modern protein can be linked to reorganization upon mutation to the ancestral local structure around the mutated site. Finally, we point out that site-specific preference conservation naturally leads to one plausible evolutionary explanation for the existence of intragenic global suppressor mutations.}, author = {Risso, Valeria and Manssour Triedo, Fadia and Delgado Delgado, Asuncion and Arco, Rocio and Barroso Deljesús, Alicia and Inglés Prieto, Álvaro and Godoy Ruiz, Raquel and Gavira, Josè and Gaucher, Eric and Ibarra Molero, Beatriz and Sánchez Ruiz, Jose}, journal = {Molecular Biology and Evolution}, number = {2}, pages = {440 -- 455}, publisher = {Oxford University Press}, title = {{Mutational studies on resurrected ancestral proteins reveal conservation of site-specific amino acid preferences throughout evolutionary history}}, doi = {10.1093/molbev/msu312}, volume = {32}, year = {2014}, } @article{1852, abstract = {To control morphogenesis, molecular regulatory networks have to interfere with the mechanical properties of the individual cells of developing organs and tissues, but how this is achieved is not well known. We study this issue here in the shoot meristem of higher plants, a group of undifferentiated cells where complex changes in growth rates and directions lead to the continuous formation of new organs [1, 2]. Here, we show that the plant hormone auxin plays an important role in this process via a dual, local effect on the extracellular matrix, the cell wall, which determines cell shape. Our study reveals that auxin not only causes a limited reduction in wall stiffness but also directly interferes with wall anisotropy via the regulation of cortical microtubule dynamics. We further show that to induce growth isotropy and organ outgrowth, auxin somehow interferes with the cortical microtubule-ordering activity of a network of proteins, including AUXIN BINDING PROTEIN 1 and KATANIN 1. Numerical simulations further indicate that the induced isotropy is sufficient to amplify the effects of the relatively minor changes in wall stiffness to promote organogenesis and the establishment of new growth axes in a robust manner.}, author = {Sassi, Massimiliano and Ali, Olivier and Boudon, Frédéric and Cloarec, Gladys and Abad, Ursula and Cellier, Coralie and Chen, Xu and Gilles, Benjamin and Milani, Pascale and Friml, Jirí and Vernoux, Teva and Godin, Christophe and Hamant, Olivier and Traas, Jan}, journal = {Current Biology}, number = {19}, pages = {2335 -- 2342}, publisher = {Cell Press}, title = {{An auxin-mediated shift toward growth isotropy promotes organ formation at the shoot meristem in Arabidopsis}}, doi = {10.1016/j.cub.2014.08.036}, volume = {24}, year = {2014}, } @inproceedings{1853, abstract = {Wireless sensor networks (WSNs) composed of low-power, low-cost sensor nodes are expected to form the backbone of future intelligent networks for a broad range of civil, industrial and military applications. These sensor nodes are often deployed through random spreading, and function in dynamic environments. Many applications of WSNs such as pollution tracking, forest fire detection, and military surveillance require knowledge of the location of constituent nodes. But the use of technologies such as GPS on all nodes is prohibitive due to power and cost constraints. So, the sensor nodes need to autonomously determine their locations. Most localization techniques use anchor nodes with known locations to determine the position of remaining nodes. Localization techniques have two conflicting requirements. On one hand, an ideal localization technique should be computationally simple and on the other hand, it must be resistant to attacks that compromise anchor nodes. In this paper, we propose a computationally light-weight game theoretic secure localization technique and demonstrate its effectiveness in comparison to existing techniques.}, author = {Jha, Susmit and Tripakis, Stavros and Seshia, Sanjit and Chatterjee, Krishnendu}, location = {Cambridge, USA}, pages = {85 -- 90}, publisher = {IEEE}, title = {{Game theoretic secure localization in wireless sensor networks}}, doi = {10.1109/IOT.2014.7030120}, year = {2014}, } @article{1854, abstract = {In this paper, we present a method for non-rigid, partial shape matching in vector graphics. Given a user-specified query region in a 2D shape, similar regions are found, even if they are non-linearly distorted. Furthermore, a non-linear mapping is established between the query regions and these matches, which allows the automatic transfer of editing operations such as texturing. This is achieved by a two-step approach. First, pointwise correspondences between the query region and the whole shape are established. The transformation parameters of these correspondences are registered in an appropriate transformation space. For transformations between similar regions, these parameters form surfaces in transformation space, which are extracted in the second step of our method. The extracted regions may be related to the query region by a non-rigid transform, enabling non-rigid shape matching. In this paper, we present a method for non-rigid, partial shape matching in vector graphics. Given a user-specified query region in a 2D shape, similar regions are found, even if they are non-linearly distorted. Furthermore, a non-linear mapping is established between the query regions and these matches, which allows the automatic transfer of editing operations such as texturing. This is achieved by a two-step approach. First, pointwise correspondences between the query region and the whole shape are established. The transformation parameters of these correspondences are registered in an appropriate transformation space. For transformations between similar regions, these parameters form surfaces in transformation space, which are extracted in the second step of our method. The extracted regions may be related to the query region by a non-rigid transform, enabling non-rigid shape matching.}, author = {Guerrero, Paul and Auzinger, Thomas and Wimmer, Michael and Jeschke, Stefan}, journal = {Computer Graphics Forum}, number = {1}, pages = {239 -- 252}, publisher = {Wiley}, title = {{Partial shape matching using transformation parameter similarity}}, doi = {10.1111/cgf.12509}, volume = {34}, year = {2014}, } @article{1862, abstract = {The prominent and evolutionarily ancient role of the plant hormone auxin is the regulation of cell expansion. Cell expansion requires ordered arrangement of the cytoskeleton but molecular mechanisms underlying its regulation by signalling molecules including auxin are unknown. Here we show in the model plant Arabidopsis thaliana that in elongating cells exogenous application of auxin or redistribution of endogenous auxin induces very rapid microtubule re-orientation from transverse to longitudinal, coherent with the inhibition of cell expansion. This fast auxin effect requires auxin binding protein 1 (ABP1) and involves a contribution of downstream signalling components such as ROP6 GTPase, ROP-interactive protein RIC1 and the microtubule-severing protein katanin. These components are required for rapid auxin-and ABP1-mediated re-orientation of microtubules to regulate cell elongation in roots and dark-grown hypocotyls as well as asymmetric growth during gravitropic responses.}, author = {Chen, Xu and Grandont, Laurie and Li, Hongjiang and Hauschild, Robert and Paque, Sébastien and Abuzeineh, Anas and Rakusová, Hana and Benková, Eva and Perrot Rechenmann, Catherine and Friml, Jirí}, journal = {Nature}, number = {729}, pages = {90 -- 93}, publisher = {Nature Publishing Group}, title = {{Inhibition of cell expansion by rapid ABP1-mediated auxin effect on microtubules}}, doi = {10.1038/nature13889}, volume = {516}, year = {2014}, } @inproceedings{1869, abstract = {Boolean controllers for systems with complex datapaths are often very difficult to implement correctly, in particular when concurrency is involved. Yet, in many instances it is easy to formally specify correctness. For example, the specification for the controller of a pipelined processor only has to state that the pipelined processor gives the same results as a non-pipelined reference design. This makes such controllers a good target for automated synthesis. However, an efficient abstraction for the complex datapath elements is needed, as a bit-precise description is often infeasible. We present Suraq, the first controller synthesis tool which uses uninterpreted functions for the abstraction. Quantified firstorder formulas (with specific quantifier structure) serve as the specification language from which Suraq synthesizes Boolean controllers. Suraq transforms the specification into an unsatisfiable SMT formula, and uses Craig interpolation to compute its results. Using Suraq, we were able to synthesize a controller (consisting of two Boolean signals) for a five-stage pipelined DLX processor in roughly one hour and 15 minutes.}, author = {Hofferek, Georg and Gupta, Ashutosh}, booktitle = {HVC 2014}, editor = {Yahav, Eran}, location = {Haifa, Israel}, pages = {68 -- 74}, publisher = {Springer}, title = {{Suraq - a controller synthesis tool using uninterpreted functions}}, doi = {10.1007/978-3-319-13338-6_6}, volume = {8855}, year = {2014}, } @inproceedings{1870, abstract = {We investigate the problem of checking if a finite-state transducer is robust to uncertainty in its input. Our notion of robustness is based on the analytic notion of Lipschitz continuity - a transducer is K-(Lipschitz) robust if the perturbation in its output is at most K times the perturbation in its input. We quantify input and output perturbation using similarity functions. We show that K-robustness is undecidable even for deterministic transducers. We identify a class of functional transducers, which admits a polynomial time automata-theoretic decision procedure for K-robustness. This class includes Mealy machines and functional letter-to-letter transducers. We also study K-robustness of nondeterministic transducers. Since a nondeterministic transducer generates a set of output words for each input word, we quantify output perturbation using setsimilarity functions. We show that K-robustness of nondeterministic transducers is undecidable, even for letter-to-letter transducers. We identify a class of set-similarity functions which admit decidable K-robustness of letter-to-letter transducers.}, author = {Henzinger, Thomas A and Otop, Jan and Samanta, Roopsha}, booktitle = {Leibniz International Proceedings in Informatics, LIPIcs}, location = {Delhi, India}, pages = {431 -- 443}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Lipschitz robustness of finite-state transducers}}, doi = {10.4230/LIPIcs.FSTTCS.2014.431}, volume = {29}, year = {2014}, } @inproceedings{1872, abstract = {Extensionality axioms are common when reasoning about data collections, such as arrays and functions in program analysis, or sets in mathematics. An extensionality axiom asserts that two collections are equal if they consist of the same elements at the same indices. Using extensionality is often required to show that two collections are equal. A typical example is the set theory theorem (∀x)(∀y)x∪y = y ∪x. Interestingly, while humans have no problem with proving such set identities using extensionality, they are very hard for superposition theorem provers because of the calculi they use. In this paper we show how addition of a new inference rule, called extensionality resolution, allows first-order theorem provers to easily solve problems no modern first-order theorem prover can solve. We illustrate this by running the VAMPIRE theorem prover with extensionality resolution on a number of set theory and array problems. Extensionality resolution helps VAMPIRE to solve problems from the TPTP library of first-order problems that were never solved before by any prover.}, author = {Gupta, Ashutosh and Kovács, Laura and Kragl, Bernhard and Voronkov, Andrei}, booktitle = {ATVA 2014}, editor = {Cassez, Franck and Raskin, Jean-François}, location = {Sydney, Australia}, pages = {185 -- 200}, publisher = {Springer}, title = {{Extensional crisis and proving identity}}, doi = {10.1007/978-3-319-11936-6_14}, volume = {8837}, year = {2014}, } @inproceedings{1875, abstract = {We present a formal framework for repairing infinite-state, imperative, sequential programs, with (possibly recursive) procedures and multiple assertions; the framework can generate repaired programs by modifying the original erroneous program in multiple program locations, and can ensure the readability of the repaired program using user-defined expression templates; the framework also generates a set of inductive assertions that serve as a proof of correctness of the repaired program. As a step toward integrating programmer intent and intuition in automated program repair, we present a cost-aware formulation - given a cost function associated with permissible statement modifications, the goal is to ensure that the total program modification cost does not exceed a given repair budget. As part of our predicate abstractionbased solution framework, we present a sound and complete algorithm for repair of Boolean programs. We have developed a prototype tool based on SMT solving and used it successfully to repair diverse errors in benchmark C programs.}, author = {Samanta, Roopsha and Olivo, Oswaldo and Allen, Emerson}, editor = {Müller-Olm, Markus and Seidl, Helmut}, location = {Munich, Germany}, pages = {268 -- 284}, publisher = {Springer}, title = {{Cost-aware automatic program repair}}, doi = {10.1007/978-3-319-10936-7_17}, volume = {8723}, year = {2014}, } @article{1876, abstract = {We study densities of functionals over uniformly bounded triangulations of a Delaunay set of vertices, and prove that the minimum is attained for the Delaunay triangulation if this is the case for finite sets.}, author = {Dolbilin, Nikolai and Edelsbrunner, Herbert and Glazyrin, Alexey and Musin, Oleg}, journal = {Moscow Mathematical Journal}, number = {3}, pages = {491 -- 504}, publisher = {Independent University of Moscow}, title = {{Functionals on triangulations of delaunay sets}}, volume = {14}, year = {2014}, } @article{1877, abstract = {During inflammation, lymph nodes swell with an influx of immune cells. New findings identify a signalling pathway that induces relaxation in the contractile cells that give structure to these organs.}, author = {Sixt, Michael K and Vaahtomeri, Kari}, journal = {Nature}, number = {7523}, pages = {441 -- 442}, publisher = {Springer Nature}, title = {{Physiology: Relax and come in}}, doi = {10.1038/514441a}, volume = {514}, year = {2014}, } @article{1884, abstract = {Unbiased high-throughput massively parallel sequencing methods have transformed the process of discovery of novel putative driver gene mutations in cancer. In chronic lymphocytic leukemia (CLL), these methods have yielded several unexpected findings, including the driver genes SF3B1, NOTCH1 and POT1. Recent analysis, utilizing down-sampling of existing datasets, has shown that the discovery process of putative drivers is far from complete across cancer. In CLL, while driver gene mutations affecting >10% of patients were efficiently discovered with previously published CLL cohorts of up to 160 samples subjected to whole exome sequencing (WES), this sample size has only 0.78 power to detect drivers affecting 5% of patients, and only 0.12 power for drivers affecting 2% of patients. These calculations emphasize the need to apply unbiased WES to larger patient cohorts.}, author = {Landau, Dan and Stewart, Chip and Reiter, Johannes and Lawrence, Michael and Sougnez, Carrie and Brown, Jennifer and Lopez Guillermo, Armando and Gabriel, Stacey and Lander, Eric and Neuberg, Donna and López Otín, Carlos and Campo, Elias and Getz, Gad and Wu, Catherine}, journal = {Blood}, number = {21}, pages = {1952 -- 1952}, publisher = {American Society of Hematology}, title = {{Novel putative driver gene mutations in chronic lymphocytic leukemia (CLL): results from a combined analysis of whole exome sequencing of 262 primary CLL aamples}}, volume = {124}, year = {2014}, } @article{1886, abstract = {Information processing in the sensory periphery is shaped by natural stimulus statistics. In the periphery, a transmission bottleneck constrains performance; thus efficient coding implies that natural signal components with a predictably wider range should be compressed. In a different regime—when sampling limitations constrain performance—efficient coding implies that more resources should be allocated to informative features that are more variable. We propose that this regime is relevant for sensory cortex when it extracts complex features from limited numbers of sensory samples. To test this prediction, we use central visual processing as a model: we show that visual sensitivity for local multi-point spatial correlations, described by dozens of independently-measured parameters, can be quantitatively predicted from the structure of natural images. This suggests that efficient coding applies centrally, where it extends to higher-order sensory features and operates in a regime in which sensitivity increases with feature variability.}, author = {Hermundstad, Ann and Briguglio, John and Conte, Mary and Victor, Jonathan and Balasubramanian, Vijay and Tkacik, Gasper}, journal = {eLife}, number = {November}, publisher = {eLife Sciences Publications}, title = {{Variance predicts salience in central sensory processing}}, doi = {10.7554/eLife.03722}, year = {2014}, } @article{1887, author = {Cremer, Sylvia}, journal = {Zoologie}, pages = {23 -- 30}, publisher = {Deutsche Zoologische Gesellschaft}, title = {{Gemeinsame Krankheitsabwehr in Ameisengesellschaften}}, year = {2014}, } @inbook{1888, abstract = {Im Rahmen meiner Arbeit mit der kollektiven Krankheitsabwehr in Ameisengesellschaften interessiert mich vor allem, wie sich die Kolonien als Ganzes gegen Krankheiten wehren können. Warum ist dieses Thema der Krankheitsdynamik in Gruppen so wichtig? Ein Vergleich von solitär lebenden Individuen mit Individuen, die in sozialen Gruppen zusammenleben, zeigt die Kosten und die Vorteile des Gruppenlebens: Einerseits haben Individuen in sozialen Gruppen aufgrund der hohen Dichte, in der die Tiere zusammenleben, den hohen Interaktionsraten, die sie miteinander haben, und der engen Verwandtschaft, die sie verbindet, ein höheres Ansteckungsrisiko. Andererseits kann die individuelle Krankheitsabwehr durch die kollektive Abwehr in den Gruppen ergänzt werden.}, author = {Cremer, Sylvia}, booktitle = {Soziale Insekten in einer sich wandelnden Welt}, pages = {65 -- 72}, publisher = {Pfeil}, title = {{Soziale Immunität: Wie sich der Staat gegen Pathogene wehrt Bayerische Akademie der Wissenschaften}}, volume = {43}, year = {2014}, } @article{1889, abstract = {We study translation-invariant quasi-free states for a system of fermions with two-particle interactions. The associated energy functional is similar to the BCS functional but also includes direct and exchange energies. We show that for suitable short-range interactions, these latter terms only lead to a renormalization of the chemical potential, with the usual properties of the BCS functional left unchanged. Our analysis thus represents a rigorous justification of part of the BCS approximation. We give bounds on the critical temperature below which the system displays superfluidity.}, author = {Bräunlich, Gerhard and Hainzl, Christian and Seiringer, Robert}, journal = {Reviews in Mathematical Physics}, number = {7}, publisher = {World Scientific Publishing}, title = {{Translation-invariant quasi-free states for fermionic systems and the BCS approximation}}, doi = {10.1142/S0129055X14500123}, volume = {26}, year = {2014}, } @article{1890, abstract = {To search for a target in a complex environment is an everyday behavior that ends with finding the target. When we search for two identical targets, however, we must continue the search after finding the first target and memorize its location. We used fixation-related potentials to investigate the neural correlates of different stages of the search, that is, before and after finding the first target. Having found the first target influenced subsequent distractor processing. Compared to distractor fixations before the first target fixation, a negative shift was observed for three subsequent distractor fixations. These results suggest that processing a target in continued search modulates the brain's response, either transiently by reflecting temporary working memory processes or permanently by reflecting working memory retention.}, author = {Körner, Christof and Braunstein, Verena and Stangl, Matthias and Schlögl, Alois and Neuper, Christa and Ischebeck, Anja}, journal = {Psychophysiology}, number = {4}, pages = {385 -- 395}, publisher = {Wiley-Blackwell}, title = {{Sequential effects in continued visual search: Using fixation-related potentials to compare distractor processing before and after target detection}}, doi = {10.1111/psyp.12062}, volume = {51}, year = {2014}, } @article{1891, abstract = {We provide theoretical tests of a novel experimental technique to determine mechanostability of proteins based on stretching a mechanically protected protein by single-molecule force spectroscopy. This technique involves stretching a homogeneous or heterogeneous chain of reference proteins (single-molecule markers) in which one of them acts as host to the guest protein under study. The guest protein is grafted into the host through genetic engineering. It is expected that unraveling of the host precedes the unraveling of the guest removing ambiguities in the reading of the force-extension patterns of the guest protein. We study examples of such systems within a coarse-grained structure-based model. We consider systems with various ratios of mechanostability for the host and guest molecules and compare them to experimental results involving cohesin I as the guest molecule. For a comparison, we also study the force-displacement patterns in proteins that are linked in a serial fashion. We find that the mechanostability of the guest is similar to that of the isolated or serially linked protein. We also demonstrate that the ideal configuration of this strategy would be one in which the host is much more mechanostable than the single-molecule markers. We finally show that it is troublesome to use the highly stable cystine knot proteins as a host to graft a guest in stretching studies because this would involve a cleaving procedure.}, author = {Chwastyk, Mateusz and Galera Prat, Albert and Sikora, Mateusz K and Gómez Sicilia, Àngel and Carrión Vázquez, Mariano and Cieplak, Marek}, journal = {Proteins: Structure, Function and Bioinformatics}, number = {5}, pages = {717 -- 726}, publisher = {Wiley-Blackwell}, title = {{Theoretical tests of the mechanical protection strategy in protein nanomechanics}}, doi = {10.1002/prot.24436}, volume = {82}, year = {2014}, } @article{1892, abstract = {Behavioural variation among conspecifics is typically contingent on individual state or environmental conditions. Sex-specific genetic polymorphisms are enigmatic because they lack conditionality, and genes causing adaptive trait variation in one sex may reduce Darwinian fitness in the other. One way to avoid such genetic antagonism is to control sex-specific traits by inheritance via sex chromosomes. Here, controlled laboratory crossings suggest that in snail-brooding cichlid fish a single locus, two-allele polymorphism located on a sex-linked chromosome of heterogametic males generates an extreme reproductive dimorphism. Both natural and sexual selection are responsible for exceptionally large body size of bourgeois males, creating a niche for a miniature male phenotype to evolve. This extreme intrasexual dimorphism results from selection on opposite size thresholds caused by a single ecological factor, empty snail shells used as breeding substrate. Paternity analyses reveal that in the field parasitic dwarf males sire the majority of offspring in direct sperm competition with large nest owners exceeding their size more than 40 times. Apparently, use of empty snail shells as breeding substrate and single locus sex-linked inheritance of growth are the major ecological and genetic mechanisms responsible for the extreme intrasexual diversity observed in Lamprologus callipterus.}, author = {Ocana, Sabine and Meidl, Patrick and Bonfils, Danielle and Taborsky, Michael}, journal = {Proceedings of the Royal Society of London Series B Biological Sciences}, number = {1794}, publisher = {Royal Society, The}, title = {{Y-linked Mendelian inheritance of giant and dwarf male morphs in shell-brooding cichlids}}, doi = {10.1098/rspb.2014.0253}, volume = {281}, year = {2014}, } @article{1893, abstract = {Phosphatidylinositol (PtdIns) is a structural phospholipid that can be phosphorylated into various lipid signaling molecules, designated polyphosphoinositides (PPIs). The reversible phosphorylation of PPIs on the 3, 4, or 5 position of inositol is performed by a set of organelle-specific kinases and phosphatases, and the characteristic head groups make these molecules ideal for regulating biological processes in time and space. In yeast and mammals, PtdIns3P and PtdIns(3,5)P2 play crucial roles in trafficking toward the lytic compartments, whereas the role in plants is not yet fully understood. Here we identified the role of a land plant-specific subgroup of PPI phosphatases, the suppressor of actin 2 (SAC2) to SAC5, during vacuolar trafficking and morphogenesis in Arabidopsis thaliana. SAC2-SAC5 localize to the tonoplast along with PtdIns3P, the presumable product of their activity. In SAC gain- and loss-of-function mutants, the levels of PtdIns monophosphates and bisphosphates were changed, with opposite effects on the morphology of storage and lytic vacuoles, and the trafficking toward the vacuoles was defective. Moreover, multiple sac knockout mutants had an increased number of smaller storage and lytic vacuoles, whereas extralarge vacuoles were observed in the overexpression lines, correlating with various growth and developmental defects. The fragmented vacuolar phenotype of sac mutants could be mimicked by treating wild-type seedlings with PtdIns(3,5)P2, corroborating that this PPI is important for vacuole morphology. Taken together, these results provide evidence that PPIs, together with their metabolic enzymes SAC2-SAC5, are crucial for vacuolar trafficking and for vacuolar morphology and function in plants.}, author = {Nováková, Petra and Hirsch, Sibylle and Feraru, Elena and Tejos, Ricardo and Van Wijk, Ringo and Viaene, Tom and Heilmann, Mareike and Lerche, Jennifer and De Rycke, Riet and Feraru, Mugurel and Grones, Peter and Van Montagu, Marc and Heilmann, Ingo and Munnik, Teun and Friml, Jirí}, journal = {PNAS}, number = {7}, pages = {2818 -- 2823}, publisher = {National Academy of Sciences}, title = {{SAC phosphoinositide phosphatases at the tonoplast mediate vacuolar function in Arabidopsis}}, doi = {10.1073/pnas.1324264111}, volume = {111}, year = {2014}, } @article{1894, abstract = {Background: Bacterial Dsb enzymes are involved in the oxidative folding of many proteins, through the formation of disulfide bonds between their cysteine residues. The Dsb protein network has been well characterized in cells of the model microorganism Escherichia coli. To gain insight into the functioning of the Dsb system in epsilon-Proteobacteria, where it plays an important role in the colonization process, we studied two homologs of the main Escherichia coli Dsb oxidase (EcDsbA) that are present in the cells of the enteric pathogen Campylobacter jejuni, the most frequently reported bacterial cause of human enteritis in the world. Methods and Results: Phylogenetic analysis suggests the horizontal transfer of the epsilon-Proteobacterial DsbAs from a common ancestor to gamma-Proteobacteria, which then gave rise to the DsbL lineage. Phenotype and enzymatic assays suggest that the two C. jejuni DsbAs play different roles in bacterial cells and have divergent substrate spectra. CjDsbA1 is essential for the motility and autoagglutination phenotypes, while CjDsbA2 has no impact on those processes. CjDsbA1 plays a critical role in the oxidative folding that ensures the activity of alkaline phosphatase CjPhoX, whereas CjDsbA2 is crucial for the activity of arylsulfotransferase CjAstA, encoded within the dsbA2-dsbB-astA operon. Conclusions: Our results show that CjDsbA1 is the primary thiol-oxidoreductase affecting life processes associated with bacterial spread and host colonization, as well as ensuring the oxidative folding of particular protein substrates. In contrast, CjDsbA2 activity does not affect the same processes and so far its oxidative folding activity has been demonstrated for one substrate, arylsulfotransferase CjAstA. The results suggest the cooperation between CjDsbA2 and CjDsbB. In the case of the CjDsbA1, this cooperation is not exclusive and there is probably another protein to be identified in C. jejuni cells that acts to re-oxidize CjDsbA1. Altogether the data presented here constitute the considerable insight to the Epsilonproteobacterial Dsb systems, which have been poorly understood so far.}, author = {Grabowska, Anna and Wywiał, Ewa and Dunin Horkawicz, Stanislaw and Łasica, Anna and Wösten, Marc and Nagy-Staron, Anna A and Godlewska, Renata and Bocian Ostrzycka, Katarzyna and Pieńkowska, Katarzyna and Łaniewski, Paweł and Bujnicki, Janusz and Van Putten, Jos and Jagusztyn Krynicka, Elzbieta}, journal = {PLoS One}, number = {9}, publisher = {Public Library of Science}, title = {{Functional and bioinformatics analysis of two Campylobacter jejuni homologs of the thiol-disulfide oxidoreductase, DsbA}}, doi = {10.1371/journal.pone.0106247}, volume = {9}, year = {2014}, } @article{1895, abstract = {Major histocompatibility complex class I (MHCI) molecules were recently identified as novel regulators of synaptic plasticity. These molecules are expressed in various brain areas, especially in regions undergoing activity-dependent synaptic plasticity, but their role in the nucleus accumbens (NAc) is unknown. In this study, we investigated the effects of genetic disruption of MHCI function, through deletion of β2-microblobulin, which causes lack of cell surface expression of MHCI. First, we confirmed that MHCI molecules are expressed in the NAc core in wild-type mice. Second, we performed electrophysiological recordings with NAc core slices from wild-type and β2-microglobulin knock-out mice lacking cell surface expression of MHCI. We found that low frequency stimulation induced long-term depression in wild-type but not knock-out mice, whereas high frequency stimulation induced long-term potentiation in both genotypes, with a larger magnitude in knock-out mice. Furthermore, we demonstrated that knock-out mice showed more persistent behavioral sensitization to cocaine, which is a NAc-related behavior. Using this model, we analyzed the density of total AMPA receptors and their subunits GluR1 and GluR2 in the NAc core, by SDS-digested freeze-fracture replica labeling. After repeated cocaine exposure, the density of GluR1 was increased, but there was no change in total AMPA receptors and GluR2 levels in wildtype mice. In contrast, following repeated cocaine exposure, increased densities of total AMPA receptors, GluR1 and GluR2 were observed in knock-out mice. These results indicate that functional deficiency of MHCI enhances synaptic potentiation, induced by electrical and pharmacological stimulation.}, author = {Edamura, Mitsuhiro and Murakami, Gen and Meng, Hongrui and Itakura, Makoto and Shigemoto, Ryuichi and Fukuda, Atsuo and Nakahara, Daiichiro}, journal = {PLoS One}, number = {9}, publisher = {Public Library of Science}, title = {{Functional deficiency of MHC class i enhances LTP and abolishes LTD in the nucleus accumbens of mice}}, doi = {10.1371/journal.pone.0107099}, volume = {9}, year = {2014}, } @article{1896, abstract = {Biopolymer length regulation is a complex process that involves a large number of biological, chemical, and physical subprocesses acting simultaneously across multiple spatial and temporal scales. An illustrative example important for genomic stability is the length regulation of telomeres - nucleoprotein structures at the ends of linear chromosomes consisting of tandemly repeated DNA sequences and a specialized set of proteins. Maintenance of telomeres is often facilitated by the enzyme telomerase but, particularly in telomerase-free systems, the maintenance of chromosomal termini depends on alternative lengthening of telomeres (ALT) mechanisms mediated by recombination. Various linear and circular DNA structures were identified to participate in ALT, however, dynamics of the whole process is still poorly understood. We propose a chemical kinetics model of ALT with kinetic rates systematically derived from the biophysics of DNA diffusion and looping. The reaction system is reduced to a coagulation-fragmentation system by quasi-steady-state approximation. The detailed treatment of kinetic rates yields explicit formulas for expected size distributions of telomeres that demonstrate the key role played by the J factor, a quantitative measure of bending of polymers. The results are in agreement with experimental data and point out interesting phenomena: an appearance of very long telomeric circles if the total telomere density exceeds a critical value (excess mass) and a nonlinear response of the telomere size distributions to the amount of telomeric DNA in the system. The results can be of general importance for understanding dynamics of telomeres in telomerase-independent systems as this mode of telomere maintenance is similar to the situation in tumor cells lacking telomerase activity. Furthermore, due to its universality, the model may also serve as a prototype of an interaction between linear and circular DNA structures in various settings.}, author = {Kollár, Richard and Bod'ová, Katarína and Nosek, Jozef and Tomáška, Ľubomír}, journal = {Physical Review E Statistical Nonlinear and Soft Matter Physics}, number = {3}, publisher = {American Institute of Physics}, title = {{Mathematical model of alternative mechanism of telomere length maintenance}}, doi = {10.1103/PhysRevE.89.032701}, volume = {89}, year = {2014}, } @article{1897, abstract = {GNOM is one of the most characterized membrane trafficking regulators in plants, with crucial roles in development. GNOM encodes an ARF-guanine nucleotide exchange factor (ARF-GEF) that activates small GTPases of the ARF (ADP ribosylation factor) class to mediate vesicle budding at endomembranes. The crucial role of GNOM in recycling of PIN auxin transporters and other proteins to the plasma membrane was identified in studies using the ARF-GEF inhibitor brefeldin A (BFA). GNOM, the most prominent regulator of recycling in plants, has been proposed to act and localize at so far elusive recycling endosomes. Here, we report the GNOM localization in context of its cellular function in Arabidopsis thaliana. State-of-the-art imaging, pharmacological interference, and ultrastructure analysis show that GNOM predominantly localizes to Golgi apparatus. Super-resolution confocal live imaging microscopy identified GNOM and its closest homolog GNOM-like 1 at distinct subdomains on Golgi cisternae. Short-term BFA treatment stabilizes GNOM at the Golgi apparatus, whereas prolonged exposures results in GNOM translocation to trans-Golgi network (TGN)/early endosomes (EEs). Malformed TGN/EE in gnom mutants suggests a role for GNOM in maintaining TGN/EE function. Our results redefine the subcellular action of GNOM and reevaluate the identity and function of recycling endosomes in plants.}, author = {Naramoto, Satoshi and Otegui, Marisa and Kutsuna, Natsumaro and De Rycke, Riet and Dainobu, Tomoko and Karampelias, Michael and Fujimoto, Masaru and Feraru, Elena and Miki, Daisuke and Fukuda, Hiroo and Nakano, Akihiko and Friml, Jirí}, journal = {Plant Cell}, number = {7}, pages = {3062 -- 3076}, publisher = {American Society of Plant Biologists}, title = {{Insights into the localization and function of the membrane trafficking regulator GNOM ARF-GEF at the Golgi apparatus in Arabidopsis}}, doi = {10.1105/tpc.114.125880}, volume = {26}, year = {2014}, } @article{1898, abstract = {Fast synaptic transmission is important for rapid information processing. To explore the maximal rate of neuronal signaling and to analyze the presynaptic mechanisms, we focused on the input layer of the cerebellar cortex, where exceptionally high action potential (AP) frequencies have been reported invivo. With paired recordings between presynaptic cerebellar mossy fiber boutons and postsynaptic granule cells, we demonstrate reliable neurotransmission upto ~1 kHz. Presynaptic APs are ultrafast, with ~100μs half-duration. Both Kv1 and Kv3 potassium channels mediate the fast repolarization, rapidly inactivating sodium channels ensure metabolic efficiency, and little AP broadening occurs during bursts of up to 1.5 kHz. Presynaptic Cav2.1 (P/Q-type) calcium channels open efficiently during ultrafast APs. Furthermore, a subset of synaptic vesicles is tightly coupled to Ca2+ channels, and vesicles are rapidly recruited to the release site. These data reveal mechanisms of presynaptic AP generation and transmitter release underlying neuronal kHz signaling.}, author = {Ritzau Jost, Andreas and Delvendahl, Igor and Rings, Annika and Byczkowicz, Niklas and Harada, Harumi and Shigemoto, Ryuichi and Hirrlinger, Johannes and Eilers, Jens and Hallermann, Stefan}, journal = {Neuron}, number = {1}, pages = {152 -- 163}, publisher = {Elsevier}, title = {{Ultrafast action potentials mediate kilohertz signaling at a central synapse}}, doi = {10.1016/j.neuron.2014.08.036}, volume = {84}, year = {2014}, } @article{1899, abstract = {Asymmetric cell divisions allow stem cells to balance proliferation and differentiation. During embryogenesis, murine epidermis expands rapidly from a single layer of unspecified basal layer progenitors to a stratified, differentiated epithelium. Morphogenesis involves perpendicular (asymmetric) divisions and the spindle orientation protein LGN, but little is known about how the apical localization of LGN is regulated. Here, we combine conventional genetics and lentiviral-mediated in vivo RNAi to explore the functions of the LGN-interacting proteins Par3, mInsc and Gα i3. Whereas loss of each gene alone leads to randomized division angles, combined loss of Gnai3 and mInsc causes a phenotype of mostly planar divisions, akin to loss of LGN. These findings lend experimental support for the hitherto untested model that Par3-mInsc and Gα i3 act cooperatively to polarize LGN and promote perpendicular divisions. Finally, we uncover a developmental switch between delamination-driven early stratification and spindle-orientation-dependent differentiation that occurs around E15, revealing a two-step mechanism underlying epidermal maturation.}, author = {Williams, Scott and Ratliff, Lyndsay and Postiglione, Maria P and Knoblich, Juergen and Fuchs, Elaine}, journal = {Nature Cell Biology}, number = {8}, pages = {758 -- 769}, publisher = {Nature Publishing Group}, title = {{Par3-mInsc and Gα i3 cooperate to promote oriented epidermal cell divisions through LGN}}, doi = {10.1038/ncb3001}, volume = {16}, year = {2014}, } @article{1900, abstract = {Epithelial cell layers need to be tightly regulated to maintain their integrity and correct function. Cell integration into epithelial sheets is now shown to depend on the N-WASP-regulated stabilization of cortical F-actin, which generates distinct patterns of apical-lateral contractility at E-cadherin-based cell-cell junctions.}, author = {Behrndt, Martin and Heisenberg, Carl-Philipp J}, journal = {Nature Cell Biology}, number = {2}, pages = {127 -- 129}, publisher = {Nature Publishing Group}, title = {{Lateral junction dynamics lead the way out}}, doi = {10.1038/ncb2913}, volume = {16}, year = {2014}, } @article{1901, abstract = {In plants, the patterning of stem cell-enriched meristems requires a graded auxin response maximum that emerges from the concerted action of polar auxin transport, auxin biosynthesis, auxin metabolism, and cellular auxin response machinery. However, mechanisms underlying this auxin response maximum-mediated root stem cell maintenance are not fully understood. Here, we present unexpected evidence that WUSCHEL-RELATED HOMEOBOX 5 (WOX5) transcription factor modulates expression of auxin biosynthetic genes in the quiescent center (QC) of the root and thus provides a robust mechanism for the maintenance of auxin response maximum in the root tip. This WOX5 action is balanced through the activity of indole-3-acetic acid 17 (IAA17) auxin response repressor. Our combined genetic, cell biology, and computational modeling studies revealed a previously uncharacterized feedback loop linking WOX5-mediated auxin production to IAA17-dependent repression of auxin responses. This WOX5-IAA17 feedback circuit further assures the maintenance of auxin response maximum in the root tip and thereby contributes to the maintenance of distal stem cell (DSC) populations. Our experimental studies and in silico computer simulations both demonstrate that the WOX5-IAA17 feedback circuit is essential for the maintenance of auxin gradient in the root tip and the auxin-mediated root DSC differentiation.}, author = {Tian, Huiyu and Wabnik, Krzysztof T and Niu, Tiantian and Li, Hongjiang and Yu, Qianqian and Pollmann, Stephan and Vanneste, Steffen and Govaerts, Willy and Rolčík, Jakub and Geisler, Markus and Friml, Jirí and Ding, Zhaojun}, journal = {Molecular Plant}, number = {2}, pages = {277 -- 289}, publisher = {Oxford University Press}, title = {{WOX5-IAA17 feedback circuit-mediated cellular auxin response is crucial for the patterning of root stem cell niches in arabidopsis}}, doi = {10.1093/mp/sst118}, volume = {7}, year = {2014}, } @article{1902, abstract = {In the 1960s-1980s, determination of bacterial growth rates was an important tool in microbial genetics, biochemistry, molecular biology, and microbial physiology. The exciting technical developments of the 1990s and the 2000s eclipsed that tool; as a result, many investigators today lack experience with growth rate measurements. Recently, investigators in a number of areas have started to use measurements of bacterial growth rates for a variety of purposes. Those measurements have been greatly facilitated by the availability of microwell plate readers that permit the simultaneous measurements on up to 384 different cultures. Only the exponential (logarithmic) portions of the resulting growth curves are useful for determining growth rates, and manual determination of that portion and calculation of growth rates can be tedious for high-throughput purposes. Here, we introduce the program GrowthRates that uses plate reader output files to automatically determine the exponential portion of the curve and to automatically calculate the growth rate, the maximum culture density, and the duration of the growth lag phase. GrowthRates is freely available for Macintosh, Windows, and Linux.We discuss the effects of culture volume, the classical bacterial growth curve, and the differences between determinations in rich media and minimal (mineral salts) media. This protocol covers calibration of the plate reader, growth of culture inocula for both rich and minimal media, and experimental setup. As a guide to reliability, we report typical day-to-day variation in growth rates and variation within experiments with respect to position of wells within the plates.}, author = {Hall, Barry and Acar, Hande and Nandipati, Anna and Barlow, Miriam}, journal = {Molecular Biology and Evolution}, number = {1}, pages = {232 -- 238}, publisher = {Oxford University Press}, title = {{Growth rates made easy}}, doi = {10.1093/molbev/mst187}, volume = {31}, year = {2014}, } @inproceedings{1903, abstract = {We consider two-player zero-sum partial-observation stochastic games on graphs. Based on the information available to the players these games can be classified as follows: (a) general partial-observation (both players have partial view of the game); (b) one-sided partial-observation (one player has partial-observation and the other player has complete-observation); and (c) perfect-observation (both players have complete view of the game). The one-sided partial-observation games subsumes the important special case of one-player partial-observation stochastic games (or partial-observation Markov decision processes (POMDPs)). Based on the randomization available for the strategies, (a) the players may not be allowed to use randomization (pure strategies), or (b) they may choose a probability distribution over actions but the actual random choice is external and not visible to the player (actions invisible), or (c) they may use full randomization. We consider all these classes of games with reachability, and parity objectives that can express all ω-regular objectives. The analysis problems are classified into the qualitative analysis that asks for the existence of a strategy that ensures the objective with probability 1; and the quantitative analysis that asks for the existence of a strategy that ensures the objective with probability at least λ (0,1). In this talk we will cover a wide range of results: for perfect-observation games; for POMDPs; for one-sided partial-observation games; and for general partial-observation games.}, author = {Chatterjee, Krishnendu}, location = {Budapest, Hungary}, number = {PART 1}, pages = {1 -- 4}, publisher = {Springer}, title = {{Partial-observation stochastic reachability and parity games}}, doi = {10.1007/978-3-662-44522-8_1}, volume = {8634}, year = {2014}, } @article{1904, abstract = {We prove a Strichartz inequality for a system of orthonormal functions, with an optimal behavior of the constant in the limit of a large number of functions. The estimate generalizes the usual Strichartz inequality, in the same fashion as the Lieb-Thirring inequality generalizes the Sobolev inequality. As an application, we consider the Schrödinger equation with a time-dependent potential and we show the existence of the wave operator in Schatten spaces.}, author = {Frank, Rupert and Lewin, Mathieu and Lieb, Élliott and Seiringer, Robert}, journal = {Journal of the European Mathematical Society}, number = {7}, pages = {1507 -- 1526}, publisher = {European Mathematical Society}, title = {{Strichartz inequality for orthonormal functions}}, doi = {10.4171/JEMS/467}, volume = {16}, year = {2014}, } @article{1905, abstract = {The unprecedented polymorphism in the major histocompatibility complex (MHC) genes is thought to be maintained by balancing selection from parasites. However, do parasites also drive divergence at MHC loci between host populations, or do the effects of balancing selection maintain similarities among populations? We examined MHC variation in populations of the livebearing fish Poecilia mexicana and characterized their parasite communities. Poecilia mexicana populations in the Cueva del Azufre system are locally adapted to darkness and the presence of toxic hydrogen sulphide, representing highly divergent ecotypes or incipient species. Parasite communities differed significantly across populations, and populations with higher parasite loads had higher levels of diversity at class II MHC genes. However, despite different parasite communities, marked divergence in adaptive traits and in neutral genetic markers, we found MHC alleles to be remarkably similar among host populations. Our findings indicate that balancing selection from parasites maintains immunogenetic diversity of hosts, but this process does not promote MHC divergence in this system. On the contrary, we suggest that balancing selection on immunogenetic loci may outweigh divergent selection causing divergence, thereby hindering host divergence and speciation. Our findings support the hypothesis that balancing selection maintains MHC similarities among lineages during and after speciation (trans-species evolution).}, author = {Tobler, Michael and Plath, Martin and Riesch, Rüdiger and Schlupp, Ingo and Grasse, Anna V and Munimanda, Gopi and Setzer, C and Penn, Dustin and Moodley, Yoshan}, journal = {Journal of Evolutionary Biology}, number = {5}, pages = {960 -- 974}, publisher = {Wiley-Blackwell}, title = {{Selection from parasites favours immunogenetic diversity but not divergence among locally adapted host populations}}, doi = {10.1111/jeb.12370}, volume = {27}, year = {2014}, } @article{1906, abstract = {In this paper, we introduce a novel scene representation for the visualization of large-scale point clouds accompanied by a set of high-resolution photographs. Many real-world applications deal with very densely sampled point-cloud data, which are augmented with photographs that often reveal lighting variations and inaccuracies in registration. Consequently, the high-quality representation of the captured data, i.e., both point clouds and photographs together, is a challenging and time-consuming task. We propose a two-phase approach, in which the first (preprocessing) phase generates multiple overlapping surface patches and handles the problem of seamless texture generation locally for each patch. The second phase stitches these patches at render-time to produce a high-quality visualization of the data. As a result of the proposed localization of the global texturing problem, our algorithm is more than an order of magnitude faster than equivalent mesh-based texturing techniques. Furthermore, since our preprocessing phase requires only a minor fraction of the whole data set at once, we provide maximum flexibility when dealing with growing data sets.}, author = {Arikan, Murat and Preiner, Reinhold and Scheiblauer, Claus and Jeschke, Stefan and Wimmer, Michael}, journal = {IEEE Transactions on Visualization and Computer Graphics}, number = {9}, pages = {1280 -- 1292}, publisher = {IEEE}, title = {{Large-scale point-cloud visualization through localized textured surface reconstruction}}, doi = {10.1109/TVCG.2014.2312011}, volume = {20}, year = {2014}, } @inproceedings{1907, abstract = {Most cryptographic security proofs require showing that two systems are indistinguishable. A central tool in such proofs is that of a game, where winning the game means provoking a certain condition, and it is shown that the two systems considered cannot be distinguished unless this condition is provoked. Upper bounding the probability of winning such a game, i.e., provoking this condition, for an arbitrary strategy is usually hard, except in the special case where the best strategy for winning such a game is known to be non-adaptive. A sufficient criterion for ensuring the optimality of non-adaptive strategies is that of conditional equivalence to a system, a notion introduced in [1]. In this paper, we show that this criterion is not necessary to ensure the optimality of non-adaptive strategies by giving two results of independent interest: 1) the optimality of non-adaptive strategies is not preserved under parallel composition; 2) in contrast, conditional equivalence is preserved under parallel composition.}, author = {Demay, Grégory and Gazi, Peter and Maurer, Ueli and Tackmann, Björn}, booktitle = {IEEE International Symposium on Information Theory}, location = {Honolulu, USA}, publisher = {IEEE}, title = {{Optimality of non-adaptive strategies: The case of parallel games}}, doi = {10.1109/ISIT.2014.6875125}, year = {2014}, } @article{1908, abstract = {In large populations, multiple beneficial mutations may be simultaneously spreading. In asexual populations, these mutations must either arise on the same background or compete against each other. In sexual populations, recombination can bring together beneficial alleles from different backgrounds, but tightly linked alleles may still greatly interfere with each other. We show for well-mixed populations that when this interference is strong, the genome can be seen as consisting of many effectively asexual stretches linked together. The rate at which beneficial alleles fix is thus roughly proportional to the rate of recombination and depends only logarithmically on the mutation supply and the strength of selection. Our scaling arguments also allow us to predict, with reasonable accuracy, the fitness distribution of fixed mutations when the mutational effect sizes are broad. We focus on the regime in which crossovers occur more frequently than beneficial mutations, as is likely to be the case for many natural populations.}, author = {Weissman, Daniel and Hallatschek, Oskar}, journal = {Genetics}, number = {4}, pages = {1167 -- 1183}, publisher = {Genetics Society of America}, title = {{The rate of adaptation in large sexual populations with linear chromosomes}}, doi = {10.1534/genetics.113.160705}, volume = {196}, year = {2014}, } @article{1909, abstract = {Summary: Phenotypes are often environmentally dependent, which requires organisms to track environmental change. The challenge for organisms is to construct phenotypes using the most accurate environmental cue. Here, we use a quantitative genetic model of adaptation by additive genetic variance, within- and transgenerational plasticity via linear reaction norms and indirect genetic effects respectively. We show how the relative influence on the eventual phenotype of these components depends on the predictability of environmental change (fast or slow, sinusoidal or stochastic) and the developmental lag τ between when the environment is perceived and when selection acts. We then decompose expected mean fitness into three components (variance load, adaptation and fluctuation load) to study the fitness costs of within- and transgenerational plasticity. A strongly negative maternal effect coefficient m minimizes the variance load, but a strongly positive m minimises the fluctuation load. The adaptation term is maximized closer to zero, with positive or negative m preferred under different environmental scenarios. Phenotypic plasticity is higher when τ is shorter and when the environment changes frequently between seasonal extremes. Expected mean population fitness is highest away from highest observed levels of phenotypic plasticity. Within- and transgenerational plasticity act in concert to deliver well-adapted phenotypes, which emphasizes the need to study both simultaneously when investigating phenotypic evolution.}, author = {Ezard, Thomas and Prizak, Roshan and Hoyle, Rebecca}, journal = {Functional Ecology}, number = {3}, pages = {693 -- 701}, publisher = {Wiley-Blackwell}, title = {{The fitness costs of adaptation via phenotypic plasticity and maternal effects}}, doi = {10.1111/1365-2435.12207}, volume = {28}, year = {2014}, } @article{1910, abstract = {angerhans cells (LCs) are a unique subset of dendritic cells (DCs) that express epithelial adhesion molecules, allowing them to form contacts with epithelial cells and reside in epidermal/epithelial tissues. The dynamic regulation of epithelial adhesion plays a decisive role in the life cycle of LCs. It controls whether LCs remain immature and sessile within the epidermis or mature and egress to initiate immune responses. So far, the molecular machinery regulating epithelial adhesion molecules during LC maturation remains elusive. Here, we generated pure populations of immature human LCs in vitro to systematically probe for gene-expression changes during LC maturation. LCs down-regulate a set of epithelial genes including E-cadherin, while they upregulate the mesenchymal marker N-cadherin known to facilitate cell migration. In addition, N-cadherin is constitutively expressed by monocyte-derived DCs known to exhibit characteristics of both inflammatory-type and interstitial/dermal DCs. Moreover, the transcription factors ZEB1 and ZEB2 (ZEB is zinc-finger E-box-binding homeobox) are upregulated in migratory LCs. ZEB1 and ZEB2 have been shown to induce epithelial-to-mesenchymal transition (EMT) and invasive behavior in cancer cells undergoing metastasis. Our results provide the first hint that the molecular EMT machinery might facilitate LC mobilization. Moreover, our study suggests that N-cadherin plays a role during DC migration.}, author = {Konradi, Sabine and Yasmin, Nighat and Haslwanter, Denise and Weber, Michele and Gesslbauer, Bernd and Sixt, Michael K and Strobl, Herbert}, journal = {European Journal of Immunology}, number = {2}, pages = {553 -- 560}, publisher = {Wiley-Blackwell}, title = {{Langerhans cell maturation is accompanied by induction of N-cadherin and the transcriptional regulators of epithelial-mesenchymal transition ZEB1/2}}, doi = {10.1002/eji.201343681}, volume = {44}, year = {2014}, } |
fc4aedae94f17528 | Lasers, chaos and bow-ties
Physics World 11 (#9), 23-24 (September 1998) -- Physics in Action
From Malvin C. Teich in the Department of Electrical and Computer Engineering, Boston University, US
Since the invention of the laser nearly 40 years ago, there has been an inexorable trend towards creating devices that are smaller, more efficient and tunable over a wider range of wavelengths. This has been achieved through continuous improvements in the three principal building blocks of the laser an active medium that provides amplification, a pump that serves as the source of energy and a resonator that provides feedback. In the latest advance, researchers from Lucent Technologies and Yale University in the US, and the Max Planck Institute for the physics of complex systems in Dresden, Germany, have tackled the limitations imposed by laser resonators in a fresh and effective way (C Gmachl et al. 1998 Science 280 1556). They have obtained strong beams of laser light from microdisk lasers, which were previously only able to produce weak laser emission.
Click for the full size image
Bow-tie models
Laser technology has come a long way since the 1960s. The very earliest lasers, such as ruby and heliumneon devices, were formidable contraptions. They relied on dilute active media with discrete energy levels such as gases or dopant ions scattered in a solid, external pumping mechanisms such as bulky coiled flashlamps or auxiliary gases excited by radio-frequency coils, and cumbersome external resonators consisting of highly flat and reflective mirrors.
The advent of the semiconductor laser in 1962 developed almost simultaneously at the MIT Lincoln Laboratory, General Electric and IBM changed all of that. The active medium, which consisted of a semiconductor pn junction, was in this case a dense three-dimensional solid. The junction could be made from various materials, so that the band-gap energy, and hence the emission wavelength, could be tuned by altering the composition.
The pumping process was simple: a current flowing through the junction created electronhole pairs that recombined to emit photons. Resonator mirrors were readily provided by the surface reflection associated with the high refractive index of the semiconductor material. Size plummeted, efficiency soared and the accessible wavelength range increased dramatically. It is not an exaggeration to view the invention of the semiconductor injection laser as the start of a new era of quantum electronics one that was ultimately to broker a fair partnership between photonics and electronics.
Many important developments in semiconductor lasers followed. Double heterostructures replaced simple pn junctions, lowering the laser threshold and increasing efficiency. Compounds of three or four semiconductors, coupled with band-gap engineering, provided an extensive palette of wavelengths. Gratings could be integrated directly with the active medium, creating distributed-feedback (DFB) lasers to complement the existing lumped-resonator lasers. The very dimensionality of the active medium was reduced; two dimensions gave birth to the quantum-well laser, while a single dimension yielded the quantum-wire laser. Even zero-dimensional devices have come to the fore in the form of quantum-dot lasers comprising just a few thousand atoms. And the vertical-cavity configuration allowed tiny lasers to be built on a single chip, providing integrated two-dimensional laser arrays.
As important as these developments were, remarkable achievements have continued to be made. In the early 1990s devices were created that did not require electronhole recombination for stimulated emission. Instead light is generated from electron transitions in a cascade of coupled quantum wells. These quantum cascade lasers (QCLs) were constructed using an exquisitely refined form of band-gap engineering made possible by molecular-beam-epitaxy technology (for a recent update see F Capasso et al. 1997 Solid State Communications 102 231). QCLs provided strong and tunable laser action in the middle infrared region of the spectrum at room temperature, promising new and important applications.
In the new work, Claire Gmachl of Lucent and her co-workers adapted the quantum cascade laser to a tiny oblong configuration (50 X 70 µm) with a very low threshold current and excellent directionality. They achieved this by turning their attention to the two-dimensional circular resonators used in microdisk semiconductor lasers. These resonators support "whispering-gallery" modes, so-named because of the ease with which an acoustic whisper can bounce along the convex surface of a church dome or gallery. Such modes rely on total internal reflection and skim around the inside rim of the resonator with an angle of incidence that is always greater than the critical angle, preventing them from refracting out of the device. Although these lasers are among the smallest in the world, light only emerges from them via evanescence (photon tunnelling), which makes the emitted light weak and cylindrically symmetric.
What Gmachl and colleagues have done is to flatten the resonator circle to create a stadium-shaped structure. This establishes preferred location angles around the perimeter of the stadium at which strong and highly directional beams are emitted. The power emitted in these beams is up to a thousand times greater than that emitted by the circular lasers.
The team has also discovered a new type of "bow-tie" mode that emerges if the structure is flattened sufficiently. This mode is so-named because of the bow-tie shape of the four-bounce, round-trip ray path within the resonator, much the same as that followed by the rays in a classical confocal resonator (see figure). These modes are sustainable because the resonator is not circular, and so provides a range of mirror curvatures. Although the effect was demonstrated using a quantum-cascade active medium, it should also be observed in other laser media with sufficiently high refractive indices.
Why should these preferred angles emerge? We are used to thinking about laser resonators in terms of solutions to the Helmholtz equation, or rays traced according to the laws of geometrical optics. It is surprising, then, that in this case the behaviour of the resonator can be explained in terms of nonlinear dynamics. The nonlinearity resides in the dependence of a given angle of incidence on its precursor incidence and location angles. When the flattening of the resonator is minimal, phase-space studies of ray-trajectories (in which the angle of incidence is plotted against location angle) in the stadium-shaped resonator show that chaotic versions of the whispering-gallery modes emerge. For more pronounced flattening, regions of stable and regular ray motion give rise to the new bow-tie laser modes. These arise from localized reflections at four particular locations on the perimeter of the resonator those that provide the curvature of a classical confocal resonator and that have adequate reflectance to exceed the laser threshold.
Moreover, because the Helmholtz equation describing optical fields is closely related to the Schrödinger equation, there is a formal relationship between their short-wavelength counterparts ray optics for the Helmholtz equation and Newtonian mechanics for the Schrödinger equation. A non-separable Helmholtz equation is associated with full or partially chaotic ray dynamics, just as a non-separable Schrödinger equation is associated with chaotic classical dynamics.
Resonator physics therefore turns out to be rich, beautiful and useful. Nonlinear dynamics and optical physics have previously intersected via the laser's active medium and pump, since nonlinearity in the laser-rate equations can give rise to chaotic time dynamics of the emitted intensity (the "green problem"). The resonator connection discussed here provides another testbed for investigating the intersection of laser physics and nonlinear dynamics.
As laser active media and pumping mechanisms have evolved in the past few decades, so too have resonator structures. The original plane-parallel mirrors that comprised one-dimensional FabryPérot resonators were soon replaced by spherical mirrors to make alignment easier. Unstable resonators and two-dimensional ring-laser structures were developed for particular applications. Distributed-feedback configurations provided an alternative to lumped resonators. And now the circular cross-section of the microdisk semiconductor laser has given way to an improved stadium-shaped version. |
c00b67ba8651b557 | Quantum cascade laser
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Quantum cascade lasers (QCLs) are semiconductor lasers that emit in the mid- to far-infrared portion of the electromagnetic spectrum and were first demonstrated by Jerome Faist, Federico Capasso, Deborah Sivco, Carlo Sirtori, Albert Hutchinson, and Alfred Cho at Bell Laboratories in 1994.[1]
Unlike typical interband semiconductor lasers that emit electromagnetic radiation through the recombination of electron–hole pairs across the material band gap, QCLs are unipolar and laser emission is achieved through the use of intersubband transitions in a repeated stack of semiconductor multiple quantum well heterostructures, an idea first proposed in the paper "Possibility of amplification of electromagnetic waves in a semiconductor with a superlattice" by R.F. Kazarinov and R.A. Suris in 1971.[2]
Intersubband vs. interband transitions[edit]
Interband transitions in conventional semiconductor lasers emit a single photon.
Within a bulk semiconductor crystal, electrons may occupy states in one of two continuous energy bands - the valence band, which is heavily populated with low energy electrons and the conduction band, which is sparsely populated with high energy electrons. The two energy bands are separated by an energy band gap in which there are no permitted states available for electrons to occupy. Conventional semiconductor laser diodes generate light by a single photon being emitted when a high energy electron in the conduction band recombines with a hole in the valence band. The energy of the photon and hence the emission wavelength of laser diodes is therefore determined by the band gap of the material system used.
A QCL however does not use bulk semiconductor materials in its optically active region. Instead it comprises a periodic series of thin layers of varying material composition forming a superlattice. The superlattice introduces a varying electric potential across the length of the device, meaning that there is a varying probability of electrons occupying different positions over the length of the device. This is referred to as one-dimensional multiple quantum well confinement and leads to the splitting of the band of permitted energies into a number of discrete electronic subbands. By suitable design of the layer thicknesses it is possible to engineer a population inversion between two subbands in the system which is required in order to achieve laser emission. Since the position of the energy levels in the system is primarily determined by the layer thicknesses and not the material, it is possible to tune the emission wavelength of QCLs over a wide range in the same material system.
In quantum cascade structures, electrons undergo intersubband transitions and photons are emitted. The electrons tunnel to the next period of the structure and the process repeats.
Additionally, in semiconductor laser diodes, electrons and holes are annihilated after recombining across the band gap and can play no further part in photon generation. However in a unipolar QCL, once an electron has undergone an intersubband transition and emitted a photon in one period of the superlattice, it can tunnel into the next period of the structure where another photon can be emitted. This process of a single electron causing the emission of multiple photons as it traverses through the QCL structure gives rise to the name cascade and makes a quantum efficiency of greater than unity possible which leads to higher output powers than semiconductor laser diodes.
Operating principles[edit]
Rate equations[edit]
Subband populations are determined by the intersubband scattering rates and the injection/extraction current.
QCLs are typically based upon a three-level system. Assuming the formation of the wavefunctions is a fast process compared to the scattering between states, the time independent solutions to the Schrödinger equation may be applied and the system can be modelled using rate equations. Each subband contains a number of electrons n_i (where i is the subband index) which scatter between levels with a lifetime \tau_{if} (reciprocal of the average intersubband scattering rate W_{if}), where i and f are the initial and final subband indices. Assuming that no other subbands are populated, the rate equations for the three level lasers are given by:
\frac{\mathrm{d}n_3}{\mathrm{d}t} = I_{\mathrm{in}} + \frac{n_1}{\tau_{13}} + \frac{n_2}{\tau_{23}} -
\frac{n_3}{\tau_{31}} - \frac{n_3}{\tau_{32}}
\frac{\mathrm{d}n_2}{\mathrm{d}t} = \frac{n_3}{\tau_{32}} + \frac{n_1}{\tau_{12}} -
\frac{n_2}{\tau_{21}} - \frac{n_2}{\tau_{23}}
\frac{\mathrm{d}n_1}{\mathrm{d}t} = \frac{n_2}{\tau_{21}} + \frac{n_3}{\tau_{31}} -
\frac{n_1}{\tau_{13}} - \frac{n_1}{\tau_{12}} - I_{\mathrm{out}}
In the steady state, the time derivatives are equal to zero and I_{\mathrm{in}}=I_{\mathrm{out}}=I. The general rate equation for electrons in subband i of an N level system is therefore:
\frac{\mathrm{d}n_i}{\mathrm{d}t} = \sum\limits_{j=1}^N\frac{n_j}{\tau_{ji}}-n_i\sum\limits_{j=1}^N\frac{1}{\tau_{ij}}+I(\delta_{iN}-\delta_{i1}),
Under the assumption that absorption processes can be ignored, (i.e. \frac{n_1}{\tau_{12}} = \frac{n_2}{\tau_{23}} = 0 , valid at low temperatures) the middle rate equation gives
\frac{n_3}{\tau_{32}} = \frac{n_2}{\tau_{21}}
Therefore if \tau_{32} > \tau_{21} (i.e. W_{21} > W_{32}) then n_3 > n_2 and a population inversion will exist. The population ratio is defined as
\frac{n_3}{n_2} = \frac{\tau_{32}}{\tau_{21}} = \frac{W_{21}}{W_{32}}
If all N steady-state rate equations are summed, the right hand side becomes zero, meaning that the system is underdetermined, and it is possible only to find the relative population of each subband. If the total sheet density of carriers N_{\mathrm{2D}} in the system is also known, then the absolute population of carriers in each subband may be determined using:
As an approximation, it can be assumed that all the carriers in the system are supplied by doping. If the dopant species has a negligible ionisation energy then N_{\mathrm{2D}} is approximately equal to the doping density.
Electron wave functions are repeated in each period of a three quantum well QCL active region. The upper laser level is shown in bold.
Active region designs[edit]
The scattering rates are tailored by suitable design of the layer thicknesses in the superlattice which determine the electron wave functions of the subbands. The scattering rate between two subbands is heavily dependent upon the overlap of the wave functions and energy spacing between the subbands. The figure shows the wave functions in a three quantum well (3QW) QCL active region and injector.
In order to decrease W_{32}, the overlap of the upper and lower laser levels is reduced. This is often achieved through designing the layer thicknesses such that the upper laser level is mostly localised in the left-hand well of the 3QW active region, while the lower laser level wave function is made to mostly reside in the central and right-hand wells. This is known as a diagonal transition. A vertical transition is one in which the upper laser level is localised in mainly the central and right-hand wells. This increases the overlap and hence W_{32} which reduces the population inversion, but it increases the strength of the radiative transition and therefore the gain.
In order to increase W_{21}, the lower laser level and the ground level wave functions are designed such that they have a good overlap and to increase W_{21} further, the energy spacing between the subbands is designed such that it is equal to the longitudinal optical (LO) phonon energy (~36 meV in GaAs) so that resonant LO phonon-electron scattering can quickly depopulate the lower laser level.
Material systems[edit]
The first QCL was fabricated in the InGaAs/InAlAs material system lattice-matched to an InP substrate.[1] This particular material system has a conduction band offset (quantum well depth) of 520 meV.[citation needed] These InP-based devices have reached very high levels of performance across the mid-infrared spectral range, achieving high power, above room-temperature, continuous wave emission.[3]
In 1998 GaAs/AlGaAs QCLs were demonstrated by Sirtori et al. proving that the QC concept is not restricted to one material system.[citation needed] This material system has a varying quantum well depth depending on the aluminium fraction in the barriers.[citation needed] Although GaAs-based QCLs have not matched the performance levels of InP-based QCLs in the mid-infrared, they have proven to be very successful in the terahertz region of the spectrum.[citation needed]
The short wavelength limit of QCLs is determined by the depth of the quantum well and recently QCLs have been developed in material systems with very deep quantum wells in order to achieve short wavelength emission. The InGaAs/AlAsSb material system has quantum wells 1.6 eV deep and has been used to fabricate QCLs emitting at 3 μm.[citation needed] InAs/AlSb QCLs have quantum wells 2.1 eV deep and electroluminescence at wavelengths as short as 2.5 μm has been observed.[citation needed]
QCLs may also allow laser operation in materials traditionally considered to have poor optical properties. Indirect bandgap materials such as silicon have minimum electron and hole energies at different momentum values. For interband optical transitions, carriers change momentum through a slow, intermediate scattering process, dramatically reducing the optical emission intensity. Intersubband optical transitions however, are independent of the relative momentum of conduction band and valence band minima and theoretical proposals for Si/SiGe quantum cascade emitters have been made.[4]
Emission wavelengths[edit]
QCLs currently cover the wavelength range from 2.75–250 μm (and extends to 355 μm with the application of a magnetic field).[citation needed]
Optical waveguides[edit]
End view of QC facet with ridge waveguide. Darker gray: InP, lighter gray: QC layers, black: dielectric, gold: Au coating. Ridge ~ 10 um wide.
End view of QC facet with buried heterostructure waveguide. Darker gray: InP, lighter gray: QC layers, black: dielectric. Heterostructure ~ 10 um wide
The first step in processing quantum cascade gain material to make a useful light-emitting device is to confine the gain medium in an optical waveguide. This makes it possible to direct the emitted light into a collimated beam, and allows a laser resonator to be built such that light can be coupled back into the gain medium.
Two types of optical waveguides are in common use. A ridge waveguide is created by etching parallel trenches in the quantum cascade gain material to create an isolated stripe of QC material, typically ~10 um wide, and several mm long. A dielectric material is typically deposited in the trenches to guide injected current into the ridge, then the entire ridge is typically coated with gold to provide electrical contact and to help remove heat from the ridge when it is producing light. Light is emitted from the cleaved ends of the waveguide, with an active area that is typically only a few micrometers in dimension.
The second waveguide type is a buried heterostructure. Here, the QC material is also etched to produce an isolated ridge. Now, however, new semiconductor material is grown over the ridge. The change in index of refraction between the QC material and the overgrown material is sufficient to create a waveguide. Dielectric material is also deposited on the overgrown material around QC ridge to guide the injected current into the QC gain medium. Buried heterostructure waveguides are efficient at removing heat from the QC active area when light is being produced.
Laser types[edit]
Although the quantum cascade gain medium can be used to produce incoherent light in a superluminescent configuration,[5] it is most commonly used in combination with an optical cavity to form a laser.
Fabry–Perot lasers[edit]
This is the simplest of the quantum cascade lasers. An optical waveguide is first fabricated out of the quantum cascade material to form the gain medium. The ends of the crystalline semiconductor device are then cleaved to form two parallel mirrors on either end of the waveguide, thus forming a Fabry–Pérot resonator. The residual reflectivity on the cleaved facets from the semiconductor-to-air interface is sufficient to create a resonator. Fabry–Pérot quantum cascade lasers are capable of producing high powers,[6] but are typically multi-mode at higher operating currents. The wavelength can be changed chiefly by changing the temperature of the QC device.
Distributed feedback lasers[edit]
A distributed feedback (DFB) quantum cascade laser[7] is similar to a Fabry–Pérot laser, except for a distributed Bragg reflector (DBR) built on top of the waveguide to prevent it from emitting at other than the desired wavelength. This forces single mode operation of the laser, even at higher operating currents. DFB lasers can be tuned chiefly by changing the temperature, although an interesting variant on tuning can be obtained by pulsing a DFB laser. In this mode, the wavelength of the laser is rapidly “chirped” during the course of the pulse, allowing rapid scanning of a spectral region.[8]
External cavity lasers[edit]
Schematic of QC device in external cavity with frequency selective optical feedback provided by diffraction grating in Littrow configuration.
In an external cavity (EC) quantum cascade laser, the quantum cascade device serves as the laser gain medium. One, or both, of the waveguide facets have an anti-reflection coating that defeats the optical cavity action of the cleaved facets. Mirrors are then arranged in a configuration external to the QC device to create the optical cavity.
If a frequency-selective element is included in the external cavity, it is possible to reduce the laser emission to a single wavelength, and even tune the radiation. For example, diffraction gratings have been used to create[9] a tunable laser that can tune over 15% of its center wavelength.
The alternating layers of the two different semiconductors which form the quantum heterostructure may be grown on to a substrate using a variety of methods such as molecular beam epitaxy (MBE) or metalorganic vapour phase epitaxy (MOVPE), also known as metalorganic chemical vapor deposition (MOCVD).
Distributed feedback (DFB) quantum cascade lasers were first commercialized in 2004,[10] and broadly-tunable external cavity quantum cascade lasers first commercialized in 2006.[11] The high optical power output, tuning range and room temperature operation make QCLs useful for spectroscopic applications such as remote sensing of environmental gases and pollutants in the atmosphere[12] and homeland security. They may eventually be used for vehicular cruise control in conditions of poor visibility,[citation needed] collision avoidance radar,[citation needed] industrial process control,[citation needed] and medical diagnostics such as breath analyzers.[13] QCLs are also used to study plasma chemistry.[14]
Their large dynamic range,[clarification needed] excellent sensitivity,[clarification needed] and failsafe operation[clarification needed] combined with the solid-state reliability should easily[original research?] overcome many of the technological hurdles[clarification needed] that impede existing technology in these markets. When used in multiple-laser systems, intrapulse QCL spectroscopy[clarification needed] offers broadband spectral coverage that can potentially be used to identify and quantify complex heavy molecules such as those in toxic chemicals, explosives, and drugs.[15]
Unguided QCL emission in the 3–5 μm atmospheric window could be used as a cheaper alternative to optical fibres for high-speed Internet access in built up areas.[citation needed]
In fiction[edit]
• The upcoming video game Star Citizen imagines external-cavity quantum cascade lasers as high-power weapons.[16]
1. ^ a b Faist, Jerome; Federico Capasso, Deborah L. Sivco, Carlo Sirtori, Albert L. Hutchinson, and Alfred Y. Cho (April 1994). "Quantum Cascade Laser" (abstract). Science 264 (5158): 553–556. Bibcode:1994Sci...264..553F. doi:10.1126/science.264.5158.553. PMID 17732739. Retrieved 2007-02-18.
2. ^ Kazarinov, R.F; Suris, R.A. (April 1971). "Possibility of amplification of electromagnetic waves in a semiconductor with a superlattice". Fizika i Tekhnika Poluprovodnikov 5 (4): 797–800.
3. ^ Razeghi, Manijeh (2009). "High-Performance InP-Based Mid-IR Quantum Cascade Lasers" (abstract). IEEE Journal of Selected Topics in Quantum Electronics 15 (3): 941–951. doi:10.1109/JSTQE.2008.2006764. Retrieved 2011-07-13.
4. ^ Paul, Douglas J (2004). "Si/SiGe heterostructures: from material and physics to devices and circuits" (abstract). Semicond. Sci. Technol. 19 (10): R75–R108. Bibcode:2004SeScT..19R..75P. doi:10.1088/0268-1242/19/10/R02. Retrieved 2007-02-18.
5. ^ Zibik, E. A.; W. H. Ng, D. G. Revin, L. R. Wilson, J. W. Cockburn, K. M. Groom, and M. Hopkinson (March 2006). "Broadband 6 µm < λ < 8 µm superluminescent quantum cascade light-emitting diodes". Appl. Phys. Lett. 88 (12): 121109. Bibcode:2006ApPhL..88l1109Z. doi:10.1063/1.2188371.
6. ^ Slivken, S.; A. Evans, J. David, and M. Razeghi (December 2002). "High-average-power, high-duty-cycle (λ ~ 6 µm) quantum cascade lasers". Applied Physics Letters 81 (23): 4321–4323. Bibcode:2002ApPhL..81.4321S. doi:10.1063/1.1526462.
7. ^ Faist, Jérome; Claire Gmachl, Frederico Capasso, Carlo Sirtori, Deborah L. Silvco, James N. Baillargeon, and Alfred Y. Cho (May 1997). "Distributed feedback quantum cascade lasers". Applied Physics Letters 70 (20): 2670. Bibcode:1997ApPhL..70.2670F. doi:10.1063/1.119208.
8. ^ "Quantum-cascade lasers smell success". Laser Focus World. PennWell Publications. 2005-03-01. Retrieved 2008-03-26.
9. ^ Maulini, Richard; Mattias Beck, Jérome Faist, and Emilio Gini (March 2004). "Broadband tuning of external cavity bound-to-continuum quantum-cascade lasers". Applied Physics Letters 84 (10): 1659. Bibcode:2004ApPhL..84.1659M. doi:10.1063/1.1667609.
10. ^ "Alpes offers CW and pulsed quantum cascade lasers". Laser Focus World. PennWell Publications. 2004-04-19. Retrieved 2007-12-01.
11. ^ "Tunable QC laser opens up mid-IR sensing applications". Laser Focus World. PennWell Publications. 2006-07-01. Retrieved 2008-03-26.
12. ^ Normand, Erwan; Howieson, Iain; McCulloch, Michael T. (April 2007). "Quantum-cascade lasers enable gas-sensing technology". Laser Focus World 43 (4): 90–92. ISSN 1043-8092. Retrieved 2008-01-25.
13. ^ Hannemann, M.; Antufjew, A.; Borgmann, K.; Hempel, F.; Ittermann, T.; Welzel, S.; Weltmann, K.D.; Völzke, H.; Röpcke, J. (2011-04-01). "Influence of age and sex in exhaled breath samples investigated by means of infrared laser absorption spectroscopy". Journal of Breath Research 5 (027101): 9. Bibcode:2011JBR.....5b7101H. doi:10.1088/1752-7155/5/2/027101.
14. ^ Lang, N.; Röpcke, J.; Wege, S.; Steinach, A. (2009-12-11). "In situ diagnostic of etch plasmas for process control using quantum cascade laser absorption spectroscopy". Eur. Phys. J. Appl. Phys. 49 (13110): 3. Bibcode:2010EPJAP..49a3110L. doi:10.1051/epjap/2009198.
15. ^ Howieson, Iain; Normand, Erwan; McCulloch, Michael T. (2005-03-01). "Quantum-cascade lasers smell success". Laser Focus World 41 (3): S3–+. ISSN 0740-2511. Retrieved 2008-01-25.
16. ^ https://robertsspaceindustries.com/comm-link/transmission/13152-Galactic-Guide-Hurston-Dynamics
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Physics (from the Greek, φύσις (phúsis), "nature" and φυσικός (phusikós), "natural"), the most fundamental physical science, is concerned with the underlying principles of the natural world. Consequently, physics deals with the elementary constituents of the Universe and their interactions, as well as the analysis of systems which are best understood in terms of these fundamental principles.
Introduction Edit
Discoveries in physics find applications throughout the other natural sciences as they regard the basic constituents of the Universe. Some of the phenomena studied in physics, such as the phenomenon of conservation of energy, are common to all material systems. These are often referred to as laws of physics. Others, such as superconductivity, stem from these laws, but are not laws themselves because they only appear in some systems. Physics is often said to be the "fundamental science" (chemistry is sometimes included), because each of the other sciences (biology, chemistry, geology, material science, engineering, medicine etc.) deals with particular types of material systems that obey the laws of physics. For example, chemistry is the science of matter (such as atoms and molecules) and the chemical substances that they form in the bulk. The structure, reactivity, and properties of a chemical compound are determined by the properties of the underlying molecules, which can be described by areas of physics such as quantum mechanics (called in this case quantum chemistry), thermodynamics, and electromagnetism. (Refer to Branches of physics)
Physics is closely related to mathematics, which provides the logical framework in which physical laws can be precisely formulated and their predictions quantified. Physical definitions, models and theories are invariably expressed using mathematical relations. A key difference between physics and mathematics is that because physics is ultimately concerned with descriptions of the material world, it tests its theories by observations (called experiments), whereas mathematics is concerned with abstract logical patterns not limited by those observed in the real world (because the real world is limited in the number of dimensions and in many other ways it does not have to correspond to richer mathematical structures). The distinction, however, is not always clear-cut. There is a large area of research intermediate between physics and mathematics, known as mathematical physics.
Physics attempts to describe the natural world by the application of the scientific method. Natural philosophy, its counterpart, is the study of the changing world by philosophy which has been also called "physics" since classical times to at least up to its separation from philosophy as a positive science in the 19th century. Mixed questions, of which solutions can be attempted through the applications of both disciplines (e.g. the divisibility of the atom) can involve natural philosophy in physics the science and vice versa.
Classical, quantum and modern physics Edit
Further information: Classical physics, Quantum physics, Modern physics, Semiclassical
Since the construction of quantum mechanics in the early twentieth century, it generally became evident to the physical community that it would be preferable for every known description of Nature to be quantized, that is, to follow the postulates of quantum mechanics. To this effect, all results that were not quantized are called classical: this includes the special and general theories of relativity. Simply because a result is classical does not mean that it was discovered before the advent of quantum mechanics. Classical theories are, generally, much easier to work with and much research is still being conducted on them without the express aim of quantization. However, there exist problems in physics in which classical and quantum aspects must be combined to attain some approximation or limit that may acquire several forms as the passage from classical to quantum mechanics is often difficult — such problems are termed semiclassical.
However, because relativity and quantum mechanics provide the most complete known description of fundamental interactions, and because the changes brought by these two frameworks to the physicist's world view were revolutionary, the term modern physics is used to describe physics which relies on these two theories. Colloquially, modern physics can be described as the physics of extremes: from systems at the extremely small (atoms, nuclei, fundamental particles) to the extremely large (the Universe) and of the extremely fast (relativity).
Branches of physics of interest to psychologistsEdit
Physics Venn diagram
Classification of physics fields by the types of effects that need to be accounted for
Physicists study a wide range of physical phenomena, from quarks to black holes, from individual atoms to the many-body systems of superconductors.
Central theories Edit
While physics deals with a wide variety of systems, there are certain theories that are used by all physicists. Each of these theories were experimentally tested numerous times and found correct as an approximation of Nature (within a certain domain of validity). For instance, the theory of classical mechanics accurately describes the motion of objects, provided they are much larger than atoms and moving at much less than the speed of light. These theories continue to be areas of active research; for instance, a remarkable aspect of classical mechanics known as chaos was discovered in the 20th century, three centuries after the original formulation of classical mechanics by Isaac Newton (16421727). These "central theories" are important tools for research into more specialized topics, and any physicist, regardless of his or her specialization, is expected to be literate in them.
Theory Major subtopics Concepts
Classical mechanics Newton's laws of motion, Lagrangian mechanics, Hamiltonian mechanics, Kinematics, Statics, Dynamics, Chaos theory, Acoustics, Fluid dynamics, Continuum mechanics Density, Dimension, Gravity, Space, Time, Motion, Length, Position, Velocity, Acceleration, Mass, Momentum, Force, Energy, Angular momentum, Torque, Conservation law, Harmonic oscillator, Wave, Work, Power, Harmonic oscillator
Electromagnetism Electrostatics, Electrodynamics, Electricity, Magnetism, Maxwell's equations, Optics Capacitance, Electric charge, Current, Electrical conductivity, Electric field, Electric permittivity, Electrical resistance, Electromagnetic field, Electromagnetic induction, Electromagnetic radiation, Gaussian surface, Magnetic field, Magnetic flux, Magnetic monopole, Magnetic permeability
Thermodynamics and Statistical mechanics Heat engine, Kinetic theory Boltzmann's constant, Conjugate variables, Enthalpy, Entropy, Equation of state, Equipartition theorem, Free energy, Heat, Ideal gas law, Internal energy, Laws of thermodynamics, Irreversible process, Partition function, Pressure, Reversible process, Spontaneous process, State function, Statistical ensemble, Temperature, Thermodynamic equilibrium, Thermodynamic potential, Thermodynamic processes, Thermodynamic state, Thermodynamic system, Viscosity
Quantum mechanics Path integral formulation, Scattering theory, Schrödinger equation, Quantum field theory, Quantum statistical mechanics Adiabatic approximation, Correspondence principle, Free particle, Hamiltonian, Hilbert space, Identical particles, Matrix Mechanics, Planck's constant, Operators, Quanta, Quantization, Quantum entanglement, Quantum harmonic oscillator, Quantum number, Quantum tunneling, Schrödinger's cat, Dirac equation, Spin, Wavefunction, Wave mechanics, Wave-particle duality, Zero-point energy, Pauli Exclusion Principle, Heisenberg Uncertainty Principle
Theory of relativity Special relativity, General relativity, Einstein field equations Covariance, Einstein manifold, Equivalence principle, Four-momentum, Four-vector, General principle of relativity, Geodesic motion, Gravity, Gravitoelectromagnetism, Inertial frame of reference, Invariance, Length contraction, Lorentzian manifold, Lorentz transformation, Metric, Minkowski diagram, Minkowski space, Principle of Relativity, Proper length, Proper time, Reference frame, Rest energy, Rest mass, Relativity of simultaneity, Spacetime, Special principle of relativity, Speed of light, Stress-energy tensor, Time dilation, Twin paradox, World line
Major fieldsEdit
Meissner effect
A magnet levitating above a high-temperature superconductor (with boiling liquid nitrogen underneath), demonstrating the Meissner effect.
Contemporary research in physics is divided into several distinct fields that study different aspects of the material world. Condensed matter physics, by most estimates the largest single field of physics, is concerned with how the properties of bulk matter, such as the ordinary solids and liquids we encounter in everyday life, arise from the properties and mutual interactions of the constituent atoms. The field of atomic, molecular, and optical physics deals with the behavior of individual atoms and molecules, and in particular the ways in which they absorb and emit light. Since the 20th century, the individual fields of physics have become increasingly specialized, and nowadays it is not uncommon for physicists to work in a single field for their entire careers. "Universalists" like Albert Einstein (18791955) who were comfortable working in multiple fields of physics, are now very rare.
Field Subfields Major theories Concepts
Gravitation physics, | Big Bang, Lambda-CDM model, Cosmic inflation, General relativity, Law of universal gravitation Gravity, Gravitational radiation, | Atomic, molecular, and optical physics Atomic physics, Molecular physics, Atomic and Molecular astrophysics, Chemical physics, Optics, Photonics Quantum optics, Quantum chemistry, Quantum information science Atom, Molecule, Diffraction, Electromagnetic radiation, Laser, Polarization, Spectral line
Particle physics Nuclear physics, Particle physics phenomenology Standard Model, Quantum field theory, Quantum chromodynamics, Electroweak theory, Effective field theory, Lattice field theory, Lattice gauge theory, Gauge theory, Supersymmetry, Grand unification theory, Superstring theory, M-theory Fundamental force (gravitational, electromagnetic, weak, strong), Elementary particle, Spin, Antimatter, Spontaneous symmetry breaking, Brane, String, Quantum gravity, Theory of everything, Vacuum energy
Condensed matter physics Solid state physics, Nanoscale and Mesoscopic physics, Polymer physics Many-body theory Phases (gas, liquid, solid, Electrical conduction, Magnetism, Self-organization, Spin, Spontaneous symmetry breaking
Theoretical and experimental physics Edit
The culture of physics research differs from the other sciences in the separation of theory and experiment. Since the 20th century, most individual physicists have specialized in either theoretical physics or experimental physics. The great Italian physicist Enrico Fermi (19011954), who made fundamental contributions to both theory and experimentation in nuclear physics, was a notable exception. In contrast, almost all the successful theorists in biology and chemistry (e.g. American quantum chemist and biochemist Linus Pauling) have also been experimentalists, though this is changing as of late.
Roughly speaking, theorists seek to develop through abstractions and mathematical models theories that can both describe and interpret existing experimental results and successfully predict future results, while experimentalists devise and perform experiments to explore new phenomena and test theoretical predictions. Although theory and experiment are developed separately, they are strongly dependent on each other. However, theoretical research in physics may further be considered to draw from mathematical physics and computational physics in addition to experimentation. Progress in physics frequently comes about when experimentalists make a discovery that existing theories cannot account for, necessitating the formulation of new theories. Likewise, ideas arising from theory often inspire new experiments. In the absence of experiment, theoretical research can go in the wrong direction; this is one of the criticisms that has been leveled against M-theory, a popular theory in high-energy physics for which no practical experimental test has ever been devised.
Phenomenology Edit
Phenomenology is intermediate between experiment and theory. It is more abstract and includes more logical steps than experiment, but is more directly tied to experiment than theory. The boundaries between theory and phenomenology, and between phenomenology and experiment, are somewhat fuzzy and to some extent depend on the understanding and intuition of the scientist describing these. An example is Einstein's 1905 paper on the photoelectric effect, "On a Heuristic Viewpoint Concerning the Production and Transformation of Light".
Applied physics Edit
Applied physics is physics that is intended for a particular technological or practical use, as for example in engineering, as opposed to basic research. This approach is similar to that of applied mathematics. Applied physics is rooted in the fundamental truths and basic concepts of the physical sciences but is concerned with the utilization of scientific principles in practical devices and systems, and in the application of physics in other areas of science. "Applied" is distinguished from "pure" by a subtle combination of factors such as the motivation and attitude of researchers and the nature of the relationship to the technology or science that may be affected by the work. [1]
Branches of Applied Physics
Accelerator physics, Acoustics, Agrophysics, Biophysics, Chemical Physics, Communication Physics, Econophysics, Engineering physics, Fluid dynamics, Geophysics, Medical physics, Nanotechnology, Optoelectronics, Photovoltaics, Physical chemistry, Physics of computation, Quantum chemistry, Quantum information science, Vehicle dynamics
Main article: History of physics
Further information: Famous physicists, Nobel Prize in physics
Francesco Hayez 001
Since antiquity, people have tried to understand the behavior of matter: why unsupported objects drop to the ground, why different materials have different properties, and so forth. The character of the Universe was also a mystery, for instance the Earth and the behavior of celestial objects such as the Sun and the Moon. Several theories were proposed, most of which were wrong. These first theories were largely couched in philosophical terms, and never verified by systematic experimental testing as is popular today. The works of Ptolemy and Aristotle, however, were also not always found to match everyday observations. There were exceptions and there are anachronisms - for example, Indian philosophers and astronomers gave many correct descriptions in atomism and astronomy, and the Greek thinker Archimedes derived many correct quantitative descriptions of mechanics and hydrostatics.
The willingness to question previously held truths and search for new answers eventually resulted in a period of major scientific advancements, now known as the Scientific Revolution of the late 17th century. The precursors to the scientific revolution can be traced back to the important developments made in India and Persia, including the elliptical model of the planets based on the heliocentric solar system of gravitation developed by Indian mathematician-astronomer Aryabhata; the basic ideas of atomic theory developed by Hindu and Jaina philosophers; the theory of light being equivalent to energy particles developed by the Indian Buddhist scholars Dignāga and Dharmakirti; the optical theory of light developed by Persian scientist Alhazen; the Astrolabe invented by the Persian Mohammad al-Fazari; and the significant flaws in the Ptolemaic system pointed out by Persian scientist Nasir al-Din al-Tusi.
As the influence of the Islamic Caliphate expanded to Europe, the works of Aristotle preserved by the Arabs, and the works of the Indians and Persians, became known in Europe by the 12th and 13th centuries. This eventually lead to the scientific revolution which culminated with the publication of the Philosophiae Naturalis Principia Mathematica in 1687 by the mathematician, physicist, alchemist and inventor Sir Isaac Newton (1643-1727).
The Scientific Revolution is held by most historians (e.g., Howard Margolis) to have begun in 1543, when the first printed copy of Nicolaus Copernicus's De Revolutionibus (most of which had been written years prior but whose publication had been delayed) was brought to the influential Polish astronomer from Nuremberg.
Sir Isaac Newton
Further significant advances were made over the following century by Galileo Galilei, Christiaan Huygens, Johannes Kepler, and Blaise Pascal. During the early 17th century, Galileo pioneered the use of experimentation to validate physical theories, which is the key idea in modern scientific method. Galileo formulated and successfully tested several results in dynamics, in particular the Law of Inertia. In 1687, Newton published the Principia, detailing two comprehensive and successful physical theories: Newton's laws of motion, from which arise classical mechanics; and Newton's Law of Gravitation, which describes the fundamental force of gravity. Both theories agreed well with experiment. The Principia also included several theories in fluid dynamics. Classical mechanics was re-formulated and extended by Leonhard Euler, French mathematician Joseph-Louis Comte de Lagrange, Irish mathematical physicist William Rowan Hamilton, and others, who produced new results in mathematical physics. The law of universal gravitation initiated the field of astrophysics, which describes astronomical phenomena using physical theories.
After Newton defined classical mechanics, the next great field of inquiry within physics was the nature of electricity. Observations in the 17th and 18th century by scientists such as Robert Boyle, Stephen Gray, and Benjamin Franklin created a foundation for later work. These observations also established our basic understanding of electrical charge and current.
File:James Clerk Maxwell.jpg
In 1821, the English physicist and chemist Michael Faraday integrated the study of magnetism with the study of electricity. This was done by demonstrating that a moving magnet induced an electric current in a conductor. Faraday also formulated a physical conception of electromagnetic fields. James Clerk Maxwell built upon this conception, in 1864, with an interlinked set of 20 equations that explained the interactions between electric and magnetic fields. These 20 equations were later reduced, using vector calculus, to a set of four equations by Oliver Heaviside.
Albert Einstein Head
Albert Einstein in 1947
In addition to other electromagnetic phenomena, Maxwell's equations also can be used to describe light. Confirmation of this observation was made with the 1888 discovery of radio by Heinrich Hertz and in 1895 when Wilhelm Roentgen detected X rays. The ability to describe light in electromagnetic terms helped serve as a springboard for Albert Einstein's publication of the theory of special relativity in 1905. This theory combined classical mechanics with Maxwell's equations. The theory of special relativity unifies space and time into a single entity, spacetime. Relativity prescribes a different transformation between reference frames than classical mechanics; this necessitated the development of relativistic mechanics as a replacement for classical mechanics. In the regime of low (relative) velocities, the two theories agree. Einstein built further on the special theory by including gravity into his calculations, and published his theory of general relativity in 1915.
One part of the theory of general relativity is Einstein's field equation. This describes how the stress-energy tensor creates curvature of spacetime and forms the basis of general relativity. Further work on Einstein's field equation produced results which predicted the Big Bang, black holes, and the expanding universe. Einstein believed in a static universe and tried (and failed) to fix his equation to allow for this. However, by 1929 Edwin Hubble's astronomical observations suggested that the universe is expanding.
From the late 17th century onwards, thermodynamics was developed by physicist and chemist Boyle, Young, and many others. In 1733, Bernoulli used statistical arguments with classical mechanics to derive thermodynamic results, initiating the field of statistical mechanics. In 1798, Thompson demonstrated the conversion of mechanical work into heat, and in 1847 Joule stated the law of conservation of energy, in the form of heat as well as mechanical energy. Ludwig Boltzmann, in the 19th century, is responsible for the modern form of statistical mechanics.
In 1895, Röntgen discovered X-rays, which turned out to be high-frequency electromagnetic radiation. Radioactivity was discovered in 1896 by Henri Becquerel, and further studied by Marie Curie, Pierre Curie, and others. This initiated the field of nuclear physics.
In 1897, Joseph J. Thomson discovered the electron, the elementary particle which carries electrical current in circuits. In 1904, he proposed the first model of the atom, known as the plum pudding model. (The existence of the atom had been proposed in 1808 by John Dalton.)
These discoveries revealed that the assumption of many physicists that atoms were the basic unit of matter was flawed, and prompted further study into the structure of atoms.
Ernest Rutherford
Ernest Rutherford
In 1911, Ernest Rutherford deduced from scattering experiments the existence of a compact atomic nucleus, with positively charged constituents dubbed protons. Neutrons, the neutral nuclear constituents, were discovered in 1932 by Chadwick. The equivalence of mass and energy (Einstein, 1905) was spectacularly demonstrated during World War II, as research was conducted by each side into nuclear physics, for the purpose of creating a nuclear bomb. The German effort, led by Heisenberg, did not succeed, but the Allied Manhattan Project reached its goal. In America, a team led by Fermi achieved the first man-made nuclear chain reaction in 1942, and in 1945 the world's first nuclear explosive was detonated at Trinity site, near Alamogordo, New Mexico.
In 1900, Max Planck published his explanation of blackbody radiation. This equation assumed that radiators are quantized, which proved to be the opening argument in the edifice that would become quantum mechanics. By introducing discrete energy elvels, Planck, Einstein, Niels Bohr, and others developed quantum theories to explain various anomalous experimental results. Quantum mechanics was formulated in 1925 by Heisenberg and in 1926 by Schrödinger and Paul Dirac, in two different ways that both explained the preceding heuristic quantum theories. In quantum mechanics, the outcomes of physical measurements are inherently probabilistic; the theory describes the calculation of these probabilities. It successfully describes the behavior of matter at small distance scales. During the 1920s Schrödinger, Heisenberg, and Max Born were able to formulate a consistent picture of the chemical behavior of matter, a complete theory of the electronic structure of the atom, as a byproduct of the quantum theory.
Quantum field theory was formulated in order to extend quantum mechanics to be consistent with special relativity. It was devised in the late 1940s with work by Richard Feynman, Julian Schwinger, Sin-Itiro Tomonaga, and Freeman Dyson. They formulated the theory of quantum electrodynamics, which describes the electromagnetic interaction, and successfully explained the Lamb shift. Quantum field theory provided the framework for modern particle physics, which studies fundamental forces and elementary particles.
Chen Ning Yang and Tsung-Dao Lee, in the 1950s, discovered an unexpected asymmetry in the decay of a subatomic particle. In 1954, Yang and Robert Mills then developed a class of gauge theories which provided the framework for understanding the nuclear forces. The theory for the strong nuclear force was first proposed by Murray Gell-Mann. The electroweak force, the unification of the weak nuclear force with electromagnetism, was proposed by Sheldon Lee Glashow, Abdus Salam and Steven Weinberg and confirmed in 1964 by James Watson Cronin and Val Fitch. This led to the so-called Standard Model of particle physics in the 1970s, which successfully describes all the elementary particles observed to date.
Quantum mechanics also provided the theoretical tools for condensed matter physics, whose largest branch is solid state physics. It studies the physical behavior of solids and liquids, including phenomena such as crystal structures, semiconductivity, and superconductivity. The pioneers of condensed matter physics include Felix Bloch, who created a quantum mechanical description of the behavior of electrons in crystal structures in 1928. The transistor was developed by physicists John Bardeen, Walter Houser Brattain and William Bradford Shockley in 1947 at Bell Telephone Laboratories.
The two themes of the 20th century, general relativity and quantum mechanics, appear inconsistent with each other. General relativity describes the universe on the scale of planets and solar systems while quantum mechanics operates on sub-atomic scales. This challenge is being attacked by string theory, which treats spacetime as composed, not of points, but of one-dimensional objects, strings. Strings have properties like a common string (e.g., tension and vibration). The theories yield promising, but not yet testable results. The search for experimental verification of string theory is in progress.
WYP2005 logo
Future directions Edit
Main article: Unsolved problems in physics
Research in physics is progressing constantly on a large number of fronts, and is likely to do so for the foreseeable future.
In condensed matter physics, the biggest unsolved theoretical problem is the explanation for high-temperature superconductivity. Strong efforts, largely experimental, are being put into making workable spintronics and quantum computers.
In particle physics, the first pieces of experimental evidence for physics beyond the Standard Model have begun to appear. Foremost amongst these are indications that neutrinos have non-zero mass. These experimental results appear to have solved the long-standing solar neutrino problem in solar physics. The physics of massive neutrinos is currently an area of active theoretical and experimental research. In the next several years, particle accelerators will begin probing energy scales in the TeV range, in which experimentalists are hoping to find evidence for the Higgs boson and supersymmetric particles.
Theoretical attempts to unify quantum mechanics and general relativity into a single theory of quantum gravity, a program ongoing for over half a century, have not yet borne fruit. The current leading candidates are M-theory, superstring theory and loop quantum gravity.
Although much progress has been made in high-energy, quantum, and astronomical physics, many everyday phenomena, involving complexity, chaos, or turbulence are still poorly understood. Complex problems that seem like they could be solved by a clever application of dynamics and mechanics, such as the formation of sandpiles, nodes in trickling water, the shape of water droplets, mechanisms of surface tension catastrophes, or self-sorting in shaken heterogeneous collections are unsolved. These complex phenomena have received growing attention since the 1970s for several reasons, not least of which has been the availability of modern mathematical methods and computers which enabled complex systems to be modeled in new ways. The interdisciplinary relevance of complex physics has also increased, as exemplified by the study of turbulence in aerodynamics or the observation of pattern formation in biological systems. In 1932, Horace Lamb correctly prophesied:
See also Edit
Notes Edit
Further reading Edit
Popular ReadingEdit
University Level TextbooksEdit
• Feynman, Richard; Leighton, Robert; Sands, Matthew (1989). Feynman Lectures on Physics, Addison-Wesley. ISBN 0201510030.
• Feynman, Richard. Exercises for Feynman Lectures Volumes 1-3, Caltech. ISBN 2356487891.
• Knight, Randall (2004). Physics for Scientists and Engineers: A Strategic Approach, Benjamin Cummings. ISBN 0805386858.
• Resnick, Robert; Halliday, David; Walker, Jearl. Fundamentals of Physics.
• Hewitt, Paul (2001). Conceptual Physics with Practicing Physics Workbook (9th ed.), Addison Wesley. ISBN 0321052021.
• Giancoli, Douglas (2005). Physics: Principles with Applications (6th ed.), Prentice Hall. ISBN 0130606200.
• Wilson, Jerry; Buffa, Anthony (2002). College Physics (5th ed.), Prentice Hall. ISBN 0130676446.
• Schiller, Christoph (2005). Motion Mountain: The Free Physics Textbook.
• H. C. Verma (2005). Concepts of Physics, Bharti Bhavan. ISBN 8177091875.
• Thornton, Stephen T.; Marion, Jerry B. (2003). Classical Dynamics of Particles and Systems (5th ed.), Brooks Cole. ISBN 0534408966.
• Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.), Prentice Hall. ISBN 013805326X.
• Wangsness, Roald K. (1986). Electromagnetic Fields (2nd ed.), Wiley. ISBN 0471811866.
• Fowles, Grant R. (1989). Introduction to Modern Optics, Dover Publications. ISBN 0486659577.
• Schroeder, Daniel V. (1999). An Introduction to Thermal Physics, Addison Wesley. ISBN 0201380277.
• Kroemer, Herbert; Kittel, Charles (1980). Thermal Physics (2nd ed.), W. H. Freeman Company. ISBN 0716710889.
• Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.), Prentice Hall. ISBN 013805326X.
• Liboff, Richard L. (2002). Introductory Quantum Mechanics, Addison-Wesley. ISBN 0805387145.
• Bohm, David (1989). Quantum Theory, Dover Publications. ISBN 0486659690.
• Eisberg, Robert; Resnick, Robert (1985). Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (2nd ed.), Wiley. ISBN 047187373X.
• Taylor, Edwin F.; Wheeler, John Archibald (1992). Spacetime Physics: Introduction to Special Relativity (2nd ed.), W.H. Freeman. ISBN 0716723271.
• Taylor, Edwin F.; Wheeler, John Archibald (2000). Exploring Black Holes: Introduction to General Relativity, Addison Wesley. ISBN 020138423X.
• Schutz, Bernard F. (1984). A First Course in General Relativity, Cambridge University Press. ISBN 0521277035.
• Bergmann, Peter G. (1976). Introduction to the Theory of Relativity, Dover Publications. ISBN 0486632822.
• Tipler, Paul; Llewellyn, Ralph (2002). Modern Physics (4th ed.), W. H. Freeman. ISBN 0716743450.
• Griffiths, David J. (1987). Introduction to Elementary Particles, Wiley, John & Sons, Inc. ISBN 0471603864.
• Perkins, Donald H. (1999). Introduction to High Energy Physics, Cambridge University Press. ISBN 0521621968.
• Povh, Bogdan (1995). Particles and Nuclei: An Introduction to the Physical Concepts, Springer-Verlag. ISBN 0387594396.
• Menzel, Donald Howard (1961). Mathematical Physics, Dover Publishications. ISBN 0486600564.
• Joos, Georg; Freeman, Ira M. (1987). Theoretical Physics, Dover Publications. ISBN 0486652270.
• Landau, L. D.; Lifshitz, E. M. (1976). Course of Theoretical Physics, Butterworth-Heinemann. ISBN 0750628960.
• Morse, Philip; Feshbach, Herman (2005). Methods of Theoretical Physics, Feshbach Publishing. ISBN 0976202123.
• Arfken, George B.; Weber, Hans J. (2000). Mathematical Methods for Physicists (5th ed.), Academic Press. ISBN 0120598256.
• Goldstein, Herbert (2002). Classical Mechanics, Addison Wesley. ISBN 0201657023.
• Jackson, John D. (1998). Classical Electrodynamics (3rd ed.), Wiley. ISBN 047130932X.
• Landau, L. D.; Lifshitz, E. M. (1972). Mechanics and Electrodynamics, Vol. 1, Franklin Book Company, Inc. ISBN 008016739X.
• Huang, Kerson (1990). Statistical Mechanics, Wiley, John & Sons, Inc. ISBN 0471815187.
• Merzbacher, Eugen (1998). Quantum Mechanics, Wiley, John & Sons, Inc. ISBN 0471887021.
• Peskin, Michael E.; Schroeder, Daniel V. (1994). Introduction to Quantum Field Theory, Perseus Publishing. ISBN 0201503972.
• Thorne, Kip S.; Misner, Charles W.; Wheeler, John Archibald (1973). Gravitation, W.H. Freeman. ISBN 0716703440.
• Wald, Robert M. (1984). General Relativity, University of Chicago Press. ISBN 0226870332.
• Weinberg, Steven (1972). Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, John Wiley & Sons. ISBN 0471925675.
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96dbb992c09a4f13 | ARRA-enabled 'Barracuda' computing cluster allows scientists to team up on larger problems
Within the Department of Energy’s (DOE) EMSL, new high-performance computing breakthroughs often are the result of combining the best of two worlds. Experimental and computational tools are integrated; suites of leading-edge hardware and software are developed in tandem; and, perhaps more than ever, scientists from different disciplines combine expertise. The addition of the Barracuda computing cluster, funded by the American Recovery and Reinvestment Act, has brought a new level of collaboration between teams of domain scientists (such as chemists) and computer scientists.
“If there wasn’t already a walking path there, we would have worn one into the grass,” said Dr. Karol Kowalski, while Dr. Sriram Krishnamoorthy agreed jokingly, referring to the expanse of lawn between EMSL and CSF, the Computational Sciences Facility at Pacific Northwest National Laboratory (PNNL).
“These frequent collaborations help us get the most out of Barracuda as we prepare a major update to NWChem [EMSL’s widely used open-source computational chemistry software application],” Kowalski continued. “The ultimate goal, of course, is to further optimize how we use computations to predict the properties of matter.”
Predicting the properties of matter has intrigued curious minds for thousands of years. Today, computational chemists like Kowalski and computer scientists like Krishnamoorthy are pushing the boundaries of such predictions using Barracuda, which is part of EMSL’s Molecular Science Computing capability. In particular, the focus is on calculating the properties and structures of molecules involved in the most societally important chemical reactions—those related to energy innovations, environmental protection, national security and human health. For example, more efficient solar panels can result from highly
advanced simulations of how electrons reorganize themselves when exposed to light.
The problem with scientifically impactful calculations is they tend to be very costly in terms of time-to-solution.
“We have a code for highly advanced theoretical formalisms for approximate solving of the Schrödinger equation that describe the properties of molecules,” Kowalski said. “But, it requires a high investment of computational resources to achieve reliable answers. Barracuda uses GPUs, or graphics processing units, a new type of architecture that can get to the solution faster. We’re working with people like Sriram, Wenjing Ma, and Oreste Villa [PNNL high-performance computing experts] to translate NWChem for use on this architecture.
“This is the first documented attempt to apply GPU-based technology to the most advanced theoretical methods.”
For researchers creating simulations and models for complex scientific problems, their implementations will translate to scientifically significant answers for larger systems, with a lower investment of resources.
A paradigm shift is happening within the high performance computing world: the move from homogeneous to heterogeneous computer architectures. In other words, rather than using multiple identical cores in parallel to solve problems, new systems are using multiple types of cores. In Barracuda’s case, it uses both CPUs (central processing units) and GPUs. With 60 nodes, each node of Barracuda consists of two quad-core Intel Xeon X5560 CPUs with 8 MB L2 cache running at 2.80 GHz.
The strategy is part of this decade’s “holy grail” quest to achieve exascale computing—a thousandfold increase in performance over today’s fastest supercomputers.
GPUs originated in the late 1990s as an innovation for the video game and computer graphics industries. Surprisingly, they have risen to prominence in broader computing applications.
“At first, the motivation was quickly manipulating a screen full of pixels,” said Krishnamoorthy. “Traditional CPUs weren’t very good at it, so a new architecture was built to move lots of data from memory to the monitor. Over time, people realized that GPUs are not only fast, but they offer significant improvements over CPUs in memory bandwidth and power efficiency. Now, these advantages are being applied far beyond graphics.”
In fact, GPU computing at its present state can bring about significant increases in overall speed, using its advantages to handle the most computationally intensive tasks and remove data bottlenecks during calculations.
Of course, a new generation of GPU-enhanced hardware is only as good as the software developed to run on it. As new supercomputers around the world are being built with GPUs, including Titan at DOE’s Oak Ridge National Laboratory, EMSL and PNNL scientists are preparing for the shift by using Barracuda to help them develop the GPU extension of NWChem—an overhaul that adds such functionality to the majority of the software application. Specifically, it improves highly accurate methods accounting for instantaneous interactions between electrons and methods designed to treat very large systems (plainwave density functional theory methods).
“We’re taking NWChem to the next level, to harness the power of GPUs for studying molecular systems,” Krishnamoorthy said. “But, the key is to keep it flexible enough to work on a large variety of heterogeneous architectures. Nvidia’s CUDA, or compute unified device architecture, is just one GPU computing engine, but there are others such as the [Khronos Group’s] optimize applications for GPU architectures, PNNL is the first DOE national laboratory to be recognized by Nvidia as a CUDA Research Center.
More information: Ma W, et al. In press. “Optimizing tensor contraction expressions for hybrid CPU-GPU execution.” Cluster Computing, Online First. DOI: 10.1007/s10586-011-0179-2
Ma W, et al. 2011. “GPU-Based Implementations of the Noniterative Regularized-CCSD(T) Corrections: Applications to Strongly Correlated Systems.” Journal of Chemical Theory and Computation 7(5):1316-1327. DOI: 10.1021/ct1007247
Ma W, et al. 2010. “Acceleration of Streamed Tensor Contraction Expressions on PGPU-Based Clusters.” In Cluster Computing (CLUSTER), Proceedings of the 2010 IEEE International Conference on Cluster Computing, pp.207-216. September 20-24, 2010, Heraklion, Crete. Institute of Electrical and Electronic Engineers, Piscataway, N.J. DOI: 10.1109/CLUSTER.2010.26 |
06c2b93c3e0d4f4d | Give to MAT
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Multimodal Representation of Quantum Mechanics:
"The Hydrogen Atom"
Professor JoAnn Kuchera-Morin, Media Arts and Technology, professor Luca Peliti, Physics Department, and PhD student Lance Putnam, Media Arts and Technology
As the sciences increasingly rely on mathematical constructs to describe the invisible processes of nature, it is important to remain cognizant of the effectiveness of empirical observation towards gaining new insights. Digital systems provide not only a means of simulating models, but also a medium for communicating through image and sound.
This work interactively visualizes and sonifies the wavefunction of an electron of a single hydrogen atom. The atomic orbitals are modeled as solutions to the time-dependent Schrödinger equation with a spherically symmetric potential given by Coulomb's law of electrostatic force. Different orbitals of the electron can be combined in superposition to observe dynamic behaviors such as photon emission and absorption.
Hydrogen Atom
Video arrow
The interactive component of the simulation allows one to fly through the atom with a probe that emits "stream particles" that follow along the largest changes in the probability current and gradient of the electron. The electron probability amplitude is sonified by scanning through groups of stream particles in the space. The pitch can be adjusted by the rate at which a particular set of stream particles is scanned across. This allows us to give the sonification procedure a certain type of musicality, by assigning specific pitches to different features in the wavefunction.
This investigation is just the beginning of an effort to multimodally represent mathematical models used in physical and theoretical sciences. By finding a common meeting ground, artists and scientists can share insights and pursue similar fundamental questions about symmetry, pattern formation, and emergence.
Hydrogen Atom
Images by Lance Putnam
More Research Spotlights: << | >> |
b5f936c04a6a23ee | Creation, Providence, and Miracle
Creation, Providence, and Miracle
Dr. William Lane Craig
In treating divine action in the world, we must distinguish between creation, providence, and miracle. Creation has typically been taken to involve God's originating the world (creatio originans) and His sustaining the world in being (creatio continuans). A careful analysis of these two notions serves to differentiate creation from conservation. Providence is God's control of the world, either through secondary causes (providentia ordinaria) or supernaturally (providentia extraordinaria). A doctrine of divine middle knowledge supplies the key to understanding God's providence over the world mediated through secondary causes. Miracles are extraordinary acts of providence which should not be conceived, properly speaking, as violations of the laws of nature, but as the production of events which are beyond the causal powers of the natural entities existing at the relevant time and place.
Source: In Philosophy of Religion, ed. Brian Davies (Washington, D.C.: Georgetown University Press, 1998), pp. 136-162
Creatio Ex Nihilo
"In the beginning God created the heavens and the earth" (Gen. 1.1). With majestic simplicity the author of the opening chapter of Genesis thus differentiated his viewpoint, not only from that of the ancient creation myths of Israel’s neighbors, but also effectively from pantheism, panentheism, and polytheism. For the author of Genesis 1, no pre-existent material seems to be assumed, no warring gods or primordial dragons are present--only God, who is said to "create" (bara, a word used only with God as its subject and which does not presuppose a material substratum) "the heavens and the earth" (et hassamayim we et ha ares, a Hebrew expression for the totality of the world or, more simply, the universe). Moreover, this act of creation took place "in the beginning" (bereshith, used here as in Is. 46.10 to indicate an absolute beginning). The author thereby gives us to understand that the universe had a temporal origin and thus implies creatio ex nihilo in the temporal sense that God brought the universe into being without a material cause at some point in the finite past.{1}
Later biblical authors so understood the Genesis account of creation.{2} The doctrine of creatio ex nihilo is also implied in various places in early extra-biblical Jewish literature.{3} And the Church Fathers, while heavily influenced by Greek thought, dug in their heels concerning the doctrine of creation, sturdily insisting, with few exceptions, on the temporal creation of the universe ex nihilo in opposition to the eternity of matter.{4} A tradition of robust argumentation against the past eternity of the world and in favor of creatio ex nihilo, issuing from the Alexandrian Christian theologian John Philoponus, continued for centuries in Islamic, Jewish, and Christian thought.{5} In 1215, the Catholic church promulgated temporal creatio ex nihilo as official church doctrine at the Fourth Lateran Council, declaring God to be "Creator of all things, visible and invisible, . . . who, by His almighty power, from the beginning of time has created both orders in the same way out of nothing." This remarkable declaration not only affirms that God created everything extra se without any material cause, but even that time itself had a beginning. The doctrine of creation is thus inherently bound up with temporal considerations and entails that God brought the universe into being at some point in the past without any antecedent or contemporaneous material cause.
At the same time, the Christian Scriptures also suggest that God is engaged in a sort of on-going creation, sustaining the universe in being. Christ "reflects the glory of God and bears the very stamp of His nature, upholding the universe by his word of power" (Heb. 1.3). Although relatively infrequently attested in Scripture in comparison with the abundant references to God’s original act of creation, the idea of continuing creation came to constitute an important aspect of the doctrine of creation as well. For Thomas Aquinas, for example, this aspect becomes the core doctrine of creation, the question of whether the world’s reception of being from God had a temporal commencement or not having only secondary importance.{6} For Aquinas creation is the immediate bestowal of being and as such belongs only to God, the universal principle of being; therefore, even if creatures have existed from eternity, they are still created ex nihilo in this metaphysical sense.
Thus, God is conceived in Christian theology to be the cause of the world both in His initial act of bringing the universe into being and in His on-going conservation of the world in being. These two actions have been traditionally classed as species of creatio ex nihilo, namely, creatio originans and creatio continuans. While this is a handy rubric, it unfortunately quickly becomes problematic if pressed to technical precision. As Philip Quinn points out{7}, if we say that a thing is created at a time t only if t is the first moment of the thing’s existence, then the doctrine of creatio continuans lands us in a bizarre form of occasionalism, according to which no persisting individuals exist. At each instant God creates a new individual, numerically distinct from its chronological predecessor, so that diachronic personal identity and agency are precluded.
Rather than re-interpret creation in such a way as to not involve a time at which a thing first begins to exist, we ought to recognize that creatio continuans is but a façon de parler and that creation needs to be distinguished from conservation. As John Duns Scotus observed,
Properly speaking . . . it is only true to say that a creature is created at the first moment (of its existence) and only after that moment is it conserved, for only then does its being have this order to itself as something that was, as it were, there before. Because of these different conceptual relationships implied by the words ‘create’ and ‘conserve’ it follows that one does not apply to a thing when the other does.{8}
Intuitively, creation involves God’s bringing something into being. Thus, if God creates some entity e (whether an individual or an event) at a time t (whether an instant or finite interval), then e comes into being at t. We can explicate this notion as follows:
E1. e comes into being at t iff (i) e exists at t, (ii) t is the first time at which e exists, and (iii) e’s existing at t is a tensed fact
E2. God creates e at t iff God brings it about that e comes into being at t
God’s creating e involves e’s coming into being, which is an absolute beginning of existence, not a transition of e from non-being into being. In creation there is no patient entity on which the agent acts to bring about its effect.{9} It follows that creation is not a type of change, since there is no enduring subject which persists from one state to another. It is precisely for this reason that conservation cannot be properly thought of as essentially the same as creation. For conservation does presuppose a subject which is made to continue from one state to another. In creation God does not act on a subject, but constitutes the subject by His action; in contrast, in conservation God acts on an existant subject to perpetuate its existence. This is the import of Scotus’s remark that only in conservation does a creature "have this order to itself as something that was, as it were, there before."
The fundamental difference between creation and conservation, then, lies in the fact that in conservation, as opposed to creation, there is presupposed a subject on which God acts. Intuitively, conservation involves God’s preservation of that subject in being over time. Conservation ought therefore to be understood in terms of God’s preserving some entity e from one moment of its existence to another. A crucial insight into conservation is that unlike creation, it does involve transition and therefore cannot occur at an instant.{10} We may therefore provide the following explication of divine conservation:
E3. God conserves e iff God acts upon e to bring about e’s existing from t until some t*>t through every sub-interval of the interval [t, t* ]
Creation and conservation thus cannot be adequately analyzed with respect to the divine act alone, but involve relations to the object of the act. The act itself (the causing of existence) may be the same in both cases, but in one case may be instantaneous and presupposes no prior object, whereas in the other case occurs over an interval and does involve a prior object.
The doctrine of creation also involves an important metaphysical feature which is rarely appreciated: it commits one to a tensed or, in McTaggart’s convenient terminology, an A-Theory of time.{11} For if one adopts a tenseless or B-Theory of time, then things do not literally come into existence. Things are then four-dimensional objects which tenselessly subsist and begin to exist only in the sense that their extension along their temporal dimension is finite in the earlier-than direction. The whole four-dimensional, space-time manifold is extrinsically (as opposed to intrinsically) timeless, existing co-eternally with God. The universe thus does not come into being on a B-Theory of time, regardless of whether it has a finite or an infinite past relative to any time. Hence, clause (iii) in E2 represents a necessary feature of creation. In the absence of clause (iii) God’s creation of the universe ex nihilo could be interpreted along tenseless lines to postulate merely the finitude of cosmic time in the earlier than direction.
Since a robust doctrine of creatio ex nihilo thus commits one to an A-Theory of time, we are brought face to face with what has been called "one of the most neglected, but also one of the most important questions in the dialogue between theology and science," namely, the relation between the concept of eternity and that of the spatio-temporal structure of the universe.{12} Since the rise of modern theology with Schleiermacher, the doctrine of creatio originans has been allowed to atrophy, while the doctrine of creatio continuans has assumed supremacy.{13} Undoubtedly this was largely due to theologians’ fear of a conflict with science, which creatio continuans permitted them to avoid by operating only within the safe harbor of metaphysics, removed from the realities of the physical, space-time world.{14} But the discovery in this century of the expansion of the universe, first predicted in 1922 by Alexander Friedman on the basis of the General Theory of Relativity, coupled with the Hawking-Penrose singularity theorems of 1968, which demonstrated the inevitability of a past, cosmic singularity as an initial boundary to space-time, forced the doctrine of creatio originans back into the spotlight.{15} As physicists Barrow and Tipler observe, "At this singularity, space and time came into existence; literally nothing existed before the singularity, so, if the Universe originated at such a singularity, we would truly have a creation ex nihilo."{16}
Of course, various and sometimes heroic attempts have been made to avert the initial cosmological singularity posited in the standard Big Bang model and to regain an infinite past. But none of these alternatives has commended itself as more plausible than the standard model. The old steady state model, the oscillating model, and vacuum fluctuation models are now generally recognized among cosmologists to have failed as plausible attempts to avoid the beginning of the universe.{17} Most cosmologists believe that a final theory of the origin of the universe must await the as yet undiscovered quantum theory of gravity. Such quantum gravity models may or may not involve an initial singularity, although attention has tended to focus on those that do not. But even those that eliminate the initial singularity, such as the Hartle-Hawking model, still involve a merely finite past and, on any physically realistic interpretation of such models, imply a beginning of the universe. This is due to the peculiar feature of such models’ employment of imaginary, rather than real, values for the time variable in the equations governing the universe during the first 10-43 sec of its existence. Imaginary quantities in science are fictional, without physical significance.{18} Thus, use of such numbers is a mathematical "trick" or auxiliary device to arrive at physically significant quantities represented by real numbers. The Euclidean four-space from which classical space-time emerges in such models is thus a mathematical fiction, a way of modeling the early universe which should not be taken as a literal description.{19}
Now it might be said that so-called "imaginary time" just is a spatial dimension and to that extent is physically intelligible and so is to be realistically construed. But now the metaphysician must surely protest the reductionistic view of time which such an account presupposes. Time as it plays a role in physics is an operationally defined quantity varying from theory to theory: in the Special Theory of Relativity it is a quantity defined via clock synchronization by light signals, in classical cosmology it is a parameter assigned to spatial hyper-surfaces of homogeneity, in quantum cosmology it is a quantity internally constructed out of the curvature variables of three-geometries. But clearly these are but pale abstractions of time itself.{20} For a series of mental events alone, a succession of contents of consciousness, is sufficient to ground time itself. An unembodied consciousness which experienced a succession of mental states, say, by counting, would be temporal; that is to say, time would in such a case exist, and that wholly in the absence of any physical processes. I take this simple consideration to be a knock-down argument that time as it plays a role in physics is at best a measure of time, rather than constitutive or definitive of time. Hence, even if one were to accept at face value the claim of quantum cosmological models that physical time really is imaginary prior to the Planck time, that is to say, is a spatial dimension, that fact says absolutely nothing at all about time itself. When it is said that such a regime exists timelessly, all that means is that our physical measures of time (which in physics are taken to define time) break down under such conditions. That should hardly surprise. But time itself must characterize such a regime for the simple reason that it is not static. I am astonished that quantum theorists can assert that the quantum regime is on the one hand a state of incessant activity or change and yet is on the other not characterized by time. If this is not to be incoherent, such a statement can only mean that our concepts of physical time are inapplicable on such a scale, not that time itself disappears. But if time itself characterizes the quantum regime, as it must if change is occurring, then one can regress mentally in time back along the imaginary time dimension through concentric circles on the spherical hyper-surface as they converge toward a non-singular point which represents the beginning of the universe and before which time did not exist. Hartle-Hawking themselves recognize that point as the origin of the universe in their model, but how that point came into being (in metaphysical, that is, ontological, time) is a question not even addressed by their theory.
Hence, even on a naive realist construal of such models, they at best show that that quantity which is defined as time in physics ceases at the Planck time and takes on the characteristics of what physics defines as a spatial dimension. But time itself does not begin at the Planck time, but extends all the way back to the very beginning of the universe. Such theories, if successful, thus enable us to model the origin of the universe without an initial cosmological singularity and, by positing a finite imaginary time on a closed surface prior to the Planck time rather than an infinite time on an open surface, actually support temporal creatio ex nihilo.
But if the spatio-temporal structure of the universe exhibits an origination ex nihilo, then the difficulty concerns how to relate that structure to the divine eternity. For given the reality of tense and God’s causal relation to the world, it is very difficult to conceive how God could remain untouched by the world’s temporality. Imagine God existing changelessly alone without creation, with a changeless and eternal determination to create a temporal world. Since God is omnipotent, His will is done, and a temporal world begins to exist. (We may lay aside for now the question whether this beginning of a temporal creation would require some additional act of intentionality or exercise of power other than God’s timeless determination.) Now in such a case, either God existed temporally prior to creation or He did not. If He did exist alone temporally prior to creation, then God is not timeless, but temporal, and the question is settled. Suppose, then, that God did not exist temporally prior to creation. In that case He exists timelessly sans creation. But once time begins at the moment of creation, God either becomes temporal in virtue of His real, causal relation to time and the world or else He exists as timelessly with creation as He does sans creation. But this second alternative seems quite impossible. At the first moment of time, God stands in a new relation in which He did not stand before (since there was no before). We need not characterize this as a change in God; but there is a real, causal relation which is at that moment new to God and which He does not have in the state of existing sans creation. At the moment of creation, God comes into the relation of causing the universe or at the very least that of co-existing with the universe, relations in which He did not before stand. Hence, even if God remains intrinsically changeless in creating the world, He nonetheless undergoes an extrinsic, or relational, change, which, if He is not already temporal prior to the moment of creation, draws Him into time at that very moment in virtue of His real relation to the temporal, changing universe. So even if God is timeless sans creation, His free decision to create a temporal world constitutes also a free decision on His part to enter into time and to experience the reality of tense and temporal becoming.
The classic Thomistic response to the above argument is, remarkably, to deny that God’s creative activity in the world implies that God is really related to the world. Aquinas tacitly agrees that if God were really related to the temporal world, then He would be temporal.{21} In the coming to be of creatures, certain relations accrue to God anew and thus, if these relations be real for God, He must be temporal in light of His undergoing extrinsic change, wholly apart from the question of whether God undergoes intrinsic change in creating the world. So Thomas denies that God has any real relation to the world. According to Aquinas, while the temporal world does have the real relation of being created by God, God does not have a real relation of creating the temporal world. Since God is immutable, the new relations predicated of Him at the moment of creation are just in our minds; in reality the temporal world itself is created with a relation inhering in it of dependence on God. Hence, God’s timelessness is not jeopardized by His creation of a temporal world.
This unusual doctrine of creation becomes even stranger when we reflect on the fact that in creating the world God does not perform some act extrinsic to His nature; rather the creature (which undergoes no change but simply begins to exist) begins to be with a relation to God of being created by God. According to this doctrine, then, God in freely creating the universe does not really do anything different than He would have, had He refrained from creating; the only difference is to be found in the universe itself: instead of God existing alone sans the universe we have instead a universe springing into being at the first moment of time possessing the property being created by God, even though God, for His part, bears no real reciprocal relation to the universe made by Him.
I think it hardly needs to be said that Thomas’s solution, despite its daring and ingenuity, is extraordinarily implausible. "Creating" clearly describes a relation which is founded on something’s intrinsic properties concerning its causal activity, and therefore creating the world ought to be regarded as a real property acquired by God at the moment of creation. It seems unintelligible, if not contradictory, to say that one can have real effects without real causes. Yet this is precisely what Aquinas affirms with respect to God and the world.
Moreover, it is the implication of Aquinas’s position that God is perfectly similar across possible worlds, the same even in worlds in which He refrains from creation as in worlds in which He creates. For in none of these worlds does God have any relation to anything extra se. In all these worlds God never acts differently, He never cognizes differently, He never wills differently; He is just the simple, unrelated act of being. Even in worlds in which He does not create, His act of being, by which creation is produced, is no different in these otherwise empty worlds than in worlds chock-full of contingent beings of every order. Thomas’s doctrine thus makes it unintelligible why the universe exists rather than nothing. The reason obviously cannot lie in God, either in His nature or His activity (which are only conceptually distinct anyway), for these are perfectly similar in every possible world. Nor can the reason lie in the creatures themselves, in that they have a real relation to God of being freely willed by God. For their existing with that relation cannot be explanatorily prior to their existing with that relation. I conclude, therefore, that Thomas’ solution, based in the denial of God’s real relation to the world, cannot succeed in hermetically sealing off God in atemporality.
The above might lead one to conclude that God existed temporally prior to His creation of the universe in a sort of metaphysical time. But while it makes sense to speak of such a metaphysical time prior to the inception of physical time at the Big Bang (think of God’s counting down to creation: . . ., 3, 2, 1, fiat lux!), the notion of an actual infinity of past events or intervals of time seems strikingly counter-intuitive. Not only would we be forced to swallow all the bizarre and ultimately contradictory consequences of an actual infinite, but we would also be saddled with the prospect of God’s having "traversed" the infinite past one moment at a time until He arrived at the moment of creation, which seems absurd. Moreover, on such an essentially Newtonian view of time, we would have to answer the difficult question which Leibniz lodged against Clarke: why did God delay for infinite time the creation of the world?{22} In view of these perplexities, it seems more plausible to adopt the Leibnizian alternative of some sort of relational view of time according to which time does not exist in the utter absence of events.{23} God existing alone sans creation would be changeless and, hence, timeless, and time would begin at the first event, which, for simplicity’s sake, we may take to be the Big Bang. God’s bringing the initial cosmological singularity into being is simultaneous (or coincident) with the singularity’s coming into being, and therefore God is temporal from the moment of creation onward. Though we might think of God as existing, say, one hour prior to creation, such a picture is, as Aquinas states, purely the product of our imagination and time prior to creation merely an imaginary time (in the phantasmagorical, not mathematical, sense!).{24}
Why, then, did God create the world? It has been said that if God is essentially characterized by self-giving love, creation becomes necessary.{25} But the Christian doctrine of the Trinity suggests another possibility. Insofar as He exists sans creation, God is not, on the Christian conception, a lonely monad, but in the tri-unity of His own being, God enjoys the full and unchanging love relationships among the persons of the Trinity. Creation is thus unnecessary for God and is sheer gift, bestowed for the sake of creatures, that we might experience the joy and fulfillment of knowing God. He invites us, as it were, into the inner-Trinitarian love relationship as His adopted children. Thus, creation, as well as salvation, is sola gratia.
The biblical worldview involves a very strong conception of divine sovereignty over the world and human affairs, even as it presupposes human freedom and responsibility. While too numerous to list here, biblical passages affirming God’s sovereignty have been grouped by D. A. Carson under four main heads: (1) God is the Creator, Ruler, and Possessor of all things, (2) God is the ultimate personal cause of all that happens, (3) God elects His people, and (4) God is the unacknowledged source of good fortune or success.{26} No one taking these passages seriously can embrace currently fashionable libertarian revisionism, which denies God’s sovereignty over the contingent events of history. On the other hand, the conviction that human beings are free moral agents also permeates the Hebrew way of thinking, as is evident from passages listed by Carson under nine heads: (1) People face a multitude of divine exhortations and commands, (2) people are said to obey, believe, and choose God, (3) people sin and rebel against God, (4) people’s sins are judged by God, (5) people are tested by God, (6) people receive divine rewards, (7) the elect are responsible to respond to God’s initiative, (8) prayers are not mere showpieces scripted by God, and (9) God literally pleads with sinners to repent and be saved.{27} These passages rule out a traditional deterministic understanding of divine providence, which precludes human freedom.
Reconciling these two streams of biblical teaching without compromising either has proven extraordinarily difficult. Nevertheless, a startling solution to this enigma emerges from the doctrine of divine middle knowledge crafted by the Counter-Reformation Jesuit theologian Luis Molina.{28} Molina proposes to furnish an analysis of divine knowledge in terms of three logical moments. Although whatever God knows, He knows eternally, so that there is no temporal succession in God’s knowledge, nonetheless there does exist a sort of logical succession in God’s knowledge in that His knowledge of certain propositions is conditionally or explanatorily prior to His knowledge of certain other propositions. In the first, unconditioned moment God knows all possibilia, not only all individual essences, but also all possible worlds. Molina calls such knowledge "natural knowledge" because the content of such knowledge is essential to God and in no way depends on the free decisions of His will. By means of His natural knowledge, then, God has knowledge of every contingent state of affairs which could possibly obtain and of what the exemplification of the individual essence of any free creature could freely choose to do in any such state of affairs that should be actual.
In the second moment, God possesses knowledge of all true counterfactual propositions, including counterfactuals of creaturely freedom. Whereas by His natural knowledge God knew what any free creature could do in any set of circumstances, now in this second moment God knows what any free creature would do in any set of circumstances. This is not because the circumstances causally determine the creature’s choice, but simply because this is how the creature would freely choose. God thus knows that were He to actualize certain states of affairs, then certain other contingent states of affairs would obtain. Molina calls this counterfactual knowledge "middle knowledge" because it stands in between the first and third moment in divine knowledge. Middle knowledge is like natural knowledge in that such knowledge does not depend on any decision of the divine will; God does not determine which counterfactuals of creaturely freedom are true or false. Thus, if it is true that If some agent S were placed in circumstances C, then he would freely perform action a, then even God in His omnipotence cannot bring it about that S would freely refrain from a if he were placed in C. On the other hand, middle knowledge is unlike natural knowledge in that the content of His middle knowledge is not essential to God. True counterfactuals are contingently true; S could freely decide to refrain from a in C, so that different counterfactuals could be true and be known by God than those that are. Hence, although it is essential to God that He have middle knowledge, it is not essential to Him to have middle knowledge of those particular propositions which He does in fact know.
Intervening between the second and third moments of divine knowledge stands God’s free decree to actualize a world known by Him to be realizable on the basis of His middle knowledge. By His natural knowledge, God knows what is the entire range of logically possible worlds; by His middle knowledge He knows, in effect, what is the proper subset of those worlds which it is feasible for Him to actualize. By a free decision, God decrees to actualize one of those worlds known to Him through His middle knowledge.
Given God’s free decision to actualize a world, in the third and final moment God possesses knowledge of all remaining propositions that are in fact true in the actual world, including future contingent propositions. Such knowledge is denominated "free knowledge" by Molina because it is logically posterior to the decision of the divine will to actualize a world. The content of such knowledge is clearly not essential to God, since He could have decreed to actualize a different world. Had He done so, the content of His free knowledge would be different.
The doctrine of middle knowledge is a doctrine of remarkable theological fecundity. Molina’s scheme would resolve in a single stroke most of the traditional difficulties concerning divine providence and human freedom. Molina defines providence as God’s ordering of things to their ends, either directly or mediately through secondary agents. By His middle knowledge God knows an infinity of orders which He could instantiate because He knows how the creatures in them would in fact freely respond given the various circumstances. He then decides by the free act of His will how He would respond in these various circumstances and simultaneously wills to bring about one of these orders. He directly causes certain circumstances to come into being and others indirectly by causally determined secondary causes. Free creatures, however, He allows to act as He knew they would when placed in such circumstances, and He concurs with their decisions in producing in being the effects they desire. Some of these effects God desired unconditionally and so wills positively that they occur, but others He does not unconditionally desire, but nevertheless permits due to His overriding desire to allow creaturely freedom and knowing that even these sinful acts will fit into the overall scheme of things, so that God’s ultimate ends in human history will be accomplished.{29} God has thus providentially arranged for everything that happens by either willing or permitting it, yet in such a way as to preserve freedom and contingency.
Molinism thus effects a dramatic reconciliation between divine sovereignty and human freedom. Before we embrace such a solution, however, we should ask what objections might be raised against a Molinist account. Surveying the literature, one discovers that the detractors of Molinism tend not so much to criticize the Molinist doctrine of providence as to attack the concept of middle knowledge upon which it is predicated. It is usually alleged that counterfactuals of freedom are not bivalent or are uniformly false or that God cannot know such counterfactual propositions. These objections have been repeatedly refuted by defenders of middle knowledge,{30} though opposition dies hard. But as Freddoso and Wierenga pointed out in an American Philosophical Association session devoted to a recent popularization of libertarian revisionism, until the opponents of middle knowledge answer the refutations of their objections--which they have yet to do,--there is little new to be said in response to their criticisms. Let us consider, then, objections, not to middle knowledge per se, but to a Molinist account of providence.
Robert Adams has recently argued that divine middle knowledge of counterfactuals of creaturely freedom is actually incompatible with human freedom. Although inspired by an argument of William Hasker for the same conclusion, Adams’s argument avoids any appeal to Hasker’s dubious--and, I should say, clearly false--premiss that on the Molinist view counterfactuals of freedom are more fundamental features of the world than are categorical facts.{31} Adams summarizes his argument "very roughly" as follows"
Suppose it is not only true that P would do A if placed in circumstances C; suppose that truth was settled, as Molinism implies, prior to God’s deciding what, if anything, to create, and it would therefore have been a truth even if P had never been in C--indeed even if P had never existed. Then it is hard to see how it can be up to P to determine freely whether P does A in C.{32}
Granted that this summary is admittedly very rough, still it is frustratingly ambiguous. The argument seems to assume as a premiss that there is a true counterfactual of creaturely freedom Ø that If P were in C, P would do A, whose antecedent is true.
Is the objection then supposed to be aimed at the imagined claim that P freely brings about the truth of Ø? Is Adams asserting that P cannot freely bring about the truth of Ø because if, posterior to God’s middle knowledge of Ø, P were not in C or did not exist at all, Ø would still be true, though P never does A in C, which is absurd? Is Adams saying that once the content of God’s middle knowledge is fixed, P is no longer free with respect to A in C? If this is the argument, then it is just the old bogey of fatalism raising its fallacious head in a new guise, as Jonathan Kvanvig points out effectively in his critique of Adams’s similar argument against the temporal pre-existence of "thisnesses."{33} Just as we have the power to act in such a way that were we to do so, future-tense propositions which were in fact true would not have been true, so things can happen differently than they will, in which case thisnesses and singular propositions which in fact exist(ed) would not have existed. Analogously, the Molinist could hold that it is within our power so to act that were we to do so, the truth of counterfactuals of creaturely freedom which is brought about by us would not have been brought about by us.
But perhaps this is not what Adams intends. Maybe the argument is that if Ø is true logically prior to God’s decree, then God still has the choice whether to instantiate worlds in which the antecedent of Ø is true or not. If, then, God decrees to actualize a world in which P is not in C or does not exist at all, Ø still remains true, being part of what Thomas Flint calls the "world type" which confronts God prior to His decree.{34} But then how can P bring about the truth of Ø, if P does not even exist? The Molinist answer to that question, however, is straightforward: P does not in that case bring about the truth of Ø. The hypothetical Molinist against whom this objection is directed holds ex hypothesi "that in the case of a true counterfactual of freedom with a true antecedent it is the agent of the free action described in the consequent who brings it about that the conditional is true."{35} That claim is consistent--though I, like Adams, cannot imagine why any Molinist should want to maintain such a claim--with the further claim that in cases of true counterfactuals of creaturely freedom lacking true antecedents, their truth is not brought about by the agents described. In my opinion, it is better to say that in all cases of true counterfactuals of creaturely freedom, the truth of a counterfactual like Ø is grounded in the obtaining in the actual world (logically prior to God’s decree) of the counterfactual state of affairs that if P were in C, then he would do A, and that any further explanation of this fact implicitly denies libertarianism.{36} Just as a true, contingent, future-tense proposition of the form It will be the case that P does A at t cannot be explained in terms of the truth of a tenseless proposition of the form P does A at t, so it is futile to try to explain true counterfactuals of creaturely freedom of the form If P were in C, P would do A in terms of categorical, indicative propositions of a form like P will do A in C. Just as irreducibly tensed facts are needed in the former case, conditional subjunctive facts are needed in the latter. Be that as it may, however, Adams’s intuitive reasoning provides no grounds for rejecting either the view that the truth of counterfactuals of creaturely freedom with true antecedents is brought about by the agents described or the view that the truth of counterfactuals of creaturely freedom of any kind is not brought about by the agents described.
Having summarized the intuitive basis of his argument, Adams develops the following more rigorous formulation:
1. According to Molinism, the truth of all true counterfactuals of freedom about us is explanatorily prior to God’s decision to create us.
2. God’s decision to create us is explanatorily prior to our existence.
3. Our existence is explanatorily prior to all of our choices and actions.
4. The relation of explanatory priority is transitive.
5. Therefore it follows from Molinism (by 1-4) that the truth of all true counterfactuals of freedom about us is explanatorily prior to all of our choices and actions.
10. It follows also from Molinism that if I freely do action A in circumstances C, then there is a true counterfactual of freedom F*, which says that if I were in C, then I would (freely) do A.
11. Therefore, it follows from Molinism that if I freely do A in C, the truth of F* is explanatorily prior to my choosing and acting as I do in C.
12. If I freely do A in C, no truth that is strictly inconsistent with my refraining from A in C is explanatorily prior to my choosing and acting as I do in C.
13. The truth of F* (which says that if I were in C, then I would do A) is strictly inconsistent with my refraining from A in C.
14. If Molinism is true, then if I freely do A in C, F* both is (by 11) and is not (by 12-13) explanatorily prior to my choosing and acting as I do in C.
15. Therefore, (by 14) if Molinism is true, then I do not freely do A in C.
In his critique of Adams’s earlier anti-Molinist argument, Alvin Plantinga charged that the argument is unsound because the dependency relation involved is not a transitive relation.{37} It seems to me that the present argument shares a similar failing. The notion of "explanatory priority" as it plays a role in the argument seems to me equivocal, and if a univocal sense can be given it, there is no reason to expect it to be transitive.
Consider the explanatory priority in (2) and (3). Here a straightforward interpretation of this notion can be given in terms of the counterfactual dependence of consequent on condition:
2’. If God had not created us, we should not exist.
3’. If we were not to exist, we should not make any of our choices and actions.
Both (2’) and (3’) are metaphysically necessary truths. But this sense of explanatory priority is inapplicable to (1), for
1’. According to Molinism, if all true counterfactuals of freedom about us were not true, God would not have decided to create us
is false. Molinism makes no such assertion, since God might still have created us even if the actually true counterfactuals of creaturely freedom were false or even, per impossible, if no such counterfactuals at all were true. The sense of explanatory priority in (1) must therefore be different than it is in (2) and (3).
The root of the difficulty seems to be a conflation of reasons and causes on Adams’s part. The priority in (2) and (3) is a sort of causal or ontic priority, but the priority in (1) is not causal or ontic, since the truth of all counterfactuals of creaturely freedom is neither a necessary nor a sufficient condition of God’s decision to create us. At best, the truth of such counterfactuals is prior to His decision in providing a partial reason for that decision. Adams’s mistake seems to be that he leaps from God’s decision in the hierarchy of reasons to God’s decision in the hierarchy of causes and by this equivocation tries to make counterfactuals of creaturely freedom explanatorily prior to our free choices.
Perhaps Adams can enunciate a univocal sense of "explanatory priority" that is applicable to (1-3). But I suspect that any such notion would be so generic that we should have to deny its transitivity or so weak that it would not be inimical to human freedom. This suspicion is borne out by Hasker’s very recent attempt to save Adams’s argument by enunciating a very broad conception of explanatory priority which is univocal in (1)-(3) and yet transitive: for contingent states of affairs p and q,
EP: p is explanatorily prior to q iff p must be included in a complete explanation of why q obtains
Hasker asserts, "It should be apparent that explanatory priority as explicated by (EP) is transitive: if p is explanatorily prior to q, and q to r, then clearly P must be included in a complete explanation of why r obtains."{38} But this is not at all clear. As Hasker observes, such a relation must also be irreflexive: "a contingent state of affairs cannot constitute an explanation (in whole or in part) of itself."{39} But if the relation described by (EP) is transitive, then it seems that the condition of irreflexivity is violated. My wife and I not infrequently find ourselves in the situation that I want to do something if she wants to do it, and she wants to do it if I want to do it. Suppose, then, that John is going to the party because Mary is going, and Mary is going to the party because John is going. It follows that if the (EP) relation is transitive, John is going to the party because John is going to the party, which conclusion is obviously wrong. Not only is such a conclusion explanatorily vacuous, but it also implies, in conjunction with (12), that John does not freely go to the party--the very conclusion Hasker wants to avoid.
Adams’s reductio also fails because (12) is false. What is undeniably true is
12’. If I freely do A in C, no truth that is strictly inconsistent with my doing A in C is explanatorily prior to my choosing and acting as I do in C.
But why would we be tempted to think that no truth which is inconsistent with my not doing A in C is explanatorily prior to my freely doing A in C? Certainly
F**. If I were in C, then I would not do A
cannot be explanatorily prior to my freely doing A in C; but why would F** not be explanatorily prior to my freely not doing A in C? Adams’s intuition seems to be that if F* were explanatorily prior to my doing A in C, then I could not refrain from A, which is a necessary condition of my doing A freely.{40} But such an assumption seems doubly wrong. First, it represents once more the fallacious reasoning of fatalism. Though F* is (ex concessionis) in fact explanatorily prior to my freely doing A in C, it is within my power to refrain from doing A in C; only if I were to do so, F* would not then be explanatorily prior to my action nor a part of God’s middle knowledge. Until Adams can show that the content of God’s middle knowledge is a "hard fact," his argument based on (12) is undercut. Second, my being able to refrain from doing A in C is not a necessary condition of my freely doing A in C. For perhaps I do A in C without any causal constraint, but it is also the case that God would not permit me to refrain from A in C. Perhaps it is true that
G. If I were to attempt to refrain from doing A in C, God would not permit me to refrain from doing A in C.
(G) is inconsistent with my refraining from doing A in C, and yet it may well be explanatorily prior to my freely doing A in C. Flint’s essay on infallibility, which appears in the same volume as Adams’s, provides a good illustration.{41} Suppose I am the Pope and A is promulgating ex cathedra only correct doctrine. God knew via His middle knowledge that if I were in C, I would freely do A. Therefore, His creative decree includes my being elected Pope. Given papal infallibility, (G) may also be true and part of God’s middle knowledge, and so is explanatorily prior to my freely doing A in C. But (G) is inconsistent with my refraining from A in C. If such a scenario is coherent--and Flint seems to have refuted all objections to it--, then (12) is false.
The sense of explanatory priority explicated in Hasker’s (EP) is so weak that even if the Molinist simply concedes the truth of (5) in this sense, then (12) is all the more obviously false. For counterfactuals concerning our free actions may be explanatorily prior to those actions only in the sense that God’s reason for creating us may have been in part that He knew we should freely do such things. But it is wholly mysterious how this sense of explanatory priority is incompatible with our performing such actions freely. In a footnote, Hasker claims that Adams’s argument can be freed from reliance on (12), referring the reader to his own argument against middle knowledge.{42} But the duly attentive reader will find in that discussion nothing but a reinteration of Hasker’s previous argument on this score with no refutation of the several objections lodged against it in the literature.{43}
Thus, it seems to me that both sides of Adams’s reductio argument are unsound. His attempt to show that counterfactuals of creaturely freedom are explanatorily prior to our actions fails due to equivocation. And even if they were in some peculiar sense explanatorily prior to our actions because they are true and known by God logically prior to categorical contingent propositions, that would not be incompatible with the freedom of our actions. In short, neither Adams nor Hasker has been able to explicate a sense of explanatory priority with respect to the truth of counterfactuals of creaturely freedom which is both transitive and inimical to human freedom. Given that the objections against a Molinist doctrine of providence thus fail, the theological power of such an account ought to prompt us to avail ourselves of it.
It hardly needs to be demonstrated that the biblical narrative of divine action in the world is a narrative replete with miraculous events. God is conceived to bring about events which natural things, left to their own resources, would not bring about. Hence, miracles are able to function as signs of divine activity.{44} "Why this is a marvel!" exclaims the man born blind, when confronted with the Pharisees’ scepticism concerning Jesus’s rectification of his sight, "Never since the world began has it been heard that any one opened the eyes of a man born blind. If this man were not from God, he could do nothing" (Jn. 9.30-33).
In order to differentiate between the customary way in which God acts and His special, miraculous action, theologians have traditionally distinguished within divine providence God’s providentia ordinaria and His providentia extraordinaria, the latter being identified with miracles. But our exposition of divine providence based on God’s middle knowledge suggests a category of non-miraculous, special providence, which it will be helpful to distinguish. One has in mind here events which are the product of natural causes but whose context is such as to suggest a special divine intention with regard to their occurrence. For example, just as the Israelites approach the Jordan River, a rockslide upstream blocks temporarily the water’s flow, enabling them to cross into the Promised Land (Josh 3. 14-17); or again, as Paul and Silas lie bound in prison for preaching the gospel, an earthquake occurs, springing the prison doors and unfastening their fetters (Acts 16.25-26). By means of His middle knowledge, God can providentially order the world so that the natural causes of such events are, as it were, ready and waiting to produce such events at the propitious time, perhaps in answer to prayers which God knew would be offered. Of course, if such prayers were not be offered or the contingent course of events were to go differently, then God would have known this and so not arranged the natural causes, including human free volitions, to produce the special providential event. Events wrought by special providence are no more outside the course and capacity of nature than are events produced by God’s ordinary providence, but the context of such events, such as their timing, their coincidental nature, and so forth, is such as to point to a special divine intention to bring them about.
If, then, we distinguish miracles from both God’s providentia ordinaria and extraordinaria, how should we characterize miracles? Since the dawning of modernity, miracles have been widely understood to be "violations of the laws of nature." In his Dictionary article on miracles, for example, Voltaire states that according to accepted usage, "A miracle is the violation of mathematical, divine, immutable, eternal laws" and is therefore a contradiction.{45} Voltaire is in fact quite right that such a definition is a contradiction, but this ought to have led him to conclude, not that miracles can thus be defined out of existence, but that the customary definition is defective. Indeed, an examination of the chief competing schools of thought concerning the notion of a natural law in fact reveals that on each theory the concept of a violation of a natural law is incoherent and that miracles need not be so defined. Broadly speaking, there are three main views of natural law today: the regularity theory, the nomic necessity theory, and the causal dispositions theory.{46}
According to the regularity theory, the "laws" of nature are not really laws at all, but just generalized descriptions of the way things happen in the world. They describe the regularities which we observe in nature. Now since on such a theory a natural law is just a generalized description of whatever occurs in nature, it follows that no event which occurs can violate such a law. Instead, it just becomes part of the description. The law cannot be violated, because it describes in a certain generalized form everything that does happen in nature.
According to the nomic necessity theory, natural laws are not merely descriptive, but tell us what can and cannot happen in the natural world. They allow us to make certain counterfactual judgments, such as "If the density of the universe were sufficiently high, it would have re-contracted long ago," which a purely descriptisivist theory would not permit. Again, however, since natural laws are taken to be universal inductive generalizations, a violation of a natural law is no more possible on this theory than on the regularity theory. So long as natural laws are universal generalizations based on experience, they must take account of anything that happens and so would be revised should an event occur which the law does not encompass.
Of course, in practice proponents of such theories do not treat natural laws so rigidly. Rather, natural laws are assumed to have implicit in them certain ceteris paribus assumptions such that a law states what is the case under the assumption that no other natural factors are interfering. When a scientific anomaly occurs, it is usually assumed that some unknown natural factors are interfering, so that the law is neither violated nor revised. But suppose the law fails to describe or predict accurately because some supernatural factors are interfering? Clearly the implicit assumption of such laws is that no supernatural factors as well as no natural factors are interfering. If the law proves inaccurate in a particular case because God is acting, the law is neither violated nor revised. If God brings about some event which a law of nature fails to predict or describe, such an event cannot be characterized as a violation of a law of nature, since the law is valid only on the assumption that no supernatural factors in addition to the natural factors come into play.
On such theories, then, miracles ought to be defined as naturally impossible events, that is to say, events which cannot be produced by the natural causes operative at a certain time and place. Whether an event is a miracle is thus relative to a time and place. Given the natural causes operative at a certain time and place, for example, rain may be naturally inevitable or necessary, but on another occasion, rain may be naturally impossible. Of course, some events, say, the resurrection, may be absolutely miraculous in that they are at every time and place beyond the productive capacity of natural causes.
According to the causal dispositions theory, things in the world have different natures or essences, which include their causal dispositions to affect other things in certain ways, and natural laws are metaphysically necessary truths about what causal dispositions are possessed by various natural kinds of things. For example, "Salt has a disposition to dissolve in water" would state a natural law. If, due to God’s action, some salt failed to dissolve in water, the natural law is not violated, because it is still true that salt has such a disposition. As a result of things’ causal dispositions, certain deterministic natural propensities exist in nature, and when such a propensity is not impeded (by God or some other free agent), then we can speak of a natural necessity. On this theory, an event which is naturally necessary must and does actually occur, since the natural propensity will automatically issue forth in the event if it is not impeded. By the same token, a naturally impossible event cannot and does not actually occur. Hence, a miracle cannot be characterized on this theory as a naturally impossible event. Rather, a miracle is an event which results from causal interference with a natural propensity which is so strong that only a supernatural agent could impede it. The concept of miracle is essentially the same as under the previous two theories, but one just cannot call a miracle "naturally impossible" as those terms are defined in this theory; perhaps we couild adopt instead the nomenclature "physically impossible" to characterize miracles under such a theory.
On none of these theories, then, should miracles be understood as violations of the laws of nature. Rather they are naturally (or physically) impossible events, events which at certain times and places cannot be produced by the relevant natural causes.
Now the question is, what could conceivably transform an event that is naturally impossible into a real historical event? Clearly, the answer is the personal God of theism. For if a transcendent, personal God exists, then He could cause events in the universe that could not be produced by causes within the universe. Given a God who created the universe, who conserves the world in being, and who is capable of acting freely, Christian theologians seem to be entirely justified in maintaining that miracles are possible. Indeed, if it is even (epistemically) possible that such a transcendent, personal God exists, then it is equally possible that He has acted miraculously in the universe. Only to the extent that one has good grounds for believing atheism to be true could one be rationally justified in denying the possibility of miracles. In this light arguments for the impossibility of miracles based upon defining them as violations of the laws of nature become fatuous.
The more interesting question is whether the identification of any event as a miracle is possible. On the one hand, it might be argued that a convincing demonstration that a purportedly miraculous event has occurred would only succeed in forcing us to revise natural law so as to accommodate the event in question. But as Swinburne has argued, a natural law is not abolished because of one exception; the counter-instance must occur repeatedly whenever the conditions for it are present.{47} If an event occurs which is, as Swinburne puts it, contrary to a law of nature and we have reasons to believe that this event would not occur again under similar circumstances, then the law in question will not be abandoned. One may regard an anomalous event as repeatable if another formulation of the natural law better accounts for the event in question, and if it is no more complex than the original law. If any doubt exists, the scientist may conduct experiments to determine which formulation of the law proves more successful in predicting future phenomena. In a similar way, one would have good reason to regard an event as a non-repeatable counter-instance to a law if the reformulated law were much more complicated than the original without yielding better new predictions or by predicting new phenomena unsuccessfully where the original formulation predicted successfully. If the original formulation remains successful in predicting all new phenomena as the data accumulate, while no reformulation does any better in predicting the phenomena and explaining the event in question, then the event should be regarded as a non-repeatable counter-instance to the law. Hence, a miraculous event would not serve to upset the natural law:
We have to some extent good evidence about what are the laws of nature, and some of them are so well-established and account for so many data that any modifications to them which suggest to account for the odd counter-instance would be so clumsy and ad hoc as to upset the whole structure of science. In such cases the evidence is strong that if the purported counter-instance occurred it was a violation of the laws of nature.{48}
Swinburne unfortunately retains the violation concept of miracle, which would invalidate his argument; but if we conceive of a miracle as a naturally impossible event, he is on target in reasoning that the admission of such an event would not lead to the abandonment of a natural law.
On the other hand, it might be urged that if a purportedly miraculous event were demonstrated to have occurred, we should conclude that the event occurred in accordance with unknown natural causes and laws. The question is, what serves to distinguish a genuine miracle from a mere scientific anomaly? Here the religio-historical context of the event becomes crucial. A miracle without a context is inherently ambiguous. But if a purported miracle occurs in a significant religio-historical context, then the chances of its being a genuine miracle are increased. For example, if the miracles occur at a momentous time (say, a man’s leprosy vanishing when Jesus speaks the words, "Be clean!") and do not recur regularly in history, and if the miracles are numerous and various, then the chances of their being the result of some unknown natural causes are reduced. In Jesus’s case, moreover, his miracles and resurrection ostensibly took place in the context of and as the climax to his own unparalleled life and teachings and produced so profound an effect on his followers that they called him Lord. The central miracle of the New Testament, the resurrection of Jesus, was, if it occurred, doubtlessly a miracle. In the first place, the resurrection so exceeds what we know of natural causes that it can only be reasonably attributed to a supernatural cause. The more we learn about cell necrosis, the more evident it becomes that such an event is naturally impossible. If it were the effect of unknown natural causes, then its uniqueness in the history of mankind becomes inexplicable. Secondly, the supernatural explanation is given immediately in the relgio-historical context in which the event occurred. Jesus’s resurrection was not merely an anomalous event, occurring without context; it came as the climax to Jesus’s own life and teachings. As Wolfhart Pannenberg explains,
The resurrection of Jesus acquires such decisive meaning, not merely because someone or anyone has been raised from the dead, but because it is Jesus of Nazareth, whose execution was instigated by the Jews because he had blasphemed against God.
Jesus’ claim to authority, through which he put himself in God’s place, was . . . blasphemous for Jewish ears. Because of this Jesus was then also slandered before the Roman Governor as a rebel. If Jesus really has been raised, this claim has been visibly and unambiguously confirmed by the God of Israel, who was allegedly blasphemed by Jesus.{49}
We should therefore have good reasons to regard Jesus’s resurrection, if it occurred, as truly miraculous. Thus, while it may, indeed, be difficult to know in some cases whether a genuine miracle has occurred, that does not imply pessimism with respect to all cases.
But perhaps the very natural impossibility of a genuine miracle precludes our ever identifying an event as a miracle. As Hume notoriously argued, perhaps it is always more rational to believe that some mistake or deception is at play than to believe that a genuine miracle has occurred.{50} This conclusion is based on Hume’s principle that it is always more probable that the testimony to a miracle is false than that the miracle occurred. But Hume’s principle incorrectly assumes that miracles are highly improbable. With respect to the resurrection of Jesus, for example, the hypothesis "God raised Jesus from the dead" is not improbable, either relative to our background information or to the specific evidence. What is improbable relative to our background information is the hypothesis "Jesus rose naturally from the dead." Given what we know of cell necrosis, that hypothesis is fantastically, even unimaginably, improbable. Conspiracy theories, apparent death theories, hallucination theories, twin brother theories--almost any hypothesis, however unlikely, seems more probable than the hypothesis that all the cells in Jesus’s corpse spontaneously came back to life again. But such naturalistic hypotheses are not more probable than the hypothesis that God raised Jesus from the dead. The evidence for the laws of nature relevant in this case makes it probable that a resurrection from the dead is naturally impossible, which renders improbable the hypothesis that Jesus rose naturally from the grave. But such evidence is simply irrelevant to the probability of the hypothesis that God raised Jesus from the dead. That hypothesis needs to be weighed in light of the specific evidence concerning such facts as the post-mortem appearances of Jesus, the vacancy of the tomb where Jesus’s corpse was laid, the origin of the original disciples’ firm belief that God had, in fact, raised Jesus, and so forth, in the religio-historical context in which the events took place and assessed in terms of the customary criteria used in justifying historical hypotheses, such as explanatory power, explanatory scope, plausibility, and so forth. When this is done, there is no reason a priori to expect that it will be more probable that the testimony is false than that the hypothesis of miracle is true.
Given the God of creation and providence described in classical theism, miracles are possible and, when occurring under certain conditions, plausibly identifiable.
Guide to Further Reading
Bilinskyji, Stephen S. "God, Nature, and the concept of Miracle." Ph.D. dissertation, University of Notre Dame, 1982.
Craig, William Lane and Smith, Quentin. Theism, Atheism, and Big Bang Cosmology. Oxford: Clarendon Press, 1993.
Freddoso, Alfred J. "The Necessity of Nature." Midwest Studies in Philosophy 11 (1986): 215-242.
Hebblethwaite, Brian and Henderson, Edward, eds. Divine Action. Edinburgh: T. & T. Clark, 1990.
Molina, Luis de. On Divine Foreknowledge: Part IV of the "Concordia". Translated with an Introduction and Notes by Alfred J. Freddoso. Ithaca, N.Y.: Cornell University Press, 1988.
Morris, Thomas V., ed. Divine and Human Action. Ithaca, N.Y.: Cornell University Press, 1988. See especially articles by Quinn, Kvanvig and McCann, Flint, and Freddoso.
Quinn, Philip L. "Creation, Conservation, and the Big Bang." In Philosophical Problems of the Internal and External Worlds, pp. 589-612. Edited by John Earman, et.al. Pittsburgh: University of Pittsburgh Press, 1993.
Swinburne, Richard. The Concept of Miracle. New York: Macmillan, 1970.
________, ed. Miracles. Philosophical Topics. New York: Macmillan Publishing Co., 1989.
Tomberlin, James E., ed. Philosophical Perspectives. Vol. 5: Philosophy of Religion. Atascadero, Calif.: Ridgeway Publishing, 1991. See especially articles by Flint, Kvanvig and McCann, and Freddoso.
{1}On Gen. 1.1 as an independent clause which is not a mere chapter title, see Claus Westermann, Genesis 1-11, trans. John Scullion (Minneapolis: Augsburg, 1984), p. 97; John Sailhammer, Genesis, Expositor’s Bible Commentary 2, ed. Frank Gaebelein (Grand Rapids, Mich.: Zondervan, 1990), p. 21.
{2}See, e.g., Prov. 8.27-9; cf. Ps. 104.5-9; also Is. 44.24; 45.18, 24; Ps. 33.9; 90.2; Jn. 1.1-3; Rom. 4.17; 11.36; I Cor. 8.6; Col. 1.16, 17; Heb. 1.2-3; 11.3; Rev. 4.11.
{3}E.g., II Maccabees 7.28; 1QS 3.15; Joseph and Aseneth 12.1-3; II Enoch 25.1ff; 26.1; Odes of Solomon 16.18-19; II Baruch 21.4. For discussion, see Paul Copan, "Is Creatio ex nihilo a Post-biblical Invention?": an Examination of Gerhard May’s Proposal," Trinity Journal 17 (1996): 77-93.
{4}Creatio ex nihilo is affirmed in the Shepherd of Hermas 1.6; 26.1 and the Apostolic Constitutions 8.12.6,8; and by Tatian Oratio ad graecos 5.3; cf.4.1ff; 12.1; Theophilus Ad Autolycum 1.4; 2.4, 10, 13; and Irenaeus Adversus haeresis 3.10.3. For discussion, see Gerhard May, Creatio ex nihilo: The Doctrine of "Creation out of Nothing" in Early Christian Thought, trans. A. S. Worrall (Edinburgh: T. & T. Clark, 1994); cf. Copan’s review article in note 3.
{5}See Richard Sorabji, Time, Creation and the Continuum (Ithaca, N.Y.: Cornell University Press, 1983), pp. 193-252; H. A. Wolfson, "Patristic Arguments against the Eternity of the World," Harvard Theological Review 59 (1966): 354-367; idem, The Philosophy of the Kalam (Cambridge, Mass.: Harvard University Press, 1976; H. A. Davidson, Proofs for Eternity, Creation and the Existence of God in Medieval Islamic and Jewish Philosophy (New York: Oxford University Press, 1987); Richard C. Dales, Medieval Discussions of the Eternity of the World, Studies in Intellectual History 18 (Leiden: E. J. Brill, 1990).
{6}Thomas Aquinas Summa theologiae 1a.2.3; Idem Summa contra gentiles 2.16; 32-38; cf. idem Summa theologiae 1a.45.1; 1a.4b.2. Though Aquinas discusses divine conservation, he does not differentiate it from creation (Idem Summa contra gentiles 3.65; Summa theologiae 1a.104.1).
{7}Philip L. Quinn, "Divine Conservation, Continuous Creation, and Human Action," in The Existence and Nature of God, ed. Alfred J. Freddoso (Notre Dame, Ind.: University of Notre Dame Press, 1983), pp. 55-79. See also idem, "Creation, Conservation, and the Big Bang," in Philosophical Problems of the Internal and External Worlds, ed. John Earman, Allen I. Janis, Gerald J. Massey, and Nicholas Rescher (Pittsburgh: University of Pittsburgh Press, 1993), pp. 589-612; idem, "Divine Conservation, Secondary Causes, and Occasionalism," in Divine and Human Action, ed. Thomas V. Morris (Ithaca, N.Y.: Cornell University Press, 1988), pp. 50-73.
{8}John Duns Scotus, God and Creatures, trans. E. Alluntis and A. Wolter (Princeton: Princeton University Press, 1975), p. 276.
{9}As noted by Alfred J. Freddoso, "Medieval Aristotelianism and the Case against Secondary Causation in Nature," in Divine and Human Action, p. 79. For the scholastics causation is a relation between substances (agents) who act upon other substances (patients) to bring about states of affairs (effects). Creatio ex nihilo is atypical because in that case no patient is acted upon.
{10}To analyze God’s conservation of e , along Quinn’s lines, as God’s re-creation of e anew at each instant of e’s existence is to run the risk of falling into the radical occasionalism of certain medieval Islamic theologians, who, out of their desire to make God not only the creator of the world, but also its ground of being, denied that the constituent atoms of things endure from one instant to another but are rather created in new states of being by God at every successive instant. There are actually two forms of occasionalism threatening Quinn: (1) the occasionalism implied by a literal creatio continuans according to which similar, but numerically distinct, individuals are created at each successive instant, and (2) the occasionalism which affirms diachronic individual identity, but denies the reality of transeunt secondary causation.
{11}On A- versus B-Theories of time: see Richard Gale, "The Static versus the Dynamic Temporal: Introduction," in The Philosophy of Time, ed. Richard M. Gale (New Jersey: Humanities Press, 1968), pp. 65-85.
{12}Wolfhart Pannenberg, "Theological Questions to Scientists," in The Sciences and Theology in the Twentieth Century, ed. A. R. Peacocke, Oxford International Symposia (Stocksfield, England: Oriel Press, 1981), p. 12.
{13}According to Schleiermacher, the original expression of the relation of the world to God, that of absolute dependence, was divided by the Church into two propositions: that the world was created and that the world is sustained. But there is no reason, he asserts, to retain this distinction, since it is linked to the Mosaic account of creation, which is the product of a mythological age. The questions of whether it is possible or necessary to conceive of God as existing apart from created things is a matter of indifference, since it has no bearing on the feeling of absolute dependence on God (F. D. E. Schleiermacher, The Christian Faith, 2d ed., ed. H. R. MacIntosh and J. S. Stewart [Edinburgh: T. & T. Clark, 1928], 36.1, 2; 41; pp. 142-143, 155).
{14}Good examples of such timorousness include Langdon Gilkey, Maker of Heaven and Earth (Garden City, N.Y.: Doubleday, 1959), pp. 310-315; Ian Barbour, Issues in Science and Religion (New York: Harper & Row, 1966), p. 383-385; Arthur Peacocke, Creation and the World of Science (Oxford: Clarendon Press, 1979), pp. 78-79.
{15}Pannenberg, "Questions," p. 12; Ted Peters, "On Creating the Cosmos," in Physics, Philosophy, and Theology: a Common Quest for Understanding, ed. R. Russell, W. Stoeger, and G. Coyne (Vatican City: Vatican Observatory, 1988), p. 291; Robert J. Russell, "Finite Creation without a Beginning: the Doctrine of Creation in Relation to Big Bang and Quantum Cosmologies," in Quantum Cosmology and the Laws of Nature, ed. R. J. Russell, N. Murphy, and C. J. Isham (Vatican City: Vatican Observatory, 1993), pp. 303-310.
{16}John D. Barrow and Frank J. Tipler, The Anthropic Cosmological Principle (Oxford: Clarendon Press, 1986), p. 442.
{17}See William Lane Craig and Quentin Smith, Theism, Atheism, and Big Bang Cosmology (Oxford: Clarendon Press, 1993) for discussion.
{18}In the case of quantum mechanics, for example, "the state vector in the Schrödinger equation is not a physical magnitude, for it is an imaginary function and such functions do not represent real physical magnitudes" (C. Liu, "The Arrow of Time in Quantum Gravity," Philosophy of Science 60 [1993]: 622). Liu contends that in the mature theory of quantum gravity a fundamental arrow of time will obtain.
{19}Hartle-Hawking’s use of imaginary numbers for the time variable allows one to redescribe a universe with an initial cosmological singularity in such a way that that point appears as a non-singular point on a curved hyper-surface. Such a re-description suppresses and also literally spatializes time, which makes evident the purely instrumental character of the model. Such a model could be of great utility to science, but it would not, as Hawking boldly asserts (Stephen Hawking, A Brief History of Time [New York: Bantam Books, 1988], pp. 140-141), eliminate the need for a Creator.
{20}See the interesting lecture by C. Rovelli, "What Does Present Day’s [sic] Physics Tell Us about Time and Space?" Lecture presented at the 1993-94 Annual Series of Lectures of the Center for Philosophy of Science of the University of Pittsburgh, September 17, 1993, p. 17, where he lists eight properties of time as characterized in natural language and compares the concepts of time found in thermodynamics, STR, GTR, and so forth; time as it is defined in quantum gravity has none of the properties usually associated with time.
{21}M.-T. Liske, "Kann Gott reale Beziehungen zu den Geschöpfen haben?" Theologie und Philosophie 68 (1993): 224.
{22}The difficulty may be formulated as follows:
1. If God delays creating at t until t’, He has good reason to do so.
2. If God existed from eternity past until creating at t’, He delayed creating at t.
3. God can have no good reason to do so.
4. \ God did not delay creating at t until t’.
5. \ God has not existed from eternity past until creating at t’.
{23}Such a view would not preclude the existence of time during hiatuses within the series of events, such as are envisioned by Sidney Shoemaker, "Time Without Change," The Journal of Philosophy 66 (1969): 363-381.
{24}Thomas Aquinas De potentia Dei 3. 1, 2.
{25}Keith Ward, Rational Theology and the Creativity of God (Oxford: Basil Blackwell, 1982), p. 86.
{26}D. A. Carson, Divine Sovereignty and Human Responsibility: Biblical Perspectives in Tension, New Foundations Theological Library (Atlanta: John Knox, 1981), pp. 24-35.
{27}Carson, Sovereignty and Responsibility, pp. 18-22. One should mention also the striking passages which speak of God’s repenting in reaction to a change in human behavior (e.g., Gen. 6.6; 1 Sam. 15.11, 35).
{28}See Luis Molina, On Divine Foreknowledge: Part IV of the "Concordia," trans. with Introduction and Notes by Alfred J. Freddoso (Ithaca, N.Y.: Cornell University Press, 1988); also William Lane Craig, The Problem of Divine Foreknowledge and Future Contingents from Aristotle to Suarez, Studies in Intellectual History 7 (Leiden: E. J. Brill, 1988), chaps. 7, 8.
{29}Molina explains,
". . . . all good things, whether produced by causes acting from a necessity of nature or by free causes, depend upon divine predetermination . . . and providence in such a way that each is specifically intended by God though His predetermination and providence, whereas the evil acts of the created will are subject as well to divine predetermination and providence to the extent that the causes from which they emanate and the general concurrence on God’s part required to elicit them are granted through divine predetermination and providence--though not in order that these particular acts should emanate from them, but rather in order that other, far different, acts might come to be, and in order that the innate freedom of the things endowed with a will might be preserved for their maximum benefit; in addition evil acts are subject to that same divine predetermination and providence to the extent that they cannot exist in particular unless God by His providence permits them in particular in the service of some greater good. It clearly follows from the above that all things without exception are individually subject to God’s will and providence, which intend certain of them as particulars and permit the rest as particulars Thus, the leaf hanging from the tree does not fall, nor does either of the two sparrows sold for a farthing fall to the ground, nor does anything else whatever happen without God’s providence and will either intending it as a particular or permitting it as a particular "(Molina On Divine Foreknowledge 4. 53. 3. 17).
On the way in which sins contribute to the eventual realization of God’s purposes, see the powerful statement in On Divine Foreknowledge 4. 53. 2. 15.
{30}Alvin Plantinga, "Reply to Robert Adams," in Alvin Plantinga, ed. James E. Tomberlin and Peter Van Inwagen, Profiles 5 (Dordrecht: D. Reidel, 1985), pp. 371-82; Jonathan L. Kvanvig, The Possibility of an All-Knowing God (New York: St. Martin’s, 1986), pp. 121-148; Alfred J. Freddoso, "Introduction," in On Divine Foreknowledge, pp. 68-78; Edward J. Wierenga, The Nature of God: an Inquiry into Divine Attributes (Ithaca, N. Y.: Cornell University Press, 1989), pp. 150-160; William Lane Craig, Divine Foreknowledge and Human Freedom, Brill’s Studies in Intellectual History 19 (Leiden: E. J. Brill, 1990), pp. 247-269; Thomas Flint, "Hasker’s God, Time, and Knowledge," Philosophical Studies 60 (1990): 103-115; William Lane Craig, "Hasker on Divine Knowledge," Philosophical Studies 67 (1992): 89-110.
{31}Hasker does attempt to re-defend his controversial premiss in William Hasker, "Middle Knowledge: a Refutation Revisited," Faith and Philosophy 12 (1995): 224-225; but his account fails to respond to any of the three objections advanced in Craig, "Hasker on Divine Knowledge," pp. 106-107, and in the end he himself concedes that ". . . the complexity of the argument . . . leaves a number of points at which doubts can arise and toward which critics can direct their fire" (Hasker, "Refutation Revisited," p. 226), so that he chooses to adopt Adams’s alternative formulation.
{32}Robert Merrihew Adams, "An Anti-Molinist Argument," in Philosophical Perspectives, vol. 5: Philosophy of Religion, ed. James E. Tomberlin (Atascadero, Calif.: Ridgeway Publishing, 1991), p. 356.
{33}Adams had argued, "My thisness, and singular propositions about me, cannot have pre-existed me because if they had, it would have been possible for them to have existed even if I had never existed, and that is not possible" (Robert Merrihew Adams, "Time and Thisness," Midwest Studies in Philosophy 11 (1986): 317). This argument is parallel to the interpretation under discussion, counterfactuals of creaturely freedom and divine middle knowledge taking the place of thisnesses and singular propositions. As Kvanvig discerns, this reasoning is susceptible to the same response as is the argument for fatalism (Jonathan L. Kvanvig, "Adams on Actualism and Presentism," Philosophy and Phenomenological Research 50 (1989): ***.
{34}Thomas P. Flint, "The Problem of Divine Freedom," American Philosophical Quarterly 20 (1983): 255-264.
{35}Adams, "Anti-Molinist Argument." p. 345.
{36}See further my Divine Foreknowledge and Human Freedom , pp. 259-262.
{37}Alvin Plantinga, "Reply to Robert Adams," p. 376.
{38}William Hasker, "Explanatory Priority: Transitive and Unequivocal, A Reply to William Craig, " Philosophy and Phenomenological Research 57 (1997): 3.
{40}He writes, ". . . (12) expresses a . . . distinctively incompatibilist intuition, that the explanatory antecedents of the totality of my choosing and doing, must leave the omission of the free action ‘open,’ at least in the sense of not being strictly inconsistent with the omission" (Adams, "Anti-Molinist Argument," p. 352).
{41}Thomas P. Flint, "Middle Knowledge and the Doctrine of Infallibility," in Philosophy of Religion, pp. 385-390.
{42}Hasker, "Explanatory Priority," p. 1. The article referenced is Hasker, "Refutation Revisited," pp. 223-236.
{43}Hasker revises the first part of his argument in deference to Adams’s version, but the second part he leaves unchanged and undefended--indeed, in footnote 17 on p. 235 he actually commends Adams’s (12) as an alternative to his argument for those "who have qualms about some of the premises in my version of the argument."
{44}It is very often said by biblical scholars anxious not to be associated with a defunct evidential apologetic use of miracles that biblical miracles function as signs, not evidence. This, however, is a false dichotomy; it is precisely because of their evidential force that miracles serve effectively as signs (see William Lane Craig, review article of Miracles and the Critical Mind, by Colin Brown, Journal of the Evangelical Theological Society 27 [1985]: 473-483).
{45}Marie François Arrouet de Voltaire, Dictionnaire philosophique (Paris: Garnler, 1967) s.v. "Miracles".
{46}For discussion see Stephen S. Bilinskyji, "God, Nature, and the Concept of Miracle" (Ph.D. dissertation, University of Notre Dame, 1982); Alfred J. Freddoso, "The Necessity of Nature," Midwest Studies in Philosophy 11 (1986): 215-242.
{47}R. G. Swinburne, "Miracles," Philosophical Quarterly 18 (1968): 321.
{48}Ibid., p. 323.
{49}Wolfhart Pannenberg, Jesus--God and Man, trans. L. L. Wilkins and D. A. Priebe (London: SCM, 1968), p. 67.
{50}David Hume, An Enquiry concerning Human Understanding, ed. L. A. Selby-Bigge, 3d ed. rev. P. H. Nidditch (Oxford: Clarendon Press, 1975), chap. 10.
Copyright (C) William Lane Craig. All Rights Reserved. |
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Measurement problem
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The measurement problem is the key set of questions that every interpretation of quantum mechanics must answer. The problem is that the wavefunction in quantum mechanics evolves according to the Schrödinger equation into a linear superposition of different states, but the actual measurements always find the physical system in a definite state, typically a position eigenstate. Any future evolution will be based on the system having the measured value at that point in time, meaning that the measurement "did something" to the process under examination. Whatever that "something" may be does not appear to be explained by the basic theory.
The best known example is the "paradox" of the Schrödinger's cat: a cat is apparently evolving into a linear superposition of basis vectors that can be characterized as an "alive cat" and states that can be described as a "dead cat". Each of these possibilities is associated with a specific nonzero probability amplitude; the cat seems to be in a "mixed" state. However, a single particular observation of the cat does not measure the probabilities: it always finds either an alive cat, or a dead cat. After that measurement the cat stays alive or dead. The measurement problem is the question: how are the probabilities converted to an actual, sharply well-defined outcome?
Different interpretations of quantum mechanics propose different solutions of the measurement problem.
• The old Copenhagen interpretation was rooted in the philosophical positivism. It claimed that the probabilities are the only quantities that should be discussed, and all other questions were considered as unscientific ones. One could either imagine that the wavefunction collapses, or one could think of the wavefunction as an auxiliary mathematical tool with no direct physical interpretation whose only role is to calculate the probabilities.
While this viewpoint was sufficient to understand the outcome of all known experiments, it did not explain why it was legitimate to imagine that the cat's wavefunction collapses once the cat is observed, but it is not possible to collapse the wavefunction of the cat or the electron before it is measured. The collapse of the wavefunction used to be linked to one of two different properties of the measurement:
• The measurement is done by a conscious being. In this specific interpretation, it was the presence of a conscious being that caused the wavefunction to collapse. However, this interpretation depends on a definition of "consciousness". Because of its spiritual flavor, this interpretation was never fully accepted as a scientific explanation.
• The measurement apparatus is a macroscopic object. Perhaps, it is the macroscopic character of the apparata that allows us to replace the logic of quantum mechanics with the classical intuition where the positions are well-defined quantities.
The latter approach was put on firm ground in the 1980s when the phenomenon of quantum decoherence was understood. The calculations of quantum decoherence allow the physicists to identify the fuzzy boundary between the quantum microworld and the world where the classical intuition is applicable. Quantum decoherence was proposed in the context of the many-worlds interpretation, but it has also become an important part of modern update of the Copenhagen interpretation that is based on consistent histories ("Copenhagen done right"). Quantum decoherence does not describe the actual process of the wavefunction collapse, but it explains the conversion of the quantum probabilities (that are able to interfere) to the ordinary classical probabilities.
Hugh Everett's relative state interpretation, also referred to as the many-worlds interpretation, attempts to avoid the problem by suggesting it is an illusion. Under this system there is only one wavefunction, the superposition of the entire universe, and it never collapses -- so there is no measurement problem. Instead the act of measurement is actually an interaction between two quantum entities, which entangle to form a single larger entity, for instance living cat/happy scientist. Everett also attempted to demonstrate the way that in measurements the probabilistic nature of quantum mechanics would appear; work later extended by Bryce DeWitt and others and renamed the many-worlds interpretation. Everett/DeWitt's interpretation posits a single universal wavefunction, but with the added proviso that "reality" from the point of view of any single observer, "you", is defined as a single path in time through the superpositions. That is, "you" have a history that is made of the outcomes of measurements you made in the past, but there are many other "yous" with slight variations in history. Under this system our reality is one of many similar ones.
The Bohm interpretation tries to solve the measurement problem very differently: this interpretation contains not only the wavefunction, but also the information about the position of the particle(s). The role of the wavefunction is to create a "quantum potential" that influences the motion of the "real" particle in such a way that the probability distribution for the particle remains consistent with the predictions of the orthodox quantum mechanics. According to the Bohm interpretation combined with the von Neumann theory of measurement in quantum mechanics, once the particle is observed, other wave-function channels remain empty and thus ineffective, but there is no true wavefunction collapse. Decoherence provides that this ineffectiveness is stable and irreversible, which explains the apparent wavefunction collapse.
References Edit
• Schlosshauer, Maximilian (2004). Decoherence, the Measurement Problem, and Interpretations of Quantum Mechanics. Rev. Mod. Phys. 76. arXiv:quant-ph/0312059
, DOI:10.1103/RevModPhys.76.1267.he:בעיית המדידה
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910a6490e142c012 | Quantum Physics
Pursuit of a Quantum Spin Liquid
Authors: George Rajna
"In a quantum spin liquid, spins continually fluctuate due to quantum effects and never enter a static ordered arrangement, in contrast to conventional magnets," Kelley said. "These states can host exotic quasiparticles that can be detected by inelastic neutron scattering." [13] An international team of researchers have found evidence of a mysterious new state of matter, first predicted 40 years ago, in a real material. This state, known as a quantum spin liquid, causes electrons-thought to be indivisible building blocks of nature-to break into pieces. [12] In a single particle system, the behavior of the particle is well understood by solving the Schrödinger equation. Here the particle possesses wave nature characterized by the de Broglie wave length. In a many particle system, on the other hand, the particles interact each other in a quantum mechanical way and behave as if they are "liquid". This is called quantum liquid whose properties are very different from that of the single particle case. [11] Quantum coherence and quantum entanglement are two landmark features of quantum physics, and now physicists have demonstrated that the two phenomena are "operationally equivalent"—that is, equivalent for all practical purposes, though still conceptually distinct. This finding allows physicists to apply decades of research on entanglement to the more fundamental but less-well-researched concept of coherence, offering the possibility of advancing a wide range of quantum technologies. [10] The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the relativistic quantum theory. The asymmetric sides are creating different frequencies of electromagnetic radiations being in the same intensity level and compensating each other. One of these compensating ratios is the electron – proton mass ratio. The lower energy side has no compensating intensity level, it is the dark energy and the corresponding matter is the dark matter.
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f461c61fc6b4cc4e | Tags: wavefunction
A wave function is a mathematical tool used in quantum mechanics. It is a function typically of space or momentum or spin and possibly of time that returns the probability amplitude of a position or momentum for a subatomic particle. Mathematically, it is a function from a space that maps the possible states of the system into the complex numbers. The laws of quantum mechanics (the Schrödinger equation) describe how the wave function evolves over time.
Learn more about quantum dots from the many resources on this site, listed below. More information on Wave Function can be found here.
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Particle in a box?
1. Mar 5, 2006 #1
If we have the case of an infinite potential well, the particle is assumed to have 0 potential energy when inside the well at all times. Why is that? Does it assume that the box itself is 0 net charge? This is because any non zero net charge and the electron will either be repelled or attracted meaning positive potential energy for some time.
Or does it imply either infinite negative or positive charge on both sides? If that is the case than why is it still 0 potential evergy at all points inside the well?
2. jcsd
3. Mar 6, 2006 #2
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Staff: Mentor
Try it yourself: assume that [itex]V = V_0[/itex] inside the box (where [itex]V_0 \ne 0[/itex]). What are the stationary-state solutions of the Schrödinger equation for this case, and their energies?
4. Mar 6, 2006 #3
Assuming potential energy V = a, the total energy levels = Energy level assuming 0 potential + a. In other words, ((hbar)(n)(pie))^2/(2ml^2) + a.
So including a potential just added that amount of energy to the total energy of the particle for each quantum state.
However, that still does not tell me why we shouldn't include a potential energy for the particle in the well. I guess I just don't have a physical understanding of why there is 0 potential energy for the particle anywhere in the well. When it reaches one end of the well and is repelled than surely it should have positive potential energy when that happens? Or is the trick that since each end has infinite potential, whenever the particle amass some potential, it is instanteously canceld from the well on the other end. So it is the infinity that is making things unintuitive?
Last edited: Mar 6, 2006
5. Mar 6, 2006 #4
The particle moves in a potential, it 'gains' potential energy when it is in a region with a higher V. So it has potential energy at the boundaries.
6. Mar 6, 2006 #5
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Staff: Mentor
Keep in mind that in physics, the "zero point" of potential energy is basically arbitrary. All that matters in terms of physically observable outcomes are differences in potential energy.
With the particle in a box, regardless of what you choose as the potential energy inside the box, a state with a given n is the same "height" above that "floor", and that's what matters, physically. The wave function comes out to be the same regardless of the value of a in your energy-level formula:
[tex]\psi(x) = \sqrt{\frac{2}{L}} \sin \left( n \pi \frac {x}{L} \right)[/tex]
So the expectation values of position, momentum, and quantities derived from them, are the same regardless of the value of a. It's purely a matter of convenience, which value you choose for a.
7. Mar 6, 2006 #6
A constant, non zero potential inside the box is not realistic. How about potential V varies as x as it should because when the electron is nearer to the walls or boundaries of the box, it should be repelled. Hence in the middle, its potential should be lowest and closer to the edges, it should be higher. so V=(x-L/2)^2 where L is the length of the box.
This is the point I was trying to make which was that potential energy of the electron should be non zero near the walls. The problem is that now I do not know how to solve this equation as V is not a constant anymore. What would the solution look like? Wouldn't this be more realistic?
8. Mar 16, 2006 #7
If it was an infinate potential well, then any charge that the electron possesses can be equalled by the potential in the well ( since it is infinate ) meaning that there would be no difference in potential of the electron and the well - in other words no potential, so the electron wouldn't repel from the walls as a result of its charge, but just simply by interacting with the walls of the " box ". much like a classical case of linear momentum. the energy levels of the electron would be constant due to the conservation of linear momentum.
9. Mar 16, 2006 #8
I have not assumed an infinite potential well. I am considering a realistic case with Potential V=(x-L/2)^2 which means that the walls are slightly negatively charged hence the electron will be repelled as it travels near the wall.
Hence the physical system is now a quantum oscillator. I suspect this is a more realistic model of the electron made out of walls that are slightly negatively charged.
10. Mar 16, 2006 #9
Anyway, I'll ignore that for now.
If you are talking about a quadratic potential then you do indeed have a harmonic oscillator (a constant potential would give the "particle in a box").
Note that if
then we would still have the usual quantum harmonic oscillator.
At this stage, considering the repelling charge of electrons at the edge of the box is not actually going to work. The reason is because you must justify the potential you claim will work. We can work out classically the potential a series of static charges will produce. However, taking a full quantum mechanical description is obviously going to be more complicated.
In your original post you considered a particle in an infinite square well, but asked why the potential in the well had to be zero. Well, as has been pointed out potential is only defined up to a constant, so this is irrelevant.
When considering a quadratic potential, you say you cannot solve the equation. Well, you can find the energy eigenvalues of a quantum harmonic oscillator, and this problem is tackled in almost every standard QM text. If you can't afford that, try http://en.wikipedia.org/wiki/Harmonic_oscillators. |
6c2ed96ade9368d3 | Category Archives: Flash Forward (TV Series)
FlashForward Cancelled – The Ultimate Blackout
Associated Content
Published May 14, 2010 by:
Robert Dougherty
Flashforward cancelled rumors were quite persistent all spring. The Flashforward cancelled outcome looked likely, given the free falling ratings ever since its premiere. However, since it was doing better overseas, some thought it had a good shot to return and cause more blackouts. But in America, the audience was shrinking and shrinking, as hopes were fading for the show to become the new Lost. Now that experiment is over, with Flashforward cancelled after just one year – which none of its infamous blackouts could foresee.
The Thursday night show premiered with a lot of promise, as the premise of the whole world seeing six months into the future intrigued many. But these days, no show other than Lost has been able to follow through on a big mystery premise. The problem wasn’t an inability to provide answers, but that less and less people cared about the endless “destiny vs. free will” debates, or the people having them.
Now with Flashforward cancelled, the Season 1 finale on May 27 will serve as the end of the show. Since the show should definitively answer if everyone’s visions come true on April 29 – and if another blackout will come afterward – there may indeed be nowhere to go after that. However, if the episode does have teasers for a Season 2, they will forever go unanswered.
Once again, ABC failed to develop the next Lost, as they had to give up just before Lost ended. With Flashforward cancelled, it may be another cautionary tale of how mystery shows are harder to make work than they look, if that’s possible.
But although Flashforward is cancelled, the network still has hopes for other mystery programs. V was their second shot at developing a new Lost, although it too met with mixed reviews and backlash after the pilot. However, they have enough faith in this sci-fi remake to give it a second season, unlike Flashforward.
Although V escaped the chopping block, others joined Flashforward in being cancelled. Scrubs’ one-year stint on ABC is over, while the critically acclaimed but low rated Better Off Ted was fired, and the short-lived Monday night comedy Romantically Challenged was let go.
With ABC’s decisions, the final cuts for fall 2010 are now under way, as shows find out if they still have a future, or have to close up shop. As one TV season ends with a slew of big finales, another is just on the horizon.
Since Flashforward is cancelled, it won’t have to see anything else on the horizon anymore – even though that was the show’s whole premise. This leaves just two episodes left for the season and series on May 20 and 27.
Flash Forward: The Antikythera Mechanism, QED and Autistic Savants (Spoiler Alert!)
As the season progresses, I think Flash Forward is getting better and better. So we now know that the anti-blackout ring is a Quantum Entanglement Device; Janice is working for the FBI, CIA and a secret group; Olivia is the key to the puzzle; Frost used autistic savants in his flash forward experiments since they have detailed memories, and the secret group wants Janice to kill Mark.
So our “secret phrase” for the night was The Antikythera Mechanism. I took this directly from their site.
The antikythera mechanism is currently housed in the Greek National Archaeological Museum in Athens and is thought to be one of the most complicated antiques in existence. At the beginning of the 20th century, divers off the island of Antikythera came across this clocklike mechanism, which is thought to be at least 2,000 years old, in the wreckage of a cargo ship. The device was very thin and made of bronze. It was mounted in a wooden frame and had more than 2,000 characters inscribed all over it. Though nearly 95 percent of these have been deciphered by experts, there as not been a publication of the full text of the inscription.
Today it is believed that this instrument was a kind of mechanical analog computer used to calculate the movements of stars and planets in astronomy. It has been estimated that the antikythera mechanism was built around 87 B.C and was lost in 76 B.C. No one has any idea about why or how it came to be on that ill-fated cargo ship. The ship was Roman though the antikythera mechanism was developed in Greece. One theory suggests that the reason it came to be on the Roman ship could be because the instrument was among the spoils of war garnered by then Roman emperor Julius Caesar.
X-rays of the device have indicated that there are at least 30 different gears present in it. British historian Derek Price has done extensive research on what the antikythera mechanism may have been used for. It was not until 1959 that Price put forth the theory that the device was used in astronomy to make calculations and predictions. In 1974, Price presented a model of how the antikythera mechanism might have functioned. When past or future dates were entered into the device it calculated the astronomical information related to the Sun, Moon, and other planets.
Some of these findings have been confirmed by more recent researches undertaken by scholars and scientists. However, the full extent of the instrument’s functions still remains unknown. Price had also suggested that the antikythera mechanism might have been on public display in a museum or a public hall. Some others have also come up with their variants of the ancient computer, based on Price’s model. Australians Allan Bromley and Frank Percival devised one such model as did Michael Wright, curator of mechanical engineering at the Science Museum, London.
A joint project is also underway to further study this astounding example of the advancements of technology in ancient times. Known as the Antikythera Mechanism Research Project, it is a collaboration between Cardiff University, the National and Kapodistrian University of Athens, the Aristotle University of Thessaloniki, the National Archaeological Museum of Athens, X-Tek Systems, UK, and Hewlett-Packard, USA. This project is funded by the Leverhulme Trust and supported by the Cultural Foundation of the National Bank of Greece. Since the study started more progress has been made. More than 80 fragments of the mechanism have now been discovered.
Flash Forward: Quantum Entanglement Device (QED) SPOILER ALERT!
Wow! What a shocker. Janis Hawk is a double mole. And, me thinks, will become impregnated by Simon. But that is not my topic of the night. We seem to have misinterpreted what QED means, so I will blog a bit on Quantum Entanglement.
Nevertheless, the fact that causality is preserved in quantum mechanics is a rigorous result in modern quantum field theories, and therefore modern theories do not allow for time travel or FTL communication. In any specific instance where FTL has been claimed, more detailed analysis has proven that to get a signal, some form of classical communication must also be used. The no-communication theorem also gives a general proof that quantum entanglement cannot be used to transmit information faster than classical signals. The fact that these quantum phenomena apparently do not allow FTL time travel is often overlooked in popular press coverage of quantum teleportation experiments. How the rules of quantum mechanics work to preserve causality is an active area of research.
Quantum Entanglement Device
Mythological godtech or clarketech communications device that supposedly employs quantum entanglement for limited FTL communicate at interstellar distances. Said to be in the hands of powers and archailects. It is commonly believed or supposed in cheap virchfiction that a few have over the centuries fallen into the hands of nearbaselines, who have however been able to use them to their useful potential. The reality is that no QED has ever existed; as with FTL it is a myth that is believed by the gullible. It has been known ever since the Atomic and Information ages of Old Earth that actual quantum entanglement cannot be used to send useful information.
More detail for those who can’t sleep
Theories involving hidden variables have been proposed in order to explain this result. These hidden variables would account for the spin of each particle, and would be determined when the entangled pair is created. It may appear then that the hidden variables must be in communication no matter how far apart the particles are, that the hidden variable describing one particle must be able to change instantly when the other is measured. If the hidden variables stop interacting when they are far apart, the statistics of multiple measurements must obey an inequality (called Bell’s inequality), which is, however, violated both by quantum mechanical theory and in experiments.
When pairs of particles are generated by the decay of other particles, naturally or through induced collision, these pairs may be termed “entangled”, in that such pairs often necessarily have linked and opposite qualities such as spin or charge. The assumption that measurement in effect “creates” the state of the measured quality goes back to the arguments of Einstein, Podolsky, and Rosen and Erwin Schrödinger (remember Schrödinger’s Cat from an earlier blog) concerning Heisenberg’s uncertainty principle and its relation to observation.
The analysis of entangled particles by means of Bell’s theorem can lead to an impression of non-locality, i.e. that there exists a connection between the members of such a pair that defies both classical and relativistic concepts of space and time. This is reasonable if it is assumed that each particle departs the location of the pair’s creation in an ambiguous state (thus yet unobserved, as per a possible interpretation of Heisenberg’s principle). In such a case, for a given observable quality of the particle, all outcomes remain a possibility and only measurement itself would precipitate a distinct value. As soon as just one of the particles is observed, its entangled pair collapses into the very same state. If each particle departs the scene of its “entangled creation” with properties that would unambiguously determine the value of the quality to be subsequently measured, then the postulated instantaneous transmission of information across space and time would not be required to account for the result of both particles having the same value for that quality. The Bohm interpretation postulates that a guide wave exists connecting what are perceived as individual particles such that the supposed hidden variables are actually the particles themselves existing as functions of that wave.
Observation of wavefunction collapse can lead to the impression that measurements performed on one system instantaneously influence other systems entangled with the measured system, even when far apart. Yet another interpretation of this phenomenon is that quantum entanglement does not necessarily enable the transmission of classical information faster than the speed of light because a classical information channel is required to complete the process.
Flash Forward: Pierre Boaistuau’s Hydra Monster (The Hoax)
In last Thursday’s episode of Flash Forward titled “Better Angels,” Mark shows Stan the Hydra Monster picture that would eventually end up on his Mosaic wall. The Hydra, as portrayed by 18th century French writer, Pierre Boaistuau, had seven heads and was eventually killed by Hercules.
Mark then segues the conversation into D. Gibbons who he now identifies as one Dyson Frost. Frost we learn is brilliant, reclusive, a Particle Physicist, trained in Engineering at MIT, minoring in Victorian Literature. He had a domineering father who only spoke to him in French even though they grew up in Wyoming. He also became a Chess Grandmaster at the age of 15 (Stan notes the White Queen chess piece they found).
Frost supposedly died in a boating accident in 1990 on a boat named Le Monstre de Boaistuau (The Monster of Boaistuau).
The Hoax of the Venetian Hydra
Many different authors discuss the hydra, among them Boaistuau, in terms of if it is real or if it is just a hoax. Through a sociohistorical analysis of the hydra in Giambattista Basile’s dragon-slayer tale “Lo mercante,” this essay challenges the universalizing interpretation of the dragon as a worthy foil for the hero. In depicting the hero’s struggle with the beast, Basile employs tropes that purposefully recall a creature that was crafted by charlatans and widely discussed in scientific texts (people in the kingdom of his story describe the hydra as having “had the crest of a cock, the head of a cat, eyes of fire, jaws of a race-hound, the winds of a bat, the claws of a bear and the tail of a serpent.”). Basile transforms the epic battle between dragon and slayer into a comic encounter in which the hero confronts a manufactured monster while playfully blurring the boundary between two seemingly disparate genres, the scientific treatise and the literary fairy tale.
Early engravings of the Hydra first appeared in Europe in Konrad Lykosthenes’ Prodigiorium ac ostentorum chronicon, Lykosthenes sought to teach Christians to recognize the divine messages that God transmitted to men through these marvellous occurrences (of the hydra). He also saw the hydra not as the bearer of a specific holy message but instead depicts the monster as the object of international trade.
Pierre Boaistuau’s Histoires prodigieuyses, similar to Lykosthenes, aimed to reform its readers through the contemplation of the prodigfies on it pages, which in turn was intended to spur the reader to expunge his or her own vice. Boaistuau cites Lykosthenes story of the hydra and muses: “If it is a true thing (as it is likely to have been, judging by the authority of the one who describes it) I believe that nature has never produced a more marvellous creature among all the monsters of this earth.”
Since Boaistuau was never able to verify that the defunct king (in Basile’s story) ever actually owned this creature, he tentatively questions its authenticity. although lacking the physical proof of the beast’s existence, Boaistuau concludes this chapter by suggesting that the monster is both a portent and a natural marvel, the most marvellous among all the monsters on earth. undoubtedly, his conclusion is motivated in part by the realization that an assertion of authenticity would be more likely to encourage his readers to reform than would be the unmasking of a hoax.
Source: Magnanini, Suzanne, Fairy-Tale Science: Monstrous Generation in the Tales of Straparola and Basile.
Flash Forward on the ropes?
Perhaps Uncle Teddy got off easy.
For two weeks in a row, FlashForward had their lowest ratings ever. Now I need to point out that they are going against the NCAA Basketball Tournaments in their time slot. You would have thought the network programmers would have been proactive and waited another month before bringing it back.
I think the two episodes this season have been very interesting. We learned that Simon was an unsuspecting “Suspect Zero.” at the stadium in Detroit. We also know that “D. Gibbons” stole Lloyd’s research and claimed it as his own. Also, we now know that Zoey actually was attending Demetri’s memorial service (he was shot by Mark three times).
Last weeks episode showed how bad ass Aaron really is.
I am hoping to see more, but the networks seem to give less time for a series to take hold. I think FlashForward is worth saving. Are you listening ABC?
Will overseas fans save ‘FlashForward’?
By Lynette Rice, Entertainment Weekly
March 2, 2010 3:28 p.m. EST
Entertainment Weekly
(Entertainment Weekly) — Sci-fi dramas like ABC’s “FlashForward” may have a tough time attracting new viewers when it returns with originals on March 18.
The freshman drama lost 43 percent of its viewers last fall while the network’s “V” was down 35 percent over four airings. (It also doesn’t help that “FlashForward” is on its third show runner now that co-creator/executive producer David Goyer has decided to leave the show to focus on his film career).
So it’s hardly surprising that none of the Big Four nets have a sci-fi show in development for the 2010-11 season (ABC’s superheroes pilot “No Ordinary Family” starring “The Shield’s” Michael Chiklis comes close).
At least the ABC-owned “FlashForward” (unlike the Warner Bros. owned “V”) has an ace in the hole: It does well in the UK, Italy and Spain, and the robust international sales could sway the network to pick up the Joseph Fiennes series for another season.
ABC may not make a decision on “FlashForward” and “V,” which returns March 30, until right before its upfront presentation in New York this May.
Several ABC dramas, in fact, have yet to receive pickups for another season, though “Desperate Housewives,” “Grey’s Anatomy,” “Private Practice” and “Brothers and Sisters” are slam dunks.
Comedies like “Cougar Town,” “The Middle” and “Modern Family” have already been picked up, while shows like “Ugly Betty” have already been cancelled.
Long shots include “Better Off Ted,” “The Deep End,” “Scrubs” and “The Forgotten.”
Flash Forward: Many-Worlds Interpretation (Hugh Everett)
Many-worlds is an interpretation of quantum mechanics that asserts the objective reality of the wavefunction, but denies the reality of wavefunction collapse. It is also known as MWI, the relative state formulation, theory of the universal wavefunction, parallel universes, many-universes interpretation or just many worlds.
The original relative state formulation is due to Hugh Everett who formulated it in 1957. Later, this formulation was popularized and renamed many-worlds by Bryce Seligman DeWitt in the 1960s and ’70s.
Proponents argue that many-worlds reconciles how we can perceive non-deterministic events, such as the random decay of a radioactive atom, with the deterministic equations of quantum physics. Prior to many-worlds, reality had been viewed as a single “world-line”. Many-worlds, rather, views reality as a many-branched tree where every possible quantum outcome is realised.
In many-worlds, the subjective appearance of wavefunction collapse is explained by the mechanism of quantum decoherence. By decoherence, many-worlds claims to resolve all of the correlation paradoxes of quantum theory, such as the EPR paradox and Schrödinger’s cat, since every possible outcome of every event defines or exists in its own “history” or “world”. In layman’s terms, there is a very large—perhaps infinite—number of universes, and everything that could possibly have happened in our past, but didn’t, has occurred in the past of some other universe or universes.
The decoherence approach to interpreting quantum theory has been further explored and developed becoming quite popular, taken as a class overall. MWI is one of many Multiverse hypotheses in physics and philosophy. It is currently considered a mainstream interpretation along with the other decoherence interpretations and the Copenhagen interpretation.
Although several versions of many-worlds have been proposed since Hugh Everett’s original work, they all contain one key idea: the equations of physics that model the time evolution of systems without embedded observers are sufficient for modelling systems which do contain observers; in particular there is no observation-triggered wavefunction collapse which the Copenhagen interpretation proposes. Provided the theory is linear with respect to the wavefunction, the exact form of the quantum dynamics modelled, be it the non-relativistic Schrödinger equation, relativistic quantum field theory or some form of quantum gravity or string theory, does not alter the validity of MWI since MWI is a metatheory applicable to all linear quantum theories, and there is no experimental evidence for any non-linearity of the wavefunction in physics. MWI’s main conclusion is that the universe (or multiverse in this context) is composed of a quantum superposition of very many, possibly even a non-denumerablely infinitely many, increasingly divergent, non-communicating parallel universes or quantum worlds.
The idea of MWI originated in Everett’s Princeton Ph.D. thesis “The Theory of the Universal Wavefunction”, developed under his thesis advisor John Archibald Wheeler, a shorter summary of which was published in 1957 entitled “Relative State Formulation of Quantum Mechanics” (Wheeler contributed the title “relative state”; Everett originally called his approach the “Correlation Interpretation”, where “correlation” refers to quantum entanglement). The phrase “many-worlds” is due to Bryce DeWitt, who was responsible for the wider popularisation of Everett’s theory, which had been largely ignored for the first decade after publication. DeWitt’s phrase “many-worlds” has become so much more popular than Everett’s “Universal Wavefunction” or Everett-Wheeler’s “Relative State Formulation” that many forget that this is only a difference of terminology; the content of all three papers is the same.
The many-worlds interpretation shares many similarities with later, other “post-Everett” interpretations of quantum mechanics which also use decoherence to explain the process of measurement or wavefunction collapse. MWI treats the other histories or worlds as real since it regards the universal wavefunction as the “basic physical entity” or “the fundamental entity, obeying at all times a deterministic wave equation”. The other decoherent interpretations, such as many histories, consistent histories, the Existential Interpretation etc, either regard the extra quantum worlds as metaphorical in some sense, or are agnostic about their reality; it is sometimes hard to distinguish between the different varieties. MWI is distinguished by two qualities: it assumes realism, which it assigns to the wavefunction, and it has the minimal formal structure possible, rejecting any hidden variables, quantum potential, any form of a collapse postulate (i.e. Copenhagenism) or mental postulates (such as the many-minds interpretation makes).
Decoherent interpretations of many-worlds use einselection to explain how a small number of classical pointer states can emerge from the enormous Hilbert space of superpositions have been proposed by Wojciech H. Zurek. “Under scrutiny of the environment, only pointer states remain unchanged. Other states decohere into mixtures of stable pointer states that can persist, and, in this sense, exist: They are einselected.” These ideas complement MWI and bring the interpretation in line with our perception of reality.
Many-worlds is often referred to as a theory, rather than just an interpretation, by those who propose that many-worlds can make testable predictions (such as David Deutsch) or is falsifiable (such as Everett) or that all the other, non-MWI, are inconsistent, illogical or unscientific in their handling of measurements; Hugh Everett argued that his formulation was a metatheory, since it made statements about other interpretations of quantum theory; that it was the “only completely coherent approach to explaining both the contents of quantum mechanics and the appearance of the world.”
Interpreting wavefunction collapse
Some versions of the Copenhagen interpretation of quantum mechanics proposed a process of “collapse” in which an indeterminate quantum system would probabilistically collapse down onto, or select, just one determinate outcome to “explain” this phenomenon of observation. Wavefunction collapse was widely regarded as artificial and ad-hoc, so an alternative interpretation in which the behavior of measurement could be understood from more fundamental physical principles was considered desirable.
Everett’s Ph.D. work provided such an alternative interpretation. Everett noted that for a composite system – for example a subject (the “observer” or measuring apparatus) observing an object (the “observed” system, such as a particle) – the statement that either the observer or the observed has a well-defined state is meaningless; in modern parlance the observer and the observed have become entangled; we can only specify the state of one relative to the the other, i.e. the state of the observer and the observed are correlated after the observation is made. This led Everett to derive from the unitary, deterministic dynamics alone (i.e. without assuming wavefunction collapse) the notion of a relativity of states.
Everett noticed that the unitary, deterministic dynamics alone decreed that after an observation is made each element of the quantum superposition of the combined subject-object wavefunction contains two “relative states”: a “collapsed” object state and an associated observer who has observed the same collapsed outcome; what the observer sees and the state of the object have become correlated by the act of measurement or observation. The subsequent evolution of each pair of relative subject-object states proceeds with complete indifference as to the presence or absence of the other elements, as if wavefunction collapse has occurred, which has the consequence that later observations are always consistent with the earlier observations. Thus the appearance of the object’s wavefunction’s collapse has emerged from the unitary, deterministic theory itself. (This answered Einstein’s early criticism of quantum theory, that the theory should define what is observed, not for the observables to define the theory). Since the wavefunction appears to have collapsed then, Everett reasoned, there was no need to actually assume that it had collapsed. And so, invoking Occam’s razor, he removed the postulate of wavefunction collapse from the theory.
A consequence of removing wavefunction collapse from the quantum formalism is that the Born rule requires derivation, since many-worlds claims to derive its interpretation from the formalism. Attempts have been made, by many-world advocates and others, over the years to derive the Born rule, rather than just conventionally assume it, so as to reproduce all the required statistical behaviour associated with quantum mechanics. There is no consensus on whether this has been successful.
Everett, Gleason and Hartle
De Witt and Graham
Bryce De Witt and his doctoral student R. Neill Graham later provided alternative (and longer) derivations to Everett’s derivation of the Born rule. They demonstrated that the norm of the worlds where the usual statistical rules of quantum theory broke down vanished, in the limit where the number of measurements went to infinity.
Deutsch et al
An information-theoretic derivation of the Born rule from Everettarian assumptions, was produced by David Deutsch (1999) and refined by Wallace (2002-2009) and Saunders (2004). Deutsch’s derivation is a two-stage proof: first he shows that the number of orthonormal Everett-worlds after a branching is proportional to the conventional probability density. Then he uses game theory to shows that these are all equally likely to be observed. The last step in particular has been criticised for circularity. Other reviews have been positive, although the status of these arguments remains highly controversial. It is fair to say that some theoretical physicists have taken them as supporting the case for parallel universes. In the New Scientist article, reviewing their presentation at a September 2007 conference, Andy Albrecht, a physicist at the University of California at Davis, is quoted as saying “This work will go down as one of the most important developments in the history of science.”
Wojciech H. Zurek (2005) has produced a derivation of the Born rule, where decoherence has replaced Deutsch’s informatic assumptions. Lutz Polley (2000) has produced Born rule derivations where the informatic assumptions are replaced by symmetry arguments.
• MWI removes the observer-dependent role in the quantum measurement process by replacing wavefunction collapse with quantum decoherence. Since the role of the observer lies at the heart of most if not all “quantum paradoxes,” this automatically resolves a number of problems; see for example Schrödinger’s cat thought-experiment, the EPR paradox, von Neumann‘s “boundary problem” and even wave-particle duality. Quantum cosmology also becomes intelligible, since there is no need anymore for an observer outside of the universe.
• MWI is realist, deterministic, local theory, akin to classical physics (including the theory of relativity), at the expense of losing counterfactual definiteness. MWI achieves this by removing wavefunction collapse, which is indeterministic and non-local, from the deterministic and local equations of quantum theory.
• MWI (or other, broader multiverse considerations) provides a context for the anthropic principle which may provide an explanation for the fine-tuned universe.
• MWI, being a decoherent formulation, is axiomatically more streamlined than the Copenhagen and other collapse interpretations; and thus favoured under certain interpretations of Ockham’s razor. Of course there are other decoherent interpretations that also possess this advantage with respect to the collapse interpretations.
Common objections and misconceptions
• MWI states that there is no special role nor need for precise definition of measurement in MWI, yet uses the word “measurement” repeatedly through out its exposition.
MWI response: “measurements” are treated a subclass of interactions, which induce subject-object correlations in the combined wavefunction. There is nothing special about measurements (they don’t trigger any wave function collapse, for example); they are just another unitary time development process. This is why no precise definition of measurement is required in Everett’s formulation.
• The many-worlds interpretation is very vague about the ways to determine when splitting happens, and nowadays usually the criterion is that the two branches have decohered. However, present day understanding of decoherence does not allow a completely precise, self contained way to say when the two branches have decohered/”do not interact”, and hence many-worlds interpretation remains arbitrary. This is the main objection opponents of this interpretation raise, saying that it is not clear what is precisely meant by branching, and point to the lack of self contained criteria specifying branching.
MWI response: the decoherence or “splitting” or “branching” is complete when the measurement is complete. In Dirac notation a measurement is complete when:
\lang O[i]|O[j]\rang = \delta_{ij}
where O[i] represents the observer having detected the object system in the i-th state. Before the measurement has started the observer states are identical; after the measurement is complete the observer states are orthonormal. Thus a measurement defines the branching process: the branching is as well- or ill- defined as the measurement is. Thus branching is complete when the measurement is complete. Since the role of the observer and measurement per se plays no special role in MWI (measurements are handled as all other interactions are) there is no need for a precise definition of what an observer or a measurement is – just as in Newtonian physics no precise definition of either an observer or a measurement was required or expected. In all circumstances the universal wavefunction is still available to give a complete description of reality.
Also, it is a common misconception to think that branches are completely separate. In Everett’s formulation, they may in principle quantum interfere (i.e. “merge” instead of “splitting”) with each other in the future, although this requires all “memory” of the earlier branching event to be lost, so no observer ever sees two branches of reality.
• There is circularity in Everett’s measurement theory. Under the assumptions made by Everett, there are no ‘good observations’ as defined by him, and since his analysis of the observational process depends on the latter, it is void of any meaning. The concept of a ‘good observation’ is the projection postulate in disguise and Everett’s analysis simply derives this postulate by having assumed it, without any discussion.
MWI response: Everett’s treatment of observations / measurements covers both idealised good measurements and the more general bad or approximate cases. Thus it is legitimate to analyse probability in terms of measurement; no circularity is present.
MWI response: Everett analysed branching using what we now call the “measurement basis“. It is fundamental theorem of quantum theory that nothing measurable or empirical is changed by adopting a different basis. Everett was therefore free to choose whatever basis he liked. The measurement basis was simply the simplest basis in which to analyse the measurement process.
• We cannot be sure that the universe is a quantum multiverse until we have a theory of everything and, in particular, a successful theory of quantum gravity. If the final theory of everything is non-linear with respect to wavefunctions then many-worlds would be invalid.
MWI response: all accepted quantum theories of fundamental physics are linear with respect to the wavefunction. Whilst quantum gravity or string theory may be non-linear in this respect there is no evidence to indicate this at the moment.
• Occam’s Razor rules against a plethora of unobservable universes – Occam would prefer just one universe; i.e. any non-MWI interpretation.
MWI response: Occam’s razor actually is a constraint on the complexity of physical theory, not on the number of universes. MWI is a simpler theory since it has fewer postulates. See the “advantages” section.
• Unphysical universes: If a state is a superposition of two states ΨA and ΨB, i.e. Ψ = (aΨA + bΨB), i.e. weighted by coefficients a and b, then if b << a, what principle allows a universe with vanishingly small probability b to be instantiated on an equal footing with the much more probable one with probability a? This seems to throw away the information in the probability amplitudes. Such a theory makes little sense.
MWI response: The magnitude of the coefficients provides the weighting that makes the branches or universes “unequal”, as Everett and others have shown, leading the emergence of the conventional probabilistic rules.
• Violation of the principle of locality, which contradicts special relativity: MWI splitting is instant and total: this may conflict with relativity, since an alien in the Andromeda galaxy can’t know I collapse an electron over here before she collapses hers there: the relativity of simultaneity says we can’t say which electron collapsed first – so which one splits off another universe first? This leads to a hopeless muddle with everyone splitting differently. Note: EPR is not a get-out here, as the alien’s and my electrons need never have been part of the same quantum, i.e. entangled.
MWI response: the splitting can be regarded as causal, local and relativistic, spreading at, or below, the speed of light (e.g. we are not split by Schrödinger’s cat until we look in the box). For spacelike separated splitting you can’t say which occured first — but this is true of all spacelike separated events, simultaneity is not defined for them. Splitting is no exception; many-worlds is a local theory.
Brief overview
Schematic representation of pair of “smallest possible” quantum mechanical systems prior to interaction: Measured system S and measurement apparatus M. Systems such as S are referred to as 1-qubit systems.
Schematic illustration of splitting as a result of a repeated measurement.
For example, consider the smallest possible truly quantum system S, as shown in the illustration. This describes for instance, the spin-state of an electron. Considering a specific axis (say the z-axis) the north pole represents spin “up” and the south pole, spin “down”. The superposition states of the system are described by (the surface of) a sphere called the Bloch sphere. To perform a measurement on S, it is made to interact with another similar system M. After the interaction, the combined system is described by a state that ranges over a six-dimensional space (the reason for the number six is explained in the article on the Bloch sphere). This six-dimensional object can also be regarded as a quantum superposition of two “alternative histories” of the original system S, one in which “up” was observed and the other in which “down” was observed. Each subsequent binary measurement (that is interaction with a system M) causes a similar split in the history tree. Thus after three measurements, the system can be regarded as a quantum superposition of 8= 2 × 2 × 2 copies of the original system S.
Relative state
The goal of the relative-state formalism, as originally proposed by Everett in his 1957 doctoral dissertation, was to interpret the effect of external observation entirely within the mathematical framework developed by Paul Dirac, von Neumann and others, discarding altogether the ad-hoc mechanism of wave function collapse. Since Everett’s original work, there have appeared a number of similar formalisms in the literature. One such idea is discussed in the next section.
The relative-state interpretation makes two assumptions. The first is that the wavefunction is not simply a description of the object’s state, but that it actually is entirely equivalent to the object, a claim it has in common with some other interpretations. The second is that observation or measurement has no special role, unlike in the Copenhagen interpretation which considers the wavefunction collapse as a special kind of event which occurs as a result of observation.
The many-worlds interpretation is DeWitt’s popularisation of Everett’s work, who had referred to the combined observer-object system as being split by an observation, each split corresponding to the different or multiple possible outcomes of an observation. These splits generate a possible tree as shown in the graphic below. Subsequently DeWitt introduced the term “world” to describe a complete measurement history of an observer, which corresponds roughly to a single branch of that tree. Note that “splitting” in this sense, is hardly new or even quantum mechanical. The idea of a space of complete alternative histories had already been used in the theory of probability since the mid 1930s for instance to model Brownian motion.
Under the many-worlds interpretation, the Schrödinger equation, or relativistic analog, holds all the time everywhere. An observation or measurement of an object by an observer is modeled by applying the wave equation to the entire system comprising the observer and the object. One consequence is that every observation can be thought of as causing the combined observer-object’s wavefunction to change into a quantum superposition of two or more non-interacting branches, or split into many “worlds”. Since many observation-like events have happened, and are constantly happening, there are an enormous and growing number of simultaneously existing states.
If a system is composed of two or more subsystems, the system’s state will be a superposition of products of the subsystems’ states. Once the subsystems interact, their states are no longer independent. Each product of subsystem states in the overall superposition evolves over time independently of other products. The subsystems states have become correlated or entangled and it is no longer possible to consider them independent of one another. In Everett’s terminology each subsystem state was now correlated with its relative state, since each subsystem must now be considered relative to the other subsystems with which it has interacted.
Successive measurements with successive splittings
Comparative properties and experimental support
One of the salient properties of the many-worlds interpretation is that observation does not require an exceptional construct (such as wave function collapse) to explain it. Many physicists, however, dislike the implication that there are infinitely many non-observable alternate universes.
as of 2006, there are no practical experiments that distinguish between Many-Worlds and Copenhagen. There may be cosmological, observational evidence.
Copenhagen interpretation
In the Copenhagen interpretation, the mathematics of quantum mechanics allows one to predict probabilities for the occurrence of various events. In the many-worlds interpretation, all these events occur simultaneously. What meaning should be given to these probability calculations? And why do we observe, in our history, that the events with a higher computed probability seem to have occurred more often? One answer to these questions is to say that there is a probability measure on the space of all possible universes, where a possible universe is a complete path in the tree of branching universes. This is indeed what the calculations give. Then we should expect to find ourselves in a universe with a relatively high probability rather than a relatively low probability: even though all outcomes of an experiment occur, they do not occur in an equal way. As an interpretation which (like other interpretations) is consistent with the equations, it is hard to find testable predictions of MWI.
Quantum suicide
There is a rather more dramatic test than the one outlined above for people prepared to put their lives on the line: use a machine which kills them if a random quantum decay happens. If MWI is true, they will still be alive in the world where the decay didn’t happen and would feel no interruption in their stream of consciousness. By repeating this process a number of times, their continued consciousness would be arbitrarily unlikely unless MWI was true, when they would be alive in all the worlds where the random decay was on their side. From their viewpoint they would be immune to this death process. Clearly, if MWI does not hold, they would be dead in the one world. Other people would generally just see them die and would not be able to benefit from the result of this experiment. See Quantum suicide.
The universe decaying to a new vacuum state
Any event that changes the number of observers in the universe may have experimental consequences. Quantum tunnelling to new vacuum state would reduce the number of observers to zero (i.e. kill all life). Some Cosmologists argue that the universe is in a false vacuum state and that consequently the universe should have already experienced quantum tunnelling to a true vacuum state. This has not happened and is cited as evidence in favour of many-worlds.
The many-worlds interpretation should not be confused with the similar many-minds interpretation which defines the split on the level of the observers’ minds.
There is a wide range of claims that are considered “many-worlds” interpretations. It is often claimed by those who do not believe in MWI that Everett himself was not entirely clear as to what he believed; however MWI adherents (such as DeWitt, Tegmark, Deutsch and others) believe they fully understand Everett’s meaning as implying the literal existence of the other worlds. Additionally Everett’s reported belief in quantum immortality, requires belief in the reality of all the many-worlds represented by the components of the uncollapsed universal wavefunction.
“Many-worlds”-like interpretations are now considered fairly mainstream within the quantum physics community. For example, a poll of 72 leading physicists conducted by the American researcher David Raub in 1995 and published in the French periodical Sciences et Avenir in January 1998 recorded that nearly 60% thought many-worlds interpretation was “true”. Max Tegmark also reports the result of a poll taken at a 1997 quantum mechanics workshop. According to Tegmark, “The many worlds interpretation (MWI) scored second, comfortably ahead of the consistent histories and Bohm interpretations.” Other such polls have been taken at other conferences: see for instance Michael Nielsen‘s blog report on one such poll. Nielsen remarks that it appeared most of the conference attendees “thought the poll was a waste of time”. MWI sceptics (for instance Asher Peres) argue that polls regarding the acceptance of a particular interpretation within the scientific community, such as those mentioned above, cannot be used as evidence supporting a specific interpretation’s validity. However, others note that science is a group activity (for instance, peer review) and that polls are a systematic way of revealing the thinking of the scientific community.
A 2005 minor poll on the Interpretation of Quantum Mechanics workshop at the Institute for Quantum Computing University of Waterloo produced contrary results, with the MWI as the least favored.
One of MWI’s strongest advocates is David Deutsch. According to Deutsch, the single photon interference pattern observed in the double slit experiment can be explained by interference of photons in multiple universes. Viewed in this way, the single photon interference experiment is indistinguishable from the multiple photon interference experiment. In a more practical vein, in one of the earliest papers on quantum computing, he suggested that parallelism that results from the validity of MWI could lead to “a method by which certain probabilistic tasks can be performed faster by a universal quantum computer than by any classical restriction of it”. Deutsch has also proposed that when reversible computers become conscious that MWI will be testable (at least against “naive” Copenhagenism) via the reversible observation of spin.
Asher Peres was an outspoken critic of MWI, for example in a section in his 1993 textbook with the title Everett’s interpretation and other bizarre theories. In fact, Peres questioned whether MWI is really an “interpretation” or even if interpretations of quantum mechanics are needed at all. Indeed, the many-worlds interpretation can be regarded as a purely formal transformation, which adds nothing to the instrumentalist (i.e. statistical) rules of the quantum mechanics. Perhaps more significantly, Peres seems to suggest that positing the existence of an infinite number of non-communicating parallel universes is highly suspect as it violates those interpretations of Occam’s Razor that seek to minimize the number of hypothesized entities. Proponents of MWI argue precisely the opposite, by applying Occam’s Razor to the set of assumptions rather than multiplicity of universes. In Max Tegmark‘s formulation, the alternative to many-worlds is the undesirable “many words”, an allusion to the complexity of von Neumann’s collapse postulate.
MWI is considered by some to be unfalsifiable and hence unscientific because the multiple parallel universes are non-communicating, in the sense that no information can be passed between them. Others claim MWI is directly testable. Everett regarded MWI as falsifiable since any test that falsifies conventional quantum theory would also falsify MWI.
According to Martin Gardner MWI has two different interpretations: real or unreal, and claims that Stephen Hawking and Steve Weinberg favour the unreal interpretation. Gardner also claims that the interpretation favoured by the majority of physicists is that the other worlds are not real in the same way as our world is real, whereas the “realist” view is supported by MWI experts David Deutsch and Bryce DeWitt. However Stephen Hawking is on record as a saying that the “other worlds are as real as ours” and Tipler reports Hawking saying that MWI is “trivially true” (scientific jargon for “obviously true”) if quantum theory applies to all reality. Roger Penrose agrees with Hawking that QM applied to the universe implies MW, although he considers the current lack of a successful theory of quantum gravity negates the claimed universality of conventional QM.
Speculative implications
Speculative physics deals with questions also discussed in science fiction.
Quantum suicide thought experiment
It has been claimed that there is a thought experiment that would clearly differentiate between the many-worlds interpretation and other interpretations of quantum mechanics. It involves a quantum suicide machine and an experimenter willing to risk death. However, at best, this would only decide the issue for the experimenter; bystanders would learn nothing. The flip side of quantum suicide is quantum immortality.
Weak coupling
Another speculation is that the separate worlds remain weakly coupled (e.g. by gravity) permitting “communication between parallel universes”. This requires that gravity be a classical force and not quantized.
Similarity to Modal Realism
The many-worlds interpretation has some similarity to modal realism in philosophy, which is the view that the possible worlds used to interpret modal claims actually exist. Unlike philosophy, however, in quantum mechanics counterfactual alternatives can influence the results of experiments, as in the Elitzur-Vaidman bomb-testing problem or the Quantum Zeno effect.
Time travel
Many-worlds in literature and science fiction
Main article: Parallel universe (fiction)
A map from Robert Sobel‘s novel For Want of a Nail, illustrates how small events – in this example the branching or point of divergence from our history is in October 1777 – can profoundly alter the course of history. According to the many-worlds interpretation every microscopic event is a branch point; all possible alternative histories actually exist.
The many-worlds interpretation (and the somewhat related concept of possible worlds) have been associated to numerous themes in literature, art and science fiction.
Some of these stories or films violate fundamental principles of causality and relativity, and are extremely misleading since the information-theoretic structure of the path space of multiple universes (that is information flow between different paths) is very likely extraordinarily complex. Also see Michael Clive Price’s FAQ referenced in the external links section below where these issues (and other similar ones) are dealt with more decisively.
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Flash Forward: Follow the Clues; Crack the Mystery
I have gotten a bit behind with my write-ups on Flash Forward since I have been out of town. I visited the Walt Disney Family Museum in the Presidio in San Francisco on Friday and will be blogging about that over the next few days.
Jeff “Doc” Jensen of Entertainment Weekly has some theories about Flash Forward. I loved to read his theories on LOST each week, so I am leaving you with his latest musings on Flash Forward.
In the new issue of Entertainment Weekly now on newsstands, you’ll find a story written by yours truly in which I geek out on my new TV obsession, the ABC sci-fi drama FlashForward. If you’re new to the show, here’s what you need to know: On Oct. 6, the planet blacked out and for 2 minutes and 17 seconds, and everyone on earth saw a brief vision of their respective futures. The saga’s center is FBI agent Mark Benford (Shakespeare In Love’s Joseph Fiennes), who during his brief quantum leap saw himself investigating an elaborate conspiracy behind mankind’s perplexing power nap. The day glimpsed in all the flashes: Thursday, April 29, 2010. (Yep, the show will air that night.) Will Mark’s faithful wife Olivia (Lost’s Sonja Walger) find herself in bed with another man? Will vaguely sinister scientist Simon Campos (Dominic Monaghan, another ex-Lostie) strangle a dude to death? And will FBI agent Demetri Noh (Star Trek’s John Cho), who saw only darkness during his flash, be (gulp) dead? “The high concept pitch is simply this: if you were given a glimpse of your future, what would you do with it?” says FF’s exec producer David S. Goyer. “If you see something bad, can you change it? If it’s good, how do you make it come true?”
One of the things I love best about the series is the explicit and implicit references to science, literature, philosophy, and pop culture. When investigated, these references suggest all sorts of possibilities about what’s really going on in the saga, or at the very least add some cool or ironic shading to the story. For example: we’ve been told that Agent Noh will be killed on March 15. That also happens to be the date of Julius Caesar’s murder. More on the nose with FlashForward: Shakespeare’s Julius Caesar, in which a seer tells the Roman leader to “beware the Ides of March,” i.e. March 15. Is that just the writers having some smarty-pants fun—or are they planting a cue that a Brutus-like colleague will betray Agent Noh?
Here’s another example—a little more tenuous, but if you follow my Doc Jensen work on Lost, you know that making pseudo-intellectual leaps are part of what I’m all about. In FlashForward’s third episode, Agent Benford traveled to Germany to interview a Nazi war criminal who claimed to know something about the true nature of the global blackout. The old Nazi was being held at Quale Prison, and as it happens, the word quale is directly linked to a philosophical term dealing with—in wikipedia’s words—“the subjective quality of conscious experience.” (The fact that FlashForward would name a prison after such a heady concept is pretty provocative. Is the show trying to suggest that objective reality is unknowable and mankind is fundamentally at odds with each other because we are locked into our unique, idiosyncratic perspectives of the external world?)
Are you with me?! I hope so, because in the weeks to come, I plan on doing even more obsessing about FlashForward here at, beginning with a complete TV Watch recap of the next episode, airing Dec. 3. Until then, here’s are some additional references (legit and perceived my crazy eyes) that FlashForward has made during its first nine episodes—and some theories about what they could mean.
Of course, Dominic Monaghan (Charlie) and Sonya Walger (Penelope) are living, breathing embodiments of Lost. Both actors played characters linked to Desmond Hume. During the show’s third season, the ex-Hatchman became super-charged with The Island’s electromagnetic energy and began having flash-forwards of Charlie’s death. He also went back in time and tried to change his destiny. David Goyer—a big Lost fan—slipped a billboard for Oceanic Airlines into the pilot, inspiring fans to wonder if both shows exist in the same universe. At the very least, they may share a similar philosophical idea: that no matter how much you try to change predestined events, fate will get what it wants.
“Across The Universe” by Rufus Wainwright
Like Lost, Cold Case and so many other shows, FF has a penchant for episode-ending slow-motion montages, set to rousing score or a thematically loaded pop song. One of my faves was Wainwright’s cover of this classic tune by The Beatles, heard in the Oct. 29 episode “Scary Monsters and Super Creeps.” In the 1999 Robert J. Sawyer novel that inspired the series, the blackout/flash-forwards are caused, in part, by an anomaly in deep space—literally “across the universe.” Coincidence? Nay! I say: Synchronicity! As in…
“Ghost In The Machine” by The Police
Agent Benford is a fan of the band and wore a T-shirt featuring this album’s artwork to an undercover operation—infiltrating an underground club catering to “ghosts,” people in the FF universe who didn’t see anything in their flash forward and thus believe they are destined to die before that date. According to band lore, the album was inspired by Sting’s fixation with Arthur Koestler, an egghead who postulated that people, events, and time are psychically linked via the concept of Synchronicity described by Carl Jung. (The Police song “Synchronicity,” from the album of the same (also inspired by Koestler), is a FlashForward theory unto itself; check out the lyrics
here.) Koestler’s books include The Ghost In The Machine, The Roots of Coincidence (a book that has had a big influence on sci-fi, fantasy, and comic book writers), and Janus: A Summing Up, an exploration of systems theory that says that a larger whole or “holarchy” is make up of individual components called “holons” that also contain systems within themselves, or something like that, or maybe nothing at all like that, and yes, I don’t have any clue what any of this means. Bookmark that Janus name—we’ll be coming back to it in a minute.
In FlashForward, Jericho is a military contractor that provides private armies to the highest bidder. Their soldiers apparently played a role in the attempted killing of Aaron Stark’s daughter, Tracy, in Afghanistan. I also suspect they are providing goons to the conspiracy that perpetrated the global blackout. Of course, Jericho was also the title of the short-lived, intensely loved cult drama that imagined the aftermath of a cataclysmic nuclear attack on the United States.
David Baldacci
Trying to make sense of Jericho’s treacherous attack on his daughter, Aaron likened the situation to a “Baldacci novel.” David Baldacci is the best-selling author of hugely successful books like The Collectors and Absolute Power, political potboilers that usually involve elaborate government conspiracies. Application to FlashForward: I’m thinking President Segovia (played by Peter Coyote)—who is (or was) tight with Assistant Director Wedeck (Courtney B. Vance)—knows much more about the global blackout than he’s telling. And remember Senator Clemente (Barbara Williams), the congresswoman who was leading the subcommittee investigation into the flash-forward event? She was no friend to Segovia and Wedeck, yet the president made her his new vice president—presumably to stifle her persecution of Wedeck’s Mosaic team. No, it’s not very realistic that a president would appoint a hated political rival as his No. 2—unless, of course, Segovia and Clemente aren’t the bitter enemies they appeared to be. I’m thinking that yes, Senator Clemente is in on the conspiracy, too. So what’s the conspiracy? Here’s the clue that sketches the big picture:
“D. Gibbons”
The “bad man” from little Charlie Benford’s spooky vision and one of many cryptic clues gleaned from Agent Benford’s flash forward. The name undoubtedly refers to Dave Gibbons, co-author of the subversive superhero saga Watchmen, whose intricate mystery plot concerns (SPOILER ALERT!) a conspiracy to encourage world peace by staging a fake alien invasion. And like FF, Watchmen stuffed coded clues and tell-tale non-sequiturs in the margins of its story. I’m thinking that the power players behind the global blackout were attempting to do something similar—usher in a new era of world peace by staging a global cataclysm designed to cause everyone to rethink their lives, the way they live their lives and the political, religious, and philosophical barriers that divide us. The bipartisan union of President Segovia and Senator Clemente is symbolic and representative of the narrative the conspiracy was/is trying to promote throughout the world. The two-faced, double-edged nature of this scheme to engineer planetary rebirth via planetary catastrophe is reflected in our next clue…
Janice is the name of the FlashForward character who saw herself having a baby on April 29… even though she’s a lesbian who is currently not in relationship and had been deeply ambivalent about even having kids until recently. Janice sounds exactly like Janus, the two-faced Roman deity of open gates and closed doors, of beginnings and endings. Janus is a deeply ironic, very paradoxical dude—both hopeful and ominous. That’s very Janice. Speaking of double-sided clues…
Beilby Porteus
Mosaic’s search for “D. Gibbons” led him to Pigeon, Utah, where he encountered a mystery man, coined “The Chess Player” by fans, in an abandoned doll factory. Before escaping, the chess player said, “He who foresees calamities, suffers them twice over.” That’s a famous quote from Porteus, an 18th-century English clergyman and noted abolitionist. His other major claim to fame was introducing something called “The Sunday Observance Act,” a “blue law” that regulated the ways people in England could spend their recreational time on Sunday. This is could be a double-faced clue. On one hand, we have an ostensible bad guy, quoting a guy linked to a righteous cause (ending slavery) and famous for forcing a righteous way of life on society (the Sunday Observance Act)—another possible proof on the aforementioned world peace conspiracy. But is The Chess Player part of The Blackout Conspiracy—or working to subvert it? The Porteus quote is darkly ironic. And coming from The Chess Player, it sounds like a warning or threat. Possibilities: If The Chess Player is trying to promote the conspiracy, he might have been trying to tell Benford that trying to solve the mystery of the blackout calamity will only produce another calamity—the ruin of its peace-promoting effect. But if The Chess Player is trying to fight the conspiracy, he might have been trying to tell Benford that flash-forward event had backfired—or will backfire when it reaches its fulfillment on April 29. FYI: Porteus is associated with another ironic quote that could be applied to FlashForward and my “Conspiracy of Peace” conspiracy theory: “War its thousands slays; peace its ten thousands.”
White Queen
The Chess Player left behind some clues for Mosaic, including a chess piece—the white queen, which provocatively intersects with all sorts of fantasy and geek pop. The White Queen is a character from Through The Looking Glass—a scatter-brained figure that lives her life backwards and struggles to live in the present. (“Jam-yesterday or Jam-tomorrow, but never Jam-today.”) That’s fitting for a show whose people got mind-scrambled during the global blackout and are now playing out futures that may have already come to pass—who are constantly being reminded and challenged to bravely defy fate by “living in the now.” However, “White Queen” is the handle of several prominent characters in comic book land, including the morally ambiguous X-Men foil Emma Frost (who can see into other people’s heads) and another baddie, Sat-Yr-9, an unhinged femme fatale from an alternate reality Earth. Yes, it is unlikely FlashForward was deliberately trying to forge a connection to the latter character. But she does embody a high concept theory in quantum physics that was explicitly referenced by Dominic Monaghan’s Simon Campos character: the idea that all possible realities actually exist. (See: the Schrödinger’s Cat thought experiment, used by Campos to seduce the blonde lady on the train.) I won’t beat this dead (Schrödinger’s) cat further by bringing this full circle and explicating the link between Alice In Wonderland and quantum mechanics, or how the whole notion of white and black chess pieces illustrate the binary either/or dynamics of alternate reality logic. However, and fittingly, I will ask you to entertain at least two possibilities: 1. That what everyone saw in their flash-forward was actually a peek into an alternate reality; and 2. That per the implications of Schrödinger’s Cat, which says that reality isn’t created until directly observed by the viewer, that the future sketched by the flash-forwards is now locked into place as a result of being directly observed by everyone in the past via the global blackout. Got that? Thought so. Also see: A Christmas Carol SPOILER ALERT! David Goyer says Charles Dickens’ classic novel—about a grinch who’s shown a vision of his (possible) future—looms large in the Dec. 3 episode.
Flash Forward: QED
While I take a couple of days to digest episode 8 “Rules of the Game” of Flash Forward, I leave you with an explanation of what QED is. While sniping back and forth during their poker game, Lloyd and Simon threw this acronym at each other.
Q.E.D. is an abbreviation of the Latin phrase quod erat demonstrandum, which literally means “which was to be demonstrated”. The phrase is written in its abbreviated form at the end of a mathematical proof or philosophical argument to signify that the last statement deduced was the one to be demonstrated; the abbreviation thus signals the completion of the proof.
Etymology and early use
The phrase is a translation into Latin from Greek ὅπερ ἔδει δεῖξαι (hoper edei deixai; abbreviated as ΟΕΔ), a phrase used by many early mathematicians, including Euclid and Archimedes. These mathematicians, in particular Euclid, are credited with founding axiomatic mathematics with its emphasis on establishing truths by logical deduction (rather than experimentation or assertion); their use of this phrase symbolizes this emphasis, as well as marking this important step in the development of mathematical philosophy.
Modern philosophy
In the European Renaissance, scholars often wrote in Latin, and phrases such as Q.E.D. were often used to conclude proofs.
Perhaps the most famous use of Q.E.D. in a philosophical argument is found in the Ethics of Baruch Spinoza, published posthumously in 1677. Written in Latin, it is considered by many to be Spinoza’s magnum opus. The style and system of the book is, as Spinoza says, “demonstrated in geometrical order”, with axioms and definitions followed by propositions. For Spinoza, this is a considerable improvement over René Descartes’s writing style in the Meditations, which follows the form of a diary.
There is another Latin phrase with a slightly different meaning, and less common in usage. Quod erat faciendum is translated as “which was to have been done”. This is usually shortened to Q.E.F. The expression quod erat faciendum is a translation of the Greek geometers’ closing ὅπερ ἔδει ποιῆσαι (hoper edei poiēsai). Euclid used this phrase to close propositions which were not proofs of theorems, but constructions. For example, Euclid’s first proposition shows how to construct an equilateral triangle given one side.
Equivalents in other languages
Q.E.D. has acquired many translations in various foreign languages. In French, German, Italian and Russian (with English, the main languages of modern Western mathematics) it is respectively C.Q.F.D., for ce qu’il fallait démontrer (or sometimes ce qui finit la démonstration), W.Z.B.W. for was zu beweisen war, C.V.D. for come volevasi dimostrare, and ч.т.д., for что и требовалось доказать. In Spanish and Portuguese it is Q.E.D. for Quedo esto demostrado. There does not appear to be a common formal English equivalent, though the end of a proof may be announced with a simple statement such as “this completes the proof” or a similar locution. Most modern math textbooks in English end proofs with a symbol, often square. (See below.) In modern Greek texts sometimes the initials ο.ε.δ. (for ὅπερ ἔδει δεῖξαι) are used at the end of a mathematical proof.
Electronic forms
When typesetting was done by a compositor with letterpress printing, complex typography such as mathematics and foreign languages were called “penalty copy” (the author paid a “penalty” to have them typeset, as it was harder than plain text). With the advent of systems such as LaTeX, mathematicians found their options more open, so there are several symbolic alternatives in use, either in the input, the output, or both. When creating TeX, Knuth provided the symbol ■ (solid black square), also called by mathematicians tombstone or Halmos symbol (after Paul Halmos, who pioneered its use). The tombstone is sometimes open: □ (hollow black square). Unicode explicitly provides the “End of Proof” character U+220E (∎), but also offers ▮ (U+25AE, black vertical rectangle) and ‣ (U+2023, triangular bullet) as alternatives. Some authors have adopted variants of this notation with other symbols, such as two forward slashes (//), or simply some vertical white space.
In popular culture
The 1982 US television series Q.E.D. starred Sam Waterston as Professor Quentin Everett Deverill, an American detective in Edwardian England who uses a style of logical deduction similar to that of Sherlock Holmes. Hence, the show’s title derives both from the protagonist’s initials (by which he is primarily known), as well as the logical proofs he presents.
Douglas Adams‘ franchise, The Hitchhiker’s Guide to the Galaxy, famously uses Q.E.D. to conclude its Babel fish proof, which determines that God no longer exists because the Babel fish is too improbable to have evolved by pure chance; therefore the Babel fish was a proof for God, and as Faith involves no proof, there was no God, QED (using humorous fallacy to mock teleology and intelligent design principles).
Source: Wikipedia |
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Open Access
Dimensional Effects on Densities of States and Interactions in Nanostructures
Nanoscale Research Letters20105:1546
Received: 15 April 2010
Accepted: 7 June 2010
Published: 2 July 2010
We consider electrons in the presence of interfaces with different effective electron mass, and electromagnetic fields in the presence of a high-permittivity interface in bulk material. The equations of motion for these dimensionally hybrid systems yield analytic expressions for Green’s functions and electromagnetic potentials that interpolate between the two-dimensional logarithmic potential at short distance, and the three-dimensional r −1 potential at large distance. This also yields results for electron densities of states which interpolate between the well-known two-dimensional and three-dimensional formulas. The transition length scales for interfaces of thickness L are found to be of order Lm/2m * for an interface in which electrons move with effective mass m *, and for a dielectric thin film with permittivity in a bulk of permittivity . We can easily test the merits of the formalism by comparing the calculated electromagnetic potential with the infinite series solutions from image charges. This confirms that the dimensionally hybrid models are excellent approximations for distances r L/2.
Density of statesCoulomb and exchange interactions in nanostructuresDielectric thin films
When we suppress motion of particles in certain directions through confining potentials, e.g. in quantum wells or quantum wires, we often model the residual low energy excitations in the system through low-dimensional quantum mechanical systems. Prominent examples of this concern layered heterostructures, and one instance where the number d of spatial dimensions enters in a manner which is of direct relevance to technology is in the density of states. In the standard parabolic band approximation, this takes the form (with two helicity or spin states)
These are densities of states per d-dimensional volume and per unit of energy. The corresponding dependence of the relation between the Fermi energy and the density n of electrons on d is
Variants of these equations (including summation over subbands) are often used for d = 2 or d = 1 to estimate carrier densities in quasi two-dimensional systems or nanowires, and the density of states plays a crucial role in all transport and optical properties of materials. Indeed, the obvious relevance for electrical conductivity properties in micro and nanotechnology implies that densities of states for d = 1, 2, or 3 are now commonly discussed in engineering textbooks, but there is another reason why I anticipate that variants of Eq. (1) will become ever more prominent in the technical literature. Densities also play a huge role in data storage, but with us still relying on binary logic switching between two stable states (spin up or down, charge or no charge, conductivity or no conductivity), data storage densities are limited by the physical densities of the systems which provide the dual states. We could (and likely will) drive information technology and integration much further if we can find ways to utilize more than just two states of a physical system to store and process information. Then, data storage densities should become proportional to energy integrals of local densities of states. Equation (1) for d = 1 or d = 2 is certainly applicable for particles which have low energies compared to the confinement energy of a nanowire or a quantum well, but how can we effectively model particles which are weakly confined to a nanowire or quantum well, or which are otherwise affected by the presence of a low-dimensional substructure? In these cases, we can devise dimensionally hybrid models [1, 2] which yield e.g. densities of states which interpolate between d = 2 and d = 3 [3, 4]. This construction will be reviewed in Sect. 2. Based on the experience gained with dimensionally hybrid Hamiltonians for massive particles, we can also construct inter-dimensional Hamiltonians for photons which should be applicable to photons in the presence of high-permittivity thin films or interfaces. These models can also be solved in terms of infinite series expansions using image charges, and the merits of this approach can easily be tested. The case of high-permittivity thin films and testing the theory against image charge solutions will be discussed in Sect. 3.
Dimensionally Hybrid Hamiltonians and Green’s Functions for Massive Particles in the Presence of Thin Films or Interfaces
We use the connection between Green’s functions and the density of states to generalize Eq. (1) for massive particles in the presence of a thin film or interface.
The energy-dependent Green’s function for a Hamiltonian H with spectrum E n and eigenstates |n, ν〉 is
Here, ν is a degeneracy index and the notation implies that continuous components in the indices (n, ν) are integrated. The first equation simply states the relation between the resolvent of the Hamiltonian and the Green’s function G(E) which is normalized as lim m→0,E→0 G(E)| d=3 = (4πr)−1.
The zero-energy Green’s function G(0) determines e.g. 2-particle correlation functions and electromagnetic interaction potentials, and the energy-dependent Green’s function G(E) determines e.g. scattering amplitudes for particles of energy E. Application for resistivity calculations is therefore another technologically relevant application of Green’s functions. However, in the present section we are interested in this function because it also determines the local density of states in a system with Hamiltonian H through the relation
Here, we explicitly included a factor 2 for the number of spin or helicity states, because the summation over degeneracy indices in (3,4) usually only involves orbital indices.
For our present investigation, the distinctive feature of the interface is that the particles move in it with an effective mass m *, while their mass in the surrounding bulk is m. We use coordinates parallel to a plane interface, which is located at z = z 0. Bold vector notation is used for quantities parallel to the interface, e.g. and .
We assume that the interface has a thickness L. If the wavenumber component orthogonal to the interface is small compared to the inverse width, |k L| 1, i.e. if the de Broglie wavelength and the incidence angle satisfy λ L|cosϑ|, we can approximate the kinetic energy of the particles through a second quantized Hamiltonian
where μ = m */L. The corresponding first quantized Hamiltonian is
The interesting aspect of the Hamiltonians (5,6) is the linear superposition of two-dimensional and three-dimensional kinetic terms. The formalism presented here could and will certainly be extended to include also kinetic terms which are linear in derivatives, in particular in the interface term. This would be motivated either by a Rashba term arising from perpendicular fields penetrating the interface [511] of from the dispersion relation in Graphene [1215]. However, for the present investigation we will use a parabolic band approximation in the bulk and in the interface.
The energy-dependent Green’s function describes scattering effects in the presence of the interface but also applies to scattering off perturbations which are not located on the interface. In an axially symmetric mixed representation
the first order approximation to scattering of an orthogonally incoming plane wave off an impurity potential
corresponds to
Green’s functions for surfaces or interfaces are commonly parametrized in an axially symmetric mixed representation like . In bra-ket notation, this corresponds for the free Green’s function G 0(E), which is also translation invariant in z direction, to
We will briefly recall the explicit form of the free Green’s function G 0(E) in the axially symmetric mixed parametrization for later comparison. The equation
To study how this is modified in the presence of the interface, we observe that the Hamiltonians (5) or (6) yield a Schrödinger equation
The corresponding equation for the Green’s function or 2-point correlation function is
The solution of this equation is described in the Appendix. In particular, we find the representation (see Eq. (27))
where the definition ℓ≡m/2μ = Lm/2m * was used. The ℓ-independent terms in (10) correspond to the free Green’s function G 0(E) (8).
The interface at z 0 breaks translational invariance in z direction, and we have with Eq. (7)
We will use the result (10) to calculate the density of states in the interface. Substitution yields
and after evaluation of the integral
This is a more complicated result than the density (1) for d = 2 or d = 3. However, it reduces to either the two-dimensional or three-dimensional density of states in the appropriate limits, see Fig. 1. For large energies, i.e. if the states only probe length scales smaller than the transition length scale ℓ, we find the two-dimensional density of states properly rescaled by a dimensional factor to reflect that it is a density of states per three-dimensional volume,
Figure 1
Figure 1
The red line is the two-dimensional limit (12). The blue line is the three-dimensional density of states. The it black line is the inter-dimensional density of states (11) for ℓ = 50 nm
For small energies, i.e. if the states probe length scales larger than ℓ, we find the three-dimensional density of states
This limiting behavior for interpolation between two and three dimensions is consistent with what is also observed for the zero-energy Green’s function in the interface, see equations (21–22) below.
Equation (11) also implies interpolating behavior for the relation between electron density and Fermi energy on the interface. The full relation is
This approximates two-dimensional behavior for ,
and three-dimensional behavior for ,
It is intuitively understandable that the presence of a layer reduces the available density of states for given energy, or equivalently increases the Fermi energy for a given density of electrons. The presence of a layer generically implies boundary or matching conditions which reduce the number of available states at a given energy.
A condition for relevance of the inter-dimensional behavior is a large transition scale compared to the layer thickness, ℓ L, see also Fig. 2. In terms of effective particle mass, this means
i.e. the energy band in the interface should be more strongly curved than in the bulk matrix for the transition to two-dimensional behavior to be observable.
Figure 2
Figure 2
The upper dotted (blue) line is the three-dimensional Green’s function (4πr)−1 in units of ℓ−1, the continuous line is the Green’s function (19) in units of ℓ−1, and the lower dotted (red) line is the two-dimensional logarithmic Green’s function ℓ·G = − (γ + ln(r/2ℓ))/(4π)
Electric Fields in the Presence of High-Permittivity Thin Films or Interfaces
The zero-energy Green’s function determines electrostatic and exchange interactions through the electrostatic potential . Here, q is an electric charge in a dielectric material of permittivity . The zero-energy Green’s function in d spatial dimensions is given by
We cannot infer from the previous section that the zero energy limit of the inter-dimensional Green’s function calculated there also yields a dimensionally hybrid potential, because we were dealing with solutions of Schrödinger’s equation instead of the Gauss law. However, we can rederive the zero energy limit of that Green’s function from the Gauss law for electromagnetic fields in the presence of a high-permittivity interface.
Suppose we have charge carriers of charge q and mass m in the presence of an interface with permittivity and permeability μ*, We continue to denote vectors parallel to the interface in bold face notation, , , etc.
If the photon wavelengths and incidence angles satisfy the condition λ L|cosϑ|, we can approximate the system with an action
Variation with respect to the electrostatic potential, , yields the Gauss law in the form
and the continuity condition E z (z 0 − 0) = E z (z 0 + 0).
We solve Eq. (16) in Coulomb gauge,
where the Green’s function has to satisfy
This equation is the zero energy limit of Eq. (9) with the substitution
We can therefore read off the solution from the results of the previous section with E = 0 and now .
Equation (10) yields in particular
with . Fourier transformation yields
The zero-energy Green’s function in the interface is given in terms of a Struve function and a Neumann function1,
This yields logarithmic behavior of interaction potentials at small distances r ℓ and 1/r behavior for large separation r ℓ of charges in high-permittivity thin films,
see also Fig. 2.
For the comparison with image charges, we set z 0 = 0 and recall that the solution for the potential of a charge q at , z = 0 proceeds through the ansatz
and symmetric continuation to z < −L/2.
This yields electric fields
and the junction conditions at z = L/2 yield for n ≥ 0 from the continuity of E r ,
and from the continuity of D z ,
These conditions can be solved through
In particular, the potential at z = 0 is
We have
and therefore for
The solution from image charges is in very good agreement with the analytic model for distances r L/2, where both the image charge solution and the analytic model show strong deviations from the bulk r −1 behavior. This is illustrated in Fig. 3 by plotting the reduced electrostatic potential for a charge q, in the interface.
Figure 3
Figure 3
Different reduced electrostatic potentials are plotted for . The upper dotted (green) line is the three-dimensional reduced potential L/(4πr). The central dotted (blue) line is the reduced potential following from the image charge solution (22). The solid (black) line is the potential from the analytic model (19). The lower dotted (red) line is the reduced logarithmic potential. The reduced potentials from our analytic model and from image charges are indistinguishable for , see also Fig. 4
It is also instructive to plot the relative deviation between the dimensionally hybrid potential which follows from (20) and the potential (23) from image charges.
Figure 4 shows that for r L/2, the dimensionally hybrid model is a very good approximation to the potential from image charges with accuracy better than 10−2 if . For , the accuracy is still better than 4 × 10−2.
Figure 4
Figure 4
The relative deviation between the dimensionally hybrid potential from (19) and the potential (22) from image charges for
An analysis of models for particles in the presence of a low effective mass interface, and for electromagnetic fields in the presence of a high-permittivity thin film, yields dimensionally hybrid densities of states (11) and electrostatic potentials (17,20) which interpolate between two-dimensional behavior and three-dimensional behavior. The analytic model for the electromagnetic fields is in very good agreement with the infinite series solution already for small distance scales r L/2, where the potential strongly deviates from the standard bulk r −1 potential. At distance scales smaller than L/2, r −1, behavior seems to dominate again for the electrostatic potential, in agreement with expectations that for distances which are small compared to the lateral extension of a dielectric slab, bulk behavior should be restored. However, note that neither the inter-dimensional analytic model nor the solution from image charges is trustworthy for very small distances, because both models rely on a continuum approximation through the use of effective permittivities, but the continuum approximation should break down at sub-nanometer scales.
The most important finding is that interfaces and thin films of width L should exhibit transitions between two-dimensional and three-dimensional distance laws for physical quantities at length scales of order Lm/2m * or , respectively. Interfaces with strong band curvature or high permittivity should provide good samples for experimental study of the transition between two-dimensional and three-dimensional behavior.
Appendix: Solution of Eq. 9
Substitution of the fourier transform
into Eq. 9 yields
This yields with (7) the condition
Fournier transformation with respect to z yields
This result implies that has the form
with the yet to be determined satisfying
For the treatment of the integrals, we should be consistent with the calculation of the free retarded Green's function (8),
This yields
And therefore
where the definition ℓm/2μ = Lm/2m*. Fourier transformation of Eq (26) with respect to k yields finally
The Green's function with only k space variables is found from the Fournier transform of Eq. 25,
and the ensuing equations
This yields
It is easily verified that Fournier transformation yields again the result (26).
1Our notations for special functions follow the conventions of Abramowitz and Stegun [16].
This research was supported by NSERC Canada.
Open Access
Authors’ Affiliations
Physics & Engineering Physics, University of Saskatchewan, Saskatoon, Canada
1. Dick R: Int. J. Theor. Phys.. 2003, 42: 569. 10.1023/A:1024446017417View ArticleGoogle Scholar
2. Dick R: Nanoscale Res. Lett.. 2008, 3: 140. Bibcode number [2008NRL.....3..140D] 10.1007/s11671-008-9126-4View ArticleGoogle Scholar
3. Dick R: Phys. E. 2008, 40: 524. 10.1016/j.physe.2007.07.025View ArticleGoogle Scholar
4. Dick R: Phys. E. 2008, 40: 2973. COI number [1:CAS:528:DC%2BD1cXntVOns7o%3D] 10.1016/j.physe.2008.02.017View ArticleGoogle Scholar
5. Bychkov YA, Rashba EI: JETP Lett.. 1984, 39: 78. Bibcode number [1984JETPL..39...78B]Google Scholar
6. Bychkov YA, Rashba EI: J. Phys. C. 1984, 17: 6039. Bibcode number [1984JPhC...17.6039B] 10.1088/0022-3719/17/33/015View ArticleGoogle Scholar
7. Cappelluti E, Grimaldi C, Marsiglio F: Phys. Rev. Lett.. 2007, 98: 167002. COI number [1:STN:280:DC%2BD2s3ptVyqsA%3D%3D]; Bibcode number [2007PhRvL..98p7002C] 10.1103/PhysRevLett.98.167002View ArticleGoogle Scholar
8. Cappelluti E, Grimaldi C, Marsiglio F: Phys. Rev. B. 2007, 76: 085334. Bibcode number [2007PhRvB..76h5334C] 10.1103/PhysRevB.76.085334View ArticleGoogle Scholar
9. Srisongmuang B, Pairor P, Berciu M: Phys. Rev. B. 2008, 78: 155317. Bibcode number [2008PhRvB..78o5317S] 10.1103/PhysRevB.78.155317View ArticleGoogle Scholar
10. Vasilopoulos P, Wang XF: Phys. E. 2008, 40: 1729. 10.1016/j.physe.2007.10.068View ArticleGoogle Scholar
11. Li S-S, Xia J-B: Nanoscale Res. Lett.. 2009, 4: 178. COI number [1:CAS:528:DC%2BD1MXht1Kntrw%3D]; Bibcode number [2009NRL.....4..178L] 10.1007/s11671-008-9222-5View ArticleGoogle Scholar
12. Semenoff GW: Phys. Rev. Lett.. 1984, 53: 2449. Bibcode number [1984PhRvL..53.2449S] 10.1103/PhysRevLett.53.2449View ArticleGoogle Scholar
13. Apalkov V, Wang XF, Chakraborty T: Mod. Phys. B. 2007, 21: 1165. COI number [1:CAS:528:DC%2BD2sXltl2jtb0%3D] 10.1142/S0217979207042604View ArticleGoogle Scholar
14. Covaci L, Berciu M: Phys. Rev. Lett. 2008, 100: 256405. Bibcode number [2008PhRvL.100y6405C] 10.1103/PhysRevLett.100.256405View ArticleGoogle Scholar
15. Li T, Zhang Z: Nanoscale Res. Lett.. 2010, 5: 169. Bibcode number [2010NRL.....5..169L] 10.1007/s11671-009-9460-1View ArticleGoogle Scholar
16. Abramowitz M., Stegun I.A. (Eds): Handbook of Mathematical Functions 9th printing. Dover Publications, New York; 1970.Google Scholar
© The Author(s) 2010 |
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A View from the Altar / Build an altar to the Lord your God on top of this rock… (Judges 6:26)
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The Secular Man’s Suicide Pact
I recently visited a place with a much larger Moslem population than my home town. That set me to thinking about things.
Leftist progress on diversity looks to be moving right along. I’m guessing they have a schedule for undoing the dispersion of Babel. We’ll see how that works out.
Trembling as I am to utter the following blasphemy against one of the Secular Man’s highest articles of faith, it appears to me that segregation is pretty much a natural thing for most people. If we don’t segregate by race, we do it by sex, religion, income, the type of work you do, your IQ, or even your status as a manager or worker bee. I’ve noted that the Ivy League humanities majors don’t pal around with the NASCAR set a whole lot, not that either side of this travesty of segregation is complaining about it, because each side is equally convinced that the other side is peopled with bigots and idiots. In fact, NASCAR people won’t even hang around NHRA. Oh, well.
So people prefer to be around folks who are like themselves. But somehow, acknowledging this obvious fact makes me a blasphemer in the eyes of the Secular Man. Like Winston said in 1984, “theyll shoot me i don’t care theyll shoot me in the back of the neck i dont care”.
There are exceptions to the general trend toward segregation, sometimes benign, sometimes not. There are cases where we let students come over here and send ours over there. No harm there. If you want a kid from France in your home for the school year, more power to you, sez I.
And there are Christian societies who send doctors, farmers, engineers and whatnot. They have an ulterior motive, of course, but it’s an open secret. They’ll treat lepers and show folks how to have safe drinking water in exchange for a chance to explain about Jesus and the cross. There is definitely no harm in that. And — speaking only of my own premillennial views — Christianity decidedly does not teach us to take over the world by force. If there is to be any force, Jesus will impose it in person when He arrives. Post-mill folks, I’ll let you speak for yourselves on this.
Other cases are not quite so benign. Caesar desegregated the Italians and the Gauls, but not to help the latter. Likewise the Assyrians in Israel, the Babylonians in Judah, and so on.
In general, I’d say anybody who thinks it’s the destiny of his group to take over the world by force is a threat. In modern times, the Communists and Fascists fit this category and were proud of it. There was a time when Americans understood this and reacted against it. The old adage about being better dead than red was a way of acknowledging the open threat posed by Communism while saying that we intended to push back with whatever force it took.
And to get back where I started from, Moslems fit this category. Islam intends to take over the world, by persuasion where it can, by force where it must.
In the vast majority of cases, your Moslem co-worker is no threat to you or anybody else. He’s just a guy trying to get by, raise his kids to be good Moslems, and keep his wife from feeling like a conspicuous fool wearing her burqa.
The problem arises when there are enough Moslems to form a society that runs along Islamic lines. Because Islam expects to own Earth and everyone on it. The Secular Man, wearing his feelings on his “coexist” bumper sticker, is just not prepared to deal with the reality of an Islam that will not rest until everyone bows to Mecca. The Secular Man’s refusal to see a threat where there clearly is one looks a lot like a suicide pact.
Will the real oligarchy please stand up
The Washington Times published an article quoting an Ivy League study saying we live in an oligarchy rather than a republic. Elites and special interests buy influence and get their way, he says, describing a government of plutocrats more than oligarchs. The Catholic News Service quotes the same study to the same effect, complaining against big corporations doing a lot of evil influence buying. The gist of it is that the rich get their way, harming the rest of us.
But the biggest source of influence over the way the government governs is the government itself.
And here’s how it works. Something like 148 million Americans are receiving some form of stipend from the government. Government is essentially buying their votes with money confiscated from the 90 million private sector workers who pay for it all. So maybe it’s true that we have an oligarchy or plutocracy settling in upon us, but the “evil” corporations are by no means the dominant players in this field. The money laundering from the government dwarfs all other forms of paid influence, whether it’s from the Koch brothers on the right or Warren Buffet and George Soros on the left.
It’s a victory for Big Irony that many of the people complaining the loudest about the influence of Big Corporations are actually part of Big Government and seem completely blind to their role in the most pervasive and ruinous corruption scheme of all.
So why are you still a Christian?
Given the general drift of western civilization, there are some on our side who are feeling like Christianity is, well, in retreat. If you could wind your time turner back 40 years, nobody living at that time would have said that America would be in the process of honoring sodomy with its own special rite of marriage. Nobody would have predicted rates of divorce and illegitimacy where they are today. Jokes about mass confusion on Fathers’ Day used to be directed at other people, not us. Now we are the joke.
Struggling families can’t find a reason to stick it out. People give up and give over to their pet sins. People born, raised, and married in the church suddenly go secular, not out of any sense of offense or hostility, but because they just don’t care any more.
So why are you still plodding along with a crowd that seems to be losing so badly? Maybe it’s because you’re part of the gray-haired set that still does all that Churchianity stuff (including Wednesday night). Maybe it’s because America isn’t the only place in the world, and in some other places like China, Christianity is growing like mad. Maybe you stick around because you’re one of those fortunate folks still plugged into a dynamite church, and you really enjoy it.
Here’s a reason for you to ponder: You should keep the faith because Jesus is alive. The only real reason to get into Christianity in the first place, if I could say it like that, is because it is true. And the central truth of Christianity is that He rose from the dead. If He rose from the dead, there is every reason in the world to continue faithful regardless of what the rest of society does.
If Jesus didn’t rise from the dead, there never was a reason to be a Christian. In the early days of Christianity, Paul said that if Christ has not risen, then “we of all men are most miserable.” Why suffer for a dead god? What sense does that make? We don’t suffer any serious persecution in America, so let’s apply the thought more accurately to us: why deny yourself the pleasures of a hedonistic life to honor the memory of a dead guy?
But if Jesus is alive, that changes everything. That would mean that He has power over death. It would mean He really is the Son of God. It would mean His church is destined to become the central focus of history. It would mean that following Jesus matters, not just to your kids or your person sense of stick-tuitiveness, or the moral tidiness of your little corner of the world, but it matters on the biggest and most cosmic scope imaginable.
And if Jesus is alive, it would mean that death is not the end of life that we all thought it was, but just a pause before we transition into something far greater He has prepared for us. It would mean that our ultimate conclusions about life stand upon a living hope. And hope is the thing that makes us keep on keeping on.
So I’m still a Christian because the tomb was emptied when Christ came out of it alive.
Ukraine and Obama’s complicated failure
Of course you’ve heard by now that Mr. Romney and Mrs. Palin both warned that Mr. Obama’s policy toward Russia would lead to the ongoing crisis in Ukraine. America should have been doing everything possible to help Ukraine establish a sturdy, free economy, a justice system free of graft and corruption, and a credible military. If we were attempting any of these things, it never made any news I could see.
But now Obama’s failures are starting to earn compound interest. Obama, who years ago helped bring about the decline of America’s manned space program — not that he did this by himself; he had plenty of help — now has another problem on his hands in the matter of Ukraine.
He can’t afford to anger the Russians too much because we depend on them to get our astronauts and supplies to/from the space station. We’re in one of those moments when you realize that great nations have to remain strong in every area. America’s space program has been the envy of the world for 60 years, that is, until the last shuttle flight. Now we have no way to get people and cargo to the space station.
So Mr. Obama will be low key in his reactions to the Russian dealings in Ukraine. It would be too embarrassing to have the Russians tell America to kiss off next time we need one of our astronauts to bum a ride in a Soyuz spacecraft.
And I feel I need to add something here. I am not ashamed of America, but I am ashamed of our self-imposed weakness and immorality after years of secular, Socialist-leaning misrule. The looming problems confronting our astronauts are the kinds of weird, unique gotchas that crop up when foreign policy is dominated by the wishful, utopian thinking of liberal academics instead of a hard-headed determination to deal with facts as they are. Russia is a powerful nation that sees itself as our rival. Mr. Putin is a smart, tough ex-KGB agent and a fierce nationalist. He plays to win and won’t hesitate to spill blood to achieve his goals. Only blind folly could have failed to see that and act accordingly. Putting our space program into a state of dependency on the Russian program is beyond naïve.
So remember that next time a liberal politician tells you America is disliked around the world, embarks on a worldwide apologize-for-America tour, and offers a former KGB agent one of those ludicrous red reset buttons.
Why I think there’s a God
Louise Antony gave her reasons for thinking there’s no God, and I dealt with those here. But what are the reasons for thinking there is one? In most Christian literature, these boil down to five.
Why There’s Something Instead of Nothing
The physical universe tells us it had a beginning. The sun isn’t merely shining; it is burning up. The world isn’t merely turning; it is spinning down to a stop. Natural processes everywhere are in a state of decay and decline. The available energy in the universe is a consumable resource. There’s an end point when the mainspring of the whole cosmos will stop ticking. Therefore, it had to have been wound up at some point in the past. So the physical universe isn’t eternal.
So something else must have been here before there was a universe. Whatever that was, it must have been eternal and must have had the capacity to bring the universe into being.
Why There’s Order Instead of Chaos
When you drive through the South and see a 1000-acre tract of pine trees planted in rows, equally spaced along the row and all the same age, you don’t have to ask if somebody did that. When I look at the far more complex arrangements of DNA, it’s obvious that a mighty intelligence made this. DNA contains the coding needed to duplicate itself. But a process capable of creating a DNA molecule from scratch simply does not exist in nature. Nothing even remotely approaching this degree of sophistication has ever been observed, not in nature, nor even in man’s most advanced laboratories.
So something eternal and powerful was there before the universe existed. And it had the capacity to bring the universe into being, wind up the spring, and then release the energy through myriads of the most intricately designed mechanisms. Such a being is intelligent beyond all the reckoning of man.
Why Things are Right and Wrong
People have a moral component to their nature. Ms. Antony shows this when she asks that we all work for peace. Nice thought, though I wish she’d explain why, on atheistic principles, peace is better than war. After all, isn’t evolution driven by conflict and winnowing away the unfit so that only the strongest and smartest survive to breed again? Here’s a case where evolutionists are better than their principles. They generally wish the world were better — and “better” is defined in moral terms.
Furthermore, there is, for lack of a better term, a genuine reality underlying morals. We aren’t merely displeased when brutes kidnap little girls and sell them into sexual slavery. No, this is really and truly evil, and wrong. And it’s not just that we feel happy about a man who would redeem little slaves out of their bondage. No, such a deed is really and truly good and right.
The fact that morality cannot be derived from nature is not an argument from gaps in our knowledge. Rather, it’s plain to see that there is no arrangement of particles and forces that can ever account for a moral right and wrong because morality involves not just an assessment of facts, but an assertion of authority. Morality is the claim, coming from outside your own head, that you ought or ought not do something. And “ought” inherently arrives in the form of a command. Morality sees what is wrong and authoritatively forbids it. Morality sees what is right and authoritatively commands it.
The origin of morality, then, is very much like the origin of the physical universe. It’s here; it’s real, and it defies natural, material explanation. It demands a source that is outside of this world, transcendent, and that was capable of implanting it in the human heart when man was first formed.
So — just building the argument — something eternal brought the universe into being, something that was powerful enough to do it, intelligent enough to design it, and this Being possessed a moral code which it then hard wired into the hearts of men.
Why We Sense the Transcendent
It’s an interesting question as to why, on naturalist/materialist principles, people should have ever evolved to be capable of wondering about what could be outside this physical dimension. Where’s the survival value in such a massive and stressful distraction? Or to take the question a level deeper, how do matter and energy interact in such a way as to produce conscious beings who ponder things higher than matter and energy?
Ms. Antony herself experiences the draw of the transcendent but drops it too soon. The real question is what a sense of transcendence is leading you to. Being a Christian, it’s obviously my opinion that God created this in us to lead us to Him. Paul told the Athenians that we “feel after Him,” (Acts 17:27) clearly expecting that even pagan men would have been open minded enough to investigate an intuition shared by virtually all people.
We Christians find our sense of transcendence filled, satisfied, yet heightened and completed by knowing our God through His Son, Jesus. People from other religions testify of their version of the same sense of transcendence. It’s not my purpose to address those experiences, only to say that whether we’re making out shapes in a fog or seeing in the full light of day, something is there, and we all sense it to some degree. And although the argument is not dispositive, I can’t frame a better explanation for a sense of transcendence than to propose that God has indeed set “eternity in our hearts” (Eccl 3:11) as a way to both prompt us to seek Him and as a way to experience Him once He is found.
The life of Jesus Christ
The chief way God chose to reveal Himself to man was through Jesus. The officers sent to arrest Him said, “Nobody ever spoke like this man.” We exhaust all the superlatives when we consider Him. His teachings set the standard for goodness even among those who reject Him. He led such a life that those who sought His ruin could accuse Him only by lying. Without money, without armies, without political connections, without allies, without any access to the levers of power, having died young, Jesus did more to change the world for good than all who ever came before or after.
And He rose from the dead. Yes, His followers reported many other miracles He did, turning water to wine, walking on water, feeding multitudes out of a sack lunch.
But the miracle of His resurrection was the story they were all, to man, willing to be tortured and die for the privilege of telling it, not because they had anything to gain by it, but because they undeniably believed it to be true. If there is a God such as I have described, and if God became a man, I would expect Him to be a man like Jesus.
So that’s it. It’s why I think God exists and has revealed Himself to us through His Son, Jesus.
Answers for an atheist
The New York Times published an interview with atheist Louise Antony who confidently affirms that there is no God. Read the linked article if you like, but her arguments against God boil down to a just handful of things.
First, Antony says, “I deny that there are beings or phenomena outside the scope of natural law.” This, of course, is no argument at all. It’s just assuming the conclusion. Presupposing materialism merely evades the debate about whether God exists. The Christian idea of God is that He is transcendent, meaning that He is “above” or “beyond” or “outside” the universe. Looking for God by material methods is like prospecting for diamonds with a metal detector. Wrong tool.
In her second argument, Antony says religious people can’t all agree on what God is, what He is like, or whether there are more gods than one. This is all true, and all irrelevant. For the sake of argument, let’s assume that all religious people are hopelessly muddled on the nature of God. Does this mean they’re all equally deceived on the existence of God? Not at all. Even in a total fog, people can know something is out there without knowing any details about it.
Antony then says she cannot reconcile the existence of evil with the existence of God. Beg pardon, but what is this “evil” she speaks of? The existence of categories like “good” and “evil” assumes a Supreme Authority who establishes what’s good and what’s not. And consider again Antony’s statement, “I deny that there are beings or phenomena outside the scope of natural law.” Yet the very categories of good and evil are outside of natural law. You cannot derive morality from Newton’s laws or the Schrödinger equation. That requires a transcendent source.
On the other hand, if good and evil are not real categories, if they’re just cultural norms or her own private intuitions, then her objection vanishes. Her argument amounts to, “I’m displeased (or we are); therefore, there is no god,” which is absurd.
But Ms. Antony is left to ponder the motions of her own heart. Why is she outraged by rape or brutality? Who cares, and why should anyone care, if orphans starve, tyrants strut, armed gangs pillage and plunder, girls are bought and sold, and all the rest of human misery is played out before our eyes? If Ms. Antony knows anything at all, she knows there’s Something Big moving out there in the fog.
And following that, Ms. Antony should be the first to accept religious experiences. After all, she’s had a big one. She’s felt the wrong of this fallen, sinful world and felt the need to put it all back right. That didn’t evolve from a big cloud of hydrogen gas. God has set eternity in our hearts, and that’s what it sounds like when people pay attention to it, even a little bit.
Kim Jung O
So now the FCC wants to install government minders in newsrooms across the country to make sure “underserved minorities” get the news they need. I guess we’ll show Kim Jung Un how it’s done. Even Mr. Obama’s lickspittle media has an eyebrow aloft. But don’t worry, lefties — if you like your freedom of the press, you can keep it!
Another foretaste of things to come
It’s no secret that Christian values are being slowly but inexorably dispossessed in America. Wedding cake bakers who refuse service to homosexual couples get sued over it, and lose. They’re told that once you open your business up to serve the public, then you have to serve whatever comes through the door.
But now a bar owner in California says he’ll refuse service to state legislators who vote for anti-gay legislation. Actually, he went a bit farther and said he’d deny them entry to his bar.
I’m thinking his valiant pro-gay stand isn’t likely to cost him a lot of money. How many Christians are clamoring to enter a gay bar in California?
Still, the principle being established here should tell every Christian that it’s past time to gird up the old loins. Christian bakers are fair game for discrimination suits if they transgress against the Secular Man’s homodoxy on the grounds that public businesses have to accept whatever the public accepts.
To borrow from Spurgeon, I’ll adventure to prophesy that anti-Christian bar owners will be immune from suits on the same grounds. Yet — lest we all forget — Californians voted against homosexual marriage, even going so far as to forbid it in their state constitution. So it’s clear that the actual public in California accepts anti-gay legislators just fine. But you can be certain that the bar owner, should he get sued for discrimination, will get a pass.
Christians should be waking up to the fact that we’re in a fight. And to paraphrase Mordecai to Esther, don’t think this won’t ever touch you.
When politics go bad
King Baasha of Israel was a drunkard. His servant Zimri murdered him while he was drunk. Short moral of story: A drunken king can’t be trusted to know who the enemy is.
Zimri took over and reigned for about a week. Another servant named Omri found out Baasha was dead and came after Zimri. Zimri neither fought nor fled, but went into his own house and burned it down upon himself. Moral: It’s easier to take over than it is to actually keep order, and once order is lost, you don’t have a lot of options.
Omri was a wicked king and plunged Israel deeper into ruinous idolatry. Moral: A guy who just wants to be in charge is about the last man you want in power.
Omri’s son Ahab eventually became king. The Bible describes Ahab as worse than all who came before him. He married Jezebel who was even worse than he was. Moral: Getting rid of drunks, killers, and tyrants doesn’t mean things are about to get better. The son might make you wish for his daddy back. And beware the tyrant’s wife.
During Ahab’s reign the prophet Elijah called for a drought that lasted for years. Moral: When the right leadership arrives, the fight isn’t over; it’s just starting, and you may dislike his methods.
At Mount Carmel, God spoke by fire from heaven. Israel, convinced, repented. They acknowledged that the Lord is God, not Baal, and executed the idolatrous priests. Then the rain came. Moral: Fixing a country starts with fixing hearts.
Baghdad Bob and ObamaCare
Kathleen Sebelius, the Baghdad Bob of ObamaCare, says job losses due to the idiotic tax scheme are a “popular myth.”
Bad timing for her announcement, though, coming right after the administration has been cheesecake grinning and doing happy hands over what a great American blessing it is to escape “job lock” by getting fired.
We should all savor this rare moment of unanimity in the political world with both Republicans and Democrats saying that ObamaCare is a failure. The GOP says it’s a failure because (among other things) it makes people lose their jobs. The Democrats say it’s a failure because ObamaCare job losses are only mythical, leaving millions of hapless citizens still locked in a job.
Is this a great country or what?
Secular Man’s smoking habits
Has anyone else noticed how smoking tobacco has been getting less legal while smoking marijuana has been getting more legal?
And isn’t it just the funniest thing that so many plain old potheads are claiming it’s for medicinal purposes?
Connecticut — nekkid and hoping you won’t notice
One of the great insights of the American Revolution is that a government’s authority derives from the consent of the governed.
The State of Connecticut passed a law saying everyone in the state must register so-called “assault” weapons and high capacity ammo magazines. Comes now the report that ten thousands of citizens in Connecticut — perhaps millions — have declined to obey. Registration schemes are plainly the first move in a game of confiscation. Many who intend not to surrender their arms are declining to register them.
This may turn out to be a very, very big deal. Failure to register a weapon in Connecticut is a class D felony. A class D felony is punishable by up to five years in prison. Despite that, gun owners in Connecticut collectively jutted their jaws and said, “Hell, no.”
How big is the problem? Connecticut estimates there are about 370,000 so-called “assault” weapons in Connecticut. Less than 50,000 have been registered. They estimate there are 2.4 million high capacity ammo magazines in the state. About 38,000 have been registered. Theoretically, Connecticut now has well over two million new felons.
You can be sure Connecticut pols see the problem just like I do. If a huge swath of the population responds with sullen defiance, the government no longer has the consent of the governed. How is it a legitimate government any more? And how do you recover that once it’s lost?
I see three options. 1) Connecticut can openly and humbly restore its legitimacy by repealing the law. 2) Officials can reduce enforcement to some low level that ruins a few people’s lives while leaving most violators untouched yet still under state threat. 3) The state can hire more SWAT teams, build way more prisons and start the crackdown.
Option 2 is most likely because the gun law was designed not to solve a problem but to make liberals feel good about themselves. Neither practicing humility nor engorging the prisons would serve that purpose, although criminalizing a bunch of rightwingers would. And if a few of them get busted, well, that’s the price one pays.
One problem: Reducing enforcement to a level that prevents serious conflict is claiming victory while hoisting a white flag. It’s like one of those dreams where you show up at work buck nekkid and nobody notices.
As Drudge says, “Developing…”
From creation clearly seen
The recent creation/evolution debate between Ken Ham and Bill Nye was pretty good. It was not excellent. The rules of the debate didn’t require the contestants to engage one another to any great extent, so the back-and-forth that challenges reasoning didn’t happen.
One of the things Mr. Ham said that begged for discussion was his remark that just doing science presupposes God and creation. Christians schooled in apologetics promptly said rah-rah, but the argument was left as a mere assertion. Nye declined to ask for explanation, and Ham obliged.
Why should there be any such thing as natural law? Why should nature be orderly and predictable? Why should gravitation behave according to a rule so precise that you can measure its effects and write a mathematical equation that tells you exactly what’s going to happen? A Christian would argue from the creation account that God intended His universe to function in an orderly way. Creatures bring forth “after their kind,” it says ten times. The motions of the earth, sun, moon, and stars provide day, night, signs, and seasons. There is order in this, and Paul tells us that the invisible things of God are clearly seen, being understood by what God made. (Rom 1:20)
But the deeper question for Mr. Nye would have been this: What is it about your thought process that leads you to look for orderliness in the first place, and why does your mind naturally recognize it and latch onto it?
Based on Nye’s frequent and brave admissions about what he doesn’t know, I can only surmise that he’d admit again that he has no idea why the Bing Bang resulted in law and order rather than sheer chaos, and he’d likely admit that he has no idea why his mind should be structured to look for order. Or he might just say it evolved this way, which is the same thing.
But the Christian can say that if we take the Word of God as our starting point, the first thing we learn is that God is, that He made the universe, and that He did it in an orderly manner. Further, God immediately set about revealing Himself to man with that revelation being set in a framework of reason and logic. The imago dei means our heads are hard-wired to look for order, to recognize it at once, and to latch onto it when it’s found.
For science to exist at all, all these Christian teachings about creation and human nature have to be assumed as prerequisites. They must be presupposed.
The questions for Mr. Nye and everyone who investigates science from a naturalistic viewpoint are these: How does the Big Bang account for the fact that the resulting cosmos functions according to fixed laws? And second, how did the mind of man come to look for such things? Christianity has an answer for these questions. Naturalism can’t do any better than offer a shrug and say that’s just the way things are — which is the opposite of true science.
Conservatives who can’t connect the dots
A few months ago while the electioneering was in full-throated roar, a “conservative” writer lamented that liberal voters seem unable to connect the dots. One quoted a low-info voter who expressed unconcern about a property tax hike because, said the voter, “I rent an apartment, so property taxes don’t affect me.” How do you connect intelligently with people this thick?
And then today, I was listening to talk radio “conservative” Mark Larsen explaining to a caller that he ‘d have no problem with the Boy Scouts changing their stance on homosexuality to go with the PeeCee flow and start accepting it. The caller wondered why the institution must change to accommodate the individual rather than the other way around, noting that the Boy Scouts have always required young men to be morally straight.
“What is morality?” wondered the blind Mr. Larsen aloud. After all, Christian denominations have differed over this or that detail. And whatever would we say to the Metropolitan churches who are openly homosexual? (Tacit premise in the question: Until you get everything perfect, you’re not allowed to say they’re wrong.)
This is a conservative, low-info talk show host who cannot connect the dots. Well, actually, Larsen says he’s libertarian, but he’s still dense on this topic and unable to connect dots, and here’s why.
Morality of any and every sort is an assertion of authority. The moment you say “ought” or “ought not,” somebody else demands, “Says who?” Morality requires an anchor. The Author, the Anchor, is God. And even though the church admittedly has quibbles a-plenty, we’re all together in relaying to you His judgment that sodomy isn’t okay.
Mr. Larsen, apparently unwilling to consider a reliable message from a capable though fallible messenger, has no anchor. How else can you even ask such a question as, “What is morality?”
And once you pull up the anchor, everything tied to it will drift away. The current debate over homosexuality didn’t spring upon America like a bolt from the sky. It started way back when Americans grew discontented with the God who insists we should keep our word. Not long thence, easy-breezy divorce became socially acceptable. A few years later, pornography began to proliferate. And then came the sexual revolution with its promiscuity, the shack-ups, the meteoric rise in illegitimacy, the loss of shame as the entertainer class breeds without commitment.
First thing you know, many major cities had whole sections of their towns devoted to sodomy, and before you can adjust to that, they’ve got us voting on whether homosexuals have a right to marry one another.
And at that point, people like Mr. Larsen cannot render a reason as to what could possibly be bad about that.
Prediction: Sometime soon our society will be debating polygamy, pedophilia, bestiality, and necrophilia, and those who (for whatever reason) disapprove of such things but who have no anchor will find themselves as tongue tied as the hapless Mr. Larsen was. Who’s to say what’s wrong, after all?
Without God as the anchor for morals, you will have no morals. He made the world where it can’t be any other way. And yes, He did that on purpose. Morals, like the rights stated in the Declaration of Independence, are derived. And just as God created us equal and endowed us with rights, so he also created us with the social, civic, and religious obligations we refer to as morality.
When you pull up the anchor, you don’t just lose your morals. You’ll start losing your rights, too. Same anchor, same God. Say good-bye to life, liberty, and the pursuit of happiness. Godless men cannot comprehend, let alone respect, the Bill of Rights. They have no clue where such things came from, no idea of what makes them special, and no sense of a higher Authority to whom all earthly authorities must give account. You can no more have rights without morality than you can have a stream without water. Both flow from the same spring, the Eternal God.
God deliver us from leaders who do not know their Maker, or even that they are made.
Lance and Oprah
The embarrassing spectacle of Lance Armstrong confessing to Oprah has failed to capture the popular imagination. For one thing, Lance is not a sympathetic character. Americans are not prone to soaring eloquence, so people call him a jerk. British writer Geoffrey Wheatcroft said of him, “Mr. Armstrong has “a voice like ice cubes,” as one French journalist puts it, and I have to admit that he reminds me of what Daniel O’Connell said about Sir Robert Peel: He has a smile like moonlight playing on a gravestone.”
Another thing is that Lance’s confession came too late. And it was lame. And it was tacky. But it fits the pattern now so familiar in no-fault America in which a famous person commits a sin, gets caught, lies about it till the lie becomes ridiculous, then finally stages a theatrical confession. The staging is usually in proportion to the fame and ego of the perpetrator. Thus, Lance. Scroll through the mental list of publicly groveling miscreants from Lance back through Anthony Weiner, Bill Clinton, South Carolina governor Mark Sanford, gay/doping preacher Ted Haggard, and a host of others.
The spiritual man can see what this is all about. Adam remains banished from Eden. The occasional rite of public humiliation is just a couple of the exiles passing by the gate and wishing for a way back in. But the gate is shut. The cherub with the flaming sword still bars the road to paradise.
A final thing about Lance’s confession is that we can all see it does no good. The public, momentarily curious, watches the ritual confessions and is vaguely aware of the hopelessness of it all. To confess seems required. A wrong was done. To admit it is demanded. We all feel the pressure of the demand. Some of us help exert it. At the same time, it’s inadequate. It’s watering a dead tree, and all the same to the tree whether it’s water or tears.
The Secular Man, two-dimensional being that he is, confesses to himself and to his peers. Who else is there? To the carnal mind, what is paradise but the pleasure he felt before his sin was found out? A degrading confession seems to be how you shake up the Etch-A-Sketch and redraw the picture.
The confession has to feature humiliation and suffering. Part of the suffering involves the rest of us smirking at the poor dumb schmuck locked in the pillory. But even when we humiliate ourselves as Lance did, the sin remains. And even if you suffer to the point of death, you’re just dead and guilty. Whether you’re confessing to Oprah or CNN, it’s still just praying to a god that cannot save. (Isa 45:20)
The riddle is solved at the cross. It is Christ’s humiliation, not ours, and His suffering and death, that brings remission of sins. It is our confession to Him, not to Oprah nor to a public filled with critics and voyeurs, that brings peace. |
f1be1509a104c621 | Superconductivity and flux quantization
This post continues my mini-series on Feynman’s Seminar on Superconductivity. Superconductivity is a state which produces many wondrous phenomena, but… Well… The flux quantization phenomenon may not be part of your regular YouTube feed but, as far as I am concerned, it may well be the most amazing manifestation of a quantum-mechanical phenomenon at a macroscopic scale. I mean… Super currents that keep going, with zero resistance, are weird—they explain how we can trap a magnetic flux in the first place—but the fact that such fluxes are quantized is even weirder.
The key idea is the following. When we cool a ring-shaped piece of superconducting material in a magnetic field, all the way down to the critical temperature that causes the electrons to condense into a superconducting fluid, then a super current will emerge—think of an eddy current, here, but with zero resistance—that will force the magnetic field out of the material, as shown below. This current will permanently trap some of the magnetic field, even when the external field is being removed. As said, that’s weird enough by itself but… Well… If we think of the super current as an eddy current encountering zero resistance, then the idea of a permanently trapped magnetic field makes sense, right? In case you’d doubt the effect… Well… Just watch one of the many videos on the effect on YouTube. 🙂 The amazing thing here is not the permanently trapped magnetic field, but the fact that it’s quantized.
trapped flux
To be precise, the trapped flux will always be an integer times 2πħ/q. In other words, the magnetic field which Feynman denotes by Φ (the capitalized Greek letter phi), will always be equal to:
Φ = 2πħ/q, with = 0, 1, 2, 3,…
Hence, the flux can be 0, 2πħ/q, 4πħ/q, 6πħ/q , and so on. The fact that it’s a multiple of 2π shows us it’s got to do with the fact that our piece of material is, effectively, a ring. The nice thing about this phenomenon is that the mathematical analysis is, in fact, fairly easy to follow—or… Well… Much easier than what we discussed before. 🙂 Let’s quickly go through it.
We have a formula for the magnetic flux. It must be equal to the line integral of the vector potential (A) around a closed loop Τ, so we write:
Now, we can choose the loop Τ to be well inside the body of the ring, so that it never gets near the surface, as illustrated below. So we know that the current J is zero there. [In case you doubt this, see my previous post.]
One of the equations we introduced in our previous post, ħθ = m·v + q·A, will then reduce to:
ħθ = q·A
Why? The v in the m·v term (the velocity of the superconducting fluid, really), is zero. Remember the analysis is for this particular loop (well inside the ring) only. So… Well… If we integrate the expression above, we get:
Combining the two expressions with the integrals, we get:
integral 2Now, the line integral of a gradient from one point to another (say from point 1 to point 2) is the difference of the values of the function at the two points, so we can write:
integral 3
Now what constraints are there on the values of θ1 and θ2? Well… You might think that, if they’re associated with the same point (we’re talking a closed loop, right?), then the two values should be the same, but… Well… No. All we can say is that the wavefunction must have the same value. We wrote that wavefunction as:
ψ = ρ(r)1/2eθ(r)
The value of this function at some point r is the same if θ changes by 2π. Hence, when doing one complete turn around the ring, the ∫∇θ·ds integral in the integral formulas we wrote down must be equal to 2π. Therefore, the second integral expression above can be re-written as:
That’s the result we wanted to explain so… Well… We’re done. Let me wrap up by quoting Feynman’s account of the 1961 experiment which confirmed London’s prediction of the effect, which goes back to 1950! It’s interesting, because… Well… It shows how up to date Feynman’s Lectures really are—or were, back in 1963, at least!feynman overview of experiment
Feynman’s Seminar on Superconductivity (2)
We didn’t get very far in our first post on Feynman’s Seminar on Superconductivity, and then I shifted my attention to other subjects over the past few months. So… Well… Let me re-visit the topic here.
One of the difficulties one encounters when trying to read this so-called seminar—which, according to Feynman, is ‘for entertainment only’ and, therefore, not really part of the Lectures themselves—is that Feynman throws in a lot of stuff that is not all that relevant to the topic itself but… Well… He apparently didn’t manage to throw all that he wanted to throw into his (other) Lectures on Quantum Mechanics and so he inserted a lot of stuff which he could, perhaps, have discussed elsewhere. :-/ So let us try to re-construct the main lines of reasoning here.
The first equation is Schrödinger’s equation for some particle with charge q that is moving in an electromagnetic field that is characterized not only by the (scalar) potential Φ but also by a vector potential A:
This closely resembles Schrödinger’s equation for an electron that is moving in an electric field only, which we used to find the energy states of electrons in a hydrogen atom: i·ħ·∂ψ/∂t = −(1/2)·(ħ2/m)∇2ψ + V·ψ. We just need to note the following:
1. On the left-hand side, we can, obviously, replace −1/i by i.
2. On the right-hand side, we can replace V by q·Φ, because the potential of a charge in an electric field is the product of the charge (q) and the (electric) potential (Φ).
3. As for the other term on the right-hand side—so that’s the −(1/2)·(ħ2/m)∇2ψ term—we can re-write −ħ2·∇2ψ as [(ħ/i)·∇]·[(ħ/i)·∇]ψ because (1/i)·(1/i) = 1/i2 = 1/(−1) = −1. 🙂
4. So all that’s left now, is that additional −q·A term in the (ħ/i)∇ − q·A expression. In our post, we showed that’s easily explained because we’re talking magnetodynamics: we’ve got to allow for the possibility of changing magnetic fields, and so that’s what the −q·A term captures.
Now, the latter point is not so easy to grasp but… Well… I’ll refer you that first post of mine, in which I show that some charge in a changing magnetic field will effectively gather some extra momentum, whose magnitude will be equal to p = m·v = −q·A. So that’s why we need to introduce another momentum operator here, which we write as:
OK. Next. But… Then… Well… All of what follows are either digressions—like the section on the local conservation of probabilities—or, else, quite intuitive arguments. Indeed, Feynman does not give us the nitty-gritty of the Bardeen-Cooper-Schrieffer theory, nor is the rest of the argument nearly as rigorous as the derivation of the electron orbitals from Schrödinger’s equation in an electrostatic field. So let us closely stick to what he does write, and try our best to follow the arguments.
Cooper pairs
The key assumption is that there is some attraction between electrons which, at low enough temperatures, can overcome the Coulomb repulsion. Where does this attraction come from? Feynman does not give us any clues here. He just makes a reference to the BCS theory but notes this theory is “not the subject of this seminar”, and that we should just “accept the idea that the electrons do, in some manner or other, work in pairs”, and that “we can think of thos−e pairs as behaving more or less like particles”, and that “we can, therefore, talk about the wavefunction for a pair.”
So we have a new particle, so to speak, which consists of two electrons who move through the conductor as one. To be precise, the electron pair behaves as a boson. Now, bosons have integer spin. According to the spin addition rule, we have four possibilities here but only three possible values:− 1/2 + 1/2 = 1; −1/2 + 1/2 = 0; +1/2 − 1/2 = 0; −1/2 − 1/2 = − 1. Of course, it is tempting to think these Cooper pairs are just like the electron pairs in the atomic orbitals, whose spin is always opposite because of the Fermi exclusion principle. Feynman doesn’t say anything about this, but the Wikipedia article on the BCS theory notes that the two electrons in a Cooper pair are, effectively, correlated because of their opposite spin. Hence, we must assume the Cooper pairs effectively behave like spin-zero particles.
Now, unlike fermions, bosons can collectively share the same energy state. In fact, they are likely to share the same state into what is referred to as a Bose-Einstein condensate. As Feynman puts it: “Since electron pairs are bosons, when there are a lot of them in a given state there is an especially large amplitude for other pairs to go to the same state. So nearly all of the pairs will be locked down at the lowest energy in exactly the same state—it won’t be easy to get one of them into another state. There’s more amplitude to go into the same state than into an unoccupied state by the famous factor √n, where n−1 is the occupancy of the lowest state. So we would expect all the pairs to be moving in the same state.”
Of course, this only happens at very low temperatures, because even if the thermal energy is very low, it will give the electrons sufficient energy to ensure the attractive force is overcome and all pairs are broken up. It is only at very low temperature that they will pair up and go into a Bose-Einstein condensate. Now, Feynman derives this √n factor in a rather abstruse introductory Lecture in the third volume, and I’d advise you to google other material on Bose-Einstein statistics because… Well… The mentioned Lecture is not among Feynman’s finest. OK. Next step.
Cooper pairs and wavefunctions
We know the probability of finding a Cooper pair is equal to the absolute square of its wavefunction. Now, it is very reasonable to assume that this probability will be proportional to the charge density (ρ), so we can write:
|ψ|= ψψ* ∼ ρ(r)
The argument here (r) is just the position vector. The next step, then, is to write ψ as the square root of ρ(r) times some phase factor θ. Abstracting away from time, this phase factor will also depend on r, of course. So this is what Feynman writes:
ψ = ρ(r)1/2eθ(r)
As Feynman notes, we can write any complex function of r like this but… Well… The charge density is, obviously, something real. Something we can measure, so we’re not writing the obvious here. The next step is even less obvious.
In our first post, we spent quite some time on Feynman’s digression on the local conservation of probability and… Well… I wrote above I didn’t think this digression was very useful. It now turns out it’s a central piece in the puzzle that Feynman is trying to solve for us here. The key formula here is the one for the so-called probability current, which—as Feynman shows—we write as:
This current J can also be written as:
Now, Feynman skips all of the math here (he notes “it’s just a change of variables” but so he doesn’t want to go through all of the algebra), and so I’ll just believe him when he says that, when substituting ψ for our wavefunction ψ = ρ(r)1/2eθ(r), then we can express this ‘current’ (J) in terms of ρ and θ. To be precise, he writes J as: current formulaSo what? Well… It’s really fascinating to see what happens next. While J was some rather abstract concept so far—what’s a probability current, really?—Feynman now suggests we may want to think of it as a very classical electric current—the charge density times the velocity of the fluid of electrons. Hence, we equate J to J = ρ·v. Now, if the equation above holds true, but J is also equal to J = ρ·v, then the equation above is equivalent to:
Now, that gives us a formula for ħθ. We write:
ħθ = m·v + q·A
Now, in my previous post on this Seminar, I noted that Feynman attaches a lot of importance to this m·v + q·A quantity because… Well… It’s actually an invariant quantity. The argument can be, very briefly, summarized as follows. During the build-up of (or a change in) a magnetic flux, a charge will pick up some (classical) momentum that is equal to p = m·v = −q·A. Hence, the m·v + q·A sum is zero, and so… Well… That’s it, really: it’s some quantity that… Well… It has a significance in quantum mechanics. What significance? Well… Think of what we’ve been writing here. The v and the A have a physical significance, obviously. Therefore, that phase factor θ(r) must also have a physical significance.
But the question remains: what physical significance, exactly? Well… Let me quote Feynman here:
“The phase is just as observable as the charge density ρ. It is a piece of the current density J. The absolute phase (θ) is not observable, but if the gradient of the phase (θ) is known everywhere, then the phase is known except for a constant. You can define the phase at one point, and then the phase everywhere is determined.”
That makes sense, doesn’t it? But it still doesn’t quite answer the question: what is the physical significance of θ(r). What is it, really? We may be able to answer that question after exploring the equations above a bit more, so let’s do that now.
The phenomenon of superconductivity itself is easily explained by the mentioned condensation of the Cooper pairs: they all go into the same energy state. They form, effectively, a superconducting fluid. Feynman’s description of this is as follows:
Frankly, I’ve re-read this a couple of times, but I don’t think it’s the best description of what we think is going on here. I’d rather compare the situation to… Well… Electrons moving around in an electron orbital. That’s doesn’t involve any radiation or energy transfer either. There’s just movement. Flow. The kind of flow we have in the wavefunction itself. Here I think the video on Bose-Einstein condensates on the French Tout est quantique site is quite instructive: all of the Cooper pairs join to become one giant wavefunction—one superconducting fluid, really. 🙂
OK… Next.
The Meissner effect
Feynman describes the Meissner effect as follows:
The math here is interesting. Feynman first notes that, in any lump of superconducting metal, the divergence of the current must be zero, so we write: ∇·J = 0. At any point? Yes. The current that goes in must go out. No point is a sink or a source. Now the divergence operator (∇·J) is a linear operator. Hence, that means that, when applying the divergence operator to the J = (ħ/m)·[θ − (q/ħ)·A]·ρ equation, we’ll need to figure out what ∇·θ = = ∇2θ and ∇·A are. Now, as explained in my post on gauges, we can choose to make ∇·A equal to zero so… Well… We’ll make that choice and, hence, the term with ∇·A in it vanishes. So… Well… If ∇·J equals zero, then the term with ∇2θ has to be zero as well, so ∇2θ has to be zero. That, in turn, implies θ has to be some constant (vector).
Now, there is a pretty big error in Feynman’s Lecture here, as it notes: “Now the only way that ∇2θ can be zero everywhere inside the lump of metal is for θ to be a constant.” It should read: ∇2θ can only be zero everywhere if θ is a constant (vector). So now we need to remind ourselves of the reality of θ, as described by Feynman (quoted above): “The absolute phase (θ) is not observable, but if the gradient of the phase (θ) is known everywhere, then the phase is known except for a constant. You can define the phase at one point, and then the phase everywhere is determined.” So we can define, or choose, our constant (vector) θ to be 0.
Hmm… We re-set not one but two gauges here: A and θ. Tricky business, but let’s go along with it. [If we want to understand Feynman’s argument, then we actually have no choice than to go long with his argument, right?] The point is: the (ħ/m)·θ term in the J = (ħ/m)·[θ − (q/ħ)·A]·ρ vanishes, so the equation we’re left with tells us the current—so that’s an actual as well as a probability current!—is proportional to the vector potential:
currentNow, we’ve neglected any possible variation in the charge density ρ so far because… Well… The charge density in a superconducting fluid must be uniform, right? Why? When the metal is superconducting, an accumulation of electrons in one region would be immediately neutralized by a current, right? [Note that Feynman’s language is more careful here. He writes: the charge density is almost perfectly uniform.]
So what’s next? Well… We have a more general equation from the equations of electromagnetism:
A and J
[In case you’d want to know how we get this equation out of Maxwell’s equations, you can look it up online in one of the many standard textbooks on electromagnetism.] You recognize this as a Poisson equation… Well… Three Poisson equations: one for each component of A and J. We can now combine the two equations above by substituting in that Poisson equation, so we get the following differential equation, which we need to solve for A:
The λ2 in this equation is, of course, a shorthand for the following constant:
Now, it’s very easy to see that both e−λr as well as e−λr are solutions for that Poisson equation. But what do they mean? In one dimension, r becomes the one-dimensional position variable x. You can check the shapes of these solutions with a graphing tool.
Note that only one half of each graph counts: the vector potential must decrease when we go from the surface into the material, and there is a cut-off at the surface of the material itself, of course. So all depends on the size of λ, as compared to the size of our piece of superconducting metal (or whatever other substance our piece is made of). In fact, if we look at e−λx as as an exponential decay function, then τ = 1/λ is the so-called scaling constant (it’s the inverse of the decay constant, which is λ itself). [You can work this out yourself. Note that for = τ = 1/λ, the value of our function e−λx will be equal to e−λ(1/λ) = e−1 ≈ 0.368, so it means the value of our function is reduced to about 36.8% of its initial value. For all practical purposes, we may say—as Feynman notes—that the field will, effectively, only penetrate to a thin layer at the surface: a layer of about 1/1/λ in thickness. He illustrates this as follows:
Moreover, he calculates the 1/λ distance for lead. Let me copy him here:
Well… That says it all, right? We’re talking two millionths of a centimeter here… 🙂
So what’s left? A lot, like flux quantization, or the equations of motion for the superconducting electron fluid. But we’ll leave that for the next posts. 🙂
Feynman’s Seminar on Superconductivity (1)
formula for B
Wave equation for A
Wave equation for potential
wave equation
The Schrödinger equation in an electromagnetic field
v = −q·A
Local conservation of probability
Induced currents
In my two previous posts, I presented all of the ingredients of the meal we’re going to cook now, most notably:
1. The formula for the torque on a loop of a current in a magnetic field, and its energy: (i) τ = μ×B, and (ii) Umech = −μ·B.
2. The Biot-Savart Law, which gives you the magnetic field that’s produced by wires carrying currents:
B formula 2
Both ingredients are, obviously, relevant to the design of an electromagnetic motor, i.e. an ‘engine that can do some work’, as Feynman calls it. 🙂 Its principle is illustrated below.
The two formulas above explain how and why the coil go around, and the coil can be made to keep going by arranging that the connections to the coil are reversed each half-turn by contacts mounted on the shaft. Then the torque is always in the same direction. That’s how a small direct current (DC) motor is made. My father made me make a couple of these thirty years ago, with a magnet, a big nail and some copper coil. I used sliding contacts, and they were the most difficult thing in the whole design. But now I found a very nice demo on YouTube of a guy whose system to ‘reverse’ the connections is wonderfully simple: he doesn’t use any sliding contacts. He just removes half of the insulation on the wire of the coil on one side. It works like a charm, but I think it’s not so sustainable, as it spins so fast that the insulation on the other side will probably come off after a while! 🙂
Now, to make this motor run, you need current and, hence, 19th century physicists and mechanical engineers also wondered how one could produce currents by changing the magnetic field. Indeed, they could use Alessandro Volta’s ‘voltaic pile‘ to produce currents but it was not very handy: it consisted of alternating zinc and copper discs, with pieces of cloth soaked in salt water in-between!
Now, while the Biot-Savart Law goes back to 1820, it took another decade to find out how that could be done. Initially, people thought magnetic fields should just cause some current, but that didn’t work. Finally, Faraday unequivocally established the fundamental principle that electric effects are only there when something is changingSo you’ll get a current in a wire by moving it in a magnetic field, or by moving the magnet or, if the magnetic field is caused by some other current, by changing the current in that wire. It’s referred to as the ‘flux rule’, or Faraday’s Law. Remember: we’ve seen Gauss’ Law, then Ampère’s Law, and then that Biot-Savart Law, and so now it’s time for Faraday’s Law. 🙂 Faraday’s Law is Maxwell’s third equation really, aka as the Maxwell-Faraday Law of Induction:
×E = −∂B/∂t
Now you’ll wonder: what’s flux got to do with this formula? ×E is about circulation, not about flux! Well… Let me copy Feynman’s answer:
Faraday's law
So… There you go. And, yes, you’re right, instead of writing Faraday’s Law as ×E = −∂B/∂t, we should write it as:
That’s a easier to understand, and it’s also easier to work with, as we’ll see in a moment. So the point is: whenever the magnetic flux changes, there’s a push on the electrons in the wire. That push is referred to as the electromotive force, abbreviated as emf or EMF, and so it’s that line and/or surface integral above indeed. Let me paraphrase Feynman so you fully understand what we’re talking about here:
When we move our wire in a magnetic field, or when we move a magnet near the wire, or when we change the current in a nearby wire, there will be some net push on the electrons in the wire in one direction along the wire. There may be pushes in different directions at different places, but there will be more push in one direction than another. What counts is the push integrated around the complete circuit. We call this net integrated push the electromotive force (abbreviated emf) in the circuit. More precisely, the emf is defined as the tangential force per unit charge in the wire integrated over length, once around the complete circuit.
So that’s the integral. 🙂 And that’s how we can turn that motor above into a generator: instead of putting a current through the wire to make it turn, we can turn the loop, by hand or by a waterwheel or by whatever. Now, when the coil rotates, its wires will be moving in the magnetic field and so we will find an emf in the circuit of the coil, and so that’s how the motor becomes a generator.
Now, let me quickly interject something here: when I say ‘a push on the electrons in the wire’, what electrons are we talking about? How many? Well… I’ll answer that question in very much detail in a moment but, as for now, just note that the emf is some quantity expressed per coulomb or, as Feynman puts it above, per unit charge. So we’ll need to multiply it with the current in the circuit to get the power of our little generator.
OK. Let’s move on. Indeed, all I can do here is mention just a few basics, so we can move on to the next thing. If you really want to know all of the nitty-gritty, then you should just read Feynman’s Lecture on induced currents. That’s got everything. And, no, don’t worry: contrary to what you might expect, my ‘basics’ do not amount to a terrible pile of formulas. In fact, it’s all easy and quite amusing stuff, and I should probably include a lot more. But then… Well… I always need to move on… If not, I’ll never get to the stuff that I really want to understand. 😦
The electromotive force
We defined the electromotive force above, including its formula:
What are the units? Let’s see… We know B was measured not in newton per coulomb, like the electric field E, but in N·s/C·m, because we had to multiply the magnetic field strength with the velocity of the charge to find the force per unit charge, cf. the F/q = v×equation. Now what’s the unit in which we’d express that surface integral? We must multiply with m2, so we get N·m·s/C. Now let’s simplify that by noting that one volt is equal to 1 N·m/C. [The volt has a number of definitions, but the one that applies here is that it’s the potential difference between two points that will impart one joule (i.e. 1 N·m) of energy to a unit of charge (i.e. 1 C) that passes between them.] So we can measure the magnetic flux in volt-seconds, i.e. V·s. And then we take the derivative in regard to time, so we divide by s, and so we get… Volt! The emf is measured in volt!
Does that make sense? I guess so: the emf causes a current, just like a potential difference, i.e. a voltage, and, therefore, we can and should look at the emf as a voltage too!
But let’s think about it some more, though. In differential form, Faraday’s Law, is just that ×E = −∂B/∂t equation, so that’s just one of Maxwell’s four equations, and so we prefer to write it as the “flux rule”. Now, the “flux rule” says that the electromotive force (abbreviated as emf or EMF) on the electrons in a closed circuit is equal to the time rate of change of the magnetic flux it encloses. As mentioned above, we measure magnetic flux in volt-seconds (i.e. V·s), so its time rate of change is measured in volt (because the time rate of change is a quantity expressed per second), and so the emf is measured in volt, i.e. joule per coulomb, as 1 V = 1 N·m/C = 1 J/C. What does it mean?
The time rate of change of the magnetic flux can change because the surface covered by our loop changes, or because the field itself changes, or by both. Whatever the cause, it will change the emf, or the voltage, and so it will make the electrons move. So let’s suppose we have some generator generating some emf. The emf can be used to do some work. We can charge a capacitor, for example. So how would that work?
More charge on the capacitor will increase the voltage V of the capacitor, i.e. the potential difference V = Φ1 − Φ2 between the two plates. Now, we know that the increase of the voltage V will be proportional to the increase of the charge Q, and that the constant of proportionality is defined by the capacity C of the capacitor: C = Q/V. [How do we know that? Well… Have a look at my post on capacitors.] Now, if our capacitor has an enormous capacity, then its voltage won’t increase very rapidly. However, it’s clear that, no matter how large the capacity, its voltage will increase. It’s just a matter of time. Now, its voltage cannot be higher than the emf provided by our ‘generator’, because it will then want to discharge through the same circuit!
So we’re talking power and energy here, and so we need to put some load on our generator. Power is the rate of doing work, so it’s the time rate of change of energy, and it’s expressed in joule per second. The energy of our capacitor is U = (1/2)·Q2/C = (1/2)·C·V2. [How do we know that? Well… Have a look at my post on capacitors once again. :-)] So let’s take the time derivative of U assuming some constant voltage V. We get: dU/dt = d[(1/2)·Q2/C]/dt = (Q/C)·dQ/dt = V·dQ/dt. So that’s the power that the generator would need to supply to charge the generator. As I’ll show in a moment, the power supplied by a generator is, indeed, equal to the emf times the current, and the current is the time rate of change of the charge, so I = dQ/dt.
So, yes, it all works out: the power that’s being supplied by our generator will be used to charge our capacitor. Now, you may wonder: what about the current? Where is the current in Faraday’s Law? The answer is: Faraday’s Law doesn’t have the current. It’s just not there. The emf is expressed in volt, and so that’s energy per coulomb, so it’s per unit charge. How much power an generator can and will deliver depends on its design, and the circuit and load that we will be putting on it. So we can’t say how many coulomb we will have. It all depends. But you can imagine that, if the loop would be bigger, or if we’d have a coil with many loops, then our generator would be able to produce more power, i.e. it would be able to move more electrons, so the mentioned power = (emf)×(current) product would be larger. 🙂
Finally, to conclude, note Feynman’s definition of the emf: the tangential force per unit charge in the wire integrated over length around the complete circuit. So we’ve got force times distance here, but per unit charge. Now, force times distance is work, or energy, and so… Yes, emf is joule per coulomb, definitely! 🙂
[…] Don’t worry too much if you don’t quite ‘get’ this. I’ll come back to it when discussing electric circuits, which I’ll do in my next posts.
Self-inductance and Lenz’s rule
We talked about motors and generators above. We also have transformers, like the one below. What’s going on here is that an alternating current (AC) produces a continuously varying magnetic field, which generates an alternating emf in the second coil, which produces enough power to light an electric bulb.
Now, the total emf in coil (b) is the sum of the emf’s of the separate turns of coil, so if we wind (b) with many turns, we’ll get a larger emf, so we can ‘transform’ the voltage to some other voltage. From your high-school classes, you should know how that works.
The thing I want to talk about here is something else, though. There is an induction effect in coil (a) itself. Indeed, the varying current in coil (a) produces a varying magnetic field inside itself, and the flux of this field is continually changing, so there is a self-induced emf in coil (a). The effect is called self-inductance, and so it’s the emf acting on a current itself when it is building up a magnetic field or, in general, when its field is changing in any way. It’s a most remarkable phenomenon, and so let me paraphrase Feynman as he describes it:
“When we gave “the flux rule” that the emf is equal to the rate of change of the flux linkage, we didn’t specify the direction of the emf. There is a simple rule, called Lenz’s rule, for figuring out which way the emf goes: the emf tries to oppose any flux change. That is, the direction of an induced emf is always such that if a current were to flow in the direction of the emf, it would produce a flux of B that opposes the change in B that produces the emf. In particular, if there is a changing current in a single coil (or in any wire), there is a “back” emf in the circuit. This emf acts on the charges flowing in the coil to oppose the change in magnetic field, and so in the direction to oppose the change in current. It tries to keep the current constant; it is opposite to the current when the current is increasing, and it is in the direction of the current when it is decreasing. A current in a self-inductance has “inertia,” because the inductive effects try to keep the flow constant, just as mechanical inertia tries to keep the velocity of an object constant.”
Hmm… That’s something you need to read a couple of times to fully digest it. There’s a nice demo on YouTube, showing an MIT physics video demonstrating this effect with a metal ring placed on the end of an electromagnet. You’ve probably seen it before: the electromagnet is connected to a current, and the ring flies into the air. The explanation is that the induced currents in the ring create a magnetic field opposing the change of field through it. So the ring and the coil repel just like two magnets with opposite poles. The effect is no longer there when a thin radial cut is made in the ring, because then there can be no current. The nice thing about the video is that it shows how the effect gets much more dramatic when an alternating current is applied, rather than a DC current. And it also shows what happens when you first cool the ring in liquid nitrogen. 🙂
You may also notice the sparks when the electromagnet is being turned on. Believe it or not, that’s also related to a “back emf”. Indeed, when we disconnect a large electromagnet by opening a switch, the current is supposed to immediately go to zero but, in trying to do so, it generates a large “back emf”: large enough to develop an arc across the opening contacts of the switch. The high voltage is also not good for the insulation of the coil, as it might damage it. So that’s why large electromagnets usually include some extra circuit, which allows the “back current” to discharge less dramatically. But I’ll refer you to Feynman for more details, as any illustration here would clutter the exposé.
Eddy currents
I like educational videos, and so I should give you a few references here, but there’s so many of this that I’ll let you google a few yourself. The most spectacular demonstration of eddy currents is those that appear in a superconductor: even back in the 1970s, when Feynman wrote his Lectures, the effect of magnetic levitation was well known. Feynman illustrates the effect with the simple diagram below: when bringing a magnet near to a perfect conductor, such as tin below 3.8°K, eddy currents will create opposing fields, so that no magnetic flux enters the superconducting material. The effect is also referred to as the Meisner effect, after the German physicist Walther Meisner, although it was discovered much earlier (in 1911) by a Dutch physicist in Leiden, Heike Kamerlingh Onnes, who got a Nobel Prize for it.
Of course, we have eddy currents in less dramatic situations as well. The phenomenon of eddy currents is usually demonstrated by the braking of a sheet of metal as it swings back and forth between the poles of an electromagnet, as illustrated below (left). The illustration on the right shows how eddy-current effect can be drastically reduced by cutting slots in the plate, so that’s like making a radial cut in our jumping ring. 🙂
eddy currentseddy currents 2
The Faraday disc
The Faraday disc is interesting, not only from a historical point of view – the illustration below is a 19th century model, so Michael Faraday may have used himself – but also because it seems to contradict the “flux of rule”: as the disc rotates through a steady magnetic field, it will produce some emf, but so there’s no change in the flux. How is that possible?
Faraday_disk_generatorFaraday disk
The answer, of course, is that we are ‘cheating’ here: the material is moving, so we’re actually moving the ‘wire’, or the circuit if you want, so here we need to combine two equations:
two laws
If we do that, you’ll see it all makes sense. 🙂 Oh… That Faraday disc is referred to as a homopolar generator, and it’s quite interesting. You should check out what happened to the concept in the Wikipedia article on it. The Faraday disc was apparently used as a source for power pulses in the 1950s. The thing below could store 500 mega-joules and deliver currents up to 2 mega-ampère, i.e. 2 million amps! Fascinating, isn’t it? 🙂800px-Homopolar_anu-MJC
Bose and Fermi
Probability amplitudes: what are they?
Instead of reading Penrose, I’ve started to read Richard Feynman again. Of course, reading the original is always better than whatever others try to make of that, so I’d recommend you read Feynman yourself – instead of this blog. But then you’re doing that already, aren’t you? 🙂
Let’s explore those probability amplitudes somewhat more. They are complex numbers. In a fine little book on quantum mechanics (QED, 1985), Feynman calls them ‘arrows’ – and that’s what they are: two-dimensional vectors, aka complex numbers. So they have a direction and a length (or magnitude). When talking amplitudes, the direction and length are known as the phase and the modulus (or absolute value) respectively and you also know by now that the modulus squared represents a probability or probability density, such as the probability of detecting some particle (a photon or an electron) at some location x or some region Δx, or the probability of some particle going from A to B, or the probability of a photon being emitted or absorbed by an electron (or a proton), etcetera. I’ve inserted two illustrations below to explain the matter.
The first illustration just shows what a complex number really is: a two-dimensional number (z) with a real part (Re(z) = x) and an imaginary part (Im(z) = y). We can represent it in two ways: one uses the (x, y) coordinate system (z = x + iy), and the other is the so-called polar form: z = reiφ. The (real) number e in the latter equation is just Euler’s number, so that’s a mathematical constant (just like π). The little i is the imaginary unit, so that’s the thing we introduce to add a second (vertical) dimension to our analysis: i can be written as 0+= (0, 1) indeed, and so it’s like a (second) basis vector in the two-dimensional (Cartesian or complex) plane.
polar form of complex number
I should not say much more about this, but I must list some essential properties and relationships:
• The coordinate and polar form are related through Euler’s formula: z = x + iy = reiφ = r(cosφ + isinφ).
• From this, and the fact that cos(-φ) = cosφ and sin(-φ) = –sinφ, it follows that the (complex) conjugate z* = x – iy of a complex number z = x + iy is equal to z* = reiφ. [I use z* as a symbol, instead of z-bar, because I can’t find a z-bar in the character set here.] This equality is illustrated above.
• The length/modulus/absolute value of a complex number is written as |z| and is equal to |z| = (x2 + y2)1/2 = |reiφ| = r (so r is always a positive (real) number).
• As you can see from the graph, a complex number z and its conjugate z* have the same absolute value: |z| = |x+iy| = |z*| = |x-iy|.
• Therefore, we have the following: |z||z|=|z*||z*|=|z||z*|=|z|2, and we can use this result to calculate the (multiplicative) inverse: z-1 = 1/z = z*/|z|2.
• The absolute value of a product of complex numbers equals the product of the absolute values of those numbers: |z1z2| = |z1||z2|.
• Last but not least, it is important to be aware of the geometric interpretation of the sum and the product of two complex numbers:
• The sum of two complex numbers amounts to adding vectors, so that’s the familiar parallelogram law for vector addition: (a+ib) + (c+id) = (a+b) + i(c+d).
• Multiplying two complex numbers amounts to adding the angles and multiplying their lengths – as evident from writing such product in its polar form: reiθseiΘ = rsei(θ+Θ). The result is, quite obviously, another complex number. So it is not the usual scalar or vector product which you may or may not be familiar with.
[For the sake of completeness: (i) the scalar product (aka dot product) of two vectors (ab) is equal to the product of is the product of the magnitudes of the two vectors and the cosine of the angle between them: ab = |a||b|cosα; and (ii) the result of a vector product (or cross product) is a vector which is perpendicular to both, so it’s a vector that is not in the same plane as the vectors we are multiplying: a×b = |a||b| sinα n, with n the unit vector perpendicular to the plane containing a and b in the direction given by the so-called right-hand rule. Just be aware of the difference.]
The second illustration (see below) comes from that little book I mentioned above already: Feynman’s exquisite 1985 Alix G. Mautner Memorial Lectures on Quantum Electrodynamics, better known as QED: the Strange Theory of Light and Matter. It shows how these probability amplitudes, or ‘arrows’ as he calls them, really work, without even mentioning that they are ‘probability amplitudes’ or ‘complex numbers’. That being said, these ‘arrows’ are what they are: probability amplitudes.
To be precise, the illustration below shows the probability amplitude of a photon (so that’s a little packet of light) reflecting from the front surface (front reflection arrow) and the back (back reflection arrow) of a thin sheet of glass. If we write these vectors in polar form (reiφ), then it is obvious that they have the same length (r = 0.2) but their phase φ is different. That’s because the photon needs to travel a bit longer to reach the back of the glass: so the phase varies as a function of time and space, but the length doesn’t. Feynman visualizes that with the stopwatch: as the photon is emitted from a light source and travels through time and space, the stopwatch turns and, hence, the arrow will point in a different direction.
[To be even more precise, the amplitude for a photon traveling from point A to B is a (fairly simple) function (which I won’t write down here though) which depends on the so-called spacetime interval. This spacetime interval (written as I or s2) is equal to I = [(x-x1)2+(y-y1)2+(z-z1)2] – (t-t1)2. So the first term in this expression is the square of the distance in space, and the second term is the difference in time, or the ‘time distance’. Of course, we need to measure time and distance in equivalent units: we do that either by measuring spatial distance in light-seconds (i.e. the distance traveled by light in one second) or by expressing time in units that are equal to the time it takes for light to travel one meter (in the latter case we ‘stretch’ time (by multiplying it with c, i.e. the speed of light) while in the former, we ‘stretch’ our distance units). Because of the minus sign between the two terms, the spacetime interval can be negative, zero, or positive, and we call these intervals time-like (I < 0), light-like (I = 0) or space-like (I > 0). Because nothing travels faster than light, two events separated by a space-like interval cannot have a cause-effect relationship. I won’t go into any more detail here but, at this point, you may want to read the article on the so-called light cone relating past and future events in Wikipedia, because that’s what we’re talking about here really.]
front and back reflection amplitude
Feynman adds the two arrows, because a photon may be reflected either by the front surface or by the back surface and we can’t know which of the two possibilities was the case. So he adds the amplitudes here, not the probabilities. The probability of the photon bouncing off the front surface is the modulus of the amplitude squared, (i.e. |reiφ|2 = r2), and so that’s 4% here (0.2·0.2). The probability for the back surface is the same: 4% also. However, the combined probability of a photon bouncing back from either the front or the back surface – we cannot know which path was followed – is not 8%, but some value between 0 and 16% (5% only in the top illustration, and 16% (i.e. the maximum) in the bottom illustration). This value depends on the thickness of the sheet of glass. That’s because it’s the thickness of the sheet that determines where the hand of our stopwatch stops. If the glass is just thick enough to make the stopwatch make one extra half turn as the photon travels through the glass from the front to the back, then we reach our maximum value of 16%, and so that’s what shown in the bottom half of the illustration above.
For the sake of completeness, I need to note that the full explanation is actually a bit more complex. Just a little bit. 🙂 Indeed, there is no such thing as ‘surface reflection’ really: a photon has an amplitude for scattering by each and every layer of electrons in the glass and so we have actually have many more arrows to add in order to arrive at a ‘final’ arrow. However, Feynman shows how all these arrows can be replaced by two so-called ‘radius arrows’: one for ‘front surface reflection’ and one for ‘back surface reflection’. The argument is relatively easy but I have no intention to fully copy Feynman here because the point here is only to illustrate how probabilities are calculated from probability amplitudes. So just remember: probabilities are real numbers between 0 and 1 (or between 0 and 100%), while amplitudes are complex numbers – or ‘arrows’ as Feynman calls them in this popular lectures series.
In order to give somewhat more credit to Feynman – and also to be somewhat more complete on how light really reflects from a sheet of glass (or a film of oil on water or a mud puddle), I copy one more illustration here – with the text – which speaks for itself: “The phenomenon of colors produced by the partial reflection of white light by two surfaces is called iridescence, and can be found in many places. Perhaps you have wondered how the brilliant colors of hummingbirds and peacocks are produced. Now you know.” The iridescence phenomenon is caused by really small variations in the thickness of the reflecting material indeed, and it is, perhaps, worth noting that Feynman is also known as the father of nanotechnology… 🙂
Light versus matter
So much for light – or electromagnetic waves in general. They consist of photons. Photons are discrete wave-packets of energy, and their energy (E) is related to the frequency of the light (f) through the Planck relation: E = hf. The factor h in this relation is the Planck constant, or the quantum of action in quantum mechanics as this tiny number (6.62606957×10−34) is also being referred to. Photons have no mass and, hence, they travel at the speed of light indeed. But what about the other wave-like particles, like electrons?
For these, we have probability amplitudes (or, more generally, a wave function) as well, the characteristics of which are given by the de Broglie relations. These de Broglie relations also associate a frequency and a wavelength with the energy and/or the momentum of the ‘wave-particle’ that we are looking at: f = E/h and λ = h/p. In fact, one will usually find those two de Broglie relations in a slightly different but equivalent form: ω = E/ħ and k = p/ħ. The symbol ω stands for the angular frequency, so that’s the frequency expressed in radians. In other words, ω is the speed with which the hand of that stopwatch is going round and round and round. Similarly, k is the wave number, and so that’s the wavelength expressed in radians (or the spatial frequency one might say). We use k and ω in wave functions because the argument of these wave functions is the phase of the probability amplitude, and this phase is expressed in radians. For more details on how we go from distance and time units to radians, I refer to my previous post. [Indeed, I need to move on here otherwise this post will become a book of its own! Just check out the following: λ = 2π/k and f = ω/2π.]
How should we visualize a de Broglie wave for, let’s say, an electron? Well, I think the following illustration (which I took from Wikipedia) is not too bad.
Let’s first look at the graph on the top of the left-hand side of the illustration above. We have a complex wave function Ψ(x) here but only the real part of it is being graphed. Also note that we only look at how this function varies over space at some fixed point of time, and so we do not have a time variable here. That’s OK. Adding the complex part would be nice but it would make the graph even more ‘complex’ :-), and looking at one point in space only and analyzing the amplitude as a function of time only would yield similar graphs. If you want to see an illustration with both the real as well as the complex part of a wave function, have a look at my previous post.
We also have the probability – that’s the red graph – as a function of the probability amplitude: P = |Ψ(x)|2 (so that’s just the modulus squared). What probability? Well, the probability that we can actually find the particle (let’s say an electron) at that location. Probability is obviously always positive (unlike the real (or imaginary) part of the probability amplitude, which oscillate around the x-axis). The probability is also reflected in the opacity of the little red ‘tennis ball’ representing our ‘wavicle’: the opacity varies as a function of the probability. So our electron is smeared out, so to say, over the space denoted as Δx.
Δx is the uncertainty about the position. The question mark next to the λ symbol (we’re still looking at the graph on the top left-hand side of the above illustration only: don’t look at the other three graphs now!) attributes this uncertainty to uncertainty about the wavelength. As mentioned in my previous post, wave packets, or wave trains, do not tend to have an exact wavelength indeed. And so, according to the de Broglie equation λ = h/p, if we cannot associate an exact value with λ, we will not be able to associate an exact value with p. Now that’s what’s shown on the right-hand side. In fact, because we’ve got a relatively good take on the position of this ‘particle’ (or wavicle we should say) here, we have a much wider interval for its momentum : Δpx. [We’re only considering the horizontal component of the momentum vector p here, so that’s px.] Φ(p) is referred to as the momentum wave function, and |Φ(p)|2 is the corresponding probability (or probability density as it’s usually referred to).
The two graphs at the bottom present the reverse situation: fairly precise momentum, but a lot of uncertainty about the wavicle’s position (I know I should stick to the term ‘particle’ – because that’s what physicists prefer – but I think ‘wavicle’ describes better what it’s supposed to be). So the illustration above is not only an illustration of the de Broglie wave function for a particle, but it also illustrates the Uncertainty Principle.
Now, I know I should move on to the thing I really want to write about in this post – i.e. bosons and fermions – but I feel I need to say a few things more about this famous ‘Uncertainty Principle’ – if only because I find it quite confusing. According to Feynman, one should not attach too much importance to it. Indeed, when introducing his simple arithmetic on probability amplitudes, Feynman writes the following about it: “The uncertainty principle needs to be seen in its historical context. When the revolutionary ideas of quantum physics were first coming out, people still tried to understand them in terms of old-fashioned ideas (such as, light goes in straight lines). But at a certain point, the old-fashioned ideas began to fail, so a warning was developed that said, in effect, ‘Your old-fashioned ideas are no damn good when…’ If you get rid of all the old-fashioned ideas and instead use the ideas that I’m explaining in these lectures – adding arrows for all the ways an event can happen – there is no need for the uncertainty principle!” So, according to Feynman, wave function math deals with all and everything and therefore we should, perhaps, indeed forget about this rather mysterious ‘principle’.
However, because it is mentioned so much (especially in the more popular writing), I did try to find some kind of easy derivation of its standard formulation: ΔxΔp ≥ ħ (ħ = h/2π, i.e. the quantum of angular momentum in quantum mechanics). To my surprise, it’s actually not easy to derive the uncertainty principle from other basic ‘principles’. As mentioned above, it follows from the de Broglie equation λ = h/p that momentum (p) and wavelength (λ) are related, but so how do we relate the uncertainty about the wavelength (Δλ) or the momentum (Δp) to the uncertainty about the position of the particle (Δx)? The illustration below, which analyzes a wave packet (aka a wave train), might provide some clue. Before you look at the illustration and start wondering what it’s all about, remember that a wave function with a definite (angular) frequency ω and wave number k (as described in my previous post), which we can write as Ψ = Aei(ωt-kx), represents the amplitude of a particle with a known momentum p = ħ/at some point x and t, and that we had a big problem with such wave, because the squared modulus of this function is a constant: |Ψ|2 = |Aei(ωt-kx)|= A2. So that means that the probability of finding this particle is the same at all points. So it’s everywhere and nowhere really (so it’s like the second wave function in the illustration above, but then with Δx infinitely long and the same wave shape all along the x-axis). Surely, we can’t have this, can we? Now we cannot – if only because of the fact that if we add up all of the probabilities, we would not get some finite number. So, in reality, particles are effectively confined to some region Δor – if we limit our analysis to one dimension only (for the sake of simplicity) – Δx (remember that bold-type symbols represent vectors). So the probability amplitude of a particle is more likely to look like something that we refer to as a wave packet or a wave train. And so that’s what’s explained more in detail below.
Now, I said that localized wave trains do not tend to have an exact wavelength. What do I mean with that? It doesn’t sound very precise, does it? In fact, we actually can easily sketch a graph of a wave packet with some fixed wavelength (or fixed frequency), so what am I saying here? I am saying that, in quantum physics, we are only looking at a very specific type of wave train: they are a composite of a (potentially infinite) number of waves whose wavelengths are distributed more or less continuously around some average, as shown in the illustration below, and so the addition of all of these waves – or their superposition as the process of adding waves is usually referred to – results in a combined ‘wavelength’ for the localized wave train that we cannot, indeed, equate with some exact number. I have not mastered the details of the mathematical process referred to as Fourier analysis (which refers to the decomposition of a combined wave into its sinusoidal components) as yet, and, hence, I am not in a position to quickly show you how Δx and Δλ are related exactly, but the point to note is that a wider spread of wavelengths results in a smaller Δx. Now, a wider spread of wavelengths corresponds to a wider spread in p too, and so there we have the Uncertainty Principle: the more we know about Δx, the less we know about Δx, and so that’s what the inequality ΔxΔp ≥ h/2π represents really.
Explanation of uncertainty principle
[Those who like to check things out may wonder why a wider spread in wavelength implies a wider spread in momentum. Indeed, if we just replace λ and p with Δλ and Δp in the de Broglie equation λ = h/p, we get Δλ = h/Δp and so we have an inversely proportional relationship here, don’t we? No. We can’t just write that Δλ = Δ(h/p) but this Δ is not some mathematical operator than you can simply move inside of the brackets. What is Δλ? Is it a standard deviation? Is it the spread and, if so, what’s the spread? We could, for example, define it as the difference between some maximum value λmax and some minimum value λmin, so as Δλ = λmax – λmin. These two values would then correspond with pmax =h/λmin and pmin =h/λmax and so the corresponding spread in momentum would be equal to Δp = pmax – pmin = h/λmin – h/λmax = h(λmax – λmin)/(λmaxλmin). So a wider spread in wavelength does result in a wider spread in momentum, but the relationship is more subtle than you might think at first. In fact, in a more rigorous approach, we would indeed see the standard deviation (represented by the sigma symbol σ) from some average as a measure of the ‘uncertainty’. To be precise, the more precise formulation of the Uncertainty Principle is: σxσ≥ ħ/2, but don’t ask me where that 2 comes from!]
I really need to move on now, because this post is already way too lengthy and, hence, not very readable. So, back to that very first question: what’s that wave function math? Well, that’s obviously too complex a topic to be fully exhausted here. 🙂 I just wanted to present one aspect of it in this post: Bose-Einstein statistics. Huh? Yes.
When we say Bose-Einstein statistics, we should also say its opposite: Fermi-Dirac statistics. Bose-Einstein statistics were ‘discovered’ by the Indian scientist Satyanendra Nath Bose (the only thing Einstein did was to give Bose’s work on this wider recognition) and they apply to bosons (so they’re named after Bose only), while Fermi-Dirac statistics apply to fermions (‘Fermi-Diraqions’ doesn’t sound good either obviously). Any particle, or any wavicle I should say, is either a fermion or a boson. There’s a strict dichotomy: you can’t have characteristics of both. No split personalities. Not even for a split second.
The best-known examples of bosons are photons and the recently experimentally confirmed Higgs particle. But, in case you have heard of them, gluons (which mediate the so-called strong interactions between particles), and the W+, W and Z particles (which mediate the so-called weak interactions) are bosons too. Protons, neutrons and electrons, on the other hand, are fermions.
More complex particles, such as atomic nuclei, are also either bosons or fermions. That depends on the number of protons and neutrons they consist of. But let’s not get ahead of ourselves. Here, I’ll just note that bosons – unlike fermions – can pile on top of one another without limit, all occupying the same ‘quantum state’. This explains superconductivity, superfluidity and Bose-Einstein condensation at low temperatures. Indeed, these phenomena usually involve (bosonic) helium. You can’t do it with fermions. Superfluid helium has very weird properties, including zero viscosity – so it flows without dissipating energy and it creeps up the wall of its container, seemingly defying gravity: just Google one of the videos on the Web! It’s amazing stuff! Bose statistics also explain why photons of the same frequency can form coherent and extremely powerful laser beams, with (almost) no limit as to how much energy can be focused in a beam.
Fermions, on the other hand, avoid one another. Electrons, for example, organize themselves in shells around a nucleus stack. They can never collapse into some kind of condensed cloud, as bosons can. If electrons would not be fermions, we would not have such variety of atoms with such great range of chemical properties. But, again, let’s not get ahead of ourselves. Back to the math.
Bose versus Fermi particles
When adding two probability amplitudes (instead of probabilities), we are adding complex numbers (or vectors or arrows or whatever you want to call them), and so we need to take their phase into account or – to put it simply – their direction. If their phase is the same, the length of the new vector will be equal to the sum of the lengths of the two original vectors. When their phase is not the same, then the new vector will be shorter than the sum of the lengths of the two amplitudes that we are adding. How much shorter? Well, that obviously depends on the angle between the two vectors, i.e. the difference in phase: if it’s 180 degrees (or π radians), then they will cancel each other out and we have zero amplitude! So that’s destructive or negative interference. If it’s less than 90 degrees, then we will have constructive or positive interference.
It’s because of this interference effect that we have to add probability amplitudes first, before we can calculate the probability of an event happening in one or the other (indistinguishable) way (let’s say A or B) – instead of just adding probabilities as we would do in the classical world. It’s not subtle. It makes a big difference: |ΨA + ΨB|2 is the probability when we cannot distinguish the alternatives (so when we’re in the world of quantum mechanics and, hence, we have to add amplitudes), while |ΨA|+ |ΨB|is the probability when we can see what happens (i.e. we can see whetheror B was the case). Now, |ΨA + ΨB|is definitely not the same as |ΨA|+ |ΨB|– not for real numbers, and surely not for complex numbers either. But let’s move on with the argument – literally: I mean the argument of the wave function at hand here.
That stopwatch business above makes it easier to introduce the thought experiment which Feynman also uses to introduce Bose versus Fermi statistics (Feynman Lectures (1965), Vol. III, Lecture 4). The experimental set-up is shown below. We have two particles, which are being referred to as particle a and particle b respectively (so we can distinguish the two), heading straight for each other and, hence, they are likely to collide and be scattered in some other direction. The experimental set-up is designed to measure where they are likely to end up, i.e. to measure probabilities. [There’s no certainty in the quantum-mechanical world, remember?] So, in this experiment, we have a detector (or counter) at location 1 and a detector/counter at location 2 and, after many many measurements, we have some value for the (combined) probability that particle a goes to detector 1 and particle b goes to counter 2. This amplitude is a complex number and you may expect it will depend on the angle θ as shown in the illustration below.
scattering identical particles
So this angle θ will obviously show up somehow in the argument of our wave function. Hence, the wave function, or probability amplitude, describing the amplitude of particle a ending up in counter 1 and particle b ending up in counter 2 will be some (complex) function Ψ1= f(θ). Please note, once again, that θ is not some (complex) phase but some real number (expressed in radians) between 0 and 2π that characterizes the set-up of the experiment above. It is also worth repeating that f(θ) is not the amplitude of particle a hitting detector 1 only but the combined amplitude of particle a hitting counter 1 and particle b hitting counter 2! It makes a big difference and it’s essential in the interpretation of this argument! So, the combined probability of a going to 1 and of particle b going to 2, which we will write as P1, is equal to |Ψ1|= |f(θ)|2.
OK. That’s obvious enough. However, we might also find particle a in detector 2 and particle b in detector 1. Surely, the probability amplitude probability for this should be equal to f(θ+π)? It’s just a matter of switching counter 1 and 2 – i.e. we rotate their position over 180 degrees, or π (in radians) – and then we just insert the new angle of this experimental set-up (so that’s θ+π) into the very same wave function and there we are. Right?
Well… Maybe. The probability of a going to 2 and b going to 1, which we will write as P2, will be equal to |f(θ+π)|indeed. However, our probability amplitude, which I’ll write as Ψ2may not be equal to f(θ+π). It’s just a mathematical possibility. I am not saying anything definite here. Huh? Why not?
Well… Think about the thing we said about the phase and the possibility of a phase shift: f(θ+π) is just one of the many mathematical possibilities for a wave function yielding a probability P=|Ψ2|= |f(θ+π)|2. But any function eiδf(θ+π) will yield the same probability. Indeed, |z1z2| = |z1||z2| and so |eiδ f(θ+π)|2 = (|eiδ||f(θ+π)|)= |eiδ|2|f(θ+π)|= |f(θ+π)|(the square of the modulus of a complex number on the unit circle is always one – because the length of vectors on the unit circle is equal to one). It’s a general thing: if Ψ is some wave function (i.e. it describes some complex amplitude in space and time, then eiδΨ is the same wave function but with a phase shift equal to δ. Huh? Yes. Think about it: we’re multiplying complex numbers here, so that’s adding angles and multiplying lengths. Now the length of eiδ is 1 (because it’s a complex number on the unit circle) but its phase is δ. So multiplying Ψ with eiδ does not change the length of Ψ but it does shift its phase by an amount (in radians) equal to δ. That should be easy enough to understand.
You probably wonder what I am being so fussy, and what that δ could be, or why it would be there. After all, we do have a well-behaved wave function f(θ) here, depending on x, t and θ, and so the only thing we did was to change the angle θ (we added π radians to it). So why would we need to insert a phase shift here? Because that’s what δ really is: some random phase shift. Well… I don’t know. This phase factor is just a mathematical possibility as for now. So we just assume that, for some reason which we don’t understand right now, there might be some ‘arbitrary phase factor’ (that’s how Feynman calls δ) coming into play when we ‘exchange’ the ‘role’ of the particles. So maybe that δ is there, but maybe not. I admit it looks very ugly. In fact, if the story about Bose’s ‘discovery’ of this ‘mathematical possibility’ (in 1924) is correct, then it all started with an obvious ‘mistake’ in a quantum-mechanical calculation – but a ‘mistake’ that, miraculously, gave predictions that agreed with experimental results that could not be explained without introducing this ‘mistake’. So let the argument go full circle – literally – and take your time to appreciate the beauty of argumentation in physics.
Let’s swap detector 1 and detector 2 a second time, so we ‘exchange’ particle a and b once again. So then we need to apply this phase factor δ once again and, because of symmetry in physics, we obviously have to use the same phase factor δ – not some other value γ or something. We’re only rotating our detectors once again. That’s it. So all the rest stays the same. Of course, we also need to add π once more to the argument in our wave function f. In short, the amplitude for this is:
eiδ[eiδf(θ+π+π)] = (eiδ)f(θ) = ei2δ f(θ)
Indeed, the angle θ+2π is the same as θ. But so we have twice that phase shift now: 2δ. As ugly as that ‘thing’ above: eiδf(θ+π). However, if we square the amplitude, we get the same probability: P= |Ψ1|= |ei2δ f(θ)| = |f(θ)|2. So it must be right, right? Yes. But – Hey! Wait a minute! We are obviously back at where we started, aren’t we? We are looking at the combined probability – and amplitude – for particle a going to counter 1 and particle b going to counter 2, and the angle is θ! So it’s the same physical situation, and – What the heck! – reality doesn’t change just because we’re rotating these detectors a couple of times, does it? [In fact, we’re actually doing nothing but a thought experiment here!] Hence, not only the probability but also the amplitude must be the same. So (eiδ)2f(θ) must equal f(θ) and so… Well… If (eiδ)2f(θ) = f(θ), then (eiδ)2 must be equal to 1. Now, what does that imply for the value of δ?
Well… While the square of the modulus of all vectors on the unit circle is always equal to 1, there are only two cases for which the square of the vector itself yields 1: (I) eiδ = eiπ = eiπ = –1 (check it: (eiπ)= (–1)ei2π = ei0 = +1), and (II) eiδ = ei2π eie= +1 (check it: ei2π)= (+1)ei4π = ei0 = +1). In other words, our phase factor δ is either δ = 0 (or 0 ± 2nπ) or, else, δ = π (or π ± 2nπ). So eiδ = ± 1 and Ψ2 is either +f(θ+π) or, else, –f(θ+π). What does this mean? It means that, if we’re going to be adding the amplitudes, then the ‘exchanged case’ may contribute with the same sign or, else, with the opposite sign.
But, surely, there is no need to add amplitudes here, is there? Particle a can be distinguished from particle b and so the first case (particle a going into counter 1 and particle b going into counter 2) is not the same as the ‘exchanged case’ (particle a going into counter 2 and b going into counter 1). So we can clearly distinguish or verify which of the two possible paths are followed and, hence, we should be adding probabilities if we want to get the combined probability for both cases, not amplitudes. Now that is where the fun starts. Suppose that we have identical particles here – so not some beam of α-particles (i.e. helium nuclei) bombarding beryllium nuclei for instance but, let’s say, electrons on electrons, or photons on photons indeed – then we do have to add the amplitudes, not the probabilities, in order to calculate the combined probability of a particle going into counter 1 and the other particle going into counter 2, for the simple reason that we don’t know which is which and, hence, which is going where.
Let me immediately throw in an important qualifier: defining ‘identical particles’ is not as easy as it sounds. Our ‘wavicle’ of choice, for example, an electron, can have its spin ‘up’ or ‘down’ – and so that’s two different things. When an electron arrives in a counter, we can measure its spin (in practice or in theory: it doesn’t matter in quantum mechanics) and so we can distinguish it and, hence, an electron that’s ‘up’ is not identical to one that’s ‘down’. [I should resist the temptation but I’ll quickly make the remark: that’s the reason why we have two electrons in one atomic orbital: one is ‘up’ and the other one is ‘down’. Identical particles need to be in the same ‘quantum state’ (that’s the standard expression for it) to end up as ‘identical particles’ in, let’s say, a laser beam or so. As Feynman states it: in this (theoretical) experiment, we are talking polarized beams, with no mixture of different spin states.]
The wonderful thing in quantum mechanics is that mathematical possibility usually corresponds with reality. For example, electrons with positive charge, or anti-matter in general, is not only a theoretical possibility: they exist. Likewise, we effectively have particles which interfere with positive sign – these are called Bose particles – and particles which interfere with negative sign – Fermi particles.
So that’s reality. The factor eiδ = ± 1 is there, and it’s a strict dichotomy: photons, for example, always behave like Bose particles, and protons, neutrons and electrons always behave like Fermi particles. So they don’t change their mind and switch from one to the other category, not for a short while, and not for a long while (or forever) either. In fact, you may or may not be surprised to hear that there are experiments trying to find out if they do – just in case. 🙂 For example, just Google for Budker and English (2010) from the University of California at Berkeley. The experiments confirm the dichotomy: no split personalities here, not even for a nanosecond (10−9 s), or a picosecond (10−12 s). [A picosecond is the time taken by light to travel 0.3 mm in a vacuum. In a nanosecond, light travels about one foot.]
In any case, does all of this really matter? What’s the difference, in practical terms that is? Between Bose or Fermi, I must assume we prefer the booze.
It’s quite fundamental, however. Hang in there for a while and you’ll see why.
Bose statistics
Suppose we have, once again, some particle a and b that (i) come from different directions (but, this time around, not necessarily in the experimental set-up as described above: the two particles may come from any direction really), (ii) are being scattered, at some point in space (but, this time around, not necessarily the same point in space), (iii) end up going in one and the same direction and – hopefully – (iv) arrive together at some other point in space. So they end up in the same state, which means they have the same direction and energy (or momentum) and also whatever other condition that’s relevant. Again, if the particles are not identical, we can catch both of them and identify which is which. Now, if it’s two different particles, then they won’t take exactly the same path. Let’s say they travel along two infinitesimally close paths referred to as path 1 and 2 and so we should have two infinitesimally small detectors: one at location 1 and the other at location 2. The illustration below (credit to Feynman once again!) is for n particles, but here we’ll limit ourselves to the calculations for just two.
Boson particles
Let’s denote the amplitude of a to follow path 1 (and end up in counter 1) as a1, and the amplitude of b to follow path 2 (and end up in counter 2) as b1. Then the amplitude for these two scatterings to occur at the same time is the product of these two amplitudes, and so the probability is equal to |a1b1|= [|a1||b1|]= |a1|2|b1|2. Similarly, the combined amplitude of a following path 2 (and ending up in counter 2) and b following path 1 (etcetera) is |a2|2|b2|2. But so we said that the directions 1 and 2 were infinitesimally close and, hence, the values for aand a2, and for band b2, should also approach each other, so we can equate them with a and b respectively and, hence, the probability of some kind of combined detector picking up both particles as they hit the counter is equal to P = 2|a|2|b|2 (just substitute and add). [Note: For those who would think that separate counters and ‘some kind of combined detector’ radically alter the set-up of this thought experiment (and, hence, that we cannot just do this kind of math), I refer to Feynman (Vol. III, Lecture 4, section 4): he shows how it works using differential calculus.]
Now, if the particles cannot be distinguished – so if we have ‘identical particles’ (like photons, or polarized electrons) – and if we assume they are Bose particles (so they interfere with a positive sign – i.e. like photons, but not like electrons), then we should no longer add the probabilities but the amplitudes, so we get a1b+ a2b= 2ab for the amplitude and – lo and behold! – a probability equal to P = 4|a|2|b|2So what? Well… We’ve got a factor 2 difference here: 4|a|2|b|is two times 2|a|2|b|2.
This is a strange result: it means we’re twice as likely to find two identical Bose particles scattered into the same state as you would assuming the particles were different. That’s weird, to say the least. In fact, it gets even weirder, because this experiment can easily be extended to a situation where we have n particles present (which is what the illustration suggests), and that makes it even more interesting (more ‘weird’ that is). I’ll refer to Feynman here for the (fairly easy but somewhat lengthy) calculus in case we have n particles, but the conclusion is rock-solid: if we have n bosons already present in some state, then the probability of getting one extra boson is n+1 times greater than it would be if there were none before.
So the presence of the other particles increases the probability of getting one more: bosons like to crowd. And there’s no limit to it: the more bosons you have in one space, the more likely it is another one will want to occupy the same space. It’s this rather weird phenomenon which explains equally weird things such as superconductivity and superfluidity, or why photons of the same frequency can form such powerful laser beams: they don’t mind being together – literally on the same spot – in huge numbers. In fact, they love it: a laser beam, superfluidity or superconductivity are actually quantum-mechanical phenomena that are visible at a macro-scale.
OK. I won’t go into any more detail here. Let me just conclude by showing how interference works for Fermi particles. Well… That doesn’t work or, let me be more precise, it leads to the so-called (Pauli) Exclusion Principle which, for electrons, states that “no two electrons can be found in exactly the same state (including spin).” Indeed, we get a1b– a2b1= ab – ab = 0 (zero!) if we let the values of aand a2, and band b2, come arbitrarily close to each other. So the amplitude becomes zero as the two directions (1 and 2) approach each other. That simply means that it is not possible at all for two electrons to have the same momentum, location or, in general, the same state of motion – unless they are spinning opposite to each other (in which case they are not ‘identical’ particles). So what? Well… Nothing much. It just explains all of the chemical properties of atoms. 🙂
In addition, the Pauli exclusion principle also explains the stability of matter on a larger scale: protons and neutrons are fermions as well, and so they just “don’t get close together with one big smear of electrons around them”, as Feynman puts it, adding: “Atoms must keep away from each other, and so the stability of matter on a large scale is really a consequence of the Fermi particle nature of the electrons, protons and neutrons.”
Well… There’s nothing much to add to that, I guess. 🙂
Post scriptum:
I wrote that “more complex particles, such as atomic nuclei, are also either bosons or fermions”, and that this depends on the number of protons and neutrons they consist of. In fact, bosons are, in general, particles with integer spin (0 or 1), while fermions have half-integer spin (1/2). Bosonic Helium-4 (He4) has zero spin. Photons (which mediate electromagnetic interactions), gluons (which mediate the so-called strong interactions between particles), and the W+, W and Z particles (which mediate the so-called weak interactions) all have spin one (1). As mentioned above, Lithium-7 (Li7) has half-integer spin (3/2). The underlying reason for the difference in spin between He4 and Li7 is their composition indeed: He4 consists of two protons and two neutrons, while Liconsists of three protons and four neutrons.
However, we have to go beyond the protons and neutrons for some better explanation. We now know that protons and neutrons are not ‘fundamental’ any more: they consist of quarks, and quarks have a spin of 1/2. It is probably worth noting that Feynman did not know this when he wrote his Lectures in 1965, although he briefly sketches the findings of Murray Gell-Man and Georg Zweig, who published their findings in 1961 and 1964 only, so just a little bit before, and describes them as ‘very interesting’. I guess this is just another example of Feynman’s formidable intellect and intuition… In any case, protons and neutrons are so-called baryons: they consist of three quarks, as opposed to the short-lived (unstable) mesons, which consist of one quark and one anti-quark only (you may not have heard about mesons – they don’t live long – and so I won’t say anything about them). Now, an uneven number of quarks result in half-integer spin, and so that’s why protons and neutrons have half-integer spin. An even number of quarks result in integer spin, and so that’s why mesons have spin zero 0 or 1. Two protons and two neutrons together, so that’s He4, can condense into a bosonic state with spin zero, because four half-integer spins allows for an integer sum. Seven half-integer spins, however, cannot be combined into some integer spin, and so that’s why Li7 has half-integer spin (3/2). Electrons also have half-integer spin (1/2) too. So there you are.
Now, I must admit that this spin business is a topic of which I understand little – if anything at all. And so I won’t go beyond the stuff I paraphrased or quoted above. The ‘explanation’ surely doesn’t ‘explain’ this fundamental dichotomy between bosons and fermions. In that regard, Feynman’s 1965 conclusion still stands: “It appears to be one of the few places in physics where there is a rule which can be stated very simply, but for which no one has found a simple and easy explanation. The explanation is deep down in relativistic quantum mechanics. This probably means that we do not have a complete understanding of the fundamental principle involved. For the moment, you will just have to take it as one of the rules of the world.” |
a47829b73293d0be | Prof. Bengt Kasemo Mar 13, ’18 23 min
Surface Science - from static to dynamic surface systems
In this Surface Science educational series in eight parts, you will be introduced to how it all began and guided through key events and important aspects in the evolution from the early days to the mature area.
The series will cover topics such as the bridging of surface physics and surface chemistry, the paradigm shift from the study of static surface systems to the exploration of dynamic ones, as well as the importance of the advances in theory without which the evolution of the field could not have happened. It will also guide you through the different analytical methods that are used in the surface science field, and dig deeper into some of the areas that have, such as biomaterials and nanotechnology.
The Evolution of Surface Science, part 2
This post about the dynamics of atom/molecule interactions with surfaces, is a natural extension of the previous post, part 1, which addressed static properties of surface systems. Static systems refer to systems not changing in time, while dynamic systems refer to systems where temporal changes are an inherent characteristic. In contrast to kinetics, which are descriptions using phenomenological rate constants, dynamics aim for a detailed atomistic description of all steps of the interaction. The relation between kinetics and dynamics is briefly articulated below and in more detail in a later post.
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The presentation describes central concepts in molecule surface dynamics such as surface scattering, potential energy surfaces(PESs), energy exchange and transfer between a surface and an incident/desorbing molecule. It talks about surface corrugation, charge transfer and adiabatic versus non-adiabatic descriptions of the interactions. Specific surface-molecule systems are not covered, except for some examples that illustrate general concepts and types of model systems. Experimental methods are only briefly described and will be covered in later posts.
The early days of surface science focused on static properties
In the early development of surface science and particularly in surface physics, the focus was on the “static properties” of simple surfaces. It was challenging enough to describe the surface, with or without adsorbates, in a “frozen” situation, without involving temporal changes and dynamics.
For many years, the determination of structural aspects of adsorbed atoms and molecules, both as single adsorbates and as dense monolayers, and the associated electron structure, was a primary field. Bond positions of atoms, bond lengths, and vibrational properties were in focus together with the associated electron structure aspects, such as bond orbitals, surface band structure and charge distributions. This development was described in the first post in this series, The evolution of surface science, part 1, and more will follow in a later post, where theoretical aspects will be described.
More complex static systems
Today the position(s), orientation(s) and electron structure of simple adsorbates, like, e.g., CO molecules, on single crystal metal surface like Ni, Cu, Pt can be described in atomistic detail and with chemical accuracy in the energies. Also “simple” compound surfaces, like metal oxides, are fairly well described. Challenges of static systems still remain as the systems addressed become more complex, like molecules with many atoms or more complex compound surfaces. Oxides, carbides, polymers and “soft” surfaces, and biological surfaces like lipid bilayers are now in focus, as are larger, e.g., organic or even biological molecules. More about this in later posts, e.g., when I will talk about biomaterial surfaces and biointerfaces.
Dynamics is the studies of temporal changes
While the static approach addresses the frozen state, the dynamics concern the time evolution of reversible or irreversible changes of the surface system. Examples are adsorption processes and diffusion, reaction, and desorption of adsorbates. Theoretically, static systems can be described by solving the time-independent Schrödinger equation, while dynamic systems require the even more challenging task to solve the time-dependent equation.
Historical remark
Historically gas surface dynamics was “born” from two “parents”; molecular collision dynamics in the gas phase, an active field already during the first part of the 20th century, and the surface community, until then focusing on static surface systems (Part 1 in this blog post series). Both communities were in the 2nd half of the 20th century looking for new areas, and challenges for their studies and they merged their knowledge, techniques, and concepts in gas surface dynamics. The molecule-molecule collision community had performed their studies by using one or two molecular beams to allow gas molecules to collide under well-controlled collisions in a vacuum, with prescribed translational and internal (electronic, vibrational and rotational) energies. This community was the main contributor of the molecular beam technique and post-scattering analytical tools, while the surface community’s main contribution was the surface knowledge, e.g., how to prepare and analyze surfaces and construct potential energy surfaces for molecule-surface systems.
Drivers towards studies of dynamic systems
One strong driver towards dynamics studies was the rich, purely basic research questions in this field; to just ask and learn how a molecule scatters against a surface, Fig. 1, and is caught or reflected by it. However, a major motivation was also the relevance for technically important surface processes and reactions like heterogeneous catalysis, dry etching, CVD (chemical vapor deposition) and oxidation. Knowledge about the static systems certainly provides significant insight also about dynamic systems, but it is insufficient to develop detailed and well-founded models and scenarios for systems developing in time. It is like comparing a few photographic snapshots (the static properties) versus a movie (dynamics) to describe a system developing in time.
Surface science Scattering.png
Figure 1. The principles of a scattering experiment. The incident particles are prepared and characterized with respect to their energies (translational and internal) and angles of incidence. Molecular beam techniques and optical excitations are major preparative tools. The scattered particles are analyzed with respect to their energies (translational and internal) and angular distributions. Mass spectrometry and optical (laser) techniques are major analytical tools.
Moving on to dynamic interactions
The overall ambition in dynamics studies is thus to understand, at an atomic level, how things happen and evolve at the surface. Typical questions asked are: “How can a molecule that bounces towards a surface lose enough energy to stick to it?", "Which are the detailed steps involved in desorption of atoms and molecules from surfaces?”, and “What causes or prevents dissociation of simple molecules like H2, CO, O2 on different surfaces?”. “How do two different atoms/molecules react on a surface to create a new molecule, as in a catalytic reaction?” Some of the elementary processes in focus here are shown in Fig. 2, and for a specific system in Fig. 3. Adsorption, desorption and diffusion events of simple atoms and molecules, like noble gas atoms and H2, CO, O2, NO, became popular model systems for studies of the collision dynamics and associated energy and charge transfer events.
Surface science Gas-Surface interaction events.png
Figure 2. Examples of gas-surface interaction events referred to in the text.
How do molecules in the gas phase interact with a surface?
Qualitatively what we want to understand in gas surface dynamics are (Fig. 2) how molecules or atoms from the gas phase interact with surfaces. For example, how do they land on the surface (sticking, adsorption, chemisorption), exchange energy with it, and then (sometimes) leave it?
As an illustration of what dynamics address and contain, we imagine the space-time path of a CO molecule approaching a partially oxygen-covered catalyst surface like Pt (Fig. 3). First, it lands on the surface (sticking) by getting rid of some of its excess energy, then it diffuses over the surface, i.e., jumps between different adsorption sites, and eventually reacts with an adsorbed oxygen atom and thereby forms a CO2 molecule. Along this path, we can identify many of the basic processes studied in gas surface dynamics; some displayed in Fig. 2. The first step is adsorption/sticking of the molecule (described below). It then most likely diffuses on the surface until it finds a pre-adsorbed oxygen atom (which was formed from a dissociated O2 molecule) with which it forms a new bond, the CO2-bond. At ambient temperature, CO2 does not (in contrast to O atoms or CO molecules) bind strongly to the surface. It, therefore, maybe after a few diffusion jumps, desorbs (see also below) from the surface.
Surface science model system.png
Figure 3. Illustration of the system CO + O2 → CO2 + O
Potential energy surfaces (PESs)
A useful concept and quantity in the discussion of dynamics is the potential energy surface (PES), Fig. 4. A point on a PES, e.g., for an atom on a real surface, corresponds to the total energy of the whole system at that point. In real space, the point is defined through all the relevant coordinates in the system and the total energy of the system at that point. For an atom, the coordinates are the atom’s distance from the surface and the x-y coordinates parallel to the surface. This gives a four-dimensional space for the atom-surface PES (xyz and energy). The x-y coordinates are necessary to define the atom's position in relation to the surface atoms in the unit cell. For a molecule, one must, in addition, define the positions of all atoms in the molecule. This means that the PES is multidimensional. For example, if a diatomic molecule rotates without changing distance from the surface, it still moves to another point on the PES. For a uniquely defined point one must specify both the azimuthal and polar coordinates of the molecular axis, and the intramolecular distance, yielding a six-dimensional PES space.
Surface science PES.png
Figure 4. A one-dimensional cut of the 6-dimensional PES for a system like O2 on Pt, where there exists a molecular adsorption well (minimum) in the entrance channel for the molecule to the surface, and then an activation barrier for dissociation of the O2 molecule and final formation of two chemisorbed O atoms. This way of drawing the 1 D cut of the 6 D PES means that the coordinate r is rather complex – it is not just the vertical distance to the surface, z, but runs through xyz coordinates of the 6 D PES such that the local minimum O2 (ad), the local maximum Ea and the absolute minimum 2O(ad) are visited. An O2 molecule from the gas phase can undergo the following events as alternatives: (i) bounce back to the gas phase against the barrier Ea (if too little energy is dissipated in the region of the O2 (ad) well, (ii) adsorb in the O2 well and then (iii) either be thermally desorbed back to the gas phase or (iv) be thermally activated to pass the barrier Ea to form two adsorbed O atoms. (v) If the kinetic energy of the incident molecule is higher than Ea it can also go directly into the O(ad) adsorption well. Note; if the molecule is trapped in the molecular adsorption well O2 (ad), and stays there for some time (length of time determined by temperature), it can perform a diffusive motion to many identical adsorption sites, before either desorbing or dissociating to 2 O(ad) (see Fig. 3).
From ground state PESs for static systems to excited state PESs for dynamics.
When addressing dynamic events on surfaces, the static ground state properties, i.e., the lowest energy state, is insufficient as a platform for the description. The static ground state is a (adiabatic) multidimensional PES, where the ground state is represented by minima. In reality, in dynamics, the system often evolves along PESs that represent excited states, rather than the ground state, for example, because of too high activation barriers between the excited PESs’ local minima and the absolute minima of the ground state PESs’. For practical time scales, the system under study may thus often be “locked” or semi-frozen in a metastable state. An example of the latter is the formation of thin, self-passivating oxide mono- or multilayers on Si and many metals like Al or Ti. Thermodynamically the whole metal piece should be oxidized, but the thin oxide scale of one or a few nm stops growing due to a too high activation barrier for atom transport and/or O2 dissociation.
The timescale of a simple molecule scattering event, with no trapping or sticking, but just a single collision, where the molecule bounces back into a vacuum, is typically 10-12 – 10-13 seconds. This is the same as one vibrational period of a molecule in free space, and can be estimated by dividing the molecule’s surface interaction range, typically of order 0.1 nm, and a typical velocity in the thermal regime, 100 – 1000 m/s. If the scattering instead involves temporal trapping on the surface, and then desorption, the time scale can be extended from many picoseconds to nanoseconds. Depending on the details of the scattering event, it can also be longer, and even go to infinity when the molecule is stuck permanently on the surface.
If a molecule has been thermally equilibrated with the surface, it can desorb due to thermal fluctuations, if the combination of temperature and binding energy to the surface are in the right range. The timescale for the desorption event, or surface "lifetime" of the molecule, can be estimated using the Arrhenius approximation (see below), where the time constant, τd, for desorption is the inverse of the desorption rate constant, τd = 1/kd = (ν exp-Ed/kBT)-1 and where ν is the so-called preexponential factor, in simple cases of order 1013. Ed is the desorption energy, which is equal to the binding energy to the surface, if there is no activation energy for adsorption. For example, when the temperature is around room temperature, binding energy of a little less than 0.8 eV gives a lifetime of order 1 s. At 100 K higher or lower T, the lifetime is of order ms and essentially infinite, respectively.
Classification of gas-surface interaction events
For classification of gas surface interaction/scattering events one frequently uses the three classes I) elastic, II) inelastic and III) reactive scattering. In the former two, all chemical bonds are preserved, and new ones are not formed, while bond-breaking and bond-formation characterize class iii). The term “scattering” is used because in the more detailed studies one always uses molecular beams, where the beam atoms or molecules scatter against a single crystal surface (Fig. 1).
I) Elastic scattering
A “cleanest” example of elastic scattering is when He atoms scatter against a single crystal surface. A unidirectional and monoenergetic beam of He atoms scatters predominantly without energy deposition, or energy pick-up, to/from the surface. One branch of studies even focuses on diffraction of He atoms to extract structural information about the scattering surface (diffraction is inherently an elastic event). As a curiosity, even H2 molecules may be diffracted by some crystalline surfaces. The strong elastic component for He and hydrogen scattering derives from the small mass ratio between the scattered atom and the surface atoms, making inelastic phonon dissipation ineffective, and from the weak electronic interaction (holds only on some surfaces for H2) – it is like throwing a ping-pong ball towards a bowling ball. When the mass ratio is reversed, it leads to stronger inelastic scattering involving phonons (see below).
II) Inelastic scattering
In inelastic scattering, energy is deposited into the surface and/or picked up from it. Possible energy conversion events are shown in Fig. 5. For atoms, a typical example is heavier noble atoms than He, e.g., Ar, and especially Xe, where kinetic (translational, T) energy of the incident atoms can be transferred to, or picked up from, the phonon bath (vibrations, V) of the surface atoms. If the incident particle is a molecule, additional energy conversion/dissipation events are possible like rotational (R) and vibrational (V) and even electronic (E) excitations. In the latter cases, the molecules in the molecular beam are prepared with specific vibrational, rotational or electronic excitations.
On the detection side, after scattering, one measures the changes in these vibrational-rotational-electronic quantum numbers by laser spectroscopy and translational energy changes by pulsed/chopped beam techniques. Large sets of data are obtained by varying the initial vibrational-rotational quantum numbers, translational energy and angles of incidence. With theoretical (molecular dynamics) tools, one then tries to make models of the surface PES and of how all the scattering events (elastic and inelastic) take place.
Surface science energy conversion events.png
Figure 5. Possible energy conversion events in molecule-surface scattering events.
T: translational energy of incident or scattered/desorbed atoms or molecules.
V: Vibrational energy of incident or scattered molecules or vibrational excitations in the surface phonons.
R: Rotational energy of incident or scattered molecules.
E: Electronic excitations in the incident or scattered atoms/molecules and/or in the surface region (e.g., electron-hole pairs).
Surface corrugation
Especially in connection with elastic and inelastic scattering the concept “surface corrugation” is often used. This concept describes how ‘corrugated’ the energy landscape of the PES under study is. As we learned above, a point on the PES is the total energy of an atom/molecule at a specific point in real space. Corrugation refers to how much this energy varies as we move over the PES, or more correctly how much that energy varies over the surface unit cell. An extreme case with low corrugation is He atoms on a metal surface which is very ‘smooth’ with little energy variation as the He atoms explore the surface unit cell. He atoms mainly interact (repulsively) with the tails of the metal valence electron wave functions spilling out into vacuum (although there is an attractive van der Waals component): The repulsive “wall” has only small variations in charge density along the surface. More polarizable noble gas atoms and especially reactive atoms/molecules can in contrast exhibit very large corrugation.
III) Reactive scattering
In reactive events chemical bonds are broken and/or new ones are formed. One of the most widely explored sub-class here is catalytic reactions of the formal type A + B → C where A, B are the original reactants and C the reaction product. The catalyst surface enhances the reaction dramatically, without being changed itself. But there are also other types of reactive events, e.g., reactive etching where A species from the gas phase may react with a surface atom, B, to form a gas phase product C, thereby removing surface material (often chemically and/or spatially selective).
Experimentally reactive scattering can be performed with one or two molecular beams (MBs). In the former case one typically first deposits some predetermined coverage of atoms/molecules, say reactant A, on the surface. Then one directs molecular beam molecules of B to the surface and detects the product molecules AB for varying incident MB parameters and for different surface temperatures. One may also detect the velocity and angular distribution of product molecules AB.
The experiment may also be reversed, i.e., by exchanging A for B. With two molecular beams, there are more opportunities for variability in the experiment. One example is that a steady state can be established with two beams, where the surface coverage stays constant, but the reaction goes on continuously. This is often made at different ratios of the A/B fluxes. One may furthermore catch reaction events involving transient states of reactants on the surface, like molecules in precursor motion.
Dry etching
Dry etching reactions constitute a sub-class of reactive scattering. Dry etching is a method used extensively in semiconductor surface processing in combination with photolithography for microelectronics. For example, a silicon surface may be dry etched with fluorine-containing molecules. A typical example is XeF2 etching of the Si surface by forming SiF4 which desorbs from the surface. The net result is a removal of one Si atom from the surface from two incident XeF2 molecules. In practice, the etching is performed in a plasma etcher, but MB experiments can help elucidate the reaction mechanisms.
Additional aspects of dynamics and special cases
Above I have outlined the essence of gas-surface dynamics. There are some additional concepts that are common in dynamics and which I briefly define and comment on below. It is also appropriate to articulate a few special cases like ‘sticking’ and thermal desorption. Let’s begin with ‘sticking’.
The ‘sticking’ of molecules and atoms on surfaces is a central concept in surface reactions. Sticking is essentially synonymous with “adsorption” as a process and leads to adsorbed/chemisorbed molecules. It connects kinetics and dynamics via the adsorption rate constant ka. The kinetic equation (no desorption) for first-order adsorption (commonly seen for adsorbing atoms) is
dθ/dt = ka (1-θ(t)) (1)
where θ(t) is the surface coverage. ka is the product of the gas impingement rate (gas pressure) and the sticking probability, s0, on the clean surface. ka contains via s0 (in an embedded form) all the atomistic details of the scattering event when a molecule hits the surface.
The simple solution to the Eq. 1 is
θ(t) = 1- e-t/τ (2)
and τ=1/ka
The solution describes how an initially clean surface, exposed to a constant flux of gas species, fills up to saturation in the same functional way as a capacitor is charged with a constant voltage. Higher gas flux and higher values of s0 shorten the time scale τ to fill up the surface with adsorbates. Typically, the time constant for filling the surface is of order a few seconds at 10-6 torr pressure and s0=1. First order adsorption is often called monoatomic adsorption but can also apply for simple molecular adsorption. If the type of adsorption is instead second order, common for dissociative adsorption of a diatomic molecule, we have dθ/dt = ka (1-θ(t))2, with a somewhat slower filling up of the surface compared to 1st order adsorption.
Sticking coefficient measurements have been subject to many studies over the years. Methods range from surface spectroscopic or desorption measurements of the coverage θ(t) for different gas doses to the surface, to sophisticated MB methods. Depending on the studied system the sticking coefficient can vary from unity, meaning that every molecule/atom striking the surface sticks to it, to many orders of magnitude lower values.
Numbers lower than unity are indicative of the existence of an activation barrier for adsorption. In such cases it may be elucidating to measure s0 at different incident energies of the molecule; an activation barrier may prevent slow molecules from sticking but not sufficiently fast ones. An activation barrier may also be overcome by internal vibrational or rotational energy of an incident molecule, due to concerted adsorption and energy conversion events (Fig. 5).
A smaller than unity sticking coefficient can, however, also be seen without an activation barrier due to inefficient energy transfer. This happens when the molecule/atom does not dissipate energy sufficiently fast in one single scattering event, to be trapped by the surface and therefore returns to the gas phase.
Thermal desorption
A sub-class of experiments in this area, not requiring molecular beams, but just a gas inlet to adsorb molecules on a surface, focuses on the desorption of molecules or atoms. The experiment starts with a surface partially or fully covered with an adsorbate that stays on the surface indefinitely, unless some external perturbation causes them to desorb. The most common perturbation is an externally applied temperature increase, leading to what is called thermal desorption, TDS or TPD (thermal desorption spectroscopy and temperature programmed desorption, respectively). Typically, in TDS, the surface is subject to a linear increase in temperature and the flux of desorbing molecules are recorded in real time by mass spectrometry until the surface is empty. By analyzing the recorded thermal desorption spectrum with some kinetic model, e.g., the Langmuir model, it is possible to determine the adsorption energy of the initially adsorbed molecules. The kinetic equation for 1st order adsorption reads
dθ/dt = – kd θ(T(t)) (3)
where kd is the desorption rate constant, and
kd=ν exp(-Ed/kT) (4)
Ed is the desorption energy, which for non-activated adsorption is essentially equal to the molecule’s binding energy to the surface. ν is the so-called preexponential factor, which in simple absolute rate theory has a value of order 1013.The applied temperature rise is an experimental parameter and enters as T(t) = ct, where c is the heating rate, and where kd (T) as we see contains the binding energy to the surface.
For diatomic molecules that dissociate upon adsorption, like O2 on many metals, the desorption is frequently found to be 2nd order in the coverage, i.e., dθ/dt = – kd θ2(T(t)). The 2nd order dependence indicates that two neighboring atoms are required to form a diatomic molecule that desorbs.
More about energy dissipation and energy exchange
A molecule outside a surface, to which it eventually binds, constitutes an excited state of the system. When a molecule-surface bond is formed, a lower energy state is formed, and in the process, the initial higher energy of the system must in some way be dissipated.
The candidates for energy dissipation are many (Fig. 5). A strong candidate is the phonons (lattice vibrations) of the solid surface. As, e.g., a heavy and energetic Xe atom hits a surface, a major energy dissipation channel is phonons. Since the energies involved are typically much larger than that of single phonons, the dissipation often involves cascades of phonons.
Another surface candidate is electronic excitations, i.e., electron-hole pairs. It is less common, but increasingly more important, as the mass ratio of incident particles and surface atoms get smaller, making phonon excitations less effective, e.g., hydrogen molecules on transition metals.
Yet another example (reactive interaction) is when electronegative molecules, like Cl2, hit an electropositive (low work function) surface. Then, electron transfer can occur from the surface to the (electron affinity level of an) approaching Cl2 molecule and lead to electron-hole pair creation as an energy dissipation channel (see more about this case below). The lifetime of electron-hole pair excitations is extremely short for metals, 10-14 – 10-15 s. Phonon lifetimes are one to two orders of magnitude longer.
But the surface excitations (phonons and electron-hole pairs) are only one set of energy dissipation channels (Fig. 5) associated with the surface. For molecules, there are also the internal molecular excitations that can pick up or give away energy, namely rotations and vibrations. In reality, a full scattering/sticking process may involve many of these elementary excitations and energy conversion events (the arrows in Fig. 5), simultaneously or sequentially. The lifetime of molecular vibrations at surfaces are in the range from tens of picoseconds to fractions of picoseconds - orders of magnitude shorter than in the gas phase, due to the efficient channeling of energy into phonons or electron-hole pairs.
It is also possible that a molecule with excess kinetic, i.e., translational energy, collides with the surface and converts translational energy to internal, rotational or vibrational energy of the same molecule, which then bounces off the surface after a translational energy-to-internal energy conversion. Fast incident molecules can thus scatter on the surface back into a vacuum or gas phase, with much higher internal vibrational-rotational excitation than in the incident molecules (and lower translational energy).
The discussion above can also be applied to the reverse process to adsorption, i.e., desorption. This step, when a molecule or atom leaves the surface, requires energy pick up from the surface (from its heat-bath of phonons, electron-hole pairs...) to convert the lower energy state of a bound molecule to a free molecule in the gas phase.
Adiabatic vs. non-adiabatic processes
Another aspect of the scattering and energy dissipation, that has generated much interest and work, is whether the energy dissipation in e.g., an adsorption event is adiabatic or non-adiabatic, which in turn affects on which PES the adsorbing particle moves.
If energy dissipation happens gradually, like in a friction process, and the energy is continuously dissipated as very low energy quanta, e.g., as phonons, i. e. heat, we call the energy dissipation and adsorption process adiabatic. However, in many cases, the system does not dissipate energy adiabatically but in larger, sudden (quantum) energy release events, like a photon, or an electron or a cascade of phonons. Then we call the energy release non-adiabatic. In PES language the latter event corresponds to first moving on an excited state PES and then making a sudden transition to a new, lower-lying (in energy) PES with a large change in energy. If the system adiabatically dissipates excess energy, the trajectory takes place on the ground state PES. In rare cases, non-adiabatic events can be detected experimentally as excited electrons or photons (electron-hole pairs, exoelectrons and surface chemiluminescence) emitted from the surface.
Gas surface dynamics and scattering event analysis
The scattering processes mentioned above are preferably studied with molecular beams. This allows well-defined translational and internal energies of the molecules in the beam. The outcome of such collision and energy conversion/dissipation events are studied by analyzing the scattered molecules with respect to angular distributions, translational energy and internal energy, and also by measuring the fraction (if there is one) of molecules stuck on the surface. Of course, also the parameters of the incident molecules (angle of incidence, translational and internal energies, etc.), are varied. The collective name of this area became gas-surface dynamics.
The scattering events take place in a multi-coordinate space. For example, to uniquely define the scattering of a two-atom molecule against a single crystal surface several aspects must be defined; (i) the direction of the molecule i. e. the azimuthal and polar angle of its trajectory to the surface, (ii) where does the molecule hit the surface in the surface unit cell (e.g., on top of a surface atom, or in the hollow between three or four surface atoms), and (iii) what is the rotational orientation of the molecule, and what is the vibrational phase. To develop a theory for the scattering event, the calculations need to include all these coordinates. Since all of them cannot be measured and some are averaged over all possible values in the experiment like the rotational orientation and vibrational phase, a challenge is to use all experimentally accessible information about the incident beam and the scattered molecules to learn about the scattering details and participating the PESs.
Charge transfer
When a strongly electronegative molecule like Cl2 approaches a metal surface with low work-function like potassium, the electron affinity level of Cl2 shifts downwards in energy due to so-called image interaction. As a result, an electron can jump or tunnel from the potassium surface to the Cl2 molecule, forming a transient Cl2- molecule (the affinity level is the lowest unoccupied electron level in the Cl2 molecule, i.e., where the weakest bound electron of a Cl2- sits.) As a result, the Cl2 molecule gets stuck to the surface and dissociates to form two KCl bonds on the surface. The timescale for the involved processes are in the range tens of fs to ps.
If the incident particle is instead very electropositive and the surface has a high work function, like K or Na incident on Pt or W, the reverse process occurs. Electrons tunnel from the alkali atom’s valence orbital into the valence band of the metal, and leads to a process called surface ionization, where Na+ or K+ ions leave the surface. These examples are extreme cases of charge transfer. Similar charge transfer can occur more smoothly involving fractional charge transfer and orbital mixing (surface and molecular orbitals) for other systems, e.g., for oxygen molecules on metal surfaces. Fractional charge transfer means that an electron orbital is partially occupied, partially empty.
In the context of adsorption/sticking, precursors and precursor motion are frequently appearing concepts. This usually refers to a motion of a molecule/atom just after it has struck the surface, but before it has dissipated its excess energy and come to rest and equilibrated with the temperature of the surface (Fig. 2c). The motion on the surface is driven by the adsorption energy and occurs before that energy has been fully dissipated.
There are several possible mechanisms for precursor motion. The molecule may in the first encounter convert some of its momentum normal to the surface into parallel momentum (elastically) or just dissipate that part of its energy (inelastically), resulting in a ‘hot’ precursor moving along the surface with higher than thermal energy, but trapped by the surface interaction. During the parallel motion, which occurs on the corresponding PES, the excess energy may be successively dissipated until the molecule is trapped in some local or absolute minimum on the PES and just becomes an adsorbed and thermally equilibrated molecule. The motion is similar to diffusion, but the latter is purely thermal in nature. A hot precursor may visit parts of the PESs not accessible via thermal diffusion.
An alternative outcome of the “hot” precursor motion on the surface is that the precursor regains energy and/or vertical momentum and scatters back to the surface. This is sometimes referred to as a trapping-desorption channel (Fig. 2f).
Another type of precursor, is when the adsorbate partially covers a surface under study and an incident molecule of the same kind lands on an area already occupied by the adsorbate. The incident adsorbate may then perform a similar parallel motion on the surface on top of adsorbates, as described above, until the molecule finds an empty site and adsorbs there (or desorbs before that). This type of precursor is sometimes referred to as an extrinsic precursor, while the type of precursor discussed above may be called “intrinsic” precursors..
Photon and electron induced dynamics
When a surface with adsorbed atoms/molecules is irradiated with electrons of a few or a few tens of eV, i.e., a little more than the surface chemical bond strengths, it may lead to a special type of desorption, called photo-induced (PID) or electron stimulated (ESD) desorption. The mechanisms involved in such events are almost always electronic excitations, either in the adsorbed molecule itself or, more common, in the surface or in the molecule-surface bond.
A common case is excitation of an electron-hole pair in the valence band of the surface. Typically, the initially excited hot electron moves to the adsorbate and into an antibonding orbital of the adsorbate – surface complex, which in turn causes the adsorbate to be repelled from the surface into a vacuum. Alternatively, it is instead the excited hole state that moves to the adsorbate and empties a bonding orbital, thereby causing desorption. Similar events may lead to dissociation of adsorbed molecules, which in turn may cause a surface reaction, if there are other adsorbates on the surface (see below for the CO + O2 case on platinum). PID studies are conceptually important also because they elucidate processes occurring in photocatalysis, of interest e.g. in photocatalytic decomposition of water to H2.
A frequently studied model system
One of the most studied model reactions in the reactive scattering category is CO reacting with an oxygen atom, the latter derived from a dissociated O2 molecule (Fig. 3) or NO2,. A major reason for the interest in this system is that it involves key reactions in automotive emission cleaning catalysis. Key reaction steps are reactions with CO and NO or NO2, in the presence of O2. Let's, therefore, return to the CO + O2 catalytic reaction on platinum (Fig. 3), which we briefly touched upon several times above.
At typical catalytic operation temperatures, say 150 – 700 C, the reaction proceeds by simultaneous adsorption of CO molecules and O2 molecules, where the latter dissociate. CO molecules stay intact on the surface with stay-times of a tenth of a second (at the lowest T) to only around a ns at the higher temperatures, before they desorb again. During this time, they diffuse rapidly on the surface, faster the higher the temperature is. The O2 molecules dissociate and form adsorbed O atoms on the surface, via the molecular intermediate state shown in Fig. 4. These O atoms are found by the diffusing CO molecules, and the two react to form CO2 molecules (the mobility of O atoms is much smaller than that of CO). CO2 molecules are by comparison very weakly bound to the surface and desorb almost instantaneously.
The CO+ O2 reaction at low temperature
This simple scenario (previous paragraph) hides some essential details of the reaction, which are discovered if one runs the experiment at a much lower temperature and in a slightly different way. When CO and O2 are co-adsorbed on Pt at sufficiently low temperature, typically well below 150 K, both molecules stay intact and non-dissociated on the surface, and they do not react. If the surface temperature is then raised monotonically with co-adsorbed CO and O2, one suddenly sees CO2 molecules leaving the surface already around 150 K, indicating reactions between the absorbed CO and oxygen. However, only a fraction of the CO molecules reacts at this temperature, and a new desorption flux of CO2 is seen when the temperature reaches around 330 K. The latter is caused by the same reaction as was just discussed above between adsorbed CO and O atoms.
The actual events that occur are that the low-temperature formation of CO2 is caused transiently by dissociating O2 molecules (thermally activated dissociation). Some of the O atom dissociation products react with neighboring CO molecules and some just form adsorbed oxygen atoms as in Fig. 3. The latter cannot react with remaining CO molecules on the surface, because of a rather high activation barrier, until the temperature reaches around 300-350 K. The latter is the ‘normal’ CO oxidation reaction on Pt, with which this paragraph began, and which has been studied extensively since the days of Langmuir. The former reaction occurring at around 150 K is in a way more exotic, but mechanistically interesting, and reveals that there is a stable adsorbed molecular O2 state that is permanently occupied at sufficiently low T and transiently at high T. This state has been spectroscopically verified.
Photoinduced CO + O2 reaction
If in the latter experiment, with co-adsorbed O2 and CO molecules, at < 150 K, where they don’t react, the temperature is not raised, but instead a flux of near UV photons is sent to the surface, CO2 molecules are also formed, without any T increase. The mechanism of this reaction is first a photoexcitation of electron-hole pairs in the Pt surface, followed by (hot) electron transfer to an antibonding orbital of the adsorbed O2 molecule, forming a transient O2-, thereby inducing its dissociation to O atoms. The latter helps to get over the activation barrier in Fig. 4. Some of these O atoms bounce into and react with neighboring CO molecules, and form CO2, in much the same way as in the TPR experiment.
In PES language, this experiment shows that there exists a ground state PES for O atoms with a deep minimum corresponding to chemisorbed O atoms, and an excited state PES for O2, with a shallow minimum for O2. It also shows that there is a local minimum for an adsorbed O2 molecule with an activation barrier towards dissociation
Concluding remarks
My ambition with this post has been to present an overview of the field called gas surface dynamics, its general concepts and phenomena and what this field is all about. I have covered scientific questions and types of studies embraced by this area. Detailed descriptions of specific model systems are beyond the scope of this post. For example, the descriptions of adatom or admolecule motion to, and on, a surface, which is dealt with in gas – surface dynamics, is also highly relevant for thin film growth by physical vapor deposition (PVD) or chemical vapor deposition (CVD), but these areas have not been covered.
As also mentioned in post 1 on static systems, I want to emphasize the important role played by theory, both for interpretation of experiments, for developing concepts and understanding and for planning experiments. The theory aspect will be discussed in a later post, as will the relevant and extensive topic of experimental methods be.
Finally, as was indicated in the introductory part of this post, dynamics of molecule-surface interactions are very far from atomistic detail for more complex systems. Let’s take as an example protein-surface interactions. Even if we choose the simplest possible, yet relevant surface, like graphite or gold, and the simplest possible protein, we are still very far from an atomistic description. The latter would require describing a very complex and multidimensional PES on a time scale from picoseconds to seconds or even minutes, keeping track of all the 100ds or more atoms in the protein and how they (re)orient, bind to the surface, break or form bonds etc., creating a full molecular-dynamics ‘movie’ of how the protein binds, orients, unfolds, (partially) denatures etc. Although we are very far from such descriptions it is the direction of future studies.
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How are anyons possible? (another version)
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I know that this question has been submitted several times (especially see How are anyons possible?), even as a byproduct of other questions, since I did not find any completely satisfactory answers, here I submit another version of the question, stated into a very precise form using only very elementary general assumptions of quantum physics. In particular I will not use any operator (indicated by $P$ in other versions) representing the swap of particles.
Assume to deal with a system of a couple of identical particles, each moving in $R^2$. Neglecting for the moment the fact that the particles are indistinguishable, we start form the Hilbert space $L^2(R^2)\otimes L^2(R^2)$, that is isomorphic to $L^2(R^2\times R^2)$. Now I divide the rest of my issue into several elementary steps.
(1) Every element $\psi \in L^2(R^2\times R^2)$ with $||\psi||=1$ defines a state of the system, where $|| \cdot||$ is the $L^2$ norm.
(2) Each element of the class $\{e^{i\alpha}\psi\:|\; \psi\}$ for $\psi \in L^2(R^2\times R^2)$ with $||\psi||=1$ defines the same state, and a state is such a set of vectors.
(3) Each $\psi$ as above can be seen as a complex valued function defined, up to zero (Lebesgue) measure sets, on $R^2\times R^2$.
(4) Now consider the "swapped state" defined (due to (1)) by $\psi' \in L^2(R^2\times R^2)$ by the function (up to a zero measure set):
$$\psi'(x,y) := \psi(y,x)\:,\quad (x,y) \in R^2\times R^2$$
(5) The physical meaning of the state represented by $\psi'$ is that of a state obtained form $\psi$ with the role of the two particles interchanged.
(6) As the particles are identical, the state represented by $\psi'$ must be the same as that represented by $\psi$.
(7) In view of (1) and (2) it must be: $$\psi' = e^{i a} \psi\quad \mbox{for some constant $a\in R$.}$$
Here physics stops. I will use only mathematics henceforth.
(8) In view of (3) one can equivalently re-write the identity above as
$$\psi(y,x) = e^{ia}\psi(x,y) \quad \mbox{almost everywhere for $(x,y)\in R^2\times R^2$}\quad [1]\:.$$
(9) Since $(x,y)$ in [1] is every pair of points up to a zero-measure set, I am allowed to change their names obtaining
$$\psi(x,y) = e^{ia}\psi(y,x) \quad \mbox{almost everywhere for $(x,y)\in R^2\times R^2$}\quad [2]$$
(Notice the zero measure set where the identity fails remains a zero measure set under the reflexion $(x,y) \mapsto (y,x)$, since it is an isometry of $R^4$ and Lebesgues' measure is invariant under isometries.)
(10) Since, again, [2] holds almost everywhere for every pair $(x,y)$, I am allowed to use again [1] in the right-hand side of [2] obtaining:
$$\psi(x,y) = e^{ia}e^{ia}\psi(x,y) \quad \mbox{almost everywhere for $(x,y)\in R^2\times R^2$}\:.$$
(This certainly holds true outside the union of the zero measure set $A$ where [1] fails and that obtained by reflexion $(x,y) \mapsto (y,x)$ of $A$ itself.)
(11) Conclusion:
$$[e^{2ia} -1] \psi(x,y)=0 \qquad\mbox{almost everywhere for $(x,y)\in R^2\times R^2$}\quad [3]$$
Since $||\psi|| \neq 0$, $\psi$ cannot vanish everywhere on $R^2\times R^2$. If $\psi(x_0,y_0) \neq 0$, $[e^{2ia} -1] \psi(x_0,y_0)=0$ implies $e^{2ia} =1 $ and so:
$$e^{ia} = \pm 1\:.$$
And thus, apparently, anyons are not permitted.
Where is the mistake?
ADDED REMARK. (10) is a completely mathematical result. Here is another way to obtain it. (8) can be written down as $\psi(a,b) = e^{ic} \psi(b,a)$ for some fixed $c \in R$ and all $(a,b) \in R^2 \times R^2$ (I disregard the issue of negligible sets). Choosing first $(a,b)=(x,y)$ and then $(a,b)=(y,x)$ we obtain resp. $\psi(x,y) = e^{ic} \psi(y,x)$ and $\psi(y,x) = e^{ic} \psi(x,y)$. They immediately produce [3] $\psi(x,y) = e^{i2c} \psi(x,y)$.
So the physical argument (4)-(7) that we have permuted again the particles and thus a further new phase may appear does not apply here.
2nd ADDED REMARK. It is clear that as soon as one is allowed to write
$\psi(x,y) = \lambda \psi(y,x)$ for a constant $\lambda\in U(1)$ and all $(x,y) \in R^2\times R^2$
the game is over: $\lambda$ turns out to be $\pm 1$ and anyons are forbidden. This is just mathematics however. My guess for a way out is that the true configuration space is not $R^2\times R^2$ but some other space whose $R^2 \times R^2$ is the universal covering.
An idea (quite rough) could be the following. One should assume that particles are indistinguishable from scratch already defining the configuration space, that is something like $Q := R^2\times R^2/\sim$ where $(x',y')\sim (x,y)$ iff $x'=y$ and $y'=x$. Or perhaps subtracting the set $\{(z,z)\:|\: z \in R^2\}$ to $R^2\times R^2$ before taking the quotient to say that particles cannot stay at the same place. Assume the former case for the sake of simplicity. There is a (double?) covering map $\pi : R^2 \times R^2 \to Q$. My guess is the following. If one defines wavefunctions $\Psi$ on $R^2 \times R^2$, he automatically defines many-valued wavefunctions on $Q$. I mean $\psi:= \Psi \circ \pi^{-1}$. The problem of many values physically does not matter if the difference of the two values (assuming the covering is a double one) is just a phase and this could be written, in view of the identification $\sim$ used to construct $Q$ out of $R^2 \times R^2$: $$\psi(x,y)= e^{ia}\psi(y,x)\:.$$ Notice that the identity cannot be interpreted literally because $(x,y)$ and $(y,x)$ are the same point in $Q$, so my trick for proving $e^{ia}=\pm 1$ cannot be implemented. The situation is similar to that of $QM$ on $S^1$ inducing many-valued wavefunctions form its universal covering $R$. In that case one writes $\psi(\theta)= e^{ia}\psi(\theta + 2\pi)$.
3rd ADDED REMARK I think I solved the problem I posted focusing on the model of a couple of anyons discussed on p.225 of this paper matwbn.icm.edu.pl/ksiazki/bcp/bcp42/bcp42116.pdf suggested by Trimok. The model is simply this one: $$\psi(x,y):= e^{i\alpha \theta(x,y)} \varphi(x,y)$$
where $\alpha \in R$ is a constant, $\varphi(x,y)= \varphi(y,x)$, $(x,y) \in R^2 \times R^2$ and $\theta(x,y)$ is the angle with respect to some fixed axis of the segment $xy$. One can pass to coordinates $(X,r)$, where $X$ describes the center of mass and $r:= y-x$. Swapping the particles means $r\to -r$. Without paying attention to mathematical details, one sees that, in fact: $$\psi(X,-r)= e^{i \alpha \pi} \psi(X,r)\quad \mbox{i.e.,}\quad \psi(x,y)= e^{i \alpha \pi} \psi(y,x)\quad (A)$$ for an anti clock wise rotation. (For clock wise rotations a sign $-$ appears in the phase, describing the other element of the braid group $Z_2$. Also notice that, for $\alpha \pi \neq 0, 2\pi$ the function vanishes for $r=0$, namely $x=y$, and this corresponds to the fact that we removed the set $C$ of coincidence points $x=y$ from the space of configurations.)
However a closer scrutiny shows that the situation is more complicated: The angle $\theta(r)$ is not well defined without fixing a reference axis where $\theta =0$. Afterwards one may assume, for instance, $\theta \in (0,2\pi)$, otherwise $\psi$ must be considered multi-valued. With the choice $\theta(r) \in (0,2\pi)$, (A) does not hold everywhere. Consider an anti clockwise rotation of $r$. If $\theta(r) \in (0,\pi)$ then (A) holds in the form $$\psi(X,-r)= e^{+ i \alpha \pi} \psi(X,r)\quad \mbox{i.e.,}\quad \psi(x,y)= e^{+ i \alpha \pi} \psi(y,x)\quad (A1)$$ but for $\theta(r) \in (\pi, 2\pi)$, and always for a anti clockwise rotation one finds $$\psi(X,-r)= e^{-i \alpha \pi} \psi(X,r)\quad \mbox{i.e.,}\quad \psi(x,y)= e^{- i \alpha \pi} \psi(y,x)\quad (A2)\:.$$ Different results arise with different conventions. In any cases it is evident that the phase due to the swap process is a function of $(x,y)$ (even if locally constant) and not a constant. This invalidate my "no-go proof", but also proves that the notion of anyon statistics is deeply different from the standard one based on the groups of permutations, where the phases due to the swap of particles is constant in $(x,y)$. As a consequence the swapped state is different from the initial one, differently form what happens for bosons or fermions and against the idea that anyons are indistinguishable particles. [Notice also that, in the considered model, swapping the initial pair of bosons means $\varphi(x,y) \to \varphi(y,x)= \varphi(x,y)$ that is $\psi(x,y)\to \psi(x,y)$. That is, swapping anyons does not mean swapping the associated bosons, and it is correct, as it is another physical operation on different physical subjects.]
Alternatively one may think of the anyon wavefunction $\psi(x,y)$ as a multi-valued one, again differently from what I assumed in my "no-go proof" and differently from the standard assumptions in QM. This produces a truly constant phase in (A). However, it is not clear to me if, with this interpretation the swapped state of anyons is the same as the initial one, since I never seriously considered things like (if any) Hilbert spaces of multi-valued functions and I do not understand what happens to the ray-representation of states. This picture is physically convenient, however, since it leads to a tenable interpretation of (A) and the action of the braid group turns out to be explicit and natural.
Actually a last possibility appears. One could deal with (standard complex valued) wavefunctions defined on $(R^2 \times R^2 - C)/\sim$ as we know (see above, $C$ is the set of pairs $(x,y)$ with $x=y$) and we define the swap operation in terms of phases only (so that my "no-go proof" cannot be applied and the transformations do not change the states):
$$\psi([(x,y)]) \to e^{g i\alpha \pi}\psi([(x,y)])$$
where $g \in Z_2$. This can be extended to many particles passing to the braid group of many particles. Maybe it is convenient mathematically but is not very physically expressive.
In the model discussed in the paper I mentioned, it is however evident that, up to an unitary transformation, the Hilbert space of the theory is nothing but a standard bosonic Hilbert space, since the considered wavefunctions are obtained from those of that space by means of a unitary map associated with a singular gauge transformation, and just that singularity gives rise to all the interesting structure! However, in the initial bosonic system the singularity was pre-existent: the magnetic field was a sum of Dirac's delta. I do not know if it makes sense to think of anyons independently from their dynamics. And I do not know if this result is general. I guess that moving the singularity form the statistics to the interaction and vice versa is just what happens in path integral formulation when moving the external phase to the internal action, see Tengen's answer.
This post imported from StackExchange Physics at 2014-04-11 15:20 (UCT), posted by SE-user V. Moretti
asked Dec 17, 2013 in Theoretical Physics by Valter Moretti (2,075 points) [ no revision ]
4 Answers
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The answer can probably be summarized in two points:
1) As discussed in a beautiful paper by Leinaas and Myrheim, the configuration space of a system of $N$ identical particles in $n$ dimensions is not $\mathbb{R}^{nN}$, but $\mathbb{R}^{nN}/S_N$ where we mod out the action of the permutation group $S_N$ (and also remove the singularities that happen when two particles occupy the same point).
2) Quantum mechanics is not about functions from the configuration space to the complex numbers, $\psi : \mathbb{R}^{nN}/S_N \to \mathbb{C}$, but, in modern terms, about sections in vector bundles on the configuration space. In classic terms, one would argue that the phase of the wave function is unobservable, and hence can be multi-valued, as discussed in Dirac's paper on magnetic monopoles.
It turns out that in $n=2$ dimensions, the configuration space of identical particles supports not just two, but many different interesting vector bundles, which corresponds to anyons.
answered Apr 16, 2016 by Greg Graviton (775 points) [ no revision ]
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In quantum mechanics on nonsimply connected spaces, we can use wave functions living on the universal covering space. In order to see that, we must remember that the symplectic potential of identical particles must include an extra piece to the free particle symplectic potential given by a flat connection represented by a magnetic vector potential. This also happens for example in the Aharonov-Bohm case. In the two dimensional identical particle case where anyons exist, this vector potential can only have the following form:
$$\mathbf{A} =\frac{\theta}{2 \pi} \frac{(-y, x )}{\left | x-y \right |^2}$$
where $ \mathbb{r_{12}} = (x,y)$ is the relative particle position.
As a form this vector potential is closed but not exact
Please see, for example, Wu's article: http://content.lib.utah.edu/utils/getfile/collection/uspace/id/4293/filename/3674.pdf
The solution of the Schrödinger equation with this type of magnetic potential is equivalent to taking following solution without the vector potential:
$$\Psi(z_1, z_2) = (z_1-z_2)^{\frac{\theta}{2\pi}}\psi(z_1, z_2)$$
where $z_{1,2}$ are complex coordinates on the plane and $\psi(z_1, z_2)$ single valued. (Please see Wu for the details).
Actually, the 2D case is actually conceptually simpler than in higher dimensions, where only bosons and fermions exist, because in higher dimensions the flat connection is torsion and has no vector potential representative in the de-Rham cohomology, but only in the Čech cohomology.
answered Apr 17, 2016 by David Bar Moshe (4,095 points) [ no revision ]
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The best way to answer the question "How are anyons possible" is to use the "dynamical" path integral formalism, rather than the "static" wave function formalism. The permutation group action on the wave function is "static" in the sense that only initial and final states are specified. It will be ambiguous if there are more than one non-equivalent ways to perform the exchange process, which is the key for the "possibility" of anyons.
Consider the amplitude from the initial state $|i\rangle$ to final state $|f\rangle$ in the path integral formalism $$\langle f|i\rangle = \int_\gamma \mathcal{D}x(t) e^{iS[x(t)]},$$ where $\gamma$ is a path from the initial configuration to the final configuration (they are set to the same). The confituration manifold will be discussed later. When two paths $\gamma_1$ and $\gamma_2$ are not equivalent to each other homotopically, we can assign a phase factor $e^{i\theta([\gamma])}$ to the path integral amplitude for each homotopy class $[\gamma]$: $$\langle f|i\rangle = \sum_{[\gamma]\in \pi_1(M)} e^{i\theta([\gamma])}\int_\gamma \mathcal{D}x(t) e^{iS[x(t)]},$$ where $\pi_1(M)$ denotes the fundamental group of the configuration space $M$. The phase factors $\{e^{i\theta([\gamma])}\}$ form an one dimensional representation of the group $\pi_1(M)$ because of the multiplication property of the propagator: $\langle f|i\rangle=\sum_m \langle f|m\rangle\langle m|i\rangle$. If we absorb the phase $\theta$ to the action $S$, it will be called a topological term as it depends only on the homotopy class.
The next task is to calculate the one dimensional representation of the fundamental group of the configuration space. For $N$ identical particles in $d$ space dimension, the configration space is $M=(\mathbb R^{Nd}\backslash D)/S_N$, where $D=\{(r_1,...,r_N)|\ \exists i\neq j,\ s.t. r_i=r_j\}$ is the space where two particles occupy the same point, and "$/S_N$" means the order of the particles is neglected.
(1) $d=1$. No exchange process can happen, and the notion of statistics is meaningless.
(2) $d=2$. $\pi_1(M)=B_N$ is the braiding group. The one dimension representation of $B_N$ is characterized by an angle $\theta$ which corresponds to the statistical angle of the Abelian anyon.
(3) $d\geq 3$. $\pi_1(M)=S_N$ is the permutation group. It means that, we only need to specify the order of particles in the initial and final states, to determine which homotopy class the path $\gamma$ belongs to. Therefore, only in this case, the wave function formalism can be used without ambiguity.
To describe the non-Abelian anyons, one only need to replace the phase factor $e^{i\theta}$ by an unitary matrix. The result is that non-Abelian anyons are determined by the higher dimension representations of the fundamental group of the configuration space.
answered Dec 19, 2013 by Tengen (105 points) [ no revision ]
May I inquire you how to prove the claim "$d=2\ \pi_1(M)=B_N$" and "$d=3\ \pi_1(M)=S_N$"
+ 1 like - 0 dislike
The point $(11)$ is not correct, by doing $2$ successive "exchanges", you may have a global phase factor, such as $\psi'(x,y) = e^{i\alpha}\psi(x,y)$. The two wave functions describe the same physical state. The correct considerations are topological, inside of considering a discrete operation, consider a continuous operation, so that it is equivalent to keep one particle fixed, and to make a rotation of the other particle of $2\pi$, the solution is in fact to look at the fundamental group ($1$st homotopy group) of $SO(d)$, where $d$ is the number of spatial dimensions (we suppose here only one time dimension). The structure and the dimension of the fundamental group (the number of different classes of paths) is correlated to the number of possible statistics. Of course, the fundamental group for $SO(d)$ with $d\geq 3$ is $\mathbb Z_2$, while the fundamental group of $SO(2)$ is $\mathbb Z$. This explains why we found different statistics (anyons) in $2$ spatial dimensions.
answered Dec 17, 2013 by Trimok (950 points) [ no revision ]
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7e4f7372300a3938 | Workshop on
Multiscale Problems in Quantum and Classical Mechanics, Averaging Techniques and Young Measures
List of Abstracts
U. Bandelow (Berlin):
Quantum-Classical Coupling in Multi Quantum Well Lasers
The presence of quantum wells in laser diodes introduces a new length scale, being in the order of the Fermi-wavelength of the carriers. This induces a mixed spectral density, where the carriers localizing within the quantum wells belong to the discrete spectrum and the free-roaming carriers belong to the continuous spectrum. Consequently, this divides the carriers into species which exhibit different properties. In particular, the latter species is viewed as a ''classical'' species, carrying classical transport on a large scale (some $\mu$m). By quantum well design the properties of the localized ''quantum'' species can be tuned and optimized for applications. Due to their localized nature the above ''quantum'' species cannot carry a current within a single-particle approach and therefore acts as a null-space for the transport. In consequence, their occupation remains fixed forever - which is contradicting to physics.
Interaction as phonon-carrier and carrier-carrier scattering will change this simplified picture and causes kinetical processes for all the species. Among others, carriers can then migrate from one species to another. Above a certain time-scale such processes can be modeled in some approximation in terms of a dynamics which effectively counts the amount of carriers being captured by the quantum wells as well as the amount of carriers escaping from the quantum wells.
M. Baro (Berlin):
Kirkner-Lent-Poisson system
We analyse a Schrödinger-Poisson system on the interval $(a,b)\subset {\mathbf R}$, i.e. the electrostatic potential $\varphi$ is determined by Poisson's equation
-\frac{d}{dx}\epsilon(x)\frac{d}{dx} \varphi(x)=
q(C(x)+u^+(x)-u^-(x)),\qquad x \in(a,b).\end{displaymath}
The densities $u^\pm$ of electrons and holes are determined by so-called Kirkner-Lent families $\{H^\pm(z)\}_{z\in{\mathbf C}_+}$, i.e. a family of Schrödinger-type operators
with transparent boundary conditions
...rac{1}{2m_\pm(b)}f'(b)=i\sqrt{\frac{z-v^\pm_b}{2m_b^\pm}}f(b), \end{displaymath}
$v^\pm_a,v^\pm_b\in{\mathbf R}$, $m_a^\pm,m_b^\pm\gt$.
This model allows to consider current carrying scattering states. Hence a current coupling with a classical drift diffusion model is possible.
We will show that the Kirkner-Lent-Poisson system always admits a solution and give some a priori estimates.
This is a joint work with P. Degond, H.-Ch. Kaiser, H. Neidhardt, and J. Rehberg.
C. Carstensen (Wien, AT):
Finite Element Approximation of Averaged Quantities in Microstructures
This talk addresses a non-convex minimization problem (M) motivated by variational models (i) for phase transformations in crystal physics -advertised, e.g., by Fonseca, Ball, James; (ii) in optimal design tasks -advertised, e.g., by Kohn; (iii) in micromagnetics -advertised, e.g., by Tartar.
In contrast to convex energy densities, a non-convex density W, typically, leads to non-attainment of the infimal energy in (M) even though W is smooth and satisfies proper growth conditions. The simplest 1D example W(x)=(x2-1)2 due to Bolza serves as a basis to explain the lack of sequentially weak lower semicontinuity and the enforced high oscillations of infimizing sequences called microstructures.
The presentation illustrates those phenomena and their impact on a finite element simulation (Mh): Strong convergence of gradients is impossible! Essentially two relaxation results from state-of-the-art calculus of variations are available to cure the problem. Young measures and (quasi-) convexification are the mathematical key concepts behind the proposed numerical alternatives (Gh) and (Qh). Recent numerical algorithms and convergence results form the center of the presentation. Mathematical difficulties encountered in enforced microstructures include non-quadratic growth conditions as well as a degenerated or even non-convex relaxed energy functional. Comments on remedies to gain strong convergence of gradients, open questions, and future developments conclude the talk.
M. Ehrhardt (Saarbrücken):
Discrete transparent boundary conditions for the Schrödinger equation: Fast calculation, approximation, and stability
This talk is concerned with transparent boundary conditions (TBCs) for the time-dependent Schrödinger equation in one and two dimensions. Discrete TBCs are introduced in the numerical simulations of whole space problems in order to reduce the computational domain to a finite region. Since the discrete TBC for the Schrödinger equation includes a convolution w.r.t. time with a weakly decaying kernel, its numerical evaluation becomes very costly for large-time simulations.
As a remedy we construct approximate TBCs with a kernel having the form of a finite sum-of-exponentials, which can be evaluated in a very efficient recursion. We prove stability of the resulting initial-boundary value scheme, give error estimates for the considered approximation of the boundary condition, and illustrate the efficiency of the proposed method on several examples.
G. Friesecke (Warwick, UK):
Long time dynamics of Fermi-Pasta-Ulam lattices: persistence of coherent modes and recurrence theorem for Fourier spectrum
In joint work with R.L.Pego (Maryland), we have recently obtained a recurrence theorem related to a fundamental question raised by Fermi, Pasta and Ulam in 1947.
F, P and U wondered numerically about the long time behaviour of the Fourier spectrum of a 1D nonlinear atomic chain with Hamiltonian
H = \sum_{j\in{\mathbb Z}} \Bigl(\frac{p_j^2}{2} + V(q_{j+1}-q_j)\Bigr). \end{displaymath}
Statistical mechanics reasoning suggests that the nonlinearity would promote asymptotically thermalized distribution of energy among the Fourier modes. (For a linear lattice, i.e. a quadratic potential V, the spectral density would be time-independent). But the numerical experiments showed strong recurrence effects, which have remained poorly understood.
Our theorem says that given any energy surface H=E with sufficiently low energy E, and given any $\varepsilon\gt$, there exists an open set of initial data whose Fourier spectral density is $\varepsilon$-recurrent. This means that after each integer multiple of some recurrence time T, the spectral density has distance at most $\epsilon$ (in the L1 norm) from the initial density.
Our construction of recurrent regions is related to the presence of solitary wave modes in the lattice. (Much of the literature has instead been viewing the FPU recurrences as KAM type effects. But it seems it has not been possible to turn this idea into a theorem. I will discuss an explanation suggested by E.Wayne.)
K. Gelfert (Dresden):
Langevin model for the slow motion in a deterministic multi-scale system
In this talk we intend to establish a novel approach for the modelling and effective simulation of systems with time-scale separation. For a class of deterministic dynamical systems where the state space variables can be divided into a group of fast and a group of slow variables it will be discussed how a fast chaotic motion can be modelled by suitable stochastic forces. Here, projection techniques, which are well known from non-equilibrium statistical mechanics, are employed to eliminate the fast motion. After an approximation step a Fokker-Planck equation governing the evolution of the density of the slow variables is derived. In this equation the diffusion term is given in terms of correlation properties of the fast motion while the drift term consists of an adiabatic average of the slow motion plus a renormalization by chaotic fluctuations.
J. Giannoulis (Stuttgart):
Young-measure solutions to a generalized Benjamin-Bona-Mahony equation
We are interested in the evolution of macroscopic properties of microscopic structure.
Starting from the viewpoint that microstructure can be modelled by spatial highly oscillatory initial data, in order to obtain its macroscopic information one is tempted to use the limit achieved as the (micro-)period tends to zero, i.e, to use Young measures. Indeed, Young measures maintain the macroscopic properties of oscillations (the simplest examples being the main value, the amplitude etc.), while they fade out the microscopic information.
Since we want to describe the evolution of the macroscopic features of our data, we have to derive a macroscopic evolution equation for Young measures from the given (microscopic) equation, the latter describing the evolution of the oscillations.
This programme will be outlined for the case of a generalization of the BBM equation (an ``alternative'' to the KdV equation) showing exemplarily the (also numerical) advantages of the application of such an approach.
S. Goedecker (Grenoble, FR):
Wavelets, Plane Waves and Multigrid Methods in the Context of Poisson's Equation
Wavelets, plane waves and multigrid methods share many central ideas. An unified point of view can help us in the construction of more efficient algorithms. In particular it will be shown how Poisson's equation discretized in a wavelet basis can be solved efficiently in a multigrid way and how multigrid algorithms can be improved by incorporating wavelet concepts.
M. Herrmann (Berlin):
Micro-macro transitions in the atomic chain
In this talk we consider micro-macro transitions in an atomic chain whose microscopic dynamics is described by a large system of Newton equations with a nonlinear but convex interaction potential. Starting form the hyperbolic scaling of space and time we discuss the thermodynamical (macroscopic) describtion of the chain for a suitable class of microscopic initial data. In particular, we identify a closure principle that describes the statistical distribution of the atoms for any macroscopic space time point. Using this closure principle we can describe the macroscopic behaviour of the chain by a system of hyperbolic pde's, namely the well known Euler system of thermodynamics. Finally, we study Riemann problems for the atomic chain.
A. Jüngel (Konstanz):
Macroscopic semiconductor modeling and simulation
The ongoing miniaturization of semiconductor devices reached nowadays a length scale at which quantum effects play a dominant role. Therefore, standard models like the classical drift-diffusion equations are physically inaccurate and have to be replaced by models which incorporate the relevant quantum effects. The state of the art in quantum semiconductor modeling ranges from microscopic models, like Schrödinger or Wigner equations, to macroscopic equations, like quantum hydrodynamic models.
This talk gives an overview of various macroscopic quantum models and presents analytical and numerical results on some of these models, in particular the (1) quantum drift-diffusion, (2) quantum hydrodynamic, and (3) viscous quantum hydrodynamic equations.
K. Kirchgässner (Stuttgart):
Travelling waves in a chain of coupled nonlinear oscillators
In a chain of nonlinear oscillators, linearly coupled to their nearest neighbors, all travelling waves of small amplitude are found as solutions of finite dimensional reversible dynamical systems. The coupling constant and the inverse wave speed form the parameter space. The ground state consists of a one-parameter family of periodic waves. It is realized in a certain parameter region containing all cases of light coupling. Beyond the border of this region the complexity of wave-forms increases via a succesion of bifurcations. An appropriate fomulation of this problem will be given, and the basic facts about the reduction to finite dimensionional systems will be indicated. We show the existence of the ground state and discuss the first bifurcation via normal form arguments. Furthermore we show the existence of nanopterons, i.e. localized waves with noncancelling periodic tails at infinity which are exponentially small in the bifurcation parameter. (Joint work with G.Iooss, CMP 211(2000)).
C. Lasser (München):
Molecular dynamics and energy level crossings
The Born-Oppenheimer approximation to quantum-mechanical molecular dynamics locally breaks down in the presence of energy level crossings. There are various types of energy level crossings, each of it associated with an own Landau-Zener formula. An asymptotic analysis of a Schrödinger equation with level crossing can be carried out by taking the solution's Wigner transform and passing to the semiclassical limit; an approach, which has been introduced to this problem class by C. Fermanian and P. Gérard. We will adopt this point of view and discuss some simple model problems with avoided crossing, codimension one and codimension two crossing.
H. Luo (Kassel):
Wavelet approximation of correlated wavefunctions
We suggest an alternative approach for the local treatment of electron corelation based on numerical methods from multiscale analysis. By this we are aming to achieve a better description of the various length- and energy-scales inherently connected with different types of electron correlations. For the first step, we take the local ansatz which corresponds to a coupled electron pair approximation (CEPA), and approximate the correlation part by means of hyperbolic wavelets. We perform diagramatic analysis for the calculation of the matrix elements.
K. Matthies (Berlin):
Atomic-scale localization of high-energy solitary waves on lattices
One-dimensional monatomic lattices with Hamiltonian $H\!=\!\sum_{n\in\mathbb Z}(\frac{1}{2}p_n^2+V(q_{n+1}-q_n))$ are known to carry localized travelling wave solutions, for generic nonlinear potentials V. In this talk we derive the asymptotic profile of these waves in the high-energy limit $H\to\infty$, for Lennard-Jones type interactions. The limit profile is proved to be a universal, highly discrete, piecewise linear wave concentrated on a single atomic spacing. This shows that dispersionless energy transport in these systems is not confined to the long-wave regime. (Joint work with G. Friesecke, Physica D, to appear)
F. Mehats (Tolouse, FR):
Analysis of a Drift-Diffusion-Schrödinger-Poisson model
(joint work with N. Ben Abdallah and N. Vauchelet)
We present the analysis of a coupled quantum-classical system, modeling the transport of a quasi bidimensional electron gas confined in a nanostructure. The coupling occurs in the momentum variable: the electrons are like point particles in the directions x parallel to the gas (classical transport) while they behave like waves in the transversal direction z (quantum description).
The transport of the gas is described by a 2D Drift-Diffusion equation, governing the evolution of a surfacic density ns. The originality of this system is that the parameters of this equation keep a trace of the quantum confinement in the transversal direction. Indeed, the effective potential which gives the drift current is calculated with the subband model through the resolution of an adiabatic Schrödinger-Poisson system. It takes into account the selfconsistent electric potential generated by the electrons and the quantification of the energy in the z variable. The system can be obtained, at least formally, as the diffusion limit of a Boltzmann-Schrödinger-Poisson system.
Let $\omega\subset {\rm I\hspace{-0.50ex}R}^2$ be a regular and bounded domain and let $\Omega =\omega \times (0,1)$. The spatial variables are $(x,z)\in \Omega$.The model studied here is the following coupled system:
\partial_t n_s -{\rm div}_x \;(\nabla_x n_s +n_s \,\nabla_x V_s)=0,\end{displaymath} (1)
...tyle\int_0^1\chi_p\,\chi_q\,dz=\delta_{pq}\,,\end{array}\right.\end{displaymath} (2)
-\Delta_{x,z} V=n,\end{displaymath} (3)
where the unknowns are the surfacic density ns(t,x), the eigen-energies $\mbox{\Large $\epsilon$}_p (t,x)$, the eigenfunctions $\chi_p(t,x;z)$,and the electrostatic potential V(t,x,z). These equations are coupled through the density n and the effective potential Vs, which are defined by
n=n_s \sum_p \frac{\displaystyle e^{-\mbox{\small $\epsilon$...
V_s=-\log \sum_{p}e^{-\mbox{\small $\epsilon$}_p}.\end{displaymath} (4)
This system (1)-(4) is completed with an initial condition and conservative boundary conditions. Using a fixed-point procedure, we have proved the existence of a unique solution to this nonlinear system.
J. M. Melenk (Leipzig):
Generalized FEM and two-scale regularity for homogenization problems
We present regularity results for the solutions of a class of elliptic boundary value problems with rapidly oscillating, periodic coefficients. At the heart is a detailed analysis of the so-called unit-cell problem. Applications of our results include mesh design principles for the generalized FEM (gFEM). The gFEM introduced by A.-M. Matache and C. Schwab is a projection method with problem-adapted ansatz space that can lead to robust convergence, i.e., the convergence is independent of the coefficients' period. In this talk, we will review the gFEM and elaborate the bearing of our regularity results on it.
A. Mielke (Stuttgart):
Macroscopic dispersive energy transport in harmonic lattices
We study oscillations of an infinite periodic lattice in one or several space dimensions. We consider the atomic distance 1/n as the small microscopic length scale and the aim is to derive macroscopic evolution laws. The simplest model displaying already most features is the one-dimensional chain
\ddot x_k=\sum_{0<\vert m\vert<M} A_{\vert m\vert} (x_{k+m}{-}x_k),\qquad k\in \mathbb Z.\end{displaymath}
From the microscopic dispersion relation $\omega^2(\theta)=-\sum_1^M 2A_m(1{-}\cos(m\theta))$we have the exact solutions $x_k(t)=\exp(\mathrm
i(\omega(\theta)t{+}\theta k))$. Using Fourier transform yields the general solutions.
Defining the macroscopic functions $X^{(n)}(\tau,y)=\sum_{\mathbb Z}
x_k \mathrm{sinc}(z{-}k)$, with sinc$\!\;t=\sin(\pi t)/(\pi t)$,in the macroscopic variables $\tau=t/n$ and y=z/n we obtain via weak convergence arguments that the associated macroscopic limit equation is the linear wave equation
\partial_\tau^2 X=\omega'(0)^2 \partial_y^2X.\end{displaymath}
The microscopic energy distribution distribution is a quadratic functional of the vleocities and the strains. It is shown that the Wigner transform can be used to describe its weak limit. The associated limit for $n\to \infty$ is a measure $\mu(\tau)\in \mathcal M(\mathbb R_y\times \mathbb S^1_\theta)$which has to satisfy the transport equation
\partial_\tau^2 \mu = \omega'(\theta)^2 \partial_y \mu\end{displaymath}
in the sense of distributions.
H. Neidhardt (Berlin):
Current coupling of drift -diffusion and Schrödinger-Poisson systems
(joint work with M.Baro, H.-Chr.Kaiser,and J.Rehberg)
Let $\hat{{\Omega}} = [a_0,b_0]$ be an closed interval on the real axis which contains the closed interval ${\Omega}= [a,b] \subset \hat{{\Omega}}$. On the semi-intervals $\tilde{{\Omega}}_a = [a_0,a)$ and $\tilde{{\Omega}}_b = (b,b_0]$, which are called classical zones, one-dimensional drift-diffusion models without generation and recombination are considered while on the closed interval ${\Omega}$, which is called the quantum zone, a description by dissipative Schrödinger-Poisson systems with given densities matrices (steady states) is assumed. Both systems are coupled to a hybrid model as follows:
The boundary coefficients of the dissipative Schrödinger-Poisson systems are determined by the quasi-Fermi levels of the drift-diffusion models in accordance with dissipative Kirkner-Lent models, see talk of M. Baro.
The current densities are constant for the coupled system, in particular, current densities for the classical and quantum zones are equal (current coupling).
It is shown that under these assumptions the hybrid model has always a solution which is in general not unique.
R. Schneider (Chemnitz):
Wavelet subspace splitting
Wavelet compression is a favourable tool for an efficient approximation of singularities. Moreover the hierarchical description admits a hyperbolic cross or sparse grid approximation, reducing the complexity of multidimensional problems. Both can be used for the numerical ab initio solution of many particle Schrödinger equation. Even Kato type singularies are resolved adaptively. However these results are limited and not satisfactory. At this point it becomes necessary to go beyond classical wavelet analysis. The complexity is too high, because the information encoded by the wavelet bases is still redundant. The idea to overcome this deficiency is to split the wavelet spaces itself into several subspaces where only a low dimensional subspace contains the essential information. This may be viewed as a better and more appropriate localization in the frequency domain. In fact, we are constructiong wavelet subspaces with higher order vanishing moments by linear combination of wavelet functions. The resulting basis functions are builting again Riesz bases. We show how to apply this construction to resolve Kato type singularities. At the end we apply the present concept to construct a stable wavelet methods which is exponetially convergent for analytic datas. By the introduction of new basis functions we have enriched our libary of basis functions. A best N-term approximation becomes now a problem of finding best bases. The strategy to find this bases depends on which a priory information is available.
S. Teufel (München):
Effective dynamics in a periodic potential: Peierls substitution and beyond
The dynamics of electrons in a crystal can be described in good approximation by the Schrödinger equation for a single particle with a periodic potential. Typically external magnetic and electric fields are weak compared to the periodic field and thus the external potentials have a slow variation on the scale set by the lattice. As a consequence of this separation of scales the macroscopic dynamics of the electron are governed by an effective Hamiltonian obtained through the famous ``Peierls substitution''. We present a perturbative scheme which allows not only for a rigorous justification for ``Peierls substitution'', but also yields corrections to arbitrary order in the small parameter describing the separation of scales.
F. Theil (Warwick, UK):
Surface energies in lattice models
Probably the best known finite scale effects in atomistic systems are surface energies. In a simplistic, two-dimensional model the total energy E associated to the atom positions $\{y(x)\}_{x \in
{\mathcal L}\subset \mathbb Z^2}$ is given by the pair sum
E(\{y\}) = \sum_{x,x' \in {\mathcal L}} V_{x-x'}(\vert y(x)-y(x')\vert)\end{displaymath}
where the potential Vx-x' is harmonic if $\vert x-x'\vert \in
[1,\sqrt{2}]$ and 0 else. This corresponds to a system of mass points coupled via linearly elastic springs. For ${\mathcal L}_R = R \Omega \cap \mathbb Z^2$ where R>0 is a scaling parameter and $\Omega \subset \mathbb R^2$ is a a continuum domain the minimum energy scales in the following way
\min_{\{y\}} E(\{y\}) = E_0 R^2 + E_1 R + o(R), \quad R \to \infty.\end{displaymath}
The surface term
E_1=\int_{\partial \Omega} \sigma_{\nu(x)} \, \mathrm{d}
\mathcal H^1(x),\end{displaymath}
where $\nu(x)$ is the surface normal at x, encodes information about the relaxation pattern of the atoms close to the surface. We show that in many cases $\sigma_\nu$ can be found by solving 1-periodic cell problems. The result can be described as a Cauchy-Born rule for surfaces.
When the surface relaxation is ignored, $\sigma_\nu$ depends smoothly on the normal vector $\nu$ except on a closed set of measure 0. The nontrivial relaxation pattern of the atom positions towards the interiour changes the qualitative behaviour of $\sigma_\nu$dramatically. Numerical studies indicate that even for small surface relaxation $\sigma_\nu$ might be nowhere smooth in $\nu$. |
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A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram called an Argand diagram, representing the complex plane. "Re" is the real axis, "Im" is the imaginary axis, and i satisfies i2 = −1.
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a solution of the equation x2 = −1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, and b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of the scientific description of the natural world.[note 1][1]
Complex numbers allow solutions to certain equations that have no solutions in real numbers. For example, the equation
has no real solution, since the square of a real number cannot be negative. Complex numbers provide a solution to this problem. The idea is to extend the real numbers with an indeterminate i (sometimes called the imaginary unit) that is taken to satisfy the relation i2 = −1, so that solutions to equations like the preceding one can be found. In this case the solutions are −1 + 3i and −1 − 3i, as can be verified using the fact that i2 = −1:
According to the fundamental theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. In contrast, some polynomial equations with real coefficients have no solution in real numbers. The 16th century Italian mathematician Gerolamo Cardano is credited with introducing complex numbers in his attempts to find solutions to cubic equations.[2]
Formally, the complex number system can be defined as the algebraic extension of the ordinary real numbers by an imaginary number i.[3] This means that complex numbers can be added, subtracted, and multiplied, as polynomials in the variable i, with the rule i2 = −1 imposed. Furthermore, complex numbers can also be divided by nonzero complex numbers. Overall, the complex number system is a field.
Geometrically, complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point (a, b) in the complex plane. A complex number whose real part is zero is said to be purely imaginary; the points for these numbers lie on the vertical axis of the complex plane. A complex number whose imaginary part is zero can be viewed as a real number; its point lies on the horizontal axis of the complex plane. Complex numbers can also be represented in polar form, which associates each complex number with its distance from the origin (its magnitude) and with a particular angle known as the argument of this complex number.
The geometric identification of the complex numbers with the complex plane, which is a Euclidean plane (), makes their structure as a real 2-dimensional vector space evident. Real and imaginary parts of a complex number may be taken as components of a vector with respect to the canonical standard basis. The addition of complex numbers is thus immediately depicted as the usual component-wise addition of vectors. However, the complex numbers allow for a richer algebraic structure, comprising additional operations, that are not necessarily available in a vector space; e.g., the multiplication of two complex numbers always yields again a complex number, and should not be mistaken for the usual "products" involving vectors, like the scalar multiplication, the scalar product or other (sesqui)linear forms, available in many vector spaces; and the broadly exploited vector product exists only in an orientation-dependent form in three dimensions.
An illustration of the complex plane. The real part of a complex number z = x + iy is x, and its imaginary part is y.
Based on the concept of real numbers, a complex number is a number of the form a + bi, where a and b are real numbers and i is an indeterminate satisfying i2 = −1. For example, 2 + 3i is a complex number.[4]
This way, a complex number is defined as a polynomial with real coefficients in the single indeterminate i, for which the relation i2 + 1 = 0 is imposed. Based on this definition, complex numbers can be added and multiplied, using the addition and multiplication for polynomials. The relation i2 + 1 = 0 induces the equalities i4k = 1 , i4k+1 = i , i4k+2 = −1 , and i4k+3 = −i , which hold for all integers k; these allow the reduction of any polynomial that results from the addition and multiplication of complex numbers to a linear polynomial in i, again of the form a + bi with real coefficients a, b.
The real number a is called the real part of the complex number a + bi; the real number b is called its imaginary part. To emphasize, the imaginary part does not include a factor i; that is, the imaginary part is b, not bi.[5][6]
Formally, the complex numbers are defined as the quotient ring of the polynomial ring in the indeterminate i, by the ideal generated by the polynomial i2 + 1 (see below).[7]
A real number a can be regarded as a complex number a + 0i whose imaginary part is 0. A purely imaginary number bi is a complex number 0 + bi whose real part is zero. As with polynomials, it is common to write a for a + 0i and bi for 0 + bi. Moreover, when the imaginary part is negative, i.e., b = −|b| < 0, it is common to write a|b|i instead of a + (−|b|)i; for example, for b = −4, 3 − 4i can be written instead of 3 + (−4)i.
Since in polynomials with real coefficients the multiplication of the indeterminate i and a real is commutative, the polynomial a + bi may be written as a + ib. This is often expedient for imaginary parts denoted by expressions, e.g., when b is a radical.[8]
The real part of a complex number z is denoted by Re(z) or ℜ(z); the imaginary part of a complex number z is denoted by Im(z) or ℑ(z). For example,
The set of all complex numbers is denoted by (upright bold) or (blackboard bold).
In some disciplines, in particular electromagnetism and electrical engineering, j is used instead of i since i is frequently used to represent electric current.[9] In these cases complex numbers are written as a + bj or a + jb.
A complex number z, as a point (red) and its position vector (blue)
A complex number z can thus be identified with an ordered pair (Re(z), Im(z)) of real numbers, which in turn may be interpreted as coordinates of a point in a two-dimensional space. The most immediate space is the Euclidean plane with suitable coordinates, which is then called complex plane or Argand diagram,[10][11] named after Jean-Robert Argand. Another prominent space on which the coordinates may be projected is the two-dimensional surface of a sphere, which is then called Riemann sphere.
Cartesian complex planeEdit
The definition of the complex numbers involving two arbitrary real values immediately suggest the use of Cartesian coordinates in the complex plane. The horizontal (real) axis is generally used to display the real part with increasing values to the right and the imaginary part marks the vertical (imaginary) axis, increasing values upwards.
A charted number may be either viewed as the coordinatized point, or as a position vector from the origin to this point. The coordinate values of a complex number z are said to give its Cartesian, rectangular, or algebraic form.
Notably, the operations of addition and multiplication take on a very natural geometric character when complex numbers are viewed as position vectors: addition corresponds to vector addition, while multiplication (see below) corresponds to multiplying their magnitudes and adding the angles they make with the real axis. Viewed in this way the multiplication of a complex number by i corresponds to rotating the position vector counterclockwise by a quarter turn (90°) about the origin
Polar complex plane Edit
Argument φ and modulus r locate a point in the complex plane.
Modulus and argumentEdit
An alternative option for coordinates in the complex plane is the polar coordinate system that uses the distance of the point z from the origin (O), and the angle subtended between the positive real axis and the line segment Oz in a counterclockwise sense. This leads to the polar form of complex numbers.
The absolute value (or modulus or magnitude) of a complex number z = x + yi is[12]
If z is a real number (that is, if y = 0), then r = |x|. That is, the absolute value of a real number equals its absolute value as a complex number.
By Pythagoras' theorem, the absolute value of complex number is the distance to the origin of the point representing the complex number in the complex plane.
The argument of z (in many applications referred to as the "phase" φ) is the angle of the radius Oz with the positive real axis, and is written as . As with the modulus, the argument can be found from the rectangular form [13] by applying the inverse tangent to the quotient of imaginary-by-real parts. By using a half-angle identity a single branch of the arctan suffices to cover the range of the arg-function, (−π, π], and avoids a more subtle case-by-case analysis
Normally, as given above, the principal value in the interval (−π, π] is chosen. Values in the range [0, 2π) are obtained by adding if the value is negative. The value of φ is expressed in radians in this article. It can increase by any integer multiple of and still give the same angle, viewed as subtended by the rays of the positive real axis and from the origin through z. Hence, the arg function is sometimes considered as multivalued. The polar angle for the complex number 0 is indeterminate, but arbitrary choice of the angle 0 is common.
The value of φ equals the result of atan2:
Together, r and φ give another way of representing complex numbers, the polar form, as the combination of modulus and argument fully specify the position of a point on the plane. Recovering the original rectangular co-ordinates from the polar form is done by the formula called trigonometric form
Using Euler's formula this can be written as
Using the cis function, this is sometimes abbreviated to
In angle notation, often used in electronics to represent a phasor with amplitude r and phase φ, it is written as[14]
Complex graphsEdit
A color wheel graph of the expression (z2 − 1)(z − 2 − i)2/z2 + 2 + 2i
When visualizing complex functions, both a complex input and output are needed. Because each complex number is represented in two dimensions, visually graphing a complex function would require the perception of a four dimensional space, which is possible only in projections. Because of this, other ways of visualizing complex functions have been designed.
In Domain coloring the output dimensions are represented by color and brightness, respectively. Each point in the complex plane as domain is ornated, typically with color representing the argument of the complex number, and brightness representing the magnitude. Dark spots mark moduli near zero, brighter spots are farther away from the origin, the gradation may be discontinuous, but is assumed as monotonous. The colors often vary in steps of π/3 for 0 to 2π from red, yellow, green, cyan, blue, to magenta. These plots are called color wheel graphs. This provides a simple way to visualize the functions without losing information. The picture shows zeros for ±1, (2+i) and poles at ±−2−2i.
Riemann surfaces are another way to visualize complex functions.[further explanation needed] Riemann surfaces can be thought of as deformations of the complex plane; while the horizontal axes represent the real and imaginary inputs, the single vertical axis only represents either the real or imaginary output. However, Riemann surfaces are built in such a way that rotating them 180 degrees shows the imaginary output, and vice versa. Unlike domain coloring, Riemann surfaces can represent multivalued functions like .
The solution in radicals (without trigonometric functions) of a general cubic equation contains the square roots of negative numbers when all three roots are real numbers, a situation that cannot be rectified by factoring aided by the rational root test if the cubic is irreducible (the so-called casus irreducibilis). This conundrum led Italian mathematician Gerolamo Cardano to conceive of complex numbers in around 1545,[15] though his understanding was rudimentary.
Work on the problem of general polynomials ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, a solution exists to every polynomial equation of degree one or higher. Complex numbers thus form an algebraically closed field, where any polynomial equation has a root.
Many mathematicians contributed to the development of complex numbers. The rules for addition, subtraction, multiplication, and root extraction of complex numbers were developed by the Italian mathematician Rafael Bombelli.[16] A more abstract formalism for the complex numbers was further developed by the Irish mathematician William Rowan Hamilton, who extended this abstraction to the theory of quaternions.
The earliest fleeting reference to square roots of negative numbers can perhaps be said to occur in the work of the Greek mathematician Hero of Alexandria in the 1st century AD, where in his Stereometrica he considers, apparently in error, the volume of an impossible frustum of a pyramid to arrive at the term in his calculations, although negative quantities were not conceived of in Hellenistic mathematics and Hero merely replaced it by its positive ( ).[17]
The impetus to study complex numbers as a topic in itself first arose in the 16th century when algebraic solutions for the roots of cubic and quartic polynomials were discovered by Italian mathematicians (see Niccolò Fontana Tartaglia, Gerolamo Cardano). It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers. As an example, Tartaglia's formula for a cubic equation of the form [note 2] gives the solution to the equation x3 = x as
At first glance this looks like nonsense. However formal calculations with complex numbers show that the equation z3 = i has solutions i, and . Substituting these in turn for in Tartaglia's cubic formula and simplifying, one gets 0, 1 and −1 as the solutions of x3x = 0. Of course this particular equation can be solved at sight but it does illustrate that when general formulas are used to solve cubic equations with real roots then, as later mathematicians showed rigorously, the use of complex numbers is unavoidable. Rafael Bombelli was the first to explicitly address these seemingly paradoxical solutions of cubic equations and developed the rules for complex arithmetic trying to resolve these issues.
The term "imaginary" for these quantities was coined by René Descartes in 1637, although he was at pains to stress their imaginary nature[18]
[...] sometimes only imaginary, that is one can imagine as many as I said in each equation, but sometimes there exists no quantity that matches that which we imagine.
([...] quelquefois seulement imaginaires c'est-à-dire que l'on peut toujours en imaginer autant que j'ai dit en chaque équation, mais qu'il n'y a quelquefois aucune quantité qui corresponde à celle qu'on imagine.)
A further source of confusion was that the equation seemed to be capriciously inconsistent with the algebraic identity , which is valid for non-negative real numbers a and b, and which was also used in complex number calculations with one of a, b positive and the other negative. The incorrect use of this identity (and the related identity ) in the case when both a and b are negative even bedeviled Euler. This difficulty eventually led to the convention of using the special symbol i in place of −1 to guard against this mistake.[citation needed] Even so, Euler considered it natural to introduce students to complex numbers much earlier than we do today. In his elementary algebra text book, Elements of Algebra, he introduces these numbers almost at once and then uses them in a natural way throughout.
In the 18th century complex numbers gained wider use, as it was noticed that formal manipulation of complex expressions could be used to simplify calculations involving trigonometric functions. For instance, in 1730 Abraham de Moivre noted that the complicated identities relating trigonometric functions of an integer multiple of an angle to powers of trigonometric functions of that angle could be simply re-expressed by the following well-known formula which bears his name, de Moivre's formula:
In 1748 Leonhard Euler went further and obtained Euler's formula of complex analysis:
by formally manipulating complex power series and observed that this formula could be used to reduce any trigonometric identity to much simpler exponential identities.
The idea of a complex number as a point in the complex plane (above) was first described by Caspar Wessel in 1799, although it had been anticipated as early as 1685 in Wallis's De Algebra tractatus.
Wessel's memoir appeared in the Proceedings of the Copenhagen Academy but went largely unnoticed. In 1806 Jean-Robert Argand independently issued a pamphlet on complex numbers and provided a rigorous proof of the fundamental theorem of algebra. Carl Friedrich Gauss had earlier published an essentially topological proof of the theorem in 1797 but expressed his doubts at the time about "the true metaphysics of the square root of −1". It was not until 1831 that he overcame these doubts and published his treatise on complex numbers as points in the plane, largely establishing modern notation and terminology. In the beginning of the 19th century, other mathematicians discovered independently the geometrical representation of the complex numbers: Buée, Mourey, Warren, Français and his brother, Bellavitis.[19]
The English mathematician G.H. Hardy remarked that Gauss was the first mathematician to use complex numbers in 'a really confident and scientific way' although mathematicians such as Niels Henrik Abel and Carl Gustav Jacob Jacobi were necessarily using them routinely before Gauss published his 1831 treatise.[20]
“If this subject has hitherto been considered from the wrong viewpoint and thus enveloped in mystery and surrounded by darkness, it is largely an unsuitable terminology which should be blamed. Had +1, -1 and √−1, instead of being called positive, negative and imaginary (or worse still, impossible) unity, been given the names say,of direct, inverse and lateral unity, there would hardly have been any scope for such obscurity.” - Gauss[21]
Augustin Louis Cauchy and Bernhard Riemann together brought the fundamental ideas of complex analysis to a high state of completion, commencing around 1825 in Cauchy's case.
The common terms used in the theory are chiefly due to the founders. Argand called the direction factor, and the modulus; Cauchy (1828) called the reduced form (l'expression réduite) and apparently introduced the term argument; Gauss used i for , introduced the term complex number for a + bi, and called a2 + b2 the norm. The expression direction coefficient, often used for , is due to Hankel (1867), and absolute value, for modulus, is due to Weierstrass.
Later classical writers on the general theory include Richard Dedekind, Otto Hölder, Felix Klein, Henri Poincaré, Hermann Schwarz, Karl Weierstrass and many others.
Relations and operationsEdit
Two complex numbers are equal if and only if both their real and imaginary parts are equal. That is, complex numbers and are equal if and only if and . If the complex numbers are written in polar form, they are equal if and only if they have the same argument and the same magnitude.
Since complex numbers are naturally thought of as existing on a two-dimensional plane, there is no natural linear ordering on the set of complex numbers. In fact, there is no linear ordering on the complex numbers that is compatible with addition and multiplication – the complex numbers cannot have the structure of an ordered field. This is because any square in an ordered field is at least 0, but i2 = −1.
Geometric representation of z and its conjugate in the complex plane
The complex conjugate of the complex number z = x + yi is given by xyi. It is denoted by either or z*.[22] This unary operation on complex numbers cannot be expressed by applying only their basic operations addition, subtraction, multiplication and division.
Geometrically, is the "reflection" of z about the real axis. Conjugating twice gives the original complex number
which makes this operation an involution. The reflection leaves both the real part and the magnitude of unchanged, that is
The imaginary part and the argument of a complex number change their sign under conjugation
For details on argument and magnitude, see the section on Polar form.
The product of a complex number and its conjugate is always a positive real number and equals the square of the magnitude of each:
This property can be used to convert a fraction with a complex denominator to an equivalent fraction with a real denominator by expanding both numerator and denominator of the fraction by the conjugate of the given denominator. This process is sometimes called "rationalization" of the denominator (although the denominator in the final expression might be an irrational real number), because it resembles the method to remove roots from simple expressions in a denominator.
The real and imaginary parts of a complex number z can be extracted using the conjugation:
Moreover, a complex number is real if and only if it equals its own conjugate.
Conjugation distributes over the basic complex arithmetic operations:
Conjugation is also employed in inversive geometry, a branch of geometry studying reflections more general than ones about a line. In the network analysis of electrical circuits, the complex conjugate is used in finding the equivalent impedance when the maximum power transfer theorem is looked for.
Addition and subtractionEdit
Addition of two complex numbers can be done geometrically by constructing a parallelogram.
Two complex numbers and are most easily added by separately adding their real and imaginary parts of the summands. That is to say:
Similarly, subtraction can be performed as
Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpretation: the sum of two complex numbers and , interpreted as points in the complex plane, is the point obtained by building a parallelogram from the three vertices , and the points of the arrows labeled and (provided that they are not on a line). Equivalently, calling these points respectively and the fourth point of the parallelogram the triangles and are congruent. A visualization of the subtraction can be achieved by considering addition of the negative subtrahend.
Since the real part, the imaginary part, and the indeterminate in a complex number are all considered as numbers in themselves, two complex numbers, given as and are multiplied under the rules of the distributive property, the commutative properties and the defining property in the following way
Reciprocal and divisionEdit
Using the conjugation, the reciprocal of a nonzero complex number z = x + yi can always be broken down to
since non-zero implies that is greater than zero.
This can be used to express a division of an arbitrary complex number by a non-zero complex number as
Multiplication and division in polar formEdit
Multiplication of 2 + i (blue triangle) and 3 + i (red triangle). The red triangle is rotated to match the vertex of the blue one and stretched by 5, the length of the hypotenuse of the blue triangle.
Formulas for multiplication, division and exponentiation are simpler in polar form than the corresponding formulas in Cartesian coordinates. Given two complex numbers z1 = r1(cos φ1 + i sin φ1) and z2 = r2(cos φ2 + i sin φ2), because of the trigonometric identities
we may derive
In other words, the absolute values are multiplied and the arguments are added to yield the polar form of the product. For example, multiplying by i corresponds to a quarter-turn counter-clockwise, which gives back i2 = −1. The picture at the right illustrates the multiplication of
Since the real and imaginary part of 5 + 5i are equal, the argument of that number is 45 degrees, or π/4 (in radian). On the other hand, it is also the sum of the angles at the origin of the red and blue triangles are arctan(1/3) and arctan(1/2), respectively. Thus, the formula
holds. As the arctan function can be approximated highly efficiently, formulas like this – known as Machin-like formulas – are used for high-precision approximations of π.
Similarly, division is given by
Square rootEdit
The square roots of a + bi (with b ≠ 0) are , where
where sgn is the signum function. This can be seen by squaring to obtain a + bi.[23][24] Here is called the modulus of a + bi, and the square root sign indicates the square root with non-negative real part, called the principal square root; also where [25]
Exponentiation Edit
Euler's formulaEdit
Euler's formula states that, for any real number x,
where e is the base of the natural logarithm. This can be proved through induction by observing that
and so on, and by considering the Taylor series expansions of eix, cos x and sin x:
Natural logarithmEdit
It follows from Euler's formula that, for any complex number z written in polar form,
where r is a non-negative real number, one possible value for the complex logarithm of z is
Because cosine and sine are periodic functions, other possible values may be obtained. For example, , so both and are two possible values for the natural logarithm of .
To deal with the existence of more than one possible value for a given input, the complex logarithm may be considered a multi-valued function, with
Alternatively, a branch cut can be used to define a single-valued "branch" of the complex logarithm.
Integer and fractional exponentsEdit
Visualisation of the square to sixth roots of a complex number z, in polar form re where φ = arg z and r = |z | – if z is real, φ = 0 or π. Principal roots are in black.
We may use the identity
to define complex exponentiation, which is likewise multi-valued:
When n is an integer, this simplifies to de Moivre's formula:
The nth roots of z are given by
for any integer k satisfying 0 ≤ kn − 1. Here nr is the usual (positive) nth root of the positive real number r. While the nth root of a positive real number r is chosen to be the positive real number c satisfying cn = r, there is no natural way of distinguishing one particular complex nth root of a complex number. Therefore, the nth root of z is considered as a multivalued function (in z), as opposed to a usual function f, for which f(z) is a uniquely defined number. Formulas such as
(which holds for positive real numbers), do in general not hold for complex numbers.
Field structureEdit
The set C of complex numbers is a field.[26] Briefly, this means that the following facts hold: first, any two complex numbers can be added and multiplied to yield another complex number. Second, for any complex number z, its additive inverse z is also a complex number; and third, every nonzero complex number has a reciprocal complex number. Moreover, these operations satisfy a number of laws, for example the law of commutativity of addition and multiplication for any two complex numbers z1 and z2:
These two laws and the other requirements on a field can be proven by the formulas given above, using the fact that the real numbers themselves form a field.
Unlike the reals, C is not an ordered field, that is to say, it is not possible to define a relation z1 < z2 that is compatible with the addition and multiplication. In fact, in any ordered field, the square of any element is necessarily positive, so i2 = −1 precludes the existence of an ordering on C.[27]
When the underlying field for a mathematical topic or construct is the field of complex numbers, the topic's name is usually modified to reflect that fact. For example: complex analysis, complex matrix, complex polynomial, and complex Lie algebra.
Solutions of polynomial equationsEdit
Given any complex numbers (called coefficients) a0, ..., an, the equation
has at least one complex solution z, provided that at least one of the higher coefficients a1, ..., an is nonzero.[28] This is the statement of the fundamental theorem of algebra, of Carl Friedrich Gauss and Jean le Rond d'Alembert. Because of this fact, C is called an algebraically closed field. This property does not hold for the field of rational numbers Q (the polynomial x2 − 2 does not have a rational root, since 2 is not a rational number) nor the real numbers R (the polynomial x2 + a does not have a real root for a > 0, since the square of x is positive for any real number x).
There are various proofs of this theorem, either by analytic methods such as Liouville's theorem, or topological ones such as the winding number, or a proof combining Galois theory and the fact that any real polynomial of odd degree has at least one real root.
Because of this fact, theorems that hold for any algebraically closed field, apply to C. For example, any non-empty complex square matrix has at least one (complex) eigenvalue.
Algebraic characterizationEdit
The field C has the following three properties: first, it has characteristic 0. This means that 1 + 1 + ⋯ + 1 ≠ 0 for any number of summands (all of which equal one). Second, its transcendence degree over Q, the prime field of C, is the cardinality of the continuum. Third, it is algebraically closed (see above). It can be shown that any field having these properties is isomorphic (as a field) to C. For example, the algebraic closure of Qp also satisfies these three properties, so these two fields are isomorphic (as fields, but not as topological fields).[29] Also, C is isomorphic to the field of complex Puiseux series. However, specifying an isomorphism requires the axiom of choice. Another consequence of this algebraic characterization is that C contains many proper subfields that are isomorphic to C.
Characterization as a topological fieldEdit
The preceding characterization of C describes only the algebraic aspects of C. That is to say, the properties of nearness and continuity, which matter in areas such as analysis and topology, are not dealt with. The following description of C as a topological field (that is, a field that is equipped with a topology, which allows the notion of convergence) does take into account the topological properties. C contains a subset P (namely the set of positive real numbers) of nonzero elements satisfying the following three conditions:
• P is closed under addition, multiplication and taking inverses.
• If x and y are distinct elements of P, then either xy or yx is in P.
Moreover, C has a nontrivial involutive automorphism xx* (namely the complex conjugation), such that x x* is in P for any nonzero x in C.
Any field F with these properties can be endowed with a topology by taking the sets B(x, p) = { y | p − (yx)(yx)* ∈ P } as a base, where x ranges over the field and p ranges over P. With this topology F is isomorphic as a topological field to C.
The only connected locally compact topological fields are R and C. This gives another characterization of C as a topological field, since C can be distinguished from R because the nonzero complex numbers are connected, while the nonzero real numbers are not.[30]
Formal constructionEdit
Construction as ordered pairsEdit
William Rowan Hamilton introduced the approach to define the set C of complex numbers[31] as the set R2 of ordered pairs (a, b) of real numbers, in which the following rules for addition and multiplication are imposed:[32]
It is then just a matter of notation to express (a, b) as a + bi.
Construction as a quotient fieldEdit
Though this low-level construction does accurately describe the structure of the complex numbers, the following equivalent definition reveals the algebraic nature of C more immediately. This characterization relies on the notion of fields and polynomials. A field is a set endowed with addition, subtraction, multiplication and division operations that behave as is familiar from, say, rational numbers. For example, the distributive law
must hold for any three elements x, y and z of a field. The set R of real numbers does form a field. A polynomial p(X) with real coefficients is an expression of the form
where the a0, ..., an are real numbers. The usual addition and multiplication of polynomials endows the set R[X] of all such polynomials with a ring structure. This ring is called the polynomial ring over the real numbers.
The set of complex numbers is defined as the quotient ring R[X]/(X 2 + 1).[33] This extension field contains two square roots of −1, namely (the cosets of) X and X, respectively. (The cosets of) 1 and X form a basis of R[X]/(X 2 + 1) as a real vector space, which means that each element of the extension field can be uniquely written as a linear combination in these two elements. Equivalently, elements of the extension field can be written as ordered pairs (a, b) of real numbers. The quotient ring is a field, because X2 + 1 is irreducible over R, so the ideal it generates is maximal.
The formulas for addition and multiplication in the ring R[X], modulo the relation X2 = −1, correspond to the formulas for addition and multiplication of complex numbers defined as ordered pairs. So the two definitions of the field C are isomorphic (as fields).
Accepting that C is algebraically closed, since it is an algebraic extension of R in this approach, C is therefore the algebraic closure of R.
Matrix representation of complex numbersEdit
Complex numbers a + bi can also be represented by 2 × 2 matrices that have the following form:
Here the entries a and b are real numbers. The sum and product of two such matrices is again of this form, and the sum and product of complex numbers corresponds to the sum and product of such matrices, the product being:
The geometric description of the multiplication of complex numbers can also be expressed in terms of rotation matrices by using this correspondence between complex numbers and such matrices. Moreover, the square of the absolute value of a complex number expressed as a matrix is equal to the determinant of that matrix:
The conjugate corresponds to the transpose of the matrix.
Though this representation of complex numbers with matrices is the most common, many other representations arise from matrices other than that square to the negative of the identity matrix. See the article on 2 × 2 real matrices for other representations of complex numbers.
Complex analysisEdit
Color wheel graph of sin(1/z). Black parts inside refer to numbers having large absolute values.
Complex exponential and related functionsEdit
The notions of convergent series and continuous functions in (real) analysis have natural analogs in complex analysis. A sequence of complex numbers is said to converge if and only if its real and imaginary parts do. This is equivalent to the (ε, δ)-definition of limits, where the absolute value of real numbers is replaced by the one of complex numbers. From a more abstract point of view, C, endowed with the metric
is a complete metric space, which notably includes the triangle inequality
for any two complex numbers z1 and z2.
Like in real analysis, this notion of convergence is used to construct a number of elementary functions: the exponential function exp(z), also written ez, is defined as the infinite series
The series defining the real trigonometric functions sine and cosine, as well as the hyperbolic functions sinh and cosh, also carry over to complex arguments without change. For the other trigonometric and hyperbolic functions, such as tangent, things are slightly more complicated, as the defining series do not converge for all complex values. Therefore, one must define them either in terms of sine, cosine and exponential, or, equivalently, by using the method of analytic continuation.
Euler's formula states:
for any real number φ, in particular
Unlike in the situation of real numbers, there is an infinitude of complex solutions z of the equation
for any complex number w ≠ 0. It can be shown that any such solution z – called complex logarithm of w – satisfies
where arg is the argument defined above, and ln the (real) natural logarithm. As arg is a multivalued function, unique only up to a multiple of 2π, log is also multivalued. The principal value of log is often taken by restricting the imaginary part to the interval (−π, π].
Complex exponentiation zω is defined as
and is multi-valued, except when is an integer. For ω = 1 / n, for some natural number n, this recovers the non-uniqueness of nth roots mentioned above.
Complex numbers, unlike real numbers, do not in general satisfy the unmodified power and logarithm identities, particularly when naïvely treated as single-valued functions; see failure of power and logarithm identities. For example, they do not satisfy
Both sides of the equation are multivalued by the definition of complex exponentiation given here, and the values on the left are a subset of those on the right.
Holomorphic functionsEdit
A function f : CC is called holomorphic if it satisfies the Cauchy–Riemann equations. For example, any R-linear map CC can be written in the form
with complex coefficients a and b. This map is holomorphic if and only if b = 0. The second summand is real-differentiable, but does not satisfy the Cauchy–Riemann equations.
Complex analysis shows some features not apparent in real analysis. For example, any two holomorphic functions f and g that agree on an arbitrarily small open subset of C necessarily agree everywhere. Meromorphic functions, functions that can locally be written as f(z)/(zz0)n with a holomorphic function f, still share some of the features of holomorphic functions. Other functions have essential singularities, such as sin(1/z) at z = 0.
Complex numbers have applications in many scientific areas, including signal processing, control theory, electromagnetism, fluid dynamics, quantum mechanics, cartography, and vibration analysis. Some of these applications are described below.
Three non-collinear points in the plane determine the shape of the triangle . Locating the points in the complex plane, this shape of a triangle may be expressed by complex arithmetic as
The shape of a triangle will remain the same, when the complex plane is transformed by translation or dilation (by an affine transformation), corresponding to the intuitive notion of shape, and describing similarity. Thus each triangle is in a similarity class of triangles with the same shape.[34]
Fractal geometryEdit
The Mandelbrot set with the real and imaginary axes labeled.
The Mandelbrot set is a popular example of a fractal formed on the complex plane. It is defined by plotting every location where iterating the sequence does not diverge when iterated infinitely. Similarly, Julia sets have the same rules, except where remains constant.
Every triangle has a unique Steiner inellipse – an ellipse inside the triangle and tangent to the midpoints of the three sides of the triangle. The foci of a triangle's Steiner inellipse can be found as follows, according to Marden's theorem:[35][36] Denote the triangle's vertices in the complex plane as a = xA + yAi, b = xB + yBi, and c = xC + yCi. Write the cubic equation , take its derivative, and equate the (quadratic) derivative to zero. Marden's Theorem says that the solutions of this equation are the complex numbers denoting the locations of the two foci of the Steiner inellipse.
Algebraic number theoryEdit
Construction of a regular pentagon using straightedge and compass.
As mentioned above, any nonconstant polynomial equation (in complex coefficients) has a solution in C. A fortiori, the same is true if the equation has rational coefficients. The roots of such equations are called algebraic numbers – they are a principal object of study in algebraic number theory. Compared to Q, the algebraic closure of Q, which also contains all algebraic numbers, C has the advantage of being easily understandable in geometric terms. In this way, algebraic methods can be used to study geometric questions and vice versa. With algebraic methods, more specifically applying the machinery of field theory to the number field containing roots of unity, it can be shown that it is not possible to construct a regular nonagon using only compass and straightedge – a purely geometric problem.
Another example are Gaussian integers, that is, numbers of the form x + iy, where x and y are integers, which can be used to classify sums of squares.
Analytic number theoryEdit
Analytic number theory studies numbers, often integers or rationals, by taking advantage of the fact that they can be regarded as complex numbers, in which analytic methods can be used. This is done by encoding number-theoretic information in complex-valued functions. For example, the Riemann zeta function ζ(s) is related to the distribution of prime numbers.
Improper integralsEdit
Dynamic equationsEdit
In differential equations, it is common to first find all complex roots r of the characteristic equation of a linear differential equation or equation system and then attempt to solve the system in terms of base functions of the form f(t) = ert. Likewise, in difference equations, the complex roots r of the characteristic equation of the difference equation system are used, to attempt to solve the system in terms of base functions of the form f(t) = rt.
In applied mathematicsEdit
Control theoryEdit
In control theory, systems are often transformed from the time domain to the frequency domain using the Laplace transform. The system's zeros and poles are then analyzed in the complex plane. The root locus, Nyquist plot, and Nichols plot techniques all make use of the complex plane.
In the root locus method, it is important whether zeros and poles are in the left or right half planes, i.e. have real part greater than or less than zero. If a linear, time-invariant (LTI) system has poles that are
Signal analysisEdit
where ω represents the angular frequency and the complex number A encodes the phase and amplitude as explained above.
This use is also extended into digital signal processing and digital image processing, which utilize digital versions of Fourier analysis (and wavelet analysis) to transmit, compress, restore, and otherwise process digital audio signals, still images, and video signals.
Another example, relevant to the two side bands of amplitude modulation of AM radio, is:
In physicsEdit
Electromagnetism and electrical engineeringEdit
In electrical engineering, the imaginary unit is denoted by j, to avoid confusion with I, which is generally in use to denote electric current, or, more particularly, i, which is generally in use to denote instantaneous electric current.
Since the voltage in an AC circuit is oscillating, it can be represented as
To obtain the measurable quantity, the real part is taken:
The complex-valued signal is called the analytic representation of the real-valued, measurable signal . [37]
Fluid dynamicsEdit
Quantum mechanicsEdit
The complex number field is intrinsic to the mathematical formulations of quantum mechanics, where complex Hilbert spaces provide the context for one such formulation that is convenient and perhaps most standard. The original foundation formulas of quantum mechanics – the Schrödinger equation and Heisenberg's matrix mechanics – make use of complex numbers.
In special and general relativity, some formulas for the metric on spacetime become simpler if one takes the time component of the spacetime continuum to be imaginary. (This approach is no longer standard in classical relativity, but is used in an essential way in quantum field theory.) Complex numbers are essential to spinors, which are a generalization of the tensors used in relativity.
Generalizations and related notionsEdit
The process of extending the field R of reals to C is known as the Cayley–Dickson construction. It can be carried further to higher dimensions, yielding the quaternions H and octonions O which (as a real vector space) are of dimension 4 and 8, respectively. In this context the complex numbers have been called the binarions.[38]
Just as by applying the construction to reals the property of ordering is lost, properties familiar from real and complex numbers vanish with each extension. The quaternions lose commutativity, i.e.: x·yy·x for some quaternions x, y, and the multiplication of octonions, additionally to not being commutative, fails to be associative: (x·yzx·(y·z) for some octonions x, y, z.
Reals, complex numbers, quaternions and octonions are all normed division algebras over R. By Hurwitz's theorem they are the only ones; the sedenions, the next step in the Cayley–Dickson construction, fail to have this structure.
The Cayley–Dickson construction is closely related to the regular representation of C, thought of as an R-algebra (an R-vector space with a multiplication), with respect to the basis (1, i). This means the following: the R-linear map
for some fixed complex number w can be represented by a 2 × 2 matrix (once a basis has been chosen). With respect to the basis (1, i), this matrix is
i.e., the one mentioned in the section on matrix representation of complex numbers above. While this is a linear representation of C in the 2 × 2 real matrices, it is not the only one. Any matrix
has the property that its square is the negative of the identity matrix: J2 = −I. Then
is also isomorphic to the field C, and gives an alternative complex structure on R2. This is generalized by the notion of a linear complex structure.
Hypercomplex numbers also generalize R, C, H, and O. For example, this notion contains the split-complex numbers, which are elements of the ring R[x]/(x2 − 1) (as opposed to R[x]/(x2 + 1)). In this ring, the equation a2 = 1 has four solutions.
The field R is the completion of Q, the field of rational numbers, with respect to the usual absolute value metric. Other choices of metrics on Q lead to the fields Qp of p-adic numbers (for any prime number p), which are thereby analogous to R. There are no other nontrivial ways of completing Q than R and Qp, by Ostrowski's theorem. The algebraic closures of Qp still carry a norm, but (unlike C) are not complete with respect to it. The completion of turns out to be algebraically closed. This field is called p-adic complex numbers by analogy.
The fields R and Qp and their finite field extensions, including C, are local fields.
See alsoEdit
1. ^ For an extensive account of the history, from initial skepticism to ultimate acceptance, See (Bourbaki 1998), pages 18-24.
2. ^ In modern notation, Tartaglia's solution is based on expanding the cube of the sum of two cube roots: With , , , u and v can be expressed in terms of p and q as and , respectively. Therefore, . When is negative (casus irreducibilis), the second cube root should be regarded as the complex conjugate of the first one.
1. ^ Penrose, Roger (2016). The Road to Reality: A Complete Guide to the Laws of the Universe (reprinted ed.). Random House. pp. 72–73. ISBN 978-1-4464-1820-8. Extract of p. 73: "complex numbers, as much as reals, and perhaps even more, find a unity with nature that is truly remarkable. It is as though Nature herself is as impressed by the scope and consistency of the complex-number system as we are ourselves, and has entrusted to these numbers the precise operations of her world at its minutest scales."
2. ^ Burton, David M. (1995), The History of Mathematics (3rd ed.), New York: McGraw-Hill, p. 294, ISBN 978-0-07-009465-9
3. ^ Bourbaki, Nicolas. "VIII.1". General topology. Springer-Verlag.
4. ^ Axler, Sheldon (2010). College algebra. Wiley. p. 262.
5. ^ Spiegel, M.R.; Lipschutz, S.; Schiller, J.J.; Spellman, D. (14 April 2009), Complex Variables (2nd Edition), Schaum's Outline Series, McGraw Hill, ISBN 978-0-07-161569-3
6. ^ Aufmann, Richard N.; Barker, Vernon C.; Nation, Richard D. (2007), "Chapter P", College Algebra and Trigonometry (6 ed.), Cengage Learning, p. 66, ISBN 978-0-618-82515-8
8. ^ See (Ahlfors 1979).
9. ^ Brown, James Ward; Churchill, Ruel V. (1996), Complex variables and applications (6th ed.), New York: McGraw-Hill, p. 2, ISBN 978-0-07-912147-9, In electrical engineering, the letter j is used instead of i.
10. ^ Pedoe, Dan (1988), Geometry: A comprehensive course, Dover, ISBN 978-0-486-65812-4
11. ^ See (Solomentsev 2001): "The plane $\R^2$ whose points are identified with the elements of $\C$ is called the complex plane"... "The complete geometric interpretation of complex numbers and operations on them appeared first in the work of C. Wessel (1799). The geometric representation of complex numbers, sometimes called the "Argand diagram" , came into use after the publication in 1806 and 1814 of papers by J.R. Argand, who rediscovered, largely independently, the findings of Wessel".
12. ^ See (Apostol 1981), page 18.
13. ^ Kasana, H.S. (2005), "Chapter 1", Complex Variables: Theory And Applications (2nd ed.), PHI Learning Pvt. Ltd, p. 14, ISBN 978-81-203-2641-5
14. ^ Nilsson, James William; Riedel, Susan A. (2008), "Chapter 9", Electric circuits (8th ed.), Prentice Hall, p. 338, ISBN 978-0-13-198925-2
15. ^ Kline, Morris. A history of mathematical thought, volume 1. p. 253.
16. ^ Katz, Victor J. (2004), "9.1.4", A History of Mathematics, Brief Version, Addison-Wesley, ISBN 978-0-321-16193-2
17. ^ Nahin, Paul J. (2007), An Imaginary Tale: The Story of −1, Princeton University Press, ISBN 978-0-691-12798-9, retrieved 20 April 2011
18. ^ Descartes, René (1954) [1637], La Géométrie | The Geometry of René Descartes with a facsimile of the first edition, Dover Publications, ISBN 978-0-486-60068-0, retrieved 20 April 2011
19. ^ Caparrini, Sandro (2000), "On the Common Origin of Some of the Works on the Geometrical Interpretation of Complex Numbers", in Kim Williams (ed.), Two Cultures, Birkhäuser, p. 139, ISBN 978-3-7643-7186-9 Extract of page 139
20. ^ Hardy, G.H.; Wright, E.M. (2000) [1938], An Introduction to the Theory of Numbers, OUP Oxford, p. 189 (fourth edition), ISBN 978-0-19-921986-5
21. ^ Extracted quotation from "A Short History of Complex Numbers", Orlando Merino, University of Rhode Island (January, 2006)
22. ^ For the former notation, See (Apostol 1981), pages 15–16.
23. ^ Abramowitz, Milton; Stegun, Irene A. (1964), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Courier Dover Publications, p. 17, ISBN 978-0-486-61272-0, Section 3.7.26, p. 17
24. ^ Cooke, Roger (2008), Classical algebra: its nature, origins, and uses, John Wiley and Sons, p. 59, ISBN 978-0-470-25952-8, Extract: page 59
25. ^ See (Ahlfors 1979), page 3.
26. ^ See (Apostol 1981), pages 15–16.
27. ^ See (Apostol 1981), page 25.
29. ^ Marker, David (1996), "Introduction to the Model Theory of Fields", in Marker, D.; Messmer, M.; Pillay, A. (eds.), Model theory of fields, Lecture Notes in Logic, 5, Berlin: Springer-Verlag, pp. 1–37, ISBN 978-3-540-60741-0, MR 1477154
30. ^ Bourbaki, Nicolas. "VIII.4". General topology. Springer-Verlag.
31. ^ Corry, Leo (2015). A Brief History of Numbers. Oxford University Press. pp. 215–16.
32. ^ See (Apostol 1981), pages 15–16.
34. ^ Lester, J.A. (1994), "Triangles I: Shapes", Aequationes Mathematicae, 52: 30–54, doi:10.1007/BF01818325
35. ^ Kalman, Dan (2008a), "An Elementary Proof of Marden's Theorem", American Mathematical Monthly, 115 (4): 330–38, doi:10.1080/00029890.2008.11920532, ISSN 0002-9890
36. ^ Kalman, Dan (2008b), "The Most Marvelous Theorem in Mathematics", Journal of Online Mathematics and its Applications
37. ^ Grant, I.S.; Phillips, W.R. (2008), Electromagnetism (2 ed.), Manchester Physics Series, ISBN 978-0-471-92712-9
38. ^ Kevin McCrimmon (2004) A Taste of Jordan Algebras, pp 64, Universitext, Springer ISBN 0-387-95447-3 MR2014924
Works citedEdit
Further readingEdit
• Bourbaki, Nicolas (1998), "Foundations of mathematics § logic: set theory", Elements of the history of mathematics, Springer
• Burton, David M. (1995), The History of Mathematics (3rd ed.), New York: McGraw-Hill, ISBN 978-0-07-009465-9
• Katz, Victor J. (2004), A History of Mathematics, Brief Version, Addison-Wesley, ISBN 978-0-321-16193-2
• Nahin, Paul J. (1998), An Imaginary Tale: The Story of , Princeton University Press, ISBN 978-0-691-02795-1
• Ebbinghaus, H. D.; Hermes, H.; Hirzebruch, F.; Koecher, M.; Mainzer, K.; Neukirch, J.; Prestel, A.; Remmert, R. (1991), Numbers (hardcover ed.), Springer, ISBN 978-0-387-97497-2 |
0b0e5af8b7eebe58 | According to the Bohr model, when an electron moves from a ground state to the excited state, it will absorb energy, but what about the distance between electrons and nucleus, will it be changed? Why ? How?
Also, if there is the distance change when from a ground state to the excited state, will there be also any change in distance when electrons move from excited state to ground state ?
• $\begingroup$ What do you want to achieve? The Bohr model is not useful to describe chemical reality. You can use it to calculate numbers, but those cannot be reproduced with other methods. And I don't understand you last sentence. You really ask is the ground state before and after excitation is the same? $\endgroup$ – Karl Oct 1 '17 at 19:49
The Bohr model shows the atom as a small, positively charged nucleus surrounded by orbiting electrons.
Electrons undergo exothermic or endothermic reactions during ground state or excited state. When the electrons move from ground state to excited state, it means that they gain more energy, resulting in higher energy levels (higher energy shells), and this causes the electrons to orbit further from the nucleus. The same explanation is applicable when electrons move from excited state to ground stage- the electrons lose energy and this results in lower energy levels (lower energy shells), causing the electrons to orbit nearer to the nucleus.
• 1
$\begingroup$ I am not sure whether the higher excited state always increases distance between electron and nucleus, especially when considering orbitals of higher angular momentum ($d, f \dots$). There might be some exceptions. $\endgroup$ – Feodoran Oct 1 '17 at 8:39
• $\begingroup$ @Feodoran Bohr knew nothing of orbitals, at least not when he devised the model named after him. And in Schrödingers model of the hydrogen atom, the different orbitals with same quantum number $n$ are degenerate. For anything else than hydrogen atoms, the original Schrödinger is quantitativly nearly as bad as Bohr (although the concept is much better of course). $\endgroup$ – Karl Oct 1 '17 at 19:59
• $\begingroup$ For Hydrogen yes, but I was thinking about atoms and molecules in general. Why do you say "Schrödinger is quantitatively nearly as bad as Bohr"? The Schrödinger equation is in principle exact (at least in terms of chemical accuracy). Whether you need to make certain approximations to make solving it feasible, is another story. $\endgroup$ – Feodoran Oct 1 '17 at 20:42
I may be wrong here as this was long ago, but if I remember correctly absorption of light and the excitation process of an electron (for example) are faster than the electronic movement and therefor appear as a straight vertical line on the energy to distance plot (Jablonski-diagram). So you start from an optimized geometry for the ground state which is not necessarily the optimized geometry for the excited that and it will relax in the time while it is being excited before it falls back to the ground state. I remember we used this electronic change to calculate pKs values of acids for example. So yes, if I remember correctly the x-axis on this diagram was the distance of the electron to the nucleus (but I could be wrong there).
There was a rule for the excitation process, I think by Fermi (?). Historically seen, when they first analyzed the emission spectrum of for example hydrogen, they found that not every possible transition is equally likely, so there has to be quantum-mechanical process which descides what transition is more likely. And as the excitation process is much faster it needs to be almost straight up, so without any change in the geometry or any distances. And essentially you will need a good overlap of those amplitudes.
If you look here for example: http://www.geoset.info/wp-content/uploads/2012/05/Electronic-spec.png
You can see that both of these curves represent the different vibrational states and those have amplitudes, which tend to be bigger and the sides and not in the middle. I think in classical mechanics you'd expect that to be somewhere around the point where the oscillating spring is relaxed. And you can show, that the absorption process is better if the overlap of those amplitudes is at it's biggest, so you start from the ground state and then you go straight upwards till you find a higher vibrational mode of the excited state where you end up in a high amplitude. This is why you end up in v'=2 in the picture. Then you still have the same geometry but it is a vibrationally exicted state and will first relax to the excited vibrational ground state v'= 0. But that curve is shifted to the right and this is how the geometry changes and I believe in you example this would be the orientation of the electron to the nucleus.
At least I hope so. If not I'm truly sorry but this was the only example I came up with right now.
• $\begingroup$ In principle you are right, but you are answering a different thing. The question was about change of electron density and you are talking about change in the nuclear geometry. $\endgroup$ – Feodoran Oct 1 '17 at 8:33
• $\begingroup$ And Fermi and Jablonski are post-Bohr. $\endgroup$ – Karl Oct 1 '17 at 19:50
Your Answer
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9abbaf39afb9d8ad | Nielsen and Chuang mention in Quantum Computation and Information that there are two kinds of measurement : general and projective ( and also POVM but that's not what I'm worried about ).
General Measurements
Quantum measurements are described by a collection $\left \{ M_{m} \right \}$ of measurement operators. These are operators acting on the state space of the system being measured. The index $m$ refers to the measurement outcomes that may occur in the experiment. If the state of the quantum system is $|\psi \rangle$ immediately before the measurement then the probability that result m occurs is given by $$ p\left ( m \right ) = \left\langle \psi | M_{m}^{\dagger}M_{m} |\psi \right\rangle $$ and the state of the system after the measurement is $$\frac{M_{m}|\psi\rangle}{\sqrt{ \left \langle \psi | M_{m}^{\dagger}M_{m} |\psi \right\rangle}}$$ The measurement operators satisfy the completeness equation $$\sum_{m} M_{m}^{\dagger}M_{m} = I$$
Projective Measurements
A projective measurement is described by an observable, $M$, a Hermitian operator on the state space of the system being observed. The observable has a spectral decomposition, $$M = \sum_{m} mP_{m}$$ where $P_{m}$ is the projector onto the eigenspace of $M$ with eigenvalue $m$. The possible outcomes of the measurement correspond to the eigenvalues, $m$, of the observable. Upon measuring the state $|\psi\rangle$, the probability of getting result $m$ is $$p(m) = \langle\psi|P_{m}|\psi\rangle$$ Given that outcome $m$ occurred, the state of the quantum system immediately after the measurement is $$\frac{P_{m}|\psi\rangle}{\sqrt{p(m)}}$$
Projective Measurements are special cases of General measurements when the measurement operators are Hermitian and orthogonal projectors.
In the introductory course I took on QM, we were introduced to measurements but were not told that they were actually projective. I am assuming that similar courses in other universities are doing the same. :(
My questions are :
• Is that the only difference between these two types of measurement?
• Is there a case where the measurement operators are not orthogonal projectors?
• What do the measurement operators intuitively mean? Where and how are they used?
I am an undergraduate student of Electrical Engineering with one semester of experience in quantum mechanics. I am currently working on a project on quantum computing with spins.
Consider the measurement operators given by
$$M_{1} = \sqrt{\frac{\sqrt{2}}{1 + \sqrt{2}}} |1\rangle\langle1|$$
$$M_{2} = \sqrt{\frac{\sqrt{2}}{1 + \sqrt{2}}} \frac{(|0\rangle - |1\rangle)(\langle 0| - \langle 1|)}{2}$$
$$M_{3} = \sqrt{I - M_{1}^{\dagger}M_{1} - M_{2}^{\dagger}M_{2}}$$
They satisfy all the conditions required for general measurement operators. But when the rules for general measurements are used to calculate the state $|\psi_{2}\rangle$ after a result "2" is obtained, $|\psi_{2}\rangle$ turns out to be given by $$|\psi_{2}\rangle = \frac{|0\rangle - |1\rangle}{\sqrt{2}} $$ which is most definitely not an eigenstate!!
• $\begingroup$ More on projective measurements in QM. $\endgroup$ – Qmechanic May 17 '15 at 12:27
• $\begingroup$ The "general measurement" as defined above looks like a simply classical average of measurements to me. It doesn't seem to introduce anything to QM that's new or different, if that is what you are asking. $\endgroup$ – CuriousOne May 17 '15 at 14:08
• $\begingroup$ @CuriousOne: I don't really how you see that this is a "simple classical average of measurements". $\endgroup$ – Martin May 17 '15 at 15:21
• $\begingroup$ It's the way they normalize the final state that gives me the idea that this is really just a linear superposition of ordinary measurement operators. I have to think about it some more, but truly, none of this can be particularly non-trivial, since they are not changing QM and everything in QM is a linear operation, so at most one can do something like a weighted average of projective operators. $\endgroup$ – CuriousOne May 17 '15 at 15:36
• 2
$\begingroup$ A POVM is a special case of general measurements where the measurement operators are not orthogonal projectors. You should figure out how POVMs are a special case of general measurements (the different outcomes are taken to orthogonal states, but the operators don't have to be projectors), and you should get more insight. $\endgroup$ – Peter Shor May 17 '15 at 22:04
Note: There is a short summary at the bottom.
This is actually also described in Nielsen&Chuang: You don't learn about general measurements, because they are completely equivalent to projective measurements + unitary time evolution + ancillary systems, all of which is described in your usual QM formalism.
The Measurement Postulate
Let's start from the beginning. Let us first formulate the usual postulate of quantum mechanics, as you know it:
Measurement Postulate (first course):
Measurements are described by projection valued measures defined by the spectral measure of an observable (self-adjoint operator). The post measurement state the projection onto the subspace of the measurement.
Now in addition to this, we have a bunch of other postulates, in particular, we have the postulate that the quantum evolution is governed by the Schrödinger equation thus time evolution is a unitary evolution. That's all very nice, but when you go to your lab, you discover that that's not what happens.
As is pointed out in Nielsen & Chuang, it seems that sometimes, the quantum state is destroyed after measurements (the measurement is not a "non-demolition-measurement"), so the state after measurement does not seem to be well-described by a projection onto this eigenspace. But also, you'll actually find out that your evolution is not according to a Hamiltonian and it is not unitary. Energy might enter the system or leave it, depending on what you do.
Why is that? The key problem to realize is that all of the postulates in your first course refer to what we call a "closed system". None of them actually state this requirement, but they all need it. Only in a closed system is energy conserved (much like in classical mechanics), so we can expect time evolution to be unitary. Just as well, only in a closed system can we expect that measurements are always described by projective measurements.
Time Evolution of Open Quantum Systems
So, what about open quantum systems, i.e. systems where in addition to our system $S$ with a Hilbert space $\mathcal{H}_S$, we have an uncontrolled environment $E$ (such as in the lab)? Let's consider time evolution as a training case, because it is much easier to understand from classical intuition - incidentally, we have the same problem in classical mechanics!
In an open system, as long as we know what the environment is doing, we can assign a Hilbert space $\mathcal{H}_E$, compute the Hamiltonian on the combined system $\mathcal{H}_S\otimes \mathcal{H}_E$, do time evolution and trace out the environment (the partial trace is the equivalent of forgetting the environment and only considering the system $S$). In other words, having prepared a state $\rho_S$ of the system and assuming it is not correlated with an environment state $\rho_E$ (this can be debated upon), the time-evolved state $\rho_S$ is given by
$$ T(\rho)= \operatorname{tr}_E(U(\rho_S\otimes \rho_E)U^*) $$
where $\operatorname{tr}_E$ is the partial trace. But this is very cumbersome. We don't always know what the environment is doing. So instead of saying that the open quantum system is part of a bigger, closed system which undergoes a unitary time evolution $U$, we can directly specify the time-evolution by specifying $T$. Then, $T$ will not be a unitary time-evolution, but a completely positive map. In classical mechanics, you do the same: Instead of considering the Lagrangian/Hamiltonian of the whole system, which you might not know, you can also try to consider only a part of that system and describe it by a master equation (this is routinely done in statistical mechanics). The same can be done in quantum mechanics, i.e. by the quantum master equation.
So what I want to argue is the following:
• Using unitary time evolution or completely positive maps is ultimately the same (mathematically).
• In the lab, you will always have noise from the environment so your system will never be closed.
• Unitary time evolutions are clumsy, because they need you to specify the environment completely, which might be hard or nearly impossible to do, so it is much nicer to only work with the open system.
• The definition of a completely positive map lets you do that. Therefore, it is a "better" postulate in a physical sense, because it eliminates key problems when applying the model to your lab.
Measurements in Open Quantum Systems
Essentially, we now have to do the exact same thing for measurements that we did for unitary time evolution. How do measurements look like if you restrict them to a subsystem?
[A small aside: Let's throw in another complication: Measurements are not really instantaneous, some of them take time. For example, suppose you have an atom with three states with different energies, one very much excited $E_3$ and two less excited states (one may be the ground state, let's call them $E_1$ and $E_2$). So you know that your system will be in either of the last states. Measuring which one of these, you can shine a laser with one of the two transition energies to the excited state, say the laser energy is $E_3-E_1$. If you get induced emission, your system was in state $E_1$, if you don't, it has to be in $E_2$. This of course takes time, so the system will evolve (and it won't be a free evolution, because the laser is doing something), so a simple measurement is not just a projective measurement, but we can hardly ever fully separate it from some time evolution. Often, this is no problem, sometimes it might be.]
What happens if we do this? How does the measurement look like on subsystems? Well, it turn out that just as completely positive maps are the restrictions of unitary time evolution, POVMs are the restrictions of measurements.
You can also see this from Naimark's dilation theorem: This theorem basically tells us that every POVM ultimately is a projective measurement if we factor in some environment. So in this sense, the POVM approach and the usual projective measurements are mathematically equivalent, if one always factors in the environment + maybe some additional unitary evolution. However, we have the same as above:
The formalism of POVMs is better suited to work with, because it does not require us to actually know or even think of the environment. We can get our measurement operators from the experiment and don't have to worry about them being projections or not (in the latter case, the system is surely not closed)
So the POVM formalism doesn't give us anything knew formally and mathematically, but it is a better way to think about actual quantum systems, which are usually not closed systems.
General Measurements and a new Postulate
Now we have POVMs. We could replace our postulate by the POVM postulate, which would cover the outcomes of experiments very well. So why don't we do it? Why don't Nielsen & Chuang do it?
Because we have actually lost something: The POVM was really only introduced to compute outcome probabilities, but if we start out with a POVM, it's not clear how we obtain a post-measurement state. Very often, we don't care, but sometimes we do, so we should think about this again (for example, when we consider "the optimal way to distinguish a set of quantum states", we at the moment don't care about the post-measurement state, so POVMs are all we need).
This "problem" of the post-measurement state can be addressed in several ways, one way is to take a POVM with effect operators $E_i$, specify a square root $M_i^*M=E$ and define a general measurement (which, in addition to the fact that for every generalized measurement $\{M_m\}_m$, $E_m:=M^*M$ defines a POVM tells you that the formalism of POVMs and general measurements is mathematically equivalent). Now, square roots are not unique, so in order to talk about the post measurement state, you'll have to refer to experiments (or specify the environment and define the measurement there, which will provide you with a unique projective measurement on the closed system).
[If you want yet another way to think about this, you can pick yet another formalism, quantum instruments which essentially does the same thing.]
So in the end, we replace our old (closed system) postulate by the general (open system) postulate:
Measurement Postulate (Nielsen&Chuang):
Measurements are described by a collection of measurement operators $\{M\}_m$ that are not necessarily projections but fulfill $\sum_m M^*_mM_m=\mathbf{1}$. The post measurement state upon measurement of $m$ is the state after application of $M_m$.
From what I have argued above, it should not come as a surprise that the two postulates are mathematically equivalent. More precisely, if we augment POVMs/general measurements by unitary time evolution and the introduction of environment systems, any such measurement should really come from a projective measurement. This was my original post:
Sketch of Proof of the Equivalence of the two postulates
This is described on page 94 to 95 in Nielsen & Chuang:
Let $\{M\}_m$ be a "general measurement" with $m=1,\ldots,n$ on a Hilbert space $\mathcal{H}$. Define $U\in \mathcal{B}(\mathcal{H}\otimes \mathbb{C}^n)$ (i.e. $U$ is a bounded operator on the composite system) via definining:
$$ U|\psi\rangle|0\rangle= \sum_{m=1}^n (M_m|\psi\rangle)|m\rangle $$
where $|m\rangle$ is the standard orthonormal basis of $\mathbb{C}^n$. Then you can show that $U$ can be extended to a unitary operation $U\in \mathcal{B}(\mathcal{H}\otimes \mathbb{C}^n)$.
Now you define the projective measurement $P$ with projections $$P_m:=\mathbf{1}_{\mathcal{H}}\otimes |m\rangle\langle m|$$
and what you can show is that first performing $U$ and then measuring the projective measurement $P$ and tracing out the system $\mathbb{C}^n$ ("forgetting" about the system) is equivalent to performing the generalized measurement $M_m$. In particular:
$$ \frac{P_m U|\psi\rangle |0\rangle}{\sqrt{\langle \psi|\langle 0|U^*P_mU|\psi\rangle|0\rangle}}= \frac{(M_m|\psi\rangle)|m\rangle}{\sqrt{\langle \psi|M_m^*M_m|\psi\rangle}} $$
and the probabilities also add up. So general measurements add nothing new.
About Closed (Quantum) Systems:
We have of course constructed the environment. Who tells us that this is the "real" physical environment or that the measurement in the real closed system is actually also projective? No one, actually. This is one other assumption that I've been making implicitly. However, I believe that this system has another deeper problem: Coming from the experimental/operational side, what actually is a closed quantum system? Unless (maybe) we consider the whole universe, we can never actually work with a completely closed system - and we can't consider the whole universe. I believe that there are actually arguments (higher level/quantum foundations) that tell us that the postulates are completely equivalent if there exists a closed quantum system, but this is philosophical.
But this means that we did add something "new": We got rid of the necessity of closed systems (if we also replace all the other axioms).
Lessons learned: (tl;dr)
So, what's the essence? I have argued that generalized measurements are nothing new, neither physically, nor mathematically, if we know about the difference of open and closed quantum systems. Therefore, they don't add anything that you didn't get from the old formalism already, so that your Quantum Mechanics 101 course is not wrong (barring problems with the definition of "closed quantum systems").
However, POVMs (or maybe general measurements) are the "right" way to think about measurements. The paradigm of open quantum systems, which is very important for real world experiments is inherently inscribed into POVMs and they also tell us why sometimes, measurements seem not repeatable in the lab. So POVMs are not some theoretical construct floating in philosophy space (closed quantum systems), but more operational descriptions of measurements. In addition, they are better to work with when describing real world situations.
As a final note: General measurements are not considered heavily in the literature. Peter Shor was so kind as to point out an (old) example of their use with this Peres, Wooters paper (paywall!). Usually however, I find that people work with POVMs instead of general measurements.
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$\begingroup$ Generalized measurements were used in this paper of Peres and Wootters, which is very important historically because thinking about its consequences led to the discovery of teleportation. $\endgroup$ – Peter Shor May 17 '15 at 22:10
• $\begingroup$ @Martin : Thanks for your answer but it doesn't really clear any of my doubts about what general measurement operators are! I have been through the same section in Nielsen and Chuang. General measurements are not trivial because the authors themselves mention, ".. it turns out that there are important problems such as the optimal way to distinguish a set of quantum states – the answer to which involves a general measurement, rather than a projective measurement". Further, projective measurements can be always repeated theoretically but such repetition may not be possible physically. $\endgroup$ – transistor May 19 '15 at 9:47
• $\begingroup$ @Sattwik: Proving that they are equivalent means that you can very well get away without knowing about them. Since you were explicitly NOT referring to POVMs, I supposed that you knew why they were interesting. Everything you quote in this section either refers to POVMs directly or is interesting for the same reason that POVMs are interesting. I'll edit my post to make all of this clearer. $\endgroup$ – Martin May 19 '15 at 10:03
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$\begingroup$ @Sattwik: when they say "... it turns out that there are important problems such as the optimal way to distinguish a set of quantum states", they are talking about a problem that can be solved using only POVMs. $\endgroup$ – Peter Shor May 19 '15 at 10:43
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$\begingroup$ @gertian: I would say that it's okay to use POVMs as main definition of a measurement, but you should not say "POVMs are measurements", but you should say "POVMs are measurements. However, they are actually implemented by some effect operators $M_i$. If you know those, you also know the post measurement state, but most of the times you won't know them." - in other words: you need to complement the definition of POVMs by effect operators to have a good main definition. $\endgroup$ – Martin Jun 15 '17 at 13:00
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55e573fc9706a717 | Say I have a molecular wavefunction as a set of molecular orbitals and want to calculate the molecule's dipole moment, but don't know how! I searched a lot but couldn't find any practical example.
$$\psi _i=\sum ^N_{i=1}C_i\mathrm e^{-\alpha _ir^2}$$
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$\begingroup$ You could use the variational theorem to determine the molecular orbital coefficients of the molecular orbitals (one electron wavefunctions). If you let you molecular wavefunctions be a linear superposition of basis atomic wavefunctions $\Psi =\sum c_i\psi_i$, with the orbital coefficients you can understand key properties of your molecule. You can rationalise trends in bond polarity too, that can't be explained with other theories! $\endgroup$ – AngusTheMan May 9 '15 at 1:19
The dipole moment $\mu$ of a molecule is a measure of charge distribution in the molecule and the polarity formed by the nuclei and electron cloud.
We can perturb our system with an external electric field $\vec E$ and gauge the response of the electron cloud and nuclei by the polarisability, i.e how much the dipole moment changes. In practice the nuclei might be so heavy that their motion is not perturbed, while electrons being light are very mobile. If we imagine that the external field is caused by some other species, and that it itself is not changing, we can call this external constant electric field $\vec E$ at least over the volume of the molecule we are considering. Imagine for arbitrary book keeping that we point it down the $z$ axis. We could also investigate how the dipole moment changes with bond vibrations to discuss IR spectroscopy or if the polarisability changes during a vibration to give Raman spectroscopy.
We can use perturbation theory to expand the wavefunction and the molecular energy in terms of small perturbations of the field. We start by Taylor expanding the energy and molecular wavefunction in terms of the electric field which acts as the perturbation parameter. \begin{equation} E(\vec E)=E^0+\bigg(\frac{\partial E}{\partial \vec E}\bigg)_0\vec E+\bigg(\frac{\partial ^2E}{\partial \vec E^2}\bigg)_0\frac{\vec E^2}{2!}+\bigg(\frac{\partial ^3E}{\partial \vec E^3}\bigg)_0\frac{\vec E^3}{3!}+\dots \end{equation} \begin{equation} \psi(\vec E)=\psi^0+\bigg(\frac{\partial \psi}{\partial \vec E}\bigg)_0\vec E+\bigg(\frac{\partial ^2\psi}{\partial \vec E^2}\bigg)_0\frac{\vec E^2}{2!}+\bigg(\frac{\partial ^3\psi}{\partial \vec E^3}\bigg)_0\frac{\vec E^3}{3!}+\dots \end{equation} If we use the notation that the wavefunction derivatives are given by; \begin{equation} \bigg(\frac {1}{i!}\bigg)\frac{\partial ^i\psi}{\partial \vec E^i}=\psi ^{(i)} \end{equation} The Hamiltonian for such a system under the influence of an electric field in the $z$ direction is $\hat H(\vec E)$. \begin{equation} \hat H(\vec E)=\hat H^0-\vec E\hat \mu _z \end{equation} Where $\hat \mu _z$ is the dipole moment operator that is a summation of the charges of the nuclei and electrons in the molecule. This is the result of Hellmann-Feynman theorem of the energy and the electric field, e.g. $\hat H(\vec E)=\hat H^0 +\hat H^1(\vec E)$, \begin{equation} \frac{dE}{d\vec E}=\bigg\langle \frac{d\hat H}{d\vec E}\bigg\rangle=\bigg\langle \frac{d(-\mu _z\vec E)}{d\vec E}\bigg\rangle \end{equation}
The time-independent Schrödinger equation is now, \begin{equation} \hat H(\vec E)\psi(\vec E)=E(\vec E)\psi (\vec E) \end{equation} With an energy; \begin{equation} E(\vec E)=\big \langle \psi (\vec E)\big|\hat H(\vec E)\big|\psi (\vec E)\big \rangle=\big\langle \psi ^{(0)}+\psi ^{(1)}\vec E+\psi ^2 \vec E^2+ \dots \big|\hat H-\vec E\hat \mu _z \big|\psi ^{(0)}+\psi ^{(1)}\vec E+\psi^2\vec E^2+\dots \big \rangle \end{equation}
With a little algebra and use of $E^{(0)}=\langle \psi ^{(0)}|\hat H^0 |\psi ^{(0)}\rangle$, as well as the Hermitian properties of the Hamiltonian. \begin{equation} E(\vec E)=E^{(0)}+2\vec E\big \langle \psi ^{(1)}\big|\hat H^{(0)}\big|\psi ^{(0)}\rangle -\vec E\big\langle \psi ^{(0)}\big|\hat \mu _z\big|\psi ^{(0)}\big \rangle +\mathcal O(\vec E^2) \end{equation} Using the Schrödinger equation $H\psi =E\psi$ and pulling the scalar energy out of the integral; \begin{equation} \big\langle \psi ^{(1)}\big|\hat H^0\big|\psi ^{(0)}\rangle =E^{(0)}\big\langle \psi ^{(1)}\big|\psi ^{(0)}\big \rangle \end{equation} Since $\langle \phi |\psi \rangle=0 $, \begin{equation} E(\vec E)=E^{(0)}-\vec E\big \langle \psi ^{(0)}\big|\hat \mu _z\big|\psi ^{(0)}\big\rangle \end{equation} Therefore the expectation value of the dipole moment along the $z$ axis for a molecular state $\psi ^{(0)}$ is $\langle \mu _z \rangle$; \begin{equation} \langle \mu _z\rangle =\big\langle \psi ^{(0)}\big|\hat \mu _z\big|\psi ^{(0)}\big\rangle \end{equation} To understand the strength of the interaction that causes the transition between the states $\psi ^{(0)}$ and $\psi ^{(1)}$ we use the transition dipole moment which is basically exactly the same but there is a wavefunction from both of the states involved, initial and final!
If you were to repeat this process but retain higher orders (Messy!) you would get (2nd order) the polarisability of the molecule which in essence is the susceptibility of the electron cloud to change with respect to an external electric field (so how the dipole moment changes. Third order would give the hyperpolarisability etc., etc...
As I said, you could also approach this from a really different angle by interpreting the molecular orbital diagram and using computational chemistry (so variational principle etc) to find the molecular orbital coefficients! That would give you a good idea of what is going on!
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$\begingroup$ Hrm. This is a quite thorough formal elaboration of the theory involved, but it doesn't actually describe how one would implement practically the dipole moment calculation. (At least, I am no closer to understanding how I would code a dipole moment calculation given a set of MO coefficients and basis functions.) $\endgroup$ – hBy2Py Sep 3 '15 at 11:57
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$\begingroup$ This whole derivation is unnecessary. Per definition the dipole moment is the expectation value of the dipole moment operator with the given wave-function. $\endgroup$ – Greg Sep 3 '15 at 14:38
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$\begingroup$ Thank you both for your comments, I will take this into account and update my answer shortly. I agree with @Brian, this answer does not give a non-specialist sufficient information on how to perform calculations to obtain values for molecules etc. However, I do not feel it is obvious that the dipole moment is the expectation value to someone who is new to this material and not as knowledgable as others. It is my experience that undergrads can struggle in this area if they can not see where something comes from. That's why I always like to give people a little background material. :) $\endgroup$ – AngusTheMan Sep 3 '15 at 15:39
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$\begingroup$ @Greg For those like myself (background in chemical engineering, and not applied mathematics / quantum physical chemistry; but with an interest in a general understanding of the inner workings of quantum computation) it is perhaps not quite as sad...? I very much appreciate the efforts of both AngusTheMan and pentavalentcarbon to lay out the details. Also, frankly: isn't this sort of exposition the entire purpose of StackExchange? $\endgroup$ – hBy2Py Sep 22 '15 at 13:56
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$\begingroup$ @Greg Practical quantum computation was the topic of the original question -- so, no, it's not at all off-topic. Also, if you're looking for mathematically rigorous developments from axiom to theorem or whatever, you're on the wrong SE site. There are Math.SE and MathOverflow for that. $\endgroup$ – hBy2Py Sep 22 '15 at 14:11
The necessary formal derivation has already been nicely done by AngusTheMan. I'll start from the last equation:
$$ \langle \mu_{z} \rangle = \langle \Psi | \hat{\mu}_{z} | \Psi \rangle $$
where $\Psi$ is the variational wavefunction; it can be any molecular state. It's important that it's variational, otherwise the expectation value approach is not exact. So, this works for SCF, CI, and MCSCF wavefunctions, but extra derivatives need to be taken for Moller-Plesset and coupled cluster wavefunctions. More work needs to be done for multideterminental wavefunctions like CI and MCSCF, but the complexity is no different for a single state in each wavefunction. There may be some MO space partitioning I'm neglecting that's required for MCSCF, so I'll restrict my work to a single-determinental wavefunction.
Expand the wavefunction as a linear combination of molecular orbitals (MOs)
$$ \Psi = \sum_{i} \psi_{i}, $$
where each molecular orbital is a linear combination of atomic orbitals (AOs)
$$ \psi_{i} = \sum_{\mu} C_{\mu i} \phi_{\mu}, $$
where $C_{\mu i}$ is the MO coefficient matrix, so our expectation value now looks like this:
$$ \langle \mu_{z} \rangle = \sum_{i}^{\textrm{occ MOs}} \sum_{\mu\nu}^{\textrm{AOs}} C_{\mu i} C_{\nu i} \langle \phi_{\mu} | \hat{\mu}_{z} | \phi_{\nu} \rangle. $$
The indices $\mu,\nu$ run over all AOs, and the index $i$ runs over the occupied MOs. There's only one index because this is a one-electron operator. I'm also neglecting any complex values here, since we almost always work with real-valued AOs and MO coefficients.
We do one last rearrangement. Replace the MO coefficients with the density matrix
$$ P_{\mu\nu} = \sum_{i}^{\textrm{occ MOs}} C_{\mu i} C_{\nu i} $$
to give the first explicit "working equation":
$$ \langle \mu_{z} \rangle = \sum_{\mu\nu}^{\textrm{AOs}} P_{\mu\nu} \langle \phi_{\mu} | \hat{\mu}_{z} | \phi_{\nu} \rangle $$
I say first for two reasons: we usually try and avoid explicit loops like this, and the expression can be broken down further, depending on what molecular properties are of interest; I'll be more clear about this later. Inside the sum there are two terms:
• The density matrix $P_{\mu\nu}$, which comes from a converged SCF calculation.
• The integral of the dipole operator over two basis functions, $\langle \phi_{\mu} | \hat{\mu}_{z} | \phi_{\nu} \rangle$. Atomic orbitals are represented as atom-centered basis functions. These can be calculated once and at any time, since the quantities here don't change over the course of a calculation.
Each of these terms is represented as a matrix. Since the index $\mu$ runs along the rows and $\nu$ runs along the columns for each matrix, "contraction" involves either a matrix product followed by the trace, or an elementwise product followed by an accumulation sum over all matrix elements. There are other details one needs to be careful about, such as what units the result should be in, which changes the prefactor (programs work internally in atomic units), and what the origin for the dipole operator is, but that's really it.
Well, sort of. I'm actually treating some of the program internals as a black box. If you're familiar with Hartree-Fock, it should be clear where $P_{\mu\nu}$ comes from, but what about the integral? For a general expectation value $\langle A \rangle$ with its corresponding operator, where does $\langle \phi_{\mu} | \hat{A} | \phi_{\nu} \rangle$ come from? If it's already available in the code, then you call a wrapper function that then calls the integral engine to do all the nasty work, and you get back a tidy matrix without having to worry about the details. If $\langle \phi_{\mu} | \hat{A} | \phi_{\nu} \rangle$ isn't present, depending on the complexity of $\hat{A}$, there can be a non-trivial amount of derivation required for the working integral equation, followed by the implementation.
Ignoring any possible contraction of primitive basis functions, expand $\langle \phi_{\mu} | \hat{A} | \phi_{\nu} \rangle$ using the definition of $\phi$ in Cartesian coordinates:
$$ \phi(\mathbf{r}; \mathbf{A}, \mathbf{a}, \zeta) = (x-A_x)^{a_x} (y-A_y)^{a_y} (z-A_z)^{a_z} e^{-\zeta |\mathbf{r} - \mathbf{A}|^2} $$
where $\mathbf{r} = (x, y, z)$ is the electron position, $\mathbf{A} = (A_x, A_y, A_z)$ is the position of the basis function (almost always atom-centered), and $\mathbf{a} = (a_x, a_y, a_z)$ are the angular momenta for each coordinate, with $l_{\textrm{max}} = a_x + a_y + a_z$ total angular momentum of the basis function. $(0,0,0)$ is an s-function, $(1,1,0)$ and $(0,0,2)$ are d-functions, and so on.
Forming the integral more explicitly gives
$$ \langle \phi_{\mu} | \hat{A} | \phi_{\nu} \rangle = \int\int d\mathbf{r}_1 d\mathbf{r}_2 \left[ (x_1-A_x)^{a_x} (y_1-A_y)^{a_y} (z_1-A_z)^{a_z} e^{-\zeta_a |\mathbf{r}_1 - \mathbf{A}|^2} \right] \\ \times\left[ \hat{A} \right]\left[ (x_2-B_x)^{b_x} (y_2-B_y)^{b_y} (z_2-B_z)^{b_z} e^{-\zeta_b |\mathbf{r}_2 - \mathbf{B}|^2} \right] $$
Before going any further, $\hat{A}$ must be defined. If $\hat{A} = 1$, this becomes an overlap integral. The dipole operator in the z-direction is given by $\hat{A} = \hat{\mu}_{z} = -ez = -e(z_3 - C_z)$, where $z_3$ is the integration coordinate and $C_z$ is the origin of the dipole in the z-direction, usually taken to be zero. Everything is kept in atomic units until after the integral/density contraction, so drop the prefactor $-e$. We can now generalize this to an arbitrary Cartesian multipole moment operator,
$$ \hat{A} = \mathfrak{M}(\mathbf{r}_3) = (x_3 - C_x)^{c_x} (y_3 - C_y)^{c_y} (z_3 - C_z)^{c_z} $$
where $(c_x, c_y, c_z)$ determine the coordinate of each multipole, and their sum is the total multipole order; for example, $(1,0,0), (0,1,0), (0,0,1)$ are the x-, y-, and z-directions of the dipole operator. The Cartesian moment operator looks just like a Gaussian basis function where $\zeta = 0$.
Once the form of an operator has been derived, it needs to be implemented as part of an integral package, each of which implement one or more algorithms for computing integrals. Each algorithm is named after the authors of the paper in which they were introduced, and are usually abbreviated. For example, the first one I know of is the Taketa, Huzinaga, O-Ohata paper (THO, DOI: 10.1143/JPSJ.21.2313), where explicit working equations are given for 2-center overlap, 2-center kinetic energy, 2-center electron-nuclear attraction, and 4-center electron repulsion integrals. A working implementation can be found in the PyQuante package. I made an IPython notebook translation of the code snippets on the front page of the official documentation.
Other, more complex algorithms are from the Pople-Hehre (PH), McMurchie-Davidson (MD, DOI: 10.1016/0021-9991(78)90092-X), Obara-Saika (OS, DOI: 10.1063/1.450106), Dupuis-Rys-King (DRK), and Head-Gordon-Pople (HGP) papers. I'm sure I'm neglecting some, including the seminal paper by Boys which introduced the use of Gaussian functions as a substitute for Slater-type functions in basis sets. A good review of these algorithms is found in a Peter Gill paper (DOI: 10.1016/S0065-3276(08)60019-2); he is the original author of the integral code in both Gaussian and Q-Chem.
To bring these things full circle, I wrote some code a few months ago to calculate the dipole moment using pyquante2 and a wrapper that calls an implementation of the Obara-Saika recursive integral algorithm. You can find it here with some comparisons to "industrial" quantum programs.
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$\begingroup$ Thanks all. I really recommend reading the Obara-Saika paper if you're at all interested in integral evaluation. The DALTON package (not sure what the primary integral algorithm is) has an impressive list of one-electron integrals that can be calculated, many of which correspond directly (expectation value times some constants) to molecular properties. $\endgroup$ – pentavalentcarbon Sep 20 '15 at 1:29
An often used approach, especially for semi-empirical methods, is to use a set of atom-centered charges (most often the Mulliken charges) to calculate the dipole moment. In that case, the molecular dipole moment is given by:
$\vec{\mu} = \sum_a \vec{r} \times q_a $
Note that if the system has a finite charge, this equation has a dependence on the position of the molecular system relative to the origin of the coordinate space. Most program use the center of mass or center of nuclear charge as origin in such cases.
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