Stochastic Models in Classical and

Quantum Mechanics∗

V.Yu. Terebizh†

Moscow State University, Russia

Crimean Astrophysical Observatory, Ukraine

27 August 2013

PACS 05.10.Gg, 03.65.-w, 42.50.Xa, 03.65.Ud.Keywords: Statistical physics, Quantum mechanics.

Abstract

Characteristic features of the stochastic models used in classicalmechanics, statistical physics, and quantum theory are discussed. Theviewpoint according to which there is no room for probabilities in Na-ture, as such, is consistently substantiated; the researcher is forcedto introduce probabilistic concepts and the corresponding models inconditions that provide only partial predictability of the phenomenabeing studied. This approach allows one to achieve a consistent inter-pretation of some important physical phenomena, in particular, therelationship between instability of processes and their irreversibilityin time, the stochastic evolution of systems in the theory of deter-ministic chaos, Boltzmann’s H-theorem, and paradoxes of quantummechanics.

∗“Advances in Quantum Systems Research”, Chapter 8. Nova Publishers, ZoheirEzziane (Ed.), ISBN: 978-1-62948-645-1.

†98409 Nauchny, Crimea, Ukraine; E-mail: valery@terebizh.ru

1

A year or so ago, while Philip Candelas (of the physicsdepartment at Texas) and I were waiting for an elevator,our conversation turned to a young theorist who had beenquite promising as a graduate student and who had thendropped out of sight. I asked Phil what had interferedwith the ex-student’s research. Phil shook his head sadlyand said, “He tried to understand quantum mechanics.”

Steven Weinberg, “Dreams of a Final Theory”, 1992

1. Introduction

Maximalism characteristic of youth so pronounced in the epigraph would unlikelyremain after several years of specific research. In the same book, Weinberg notes,“Most physicists use quantum mechanics every day in their working lives withoutneeding to worry about the fundamental problem of its interpretation... But Iadmit to some discomfort in working all my life in a theoretical framework thatno one fully understands.”

Another outstanding physicist, Richard Feynman [1985], conveyed his attitudetoward the orthodox quantum theory even more emotionally, “I can’t explain whyNature behaves in this peculiar way. . . The theory of quantum electrodynamicsdescribes Nature as absurd from the point of view of common sense. And it agreesfully with experiment. So I hope you can accept Nature as She is – absurd.” Thenegative attitude of Albert Einstein and Erwin Schrodinger in this respect is wellknown.

What did not suit the scientists to whom physics owes the introduction of theconcept of photons, the fundamental equation for the wave function, the over-coming of serious difficulties of quantum electrodynamics, and the creation of aunified theory of electroweak interactions? (Nobel Prizes were awarded for all fourcontributions.) It is often said that Einstein did not accept the probabilistic inter-pretation of the reality that quantum mechanics brought in. However, it should beremembered that even in his first works, Einstein gave a theory of Brownian mo-tion, independently built a basic framework of statistical mechanics, anticipatedan important probabilistic expansion that many years later was associated withnames of Karhunen and Loeve, introduced the probabilities of transitions in atoms,and performed a number of other studies clearly showing that he subtly under-stood the probabilistic problems and had a good command of the correspondingtechnique. Of course, that’s not the point; what matters is the understanding ofthe researcher–Nature interrelationship expressed in Einstein’s famous aphorism,“I cannot believe that God plays dice.”

2

It is natural to suppose that the task of science is to construct the simplestmodel for the range of phenomena being studied that is consistent with experi-mental data and that has a forecasting power. Previously, the term theory wascommonly used instead of the term model ; the latter term reflects better the in-complete, transient character of our understanding of Nature. Why on earth werethe probabilistic models of classical physics accepted by the physical community1,while there is no universally accepted interpretation of quantum mechanics as yet?And this is despite the fact that extensive literature, including an in-depth anal-ysis of real and thought experiments, is devoted to the analysis of probabilisticfoundations of physics and the corresponding interpretation of quantum theory(see, in particular, Bohr 1935; Mandelstam 1950; Fok 1957; Wigner 1960; Born1963; Chirikov 1978; Kravtsov 1989; Alimov and Kravtsov 1992; Mermin 1998).

The approach suggested by quantum mechanics will become clearer if we ini-tially trace the emergence of probabilistic models in classical physics. Even anexamination of a simple experiment with die gives clear evidence of this kind. Theobjective of this chapter is, as far as possible, to facilitate familiarization withthe probabilistic problems of physics to avoid situations like that described in theepigraph2. We consistently substantiate the viewpoint according to which there isno room for probabilities in Nature, as such; the researcher is forced to introduceprobabilistic concepts and the corresponding models in conditions that provideonly partial predictability of the phenomena being studied in both classical andquantum mechanics. Light was diffracted by holes, atoms retained stability, anddice and systems consisting of a large number of molecules demonstrated certainregularities long before Christiaan Huygens, Blaise Pascal, Pierre Fermat, andJakob Bernoulli laid the foundations of the probability theory.

For coherence, we had to cursorily touch on some of the facts that are cov-ered well in textbooks. From the probability theory, to understand the subsequentmaterial, it will suffice to know that a discrete random variable ξ is specified bya set of values x1, x2, . . . , xN that it can take in an experiment and by the cor-responding probabilities of these values p1, p2, . . . , pN . The number of values Ncan be infinitely large; the probabilities must be nonnegative and add up to one.It is very useful not to confuse the notations of the random variable itself and itspossible values. In applied research, such confusion often leads to misunderstand-ings; experts on the probability theory occasionally agree to that when there is ashortage of symbols.

1Not always unconditionally, though; it will suffice to recall the long-term debate con-cerning Boltzmann’s H-theorem.

2See also the preliminary paper by Terebizh [2010].

3

2. Throws of a die and other unstable pro-

cesses

Consider once again the old problem of the throws of a die by focusing attentionon the physical aspect of the situation. Suppose that such throws are made inthe following conditions: (1) the die has the shape of a cube whose edges andangles are identical with a microscopic accuracy; (2) the inscriptions 1, 2, . . . , 6that distinguish the faces from one another are made in such a way that theirmass is smaller than that of the die by several orders of magnitude; (3) the die andthe table on which it falls are made of an elastic material; (4) the initial orientationof the die is always the same, say, face ‘6’ is directed upward and face ‘4’ is orientednorthward3; (5) the die is released from fixation in some delicate way that is nownot concretized; (6) the number written on the upper face of the stopped die istaken as the result of a single throw. The problem consists in predicting the resultof a single throw of the die. We do not go into further details of the experiment –the aforesaid is enough to understand the essence of the problem.

First, let the initial height h of the die above the table not exceed the cube edgelength a. Clearly, at such a small height, ‘6’ will prevail in a series of throws; thefaces that have been initially sideways will occur rarely, while many throws wouldhave to be made until the occurrence of ‘1’. In these conditions, the theory (model)that allows the results of the experiment to be predicted successfully does not needto invoke any concepts of the branch of mathematics called the probability theory.We can just reckon ‘6’ to be the only possible result, consider the occurrence ofone of the side faces as a consequence of inexact adherence to the experimentalconditions, and consider the occurrence of ‘1’ as an exceptional event that requiresan additional study.

At h/a ≃ 1 − 2, the results of experiments will become more varied. Fortheir interpretation, we can invoke sophisticated means of recording the initialposition of the die and then calculate its motion on a supercomputer using thecorresponding equations of aerodynamics and elasticity theory. Obviously, in thisway, we will be able to achieve such a good accuracy of predictions that thedynamical model of the process will be considered acceptable.

However, a much simpler, probabilistic, model turns out to be also useful evenhere. It does not require a high accuracy of the information about the initialposition of the die and laborious calculations. This model postulates that thenumber written on the upper face is a random variable ξ that takes one of thevalues 1, 2, . . . , 6 with probabilities p1, p2, . . . , p6. Analysis of the results of a series

3The die is assumed to be numbered in a standard way, so that the sums of the numberson opposite faces are equal to 7.

4

of throws of the die from a fixed initial height shows that for the subsequentresults to be predicted successfully, we must assign the highest probability, p6,to the upper face, slightly lower probabilities, p2 = p3 = p4 = p5, to the fourside faces, and the minimum probability, p1, to the lower face. Of course, allof the introduced probabilities depend on the initial height of the die. As theratio h/a increases, the dynamical model leads to errors increasingly often, whilethe probabilistic model retains its efficiency, when the probability distribution{pk(h)}6k=1

is properly redefined. In particular, if the observed frequencies ofoccurrence of the faces at h/a ≫ 1 do not differ from 1/6 within the boundariesprescribed by mathematical statistics4, we have no reason to suspect asymmetryof the die. Otherwise, both models require a scrupulous study of the degree ofhomogeneity of the die material and the experimental conditions.

Why are we forced to abandon the deterministic model at a significant initialheight of the die? The reason is that as h/a increases, the factors that we dis-regarded play an increasingly important role: a small spread in initial positionsof the die, nonidentical operations of the die release mechanism, the influence ofthe environment, etc. In those cases where the result of the experiment changessignificantly with small variations in initial conditions and behavior of the process,the latter is called unstable. In classical mechanics, instability had not been con-sidered as a fundamental difficulty for a long time: the problem seemed to consistonly in a large volume of required information on the initial state of the system.Present-day studies have shown that the development of instability with time isoften exponential, so that extremely small deviations rapidly reach a significantvalue (see, e.g., Lichtenberg and Lieberman 1983; Zaslavsky 1984, 1999; Schuster1984). Since some uncertainty in the initial and boundary conditions is unavoid-able in view of the atomic structure of the material, many processes of the worldthat surrounds us are fundamentally unstable (these issues are discussed in moredetail in the next section).

The applicability of probabilistic models to the description of processes similarto the throwing of a die caused no particular disagreement. The stochasticity ofthe quantum behavior, about which we will talk below, apart, it could be assumedthat as all details of the process are refined (at least mentally), its result becomesincreasingly definite. The Brownian motion of particles ∼ 1µm in size in a liquidcan serve as an example. By invoking the simplest model of a random walk todescribe the motion of an individual particle, we draw useful probabilistic con-clusions about the behavior of an ensemble of many such particles. At the sametime, analysis of the long-term observations of one specific particle using a refinedtheory of the motion of small particles in a liquid allows us to take into account

4This discipline ascertains the results of imaginary and, hence, completely ‘pure’ ex-periments for samples of an arbitrarily large size.

5

the particle inertia and the viscosity effects and, hence, to predict its behaviorwith a greater certainty than that of the prediction provided by the model of apurely random walk (Chandrasekhar 1943). Similarly, it is impossible to create afinal, complete theory of turbulence; various probabilistic models of this complexphenomenon reflect to some degree the properties of viscous fluid flow.

The traditional description of traffic in big cities may be considered as a goodexample of natural transition to a probabilistic model. Any driver will undoubtedlyreject the assumption that his trips on Friday are random: he initially drove towork, subsequently visited several predetermined shops, and, finally, taking hisfamily, drove to the country cottage. In principle, municipal authorities couldcollect information about the plans of each car owner beforehand, but such anextensive picture on the scales of the city is not required at all – it will sufficeto introduce a stochastic model of traffic along major highways, correcting theparameters of the corresponding probability distribution depending on the time ofthe day.

3. Statistical mechanics

3.1. Instability of motion and irreversibility

The irreversibility of the evolution of large ensembles of particles is usually illus-trated by an example of a set of molecules of two types (by convention, ‘white’and ‘blue’) that fill a closed vessel. Initially, the white and blue molecules areseparated by a baffle. Experiment shows that after the removal of the baffle, bothtypes mix between themselves, so that the contents of the vessel appear ‘cyan’ afterthe relaxation time. Why has the inverse process, when the initial cyan mixtureis separated with time and the white molecules are in one part of the vessel, whilethe blue molecules are in its other part, been never observed?

To get a convincing answer to this question, let us first turn to a very simplethought experiment. Suppose that small identical balls move in a closed box inthe shape of a rectangular parallelepiped. At the initial time, all balls touch oneof the walls, their velocities are directed exactly perpendicular to this wall, andthe minimum separation between the centers of the balls exceeds their diameter.The initial velocities need not be identical; it is interesting to choose them to bedifferent in accordance with some law. We assume the wall surfaces, along withthe ball surfaces, to be perfectly smooth and the collisions to be absolutely elastic;there is no mutual attraction. Classical mechanics easily predicts the behavior ofthis system for an arbitrarily long time interval: each of the balls independentlyof others vibrates along a straight line segment perpendicular to the two walls. Ifthe initial velocities were not chosen in a special way (all such ways can be easily

6

specified in advance), then in the course of time the set of balls will scatter more orless uniformly between the two walls of the box. At the same time, if all velocitiesare reversed at an arbitrary instant of time, then the set of balls will return exactlyto its initial state. In a similar way, having mixed after many laps on the stadium,the long-distance runners will simultaneously return to the starting line if theyturn back by the signal of the referee and each will retain his speed.

Let us complicate the experimental conditions only slightly: suppose that thereis a small asymmetric convexity on the box wall at the point of collision of one ofthe balls. At the very first impact on the wall, the ball under consideration willbe reflected slightly sideways and, some finite time later, will hit one of the sidewalls. Generally, the subsequent trajectory of the ball is very complex; speciallychosen initial conditions apart, it can be asserted that, after a lapse of time, theballs will begin to collide not only with the box walls but also between themselves.Nevertheless, classical mechanics insists on the possibility of predicting the stateof the system after an arbitrarily long time interval as accurately as is wished;however, this requires knowing the initial positions and velocities of the balls withan infinitely high accuracy (Krylov 1950; Sinai 1970). The fully developed stateof such models is called deterministic chaos (Schuster 1984). Determinacy is at-tributable to the complete absence of random factors (noise), so that the evolutionis described by a system of differential or difference equations, while the allusionto chaos is attributable to extreme entanglement of the trajectories even in thecase of systems with only a few degrees of freedom. We emphasize: there is nochaos here in the true sense of this word, which implies dominant stochasticity ofthe behavior.

It should be kept in mind that states appropriately called illusory chaos can berealized in some macroscopic systems. Such systems have a long-term ‘memory’of their past states. We will give an example of a real experiment from Brewerand Hahn [1984]: “A viscous fluid is placed in the ring-shaped space between twoconcentric plastic cylinders. Whereas the outer cylinder is stationary, the innerone is free to rotate about its axis. A streak of colored dye, representing an initialalignment of particles, is injected into the fluid. When the inner cylinder is turned,the dye disperses throughout the liquid. If one were to show the volume betweenthe cylinders to a thermodynamicist, he or she would say that the dye is completelydisordered... Reversal of the rotation of the inner cylinder reverses the mixingprocess; after an equal number of reverse rotations the dye streak reappears.” Thecited paper includes impressive photographs. Obviously, a situation similar to theabove example with runners is realized here.

The phenomenon of nuclear spin echo discovered by E.L. Hahn in 1950 alsodemonstrates the possibility of a long-lived memory in systems with stable motionbut now on the atomic scale: “A sample of glycerin was placed in a magnetic

7

field and exposed to two short bursts of electromagnetic radio-frequency radiation,separated by an interval t of a few hundredths of a second. The sample retained amemory of the pulse sequence, and at time 2t seconds after the first radio-frequencypulse the sample itself emitted a third pulse, an echo” (Brewer and Hahn 1984).The experiment can be explained as follows. Glycerin is initially prepared byorienting the proton spins parallel to the external magnetic field. The first pulsetriggers a complex precession of the proton spins and the second pulse turns thespins through 180◦, so that all spins are again oriented identically after time t. Atthis moment, the atoms emit the echo pulse of radiation.

The evolution of the system in the last two experiments may be called regular,stable, because small changes in initial or external conditions do not lead to asignificant change in its state after a long time interval. The very possibility ofa regular evolution of macroscopic systems is of fundamental importance for theunderstanding of statistical mechanics: Generally speaking, we do not always haveto refer to probabilistic models.

On the other hand, even the early works by Henri Poincare (1892) and JacquesHadamard (1898) and, presently, many of the studies that followed the paper byE.N. Lorenz (1963) showed that “...stable regular classical motion is the exception,contrary to what is implied in many texts” (Schuster 1984). As a rule, even strictlyclassical systems with a small number of degrees of freedom in the absence of noiseexhibit instability of motion (generally, behavior): negligibly small variations ininitial conditions lead to a radical difference of the final pictures. In contrastto regular evolution, where the divergence of the phase elements with time is nohigher than the linear one, the divergence in unstable systems is very fast – itis exponential. Any finite accuracy of specifying the initial conditions guaranteesthe possibility of keeping track of the evolution of an unstable system only for ashort time; its subsequent behavior is indistinguishable from the evolution of asystem with a different initial state. The questions of what the relaxation timeis, whether a stationary, on average, density distribution will be established, andwhether this distribution will be uniform, or the regions of avoidance will remain,as well as many other problems concerning deterministic chaos, have been solvedonly partially.

We are forced to conclude that there are no tools at the disposal of classicalmechanics that would allow one to keep track of the evolution of an even idealsystem with a small number of degrees of freedom for a long time if it is unstable.This is all the more true with regard to real systems. For example, in the exper-iment we consider, not only the very complex pattern of roughness of the wallsand ball surfaces but also – since their constituent atoms move – the variabilityof these characteristics with time, the inelasticity of the impacts, and many otherphenomena should have been taken into account. Only recently the role of yet

8

another factor that effectively influences the behavior of a classical gas in a closedvessel has been assessed, the interaction of molecules with the radiation field. Itcan still be imagined how to isolate the gas from thermal radiation from the ves-sel walls, but the collisions between molecules even at moderate temperatures areinevitably accompanied by low-frequency electromagnetic radiation. Gertsensteinand Kravtsov [2000] showed that this mechanism leads to a significant deviationof the trajectory of a molecule from the results of purely Newtonian calculationsin an astonishingly short time, which only a few times longer the mean free-flighttime of the molecules. So, with the aid of classical mechanics, only some generalfeatures of the evolution of a many-particle system can be established and othermodels should be invoked to create a real picture.

Two factors necessarily considered jointly – the character of the initial state andthe instability of evolution – allow us to answer the question of why we see so manyirreversible processes in the world that surrounds us, say, the sea waves breakingagainst the shore do not recover their shape and do not go back. Many systemswere initially prepared – ordered – by Nature or man and then the instability ofevolution intervened. The same roughly ordered sequence of sea waves is generatedin a natural way – by a strong wind, whereas the recovery of a regular system ofwaves requires enormous purposeful work. As Richard Feynman [1965a] believed,the hypothesis that the Universe was more ordered in the past should be added tothe known physical laws.

Now, we are ready to return to the experiment with two types of moleculesin a closed vessel. Clearly, in conditions that approach the real ones at least inpart, the motion of an arbitrarily chosen molecule will be unstable. For example,if we launch several realizations of the mixing process from the same – withinthe theoretical possibilities of the experiment – initial state, with the externalconditions being retained as scrupulously as possible, then this molecule will be incompletely different regions of the vessel after a finite time in different realizationsof the process.

So far, when discussing the thought experiments concerning the evolution ofmany-particle systems, we have said no word about probabilities. Their intro-duction is inevitable, because the evolution of typical systems is unstable; for thisreason, real processes evolve in a way unpredictable for the researcher. (Einsteinsaid, “God does not care about our mathematical difficulties. He integrates empir-ically.”) In these conditions, which are much more complicated than those in theexperiment with the throw of a die, the researcher is forced to invoke a particularprobabilistic model by specifying an appropriate stochastic apparatus. Only af-terward and only within the framework of the adopted stochastic model do we havethe right to say that the initially separated state will eventually become uniformlycolored with a particular probability and the latter can return to the original state

9

with a low probability. Even in the model of an ideal gas, reversing the velocities ofall molecules will not return the system to the separated state because arbitrarilyweak stochasticity admitted by the researcher will not allow this to happen. Thisis all the more true for the models that represent real systems with a distinctinstability of motion.

As an example of a successful probabilistic model, we will mention the ‘dog-and-flea model’ suggested by Boltzmann and considered in detail by Paul andTatyana Ehrenfest in 1907 (see Kac 1957). The model illustrates the transitionto statistical equilibrium of the gas that nonuniformly fills a closed volume. Ini-tially, 2N numbered balls are distributed among two boxes in some specified way.Subsequently, a random number generator creates an integer that is uniformlydistributed on the set 1, 2, . . . , 2N ; the ball with this number is moved from thebox where it lies to the other box. The procedure is repeated many times. Thissimple model admits an exhaustive analytical study; it is now easy to perform anumber of corresponding computer realizations of the process as well. In partic-ular, it is curious to trace its evolution from the state when all balls were in thesame box. Above, we deliberately emphasized that the distribution of the randomball number is uniform: in general, some different discrete probability distributioncan also be specified. In this case, a new probabilistic model of the process willbe introduced which may describe better the actual behavior of the specific gassample. Obviously, at N ≫ 1, the system that reached statistical equilibrium inthe adopted probabilistic model will return to the original state only with a verylow probability.

It is easy to continue examples similar to those considered above, by succes-sively passing from simple situations to more complicated ones, from Boltzmann’sideas to Gibbs’ ensembles (see, in particular, the lectures by Uhlenbeck and Ford1963). Analysis convincingly indicates that the deterministic model proposed byclassical mechanics is unproductive when systems with an unstable behavior arestudied. The main objective of statistical physics is to construct adequate proba-bilistic models of such phenomena. This is also suggested by the name of this fieldof physics itself5.

5However, the word ‘statistical’ is illegitimately used in the literature instead of ‘prob-abilistic’ or ‘stochastic’. The latter terms emphasize the presence of random factors in themodel and must be contrasted with the term ‘deterministic’. In contrast, mathematicalstatistics deals with the problem that is inverse to the main range of problems of theprobability theory, namely, the reconstruction of information about the probabilistic lawsfrom a specified random realization. Therefore, one cannot say that the behavior of thesystem has a ‘statistical’ character.

10

3.2. Boltzmann’s H-theorem

The best known case of prolonged debates in classical physics spawned by theprobabilistic treatment of the phenomenon is related to the H-theorem. In 1872,Ludwig Boltzmann concluded that some function of time H(t) that characterizesthe state of a rarefied gas either decreases with time or retains its value6. In thisform, the H-theorem is in conflict with the symmetry of the laws of mechanicsrelative to time reversal and with the theorem proven by Henri Poincare accord-ing to which a closed mechanical system returns to an arbitrarily small vicinityof almost any initial state after a fairly long time interval. These contradictionscalled the Loschmidt and Zermelo paradoxes forced Boltzmann to turn to theprobabilistic treatment of the process of approaching equilibrium: the change inH(t) describes only the most probable evolution of the system. The present-dayformulation (Huang 1963) of the H-theorem also includes an important refinementof the initial state: “If at a given instant t the state of the gas satisfies the as-sumption of molecular chaos, then with an overwhelming probability at the instantt+ ǫ (ǫ→ 0) dH/dt ≤ 0.”

The aforesaid clearly shows that the H-theorem says not about the behavior ofan ensemble of classical particles but only about some probabilistic model intendedto sufficiently describe the approach of the system being studied to equilibrium.The physical mechanism that destroys the order is the instability of motion, whilethe law of entropy increase is only our statement of this objective phenomenon.Entropy is a useful model concept but by no means an objective property of Nature.This is evidenced at least by the fact that the value of entropy depends on theadopted discretization of the phase µ-space into cells: as the cell sizes decrease,the entropy decreases (see, e.g., Pauli 1954).

As long as one talks about the behavior of a deterministic classical gas, thearguments of Loschmidt and Zermelo remain valid. In contrast, the ‘gas’ intro-duced in the appropriate stochastic model need not obey all the laws of classicalmechanics. A good model developed for a specific situation has the right to disre-gard in full extent the classical reversibility in time, thus, not to obey Poincare’srecurrent theorem (especially since the corresponding cycle is excessively long!).For example, the H-theorem invokes a specific model of a rarefied gas that suf-ficiently describes its evolution from the hypothetical state of molecular chaosbeing realized in practice only with a limited accuracy. Precisely “This statisti-cal assumption [about molecular chaos] introduces irreversibility in time” (Uhlen-beck and Ford 1963). Present-day models of statistical physics, in particular, the

6For a rarefied gas, H(t) coincides with entropy taken with the opposite sign, so theH-theorem may be considered as a special case of the law of entropy increase in a closedsystem.

11

well-known Bogolyubov-Born-Green-Kirkwood-Yvon approach, introduce similarprobabilistic assumptions but on a deeper level than that of Boltzmann’s theory.

Clearly, being forced to choose some probabilistic model, we can no longer pur-port to describe in detail the process, which we hoped to do within the frameworkof classical dynamics.

3.3. Arrow of time

The widely discussed question of the difference between the past and the futureis usually assigned to the same range of problems of statistical physics. In viewof aforesaid it seems obvious that it is illegitimate to condition the directionalityof time (an ‘Arrow of time’, according to Arthur Eddington) by the behavior oflarge ensembles of particles described by statistical mechanics. Let us add thatotherwise we would have to observe not only fluctuations of the characteristics ofmany-particle systems, but time also. On the contrary, just the flow of time allowsus to observe successive states of an unstable system and to describe its evolutionusing probabilistic models. As was noted by St. Augustine [398], “Let no manthen tell me that the motions of the heavenly bodies are times, because, when atthe prayer of one the sun stood still in order that he might achieve his victoriousbattle, the sun stood still, but time went on.”

Clarifying the nature of time and, in particular, its directionality is a muchdeeper problem of physics.

4. Characteristic features of classical stochas-

tic models

Let us summarize the conclusions that follow from the analysis of probabilisticmodels of classical physics in order to subsequently trace the corresponding changesin quantum theory7.

It is well known that any model of the phenomenon under study is neitherunique nor exhaustive. Even a very successful model has boundaries within whichit is more preferable than other models, but a more perfect model usually replacesit when the range of studies is extended. For example, Eintein’s general theory ofrelativity replaced Newton’s theory of gravitation; the latter retained its efficiencyin the case of low velocities and weak gravitational fields. The choice of one ofthe many possible models is determined by the principle formulated by William

7The author hopes that he will not be rebuked for “playing with words specially madeup for this”.

12

Occam in the 14th century: The researcher must prefer the simplest model from anumber of alternatives that give an explanation for the experimental results.

Obviously, the simplicity of a model should be understood in a comparativecontext. Quite often, one has to use sophisticated constructions even within theframework of the simplest possible models. Say, when the distribution of people inheight is studied, the histogram is approximated by a Gaussian function that con-tains the seemingly irrelevant constant π (see Wigner’s 1960 well-known essay).Present-day models for the structure of matter invoke incomparably more com-plicated tools; as a rule, abstract mathematical constructions are used in them.Technically, the general theory of relativity is much more complex than Newto-nian theory, but the former invokes fewer a priory assumptions, which justifies itscomparative simplicity.

If the model is simple and productive and especially when it works for decadesor even centuries (as is now commonly said, became a paradigm), people beginto assign the concepts with which their theories operate to Nature itself. Forexample, in the past, a wide debate was caused by the gravitational paradox – thedivergence of the Newtonian potential in an unbounded homogeneous Universe.The divergence of the potential was believed to reject the infinite model. Asusual, the paradox only emphasized the conditionality of the tool we choose; if wetalk about the gravitational force at an arbitrary point of an unbounded and, onaverage, homogeneous Universe, then it is finite and obeys Holtzmark’s probabilitydistribution. To be more precise, if the spatial distribution of stars is described bya Poisson model, then the probability that the value of the force is larger than agiven value F rapidly approaches zero as F increases (see Chandrasekhar 1943).

Other concepts that became habitual in the working models of phenomena,for example, entropy, are also conditional. The subjectivity of any model is worthnoting once again, because it plays an important role in the picture of the worldto which quantum mechanics leads (see Section 9).

Strange as it may seem, many debates were caused by the confusion of theconcepts of ‘randomness’ and ‘unpredictability’. Unless you know the scheduleof a bus, its arrival is unpredictable for you, but you are free either to choosesome probabilistic waiting model or to call the dispatcher. The fact that we areunable to predict the position of a Brownian particle cannot serve as a reason forconsidering its motion to be objectively random; only the economical model wechose is such.

Similarly, the fact that we can not predict the sequence of digits in the infinitedecimal expansion of the number π = 3.14 . . . does not mean that they appear ran-domly. They are completely determined by Nature, but for a non-specialist in num-ber theory (such as the author) their seemingly random and plausible equiprobableappearance is a fairly good probabilistic model.

13

The above discussion of the procedure for throwing die, the models for theBrownian motion of small particles, the traffic in a big city, and typical models ofstatistical mechanics clearly illustrate the consistent viewpoint of classical physicson the fundamental question of how probabilities appear in the theory:

(A) There is no room for probabilities in Nature itself. The researcheris forced to invoke probabilistic models in describing unstable pro-cesses, more specifically, when the initial data and external conditionsfor the experiment are known only approximately, while its results de-pend on these circumstances so significantly that they become partlyunpredictable.

Einstein may have had something of this kind in mind in his aphorism about Godplaying dice: It seemed that, in contrast to classical physics, quantum mechanicsprescribes a probabilistic behavior to the objects of investigation themselves. Theincentives for such a radical assertion can be understood by considering an exper-iment with the passage of light through two slits, but we will defer this discussionuntil Section 6. In the next section, we consider an intermediate situation wherethe quantum behavior is consistent with the conclusion A in an obvious way.

5. Passage of polarized light through a crystal

of tourmaline

At the turn of the 19th and 20th centuries, a fundamental property of the recordingof light, its discreteness, was found. If the brightness of a light source is graduallyreduced, then the picture being recorded becomes increasingly ‘grainy’. It is easyto make the light source so weak that individual flashes, photoevents, separatedby long time intervals, say, more than an hour, are recorded. Significantly, whenmonochromatic light is used, the same portion of energy is recorded each time.This phenomenon can be explained by invoking the hypothesis put forward byEinstein in 1905: light consists of spatially localized quanta, photons, that haveenergy and momentum. Einstein obtained further evidence for the discrete modelof light 4 years later, when he analytically found an expression describing thefluctuations of radiation energy in a closed volume.

The above explanations were needed in connection with the description of anexperiment on the passage of polarized light through a crystal of tourmaline givenby Paul Dirac [1958]:

“It is known experimentally that when plane-polarized light is used for ejectingphoto-electrons, there is a preferential direction for the electron emission. Thus the

14

polarization properties of light are closely connected with its corpuscular propertiesand one must ascribe a polarization to the photons.

Suppose we have a beam of light passing through a crystal of tourmaline,which has the property of letting through only light plane-polarized perpendicularto its optic axis. Classical electrodynamics tells us what will happen for any givenpolarization of the incident beam. If this beam is polarized perpendicular to theoptical axis, it will all go through the crystal; if parallel to the axis, none of it willgo through; while if polarized at an angle α to the axis, a fraction sin2 α will gothrough. How are we to understand these results on a photon basis?

A beam that is plane-polarized in a certain direction is to be pictured as madeup of photons each plane-polarized in that direction... A difficulty arises, however,in the case of the obliquely polarized incident beam. Each of the incident photonsis then obliquely polarized and it is not clear what will happen to such a photonwhen it reaches the tourmaline.

A question about what will happen to a particular photon under certain con-ditions is not really very precise. To make it precise one must imagine someexperiment performed having a bearing on the question and inquire what will bethe result of the experiment. Only questions about the results of experimentshave a real significance and it is only such questions that theoretical physics hasto consider.

In our present example the obvious experiment is to use an incident beamconsisting of only a single photon and to observe what appears on the back sideof the crystal. According to quantum mechanics the result of this experiment willbe that sometimes one will find a whole photon, of energy equal to the energyof the incident photon, on the back side and other times one will find nothing.When one finds a whole photon, it will be polarized perpendicular to the opticaxis. One will never find only a part of a photon on the back side. If one repeatsthe experiment a large number of times, one will find the photon on the back sidein a fraction sin2 α of the total number of times. Thus we may say that the photonhas a probability sin2 α of passing through the tourmaline and appearing on theback side polarized perpendicular to the axis and a probability cos2 α of beingabsorbed. These values for the probabilities lead to the correct classical results foran incident beam containing a large number of photons.

In this way we preserve the individuality of the photon in all cases. We areable to do this, however, only because we abandon the determinacy of the classicaltheory. The result of an experiment is not determined, as it would be according toclassical ideas, by the conditions under the control of the experimenter. The mostthat can be predicted is a set of possible results, with a probability of occurrencefor each.”

The process of creating a probabilistic model for the phenomenon is described

15

very clearly in these words. As in the case of macroscopic objects, the necessity ofa probabilistic model in the experiment considered stems from the fact that “theresult of an experiment is not determined... by the conditions under the controlof the experimenter.” However, there is also a fundamental difference betweenthe situations. Whereas in classical physics we can still hope for a refinementof the conditions in which the experiment is carried out, in the microworld theexperimenter influences the process under study so significantly that turning to aprobabilistic model becomes inevitable. Having stepped on an anthill, one shouldnot be surprised by the fussiness of its inhabitants.

Nevertheless, if this was the only peculiarity of quantum phenomena, then thedifficulties with the perception of quantum mechanics that were mentioned in theIntroduction would remain incomprehensible. The problem consists not in the re-jection of the probabilistic description of the results of measurements; the point isthat the probabilistic model suggested by quantum mechanics is considered not asan approximate description of some deeper picture of microworld phenomena butis given as a fundamental property of Nature. The situation is often characterizedby asserting that there are no ‘hidden parameters’ in quantum theory that couldgive a more complete description (see, e.g., Faddeev and Yakubovsky 1980, pp.37-38). Such a strange model emerged in the course of painful attempts to explainthe peculiar features of the quantum behavior and, first of all, the seeming non-locality of quantum interaction. These features have been repeatedly illustratedusing thought and real experiments on the interference of light, to the descriptionof which we will now turn. For the subsequent comparison with the quantum pro-cedure, we will briefly repeat an elementary derivation of the expression for theintensity of light in the interference pattern.

6. Interference of light according to the clas-

sical wave theory

Let us first assume (Fig. 1) that there is only one narrow slit in an opaque screenon which the light from a bright source O is incident; the radiation passed throughthe slit is recorded by the detector located on the other side of the screen. If thelight is bright and the detector resolution is low, then a continuous flux distributionwith one maximum at point C lying on the axial line is recorded.

The experiment with one narrow slit imposes no serious constraints on themodel invoked for its interpretation. In contrast, Young’s experiment with twonarrow slits performed in 1801 (Fig. 2) required significant concretization of themodel, namely, the wave theory of light that includes the concept of wave inter-ference. It was necessary to explain the peculiar distribution of the light flux on

16

Figure 1: Light recording in the case of one slit in the screen.

the detector characterized by several maxima gradually decreasing with increas-ing distance from the axial line. The classical wave theory of light proposed byThomas Young, Christiaan Huygens, and Augustin Fresnel excellently coped withthis.

According to the wave model in the form that it attained by the late 19thcentury, light is a set of harmonic ether oscillations with various temporal periods Tand spatial wavelengths λ. Outside an ordinary material, the oscillation frequencyν ≡ 1/T is related to the wavelength by the relation νλ = c, where c is the speedof light. In the experiment considered in Fig. 2, the primary wave from source O isincident on the screen and generates electron oscillations in it; this is equivalent tothe fact that each of the slits serves as a source of coherent secondary waves. Theoscillation amplitude at some point Q of the detector is determined by the phasedifference between the waves arriving at this point from both slits. The reactionof the detector at this point depends on the local light flux, which is proportionalto the wave amplitude squared.

Consider, for simplicity, a monochromatic light source located at equal dis-tances from slits A and B. Near the slits, the source generates harmonic oscil-lations a sinωt with amplitude a and angular frequency ω = 2πν = 2πc/λ. LetD be the separation between the slits and z be the distance of the detector fromthe screen; we are interested in the oscillation amplitude at point Q located atdistance x from the axis. If we neglect the difference in the degree of attenuationof the secondary waves on their way from the screen to the detector due to thedistances ℓ1(x) and ℓ2(x) of the point of observation from the slits being unequal,then the waves of equal amplitudes but with different time delays ℓ1/c and ℓ2/ccan be assumed to arrive at point Q. The combined oscillation is proportional to

sin[ω(t− ℓ1/c)] + sin[ω(t− ℓ2/c)] =2 cos[π(ℓ2 − ℓ1)/λ] sin[ωt− π(ℓ1 + ℓ2)/λ].

(1)

This is a harmonic oscillation with amplitude 2 cos[π(ℓ2 − ℓ1)/λ]. As was said, the

17

Figure 2: Light interference in the case of two slits in the screen.

light flux F at distance x from the axis is proportional to the amplitude squared:

F (x) = 4 cos2(δφ/2) = 2[1 + cos(δφ)], (2)

whereδφ(x) = 2π(ℓ2 − ℓ1)/λ (3)

is the phase difference between the secondary waves. At x≪ z, we can assumethat ℓ2 − ℓ1 ≃ Dx/z, so that

F (x) ≃ 2

[

1 + cos

(

2πDx

λz

)]

. (4)

Formula (4) describes a periodic flux distribution on the detector. The spatialperiod of the pattern that specifies the linear resolution when the structure of thelight source is studied is

∆x = λz/D, (5)

and the corresponding angular resolution is ∆θ ≡ ∆x/z = λ/D.In reality, the height of the maxima decreases with increasing distance from

the symmetry axis, which necessitates a more developed theory. In particular, itshould take into account the increase in the distance of the point of observationfrom the slits and the finiteness of their width. Allowance for the latter factorleads to the expression

F (x) ≃ 2 sinc2(bx/λz)

[

1 + cos

(

2πDx

λz

)]

, (6)

where the function sinc(t) ≡ sin(πt)/(πt) and b denotes the width of each slit. Theadditional – compared to Eq. (4) – factor sinc2(bx/λz) predicts the decrease inbrightness with increasing distance from the axis to zero at

x0 = λz/b, (7)

18

and, since we assume that b≪ D, we have x0 ≫ ∆x and the resolution of thepattern is still specified by Eq. (5). A consistent allowance for other featuresof the experiment within the framework of a classical model leads to a detailedinterpretation of the observed pattern, with the exception of one significant fact –discreteness of the detector counts. We will discuss this phenomenon in the nextsection.

Let us give an example. Assume that the light wavelength be λ = 0.5µm,the slit width be 10 wavelengths, i.e., b = 5µm, the separation between the slitsbe D = 150µm, and the distance of the detector from the screen be z = 1 m.According to Eqs. (5) and (7), the period of the interference pattern is ∆x ≃3.3 mm and the characteristic size of the image modulation due to the finite slitwidth is x0 = 100 mm.

Thus, the classical wave theory of light gives a satisfactory description for thepattern in Young’s experiment averaged over a long time interval. This theory wasalso successful in interpreting an enormous number of other experiments.

7. Wave-particle duality

Let us now consider how the interference of light is explained by the models thattake into account its quantum nature.

First, let us turn to the passage of light through a single slit (Fig. 1). Ashas already been said, the observed distribution of photo-counts in the shape ofa single-humped curve increasingly exhibits granularity due to the absorption ofindividual photons with decreasing brightness of the light source. These photonsare simply like the classical particles (micropellets), because both the discretenessof counts and the overall shape of their distribution on the light detector can beexplained in this case.

In Young’s experiment with two slits (Fig. 2), the decrease in the brightnessof the light source also reveals the discreteness of photo-counts, but the classicalview of photons as micropellets turns out to be untenable. Indeed, the micropel-lets must come to the detector either through slit A or through slit B, so thatthe expected image is a superposition of two single-humped distributions shiftedrelative to the axis – basically, the projections of slits A and B onto the screenfrom point O. This pattern clearly differs from the observed distribution, whichis characterized by sharp intensity variations. The real distribution is formed byidentical photo-events, but the arrival of a photon at a given point of the detectordepends significantly on the presence of a second slit. If we close one of the slits,then we will see a smooth single-humped distribution shifted relative to the axis,while an interference pattern with a distinct alternation of extrema appears when

19

both slits are open. Try to imagine how opening the second slit in the modeloperating with photons-micropellets can reduce the frequency of their falling tosome place of the detector!

Thus, we get the impression that the propagation of light obeys the laws ofthe wave theory, while it interacts with the detector as if it consisted of local-ized particles. It is this situation that is usually characterized by the concept ofwave-particle duality. The brightness of the light source can be made so low thatindependent photon recording events will be recorded with a certainty. The dis-tribution of photo-events accumulated over a long time closely coincides with theinterference pattern that is observed in the case of a bright source. To explainthis fact, we have to assume that “each photon then interferes only with itself.Interference between two different photons never occurs.” (Dirac 1958, p. 9).

This conclusion emphasizes perhaps the most striking peculiarity of the quan-tum behavior. Indeed, the slits can be separated very far from each other on thescale of the light wavelength and, nevertheless, both the interference pattern andthe discreteness of counts are retained. In particular, for the example describedat the end of Section 6, the separation between the slits D was 300 wavelengths.Using lasers allows the ratio D/λ to be increased even more, by many times. Thetelescopes constituting stellar interferometers are tens and hundreds of metersapart.

Albert Einstein was the first to realize the inevitability of wave-particle dualityin describing light in 1908: “I already attempted earlier to show that our currentfoundations of the radiation theory have to be abandoned... It is my opinion thatthe next phase in the development of theoretical physics will bring us a theory oflight which can be interpreted as a kind of fusion of the wave and the emissiontheory.” Recall that the ‘emission theory’ that considered light as a flux of veryunusual particles was suggested by Isaac Newton.

8. Behavior of material particles

However difficult it is to imagine the diffraction of a single photon, but the idea ofelectromagnetic waves, which can naturally reach simultaneously two slits spacedfar apart, helps us in the case of light. However, a similar diffraction pattern isobserved if light is replaced by a flux of electrons or other particles with a nonzerorest mass (for brevity, such particles are often called ‘material’ ones), or even wholeatoms! This is evidenced by various experiments, the first of which was carriedout in the 1920s. In particular, the diffraction of uncharged particles, neutrons,by the crystal lattice formed by the atomic nuclei of a solid body is observed. Asatisfactory model of this phenomenon suggests the diffraction of neutron waves

20

by a crystal lattice whose spacing exceeds considerably the neutron wavelength.According to Louis de Broglie (1924), a wave process with the following wave-

length is associated with any particle whose rest mass m0 is nonzero:

λ =h

m0v

√

1− v2/c2 , (8)

where h ≃ 6.626 · 10−27 ergs·sec is Planck’s constant and v is the particle velocity.For example, when an electron (its rest mass is m0 ≃ 9.11 · 10−28 g) is acceleratedin an electric field with a potential difference of 1 kilovolt, it reaches speed v ≃1.9 · 109 cm/s; in this case, the de Broglie wavelength is λ ≃ 0.4 · 10−8 cm ≃ 0.4 A.If such a beam of electrons is directed to a nickel single crystal, for which thelattice spacing is about 2 A, then the appearance of interference extrema in thedistribution of scattered electrons should be expected. This pattern was firstobserved in 1927 by C. Davisson and L. Germer.

Thus, the experimental data suggest the wave nature of not only photons butalso material particles. Since the electron can be likened neither to a micropelletnor to a wave, we have to abandon the seemingly only possible alternatives whenconsidering the diffraction of electrons by two slits: (1) the electron passes eitherthrough one slit or through the other; (2) it passes through both slits simultane-ously. The point is that we unjustifiably transfer the concept of body ‘trajectory’worked out by macroscopic experience to the quantum world. This is discussed inmore detail in the next section.

The wave-particle duality has been discussed for about a century. Excellent ex-planations in this connection were given by E.V. Shpol’skiy [1974] in his course onatomic physics: “Since the properties of particles and waves are not only too differ-ent but also, in many respects, exclude each other and the electrons undoubtedlyhave a single nature, we have to conclude that the electrons are actually neitherthe former nor the latter, so that the pictures of waves and particles are suitablein some cases and unsuitable in other cases. The properties of microparticles areso peculiar and their behavior is so different from that of the macroscopic bodiesthat surround us in everyday life that we have no suitable images for them. How-ever, it is clear that since we are forced to use both wave and particle pictures todescribe the same objects, we can no longer ascribe all properties of particles andall properties of waves to these objects.”

New names have been repeatedly proposed for microworld objects. For exam-ple, Feynman once used the term wavicles, a derivative of the words waves andparticles. Unfortunately, none of the names was so apt to be widely used in theliterature. Habitually, quantum structures are most often called microparticles,but the aforesaid should always be kept in mind.

21

9. Quantum mechanics

Attempts to more deeply understand the nature of the wave-particle duality led inthe mid-1920th to the creation of quantum mechanics – the theory of phenomenain describing which Planck’s constant plays a crucial role. Quantum mechanicsand its development including the special theory of relativity – quantum fieldtheory – give a satisfactory description of the entire set of phenomena in theworld that surrounds us, with the exception of gravity. Some of the aspects ofthis description achieved remarkable agreement with experimental data, ∼ 10−10,which is indicative of a high efficiency of the model created by Erwin Schrodinger,Werner Heisenberg, Max Born, and Paul Dirac. At the same time, the words ofoutstanding physicists given at the beginning of this chapter suggest that, whileproviding a consistent formal procedure for calculating the results of experimentsin the domain of atomic phenomena, quantum mechanics raises difficult concep-tual questions. John Bell [1987], who proposed well-known experiments to testquantum mechanics, reached a bitter conclusion: “When I look at quantum me-chanics I see that it’s a dirty theory: You have a theory which is fundamentallyambiguous.”

9.1. Peculiarities of the apparatus

Consider the explanation of the experiment on the diffraction of particles, say,electrons, by two narrow slits proposed by quantum mechanics (this descriptioncan also be equally extended to the experiment with photons discussed in Section6). Analysis of this experiment allows the formal aspect of the quantum-theorycalculations to be perceived. For simplicity, consider a stationary process wheresource O provides, on average, a constant number of electrons in unit time (Fig. 2).It is required to find the mean particle flux at point Q located at distance x fromthe axis. In the case of two open slits, the calculation is as follows.

With the aid of the procedure briefly described below two complex numbersshould be formed: ϕ1(x) and ϕ2(x) – the amplitudes of the probability of electronpassage through slits A and B, respectively, followed by their detecting atQ(x). Bydefinition, the total amplitude of the probability of detecting at Q that we denoteby ϕ(x) is the sum of the amplitudes corresponding to all mutually exclusive paths;in our case,

ϕ(x) = ϕ1(x) + ϕ2(x). (9)

Knowledge of the amplitude allows us to find the probability f(x)dx that an ar-bitrary particle emerged from O will fall into an infinitely small interval of widthdx near point Q:

f(x)dx = |ϕ(x)|2dx. (10)

22

The electron flux near point Q and the number of photo-events per unit time areproportional to the probability density f(x). It follows from the two previousformulas that

f(x) = |ϕ1(x)|2 + |ϕ2(x)|2 + 2ℜ[ϕ∗

1(x)ϕ2(x)], (11)

where the symbols ℜ and ∗ correspond to the separation of the real part of thecomplex number and complex conjugation, respectively. The classical descriptionwould be restricted to the first two terms in Eq. (11), which define the probability ofpassage either through the first slit or through the second one. Quantum mechanicsintroduces the third term dependent on the phases; it is the relationship betweenthe phases ϕ1(x) and ϕ2(x) that allows the interference of microparticles to beproperly described. All of this resembles the operations that we performed inSection 6 when analyzing the diffraction of light by two slits, but the probabilityamplitudes cannot be treated as waves in ordinary space.

Obviously, before we turn to Eq. (11), we should specify how the probabilityamplitudes are calculated and what the rules for handling these quantities are.We will only cursorily touch on these questions here within the framework ofnonrelativistic quantum mechanics; for a detailed description, see the lectures byFeynman [1965b] and textbooks on quantum mechanics. Feynman showed thatthe probability amplitude ϕ for some path is defined by the action S, i.e., thetime integral of the difference between the kinetic and potential energies alongthis path:

ϕ ∝ eiS/h, (12)

where i is the imaginary unit and h is Planck’s constant divided by 2π. Thetransition amplitude, considered as a function of the final state, is the famous ‘psi-function’ ψ(x, t), which forms the basis for the adopted description of quantumphenomena. To find ψ(x, t), one should either solve Schrodinger’s equation orcalculate the action S and use Eq. (12).

Note, incidentally, that representation (12) elucidates the nature of the well-known principle of least action. Since Planck’s constant is small, paths with greatlydiffering S are characterized by an enormous phase difference and, hence, theircontributions cancel each other out; only for the paths near the S extremum is thephase variation small, so that the amplitudes are added constructively. Therefore,material particles, like photons, ‘choose’ the paths on which the action is extremal.

The main rule for handling the probability amplitudes says: If a given finitestate is attainable along several independent paths, then the total probabilityamplitude of the process is the sum of the amplitudes for all paths consideredseparately. We emphasize that the linearity of the system holds for the probabilityamplitudes, while the probabilities themselves are related to the amplitudes in aquadratic way. Strictly speaking, in the problem of particle diffraction by two slits

23

of finite width, we should have added the amplitudes for the set of all paths fromO to Q enclosed by the slits, but, as a first approximation, when the slit width issmall, each of the beams can be replaced by one path. It is this approximationthat is implied in Eq. (9).

When the probability amplitude is calculated for a path of complicated shape,the following rule turns out to be useful: for any route, the probability ampli-tude can be represented as the product of the amplitudes corresponding to themotion along separate parts of this route. For example, in our problem, the am-plitude ϕ1(x) is the product of ϕ(O → A), the transition amplitude from O to slitA, and ϕ(A → Q), the transition amplitude from slit A to point Q(x). SolvingSchrodinger’s equation or performing calculations according to Eq. (12) for sepa-rate paths of two possible transition ways from the source to the detector (pointof interest), and then multiplying the corresponding partial amplitudes, we canfind the distribution of the mean particle flux along the detector with the aid ofEq. (11).

9.2. Interpretation

The above brief description is intended only to emphasize the most importantfeatures of the quantum mechanical approach:

(B) The basic picture of phenomena is described in the language ofprobability amplitudes – new concepts that have no classical analogue.The complex probability amplitudes are calculated according to thespecified set of rules. It is possible then to find the probabilities ofvarious events that are defined as the squares of the absolute valuesof the corresponding amplitudes.

Thus, according to Max Born (1926), quantum mechanics initially dealt with theprobabilistic picture of physical phenomena. This inference, per se, is consistentwith conclusion A in Section 4: If probabilistic models are needed even for thedescription of classical unstable models, they are all the more inevitable in describ-ing microworld phenomena, where the predictability of measurements is restrictedby several fundamental circumstances, in particular, by Heisenberg’s uncertaintyrelations. For example, when the experiment in which the position x of a particleand the conjugate momentum p are measured simultaneously is carried out, thestandard deviations σx and σp of the measured quantities obey the inequality

σxσp ≥ h/2. (13)

Restrictions of this type are formally included in the apparatus of quantum me-chanics; their necessity becomes clear from the analysis of simple thought exper-

24

iments (see Bohr 1935; Born 1963, Ch. 4, Sect. 7; Shpol’skiy 1974, Sect. 148,149).

The actually significant difference between quantum mechanics and classicalmodels is attributable not to the probabilistic way of reasoning but primarily tothe peculiarity of the quantum behavior itself (above all, nonlocal – in the clas-sical sense – interaction), the abandonment of searches for a deeper deterministicunderlying picture, and the clarification of the key role of experimental condi-tions for the possibilities of describing the phenomenon under study. Collectively,these features required introducing new concepts, including the concept of wavefunction.

Classical physics invoked a probabilistic model for some phenomenon in a sit-uation where the impossibility of describing it exhaustively was obvious either inview of the instability of its behavior, or due to the extreme complexity of theaccompanying processes. Such is the origin of the models related to the throwsof die, the theory of Brownian motion, and the ensembles of statistical physics.The reality of the deterministic, in principle, behavior of the system under studyhas always been implied, even if it was not possible to give its detailed descrip-tion8. Quantum mechanics considers a probabilistic model not as an approximatedescription of some deeper picture of microworld phenomena but as a fundamen-tal property of Nature. This statement has repeatedly appeared in various formsthroughout the history of development of quantum theory; it will suffice to citethe opinion of Wolfgang Pauli [1954]: “It was wave or quantum mechanics thatwas first able to assert the existence of primary probabilities in the laws of nature,which accordingly do not admit of reduction to deterministic natural laws by aux-iliary hypotheses, as do for example the thermodynamic probabilities of classicalphysics. This revolutionary consequence is regarded as irrevocable by the greatmajority of modern theoretical physicists – primarily by M. Born, W. Heisenbergand N. Bohr, with whom I also associate myself”.

In the early 1950s, when this authoritative statement was made, the crucialrole of the instability of motion in classical physics was not yet so clear; now,the allusion to the possibility of reducing the probabilistic models of statisticalphysics and thermodynamics to deterministic laws seems untenable. As for the“primary probabilities” of quantum mechanics, Pauli most likely had in mindonly the inevitability of the probabilistic description of the microworld but notGod playing dice. The latter would become inevitable if we were dealing onlywith Nature, as such. In reality, however, we are always forced to give a jointdescription of the phenomenon and the experimental setup. In this connection,

8Einstein also had this in mind in 1916 in his theory of interaction of the radiation fieldwith atoms. However, the Einsteinian transition probabilities were the proclaimers of anew quantum theory...

25

Max Born [1963] noted that before the creation of the relativity theory, the conceptof ‘simultaneity’ of two spatially separated events was also considered self-evident,and only the analysis of the experimental foundations of this concept performed byEinstein showed its dependence on the frame of reference. V.A. Fok characterizedthe situation in the microworld as “relativity to the means of observations”.

The following two opinions separated by seven decades clarify well the essenceof the matter.

In his introductory article to the debate between Einstein and Bohr on com-pleteness of the quantum mechanical description of reality, Fok [1936] gave thefollowing clarifications9: “Quantum mechanics actually studies the objective prop-erties of Nature in the sense that its laws are dictated by Nature itself, not byhuman imagination. However, the concept of state in the quantum sense is notamong the objective concepts. In quantum mechanics, the concept of state mergeswith the concept of ‘information about the state obtained through a certain max-imally accurate experiment’. The wave function in it describes not the state inan ordinary sense but this ‘information about the state’... By the maximally ac-curate experiment we mean such an experiment that allows all of the quantitiesthat can be known simultaneously to be found. This definition is applicable toboth classical and quantum mechanics. However, in classical mechanics, there wasbasically one maximally accurate experiment, namely, the experiment that gavethe values of all mechanical quantities, in particular, the positions and momentumcomponents. It is because any two maximally accurate experiments in classicalmechanics give the same information about the system that one could talk aboutthe state of the system there as about something objective, without specifyingthrough which experiment the information was obtained”.

The second extract allows the viewpoint of a modern researcher, David Mer-min, to be judged: “A wave function is a human construction. Its purpose is toenable us to make sense of our macroscopic observations. My point of view isexactly the opposite of the many-worlds interpretation. Quantum mechanics isa device for enabling us to make our observations coherent, and to say that weare inside of quantum mechanics and that quantum mechanics must apply to ourperceptions is inconsistent.” (a quotation from the paper by Byrne 2007).

The many-worlds interpretation mentioned by Mermin concerns the interpreta-tion of the measurement process proposed by H. Everett in the mid-1950s (for thehistory and references, see Byrne 2007). Everett’s dissertation was discussed be-fore its publication in Copenhagen by leading physicists; the reaction was negative.Fok’s clarifications and Mermin’s remark clearly reveal the reason why Everett’sidea cannot be considered acceptable (see also comments by Feynman regarding

9Now, one says ‘complete experiment’ instead of ‘maximally accurate experiment’.

26

the role of an observer cited at the end of this section).The term ‘objectivity’ concerning the concept of a quantum system’s state is

occasionally perceived not quite unambiguously; therefore, the following examplemay prove to be useful. Suppose that we have a set of dice at our disposal, each ofwhich is made asymmetric in some known way. Say, one of the dice is made so thatnumber ‘6’ occurs very rarely. Obviously, the result of a throw is determined bothby the (subjective) choice of a die from the set and by the (objective) structureof this die. Similarly, the result of a quantum mechanical experiment dependsboth on the ‘objective state’ of the system unknown to us and on the character ofthe question asked by the experimenter by choosing the experimental conditions.As John Wheeler noted, “Schrodinger’s wave function bears to (the unknowable)physical reality the same relationship that a weather forecast bears to the weather.”

The above interpretation of the wave function, basically kept within the frame-work of the orthodox interpretation of quantum mechanics, allows some of theknown paradoxes to be avoided10. For example, putting the wave function in thelist of objects of physical reality led to a discussion of the problem of its ‘collapse’with a superluminal speed (!) as the result of carrying out an experiment. How-ever, we should then also say that the probability distribution {pk} of the possibleresults of throwing a die collapses similarly when the die stops. This did not hap-pen; paradoxes appear when the concepts introduced by us are ascribed to Natureitself.

A reasonable interpretation of the experiment on the diffraction of electrons bytwo slits should be sought in the same direction. The experimental data stronglysuggest that separate electrons are diffracted as if each of them came to the de-tector through both slits at the same time. Emphasizing the inevitability of ajoint description of the physical process under study and the recording, necessar-ily macroscopic, instrument, quantum mechanics requires particular thoroughnessin choosing the words and concepts used. In the case under consideration, theposed question is formulated as follows: Can the interference pattern and thepassage of electrons through a particular slit be observed simultaneously in someexperiment? A comprehensive analysis shows that using any means that allows theelectron trajectory to be established immediately destroys the interference (see, inparticular, the lectures on quantum mechanics by Feynman 1965b). Therefore,the question of whether an electron passes through both slits in the experiment oninterference is empty to the same extent as the question about the number of devilsfit at the needle tip widely debated in the Middle Ages. No physical experimentcan answer questions of this kind, hence, physics is forced to abandon the classical

10We everywhere understand a ‘paradox’ as an ‘imaginary contradiction’. Feynmanemphasized: “In physics there are never any real paradoxes because there is only onecorrect answer... So in physics a paradox is only a confusion in our own understanding.”

27

concept of ‘trajectory’ in the cases where the result can be achieved by variouspaths to which markedly differing probability amplitudes correspond.

Reality can be successfully described using the rule of addition of the proba-bility amplitudes and the fact that the laws of quantum mechanics seem absurdto us only says about the degree of discrepancy between our everyday experienceand the microworld laws.

It should be said that, apart from the underestimation of the macroscopicexperimental conditions specified by an observer, the viewpoint that clearly over-estimates the role of an observer in studying quantum phenomena is also fairlypopular. More specifically, it is believed that Nature is real only to the extentto which it appears before the observer. The feelings of an acting physicist thatcontinuously ponders new experimental data was vividly expressed by RichardFeynman [1995]: “This is a horrible viewpoint. Do you seriously entertain thethought that without the observer there is no reality? Which observer? Any ob-server? Is a fly an observer? Is a star an observer? Was there no reality in theuniverse before 109 B.C. when life began? Or are you the observer? Then there isno reality to the world after you are dead?”

9.3. Some probabilistic aspects of the uncertainty rela-

tion

Let us note, referring to the physical meaning of the inequality (13), that quite of-ten met ambiguity in interpretation of this relation caused largely by use of vagueconcept of ‘measurements error’. Meanwhile, the inequality concerns only therelationship between the characteristic widths σX and σP of the probability dis-tributions of X and its conjugate momentum P , which are interpreted as randomvariables.

For definiteness, we will continue discussion in a frame of experiment on diffrac-tion of weak flux of electrons on a narrow slit. Obviously, at registration of anyelectron, its position is measured with accuracy of an order of width of the slit,whereas its transverse momentum is defined by accuracy of an estimation the de-viation of impact point from an axial line. We can make width of the very massiveslit arbitrarily small, and the measurements with the detector arbitrarily detailed,so the accuracy of measurement of realizations (x1, p1), (x2, p2),. . ., (xN , pN ) in Nconsecutive passages of electrons through the slit is defined only by devices whichare used in experiment. The actually important feature of quantum phenomenais that narrowing a slit leads to increase of average width of a whole set of impactpoints on the detector. Thus, the true uncertainty of experiment is limited notby accuracy of individual measurements, but the wave nature of microparticlesgenerating the diffraction phenomenon.

28

From the formal point of view, the inequality (13) is a consequence of thefact that the coordinate density distribution f(x) and the conjugate momentumdensity distribution g(p) are interdependent. Really, for any state |ψ〉 the proba-bility amplitude in coordinate representation, 〈x|ψ〉 ≡ ψ(x), and the probabilityamplitude in momentum representation, 〈p|ψ〉 ≡ η(p), are connected by Fouriertransform:

η(p) =1√2πh

∫

e−ipx/hψ(x) dx, ψ(x) =1√2πh

∫

eipx/hη(p) dp. (14)

The mentioned interrelation of probability densities is caused by their quadraticdefinition through the probability amplitudes:

f(x) = |ψ(x)|2, g(p) = |η(p)|2. (15)

Hermann Weyl has shown that Heisenberg’s inequality (13) is a direct consequenceof equations (14) and (15); corresponding derivation can be found in the book byFok [2007], P. II, Ch. I, §7.

Said above suggests that conjugate random variables X and P possess not onlythe particular probability densities, but also a joint probability density f(x, p).Nevertheless, searches for a physically sensible joint density, the beginning towhich has put Wigner [1932], were unsuccessful. This circumstance deservs tobe included in the list of many ‘oddities’ of quantum mechanics. Ballentine [1998]has devoted to the quantum mechanics in phase space a special chapter of hisrecent monograph.

9.4. Quantum chaos

In classical mechanics, the concept of ‘chaos’ is associated with a distinct insta-bility of particle trajectories under small variations in initial data and ambientconditions. As we saw, it is the instability that impels us to introduce proba-bilistic models of classical phenomena. Therefore, the following question is quitelegitimate: Can the probabilistic nature of the quantum theory be attributable toa similar instability?

Obviously, the classical concept of chaos cannot be extended directly to quan-tum mechanics, which not only insists on the fundamental impossibility of a simul-taneous exact measurement of the conjugate coordinates and momenta of particlesbut also dispenses with the word ‘trajectory’. If the complete description of realityis assumed to be given by a wave function, then the concept of ‘quantum chaos’could be attempted to associate with the manner of time variation of the wavefunction defined by Schrodinger’s equation. However, the latter is a first-order

29

equation that always specifies a stable time evolution: two close, in some appro-priate sense, states remain so during the entire subsequent evolution11. As DavidPoulin [2002] points out, this conclusion follows even from the requirement thatthe system’s energy be real. In contrast, the description of chaotic systems in clas-sical physics is accompanied by differential equations of the second or higher orders(of course, invoking such equations is not a sufficient condition for the describedphenomenon being chaotic). Thus, a productive definition of quantum chaos, ifthere is a need for this concept, should be searched for in a different direction.

These searches are being conducted. In particular, it appears that a smalldifference, but now of the Hamiltonians rather than the initial states, can lead toan exponential divergence of initially close systems. Another important directionis the development of decoherence of quantum systems due to the influence of theenvironment. In view of the peculiar behavior of chaotic systems, we will also men-tion that it is desirable to refine Ehrenfest’s theorem concerning the passage to theclassical limit. All these studies are of interest in their own right, but the questionabout the origins of the probabilistic interpretation of quantum mechanics shouldbe associated not with them. We have in mind the “relativity to the means ofobservations” and the peculiarity of the concept of ‘state’ in quantum mechanicsthat were discussed in the previous section. Schrodinger’s equation describes notthe evolution of the system’s state that exists independently of the experimen-tation but, speaking somewhat simplified, the time behavior of the potentialitiesof a specific experiment with regard to the system under study. Therefore, it isgenerally illegitimate to expect that the variation in wave function will resemblethe behavior of classical particles.

On the other hand, the instability of evolution is important not in itself; inclassical physics, it determines the fact that serves as a real basis for turning toprobabilistic models, namely, it produces situations where, as Dirac said, “Theresult of an experiment is not determined... by the conditions under the controlof the experimenter.” But the latter, for a number of reasons, is also highlycharacteristic of our experience in microphysics. It is not only a matter of theinfluence of a classical observer on quantum processes. The inappropriateness ofthe concept of particle trajectory to the results of quantum experiments reflectsonly one aspect of a more general concept – the microparticle identity principle.Indistinguishability of two particles of a given class (say, electrons) themselves andtheir trajectories serves as a basis for the amplitude addition rule, which determinesthe characteristic features of the quantum theory.

So far we have restricted ourselves to the discussion of nonrelativistic quantummechanics, because the inevitability of introducing a probabilistic model is clear

11The same is true with regard to the systems for which only the density matrix exists.

30

even within the framework of this theory. However, no interpretation of a widerange of experimental data is possible without invoking the special theory of rel-ativity. The picture of the world painted by the corresponding model – quantumfield theory – is much richer in colors. For example, it appears that the smallerthe interaction scale, the more intensely the physical vacuum ‘boils’: new parti-cles incessantly ‘evaporate’ from the vacuum and annihilate almost immediately.Nevertheless, virtual particles often have time to interact with real particles andatoms, which determines the observed effects (in particular, Einsteinian sponta-neous transition coefficient). This picture, which more deserves the name quantumchaos, clearly suggests that the probabilistic approach in describing the microworldis quite natural.

10. Spooky action at a distance

The words in the title of this section were used by Einstein, who was the first to seestriking corollaries of the quantum theory. In classical physics, we got accustomedto the fact that the interrelation between events is attributable either to the actionof one of them on the other or to their common past history. In the opinion ofEinstein, Podolsky, and Rosen [1935], the orthodox quantum theory introduces anew type of interaction that can manifest itself in the parts of the system thathave already ceased to influence each other in an ordinary sense. This paper isso often referred to that the abbreviation EPR became common for it and for thecorresponding effect.

In Schrodinger’s papers that followed the EPR work in the same year, the term‘entanglement’ of the properties of noninteracting systems was used. Over the lastquarter of the century, this phenomenon was subjected to thorough experimentaltesting; we will consider an idealized version of the real experiment that retainsthe essence of the original (Bohm 1952, Ch. 22). To abstract from insignificant,in this context, features of the experiment related to the particle charge, we willdeal with a neutral particle, say, a neutron.

10.1. Thought experiment by Einstein, Podolsky, and

Rosen

First, recall some of the peculiarities of quantum measurements using a specificexample. Let it be required to measure the spin of a neutron – its intrinsic me-chanical moment and the related magnetic moment. This can be done using theStern-Gerlach setup, in which the microparticles fly between the poles of a magnetthat produces a strong and, what is important, nonuniform magnetic field. Since

31

Figure 3: Scheme of the thought experiment on measuring the spins of twoneutrons emerging from source O. Detectors A and B are separated by animpermeable baffle (hatched), numbers 1, 2, and 3 mark the directions inwhich the spin is measured.

the spin of a free neutron is an arbitrarily directed vector, one could expect thatwhen measuring its projection in the direction specified by the arrangement of themagnet poles, we would obtain a certain value from a continuous range of val-ues. However, experiment shows that the neutron spin projection vector is alwaysequal in magnitude to h/2 and is directed to one or the other pole of the magnet.The appearance of Planck’s constant here suggests that the microparticle spin is apurely quantum property; the angular momentum of a body about an axis passingthrough the center of inertia represents its indirect analogy in classical mechanics.According to the interpretation adopted in quantum mechanics, the choice of oneof the two possible neutron spin projection directions is random. In the experimentunder consideration, the corresponding probabilities are

p+ = cos2(θ/2), p− = sin2(θ/2), (16)

where θ ∈ [0, π] is the angle between the spin vector and the direction in which it ismeasured. As we see from Eqs. (16), if the spin was initially directed to one of thepoles, then its measured projection would retain its direction. These peculiaritiesof the experiment are a special case of a general principle of quantum mechanics:when an observable quantity is measured in a closed system, one of the eigenvaluescorresponding to this quantity will be obtained (Dirac 1958, Sect. 10). In our case,the eigenvalues of the spin are ±h/2.

Let us now turn to the critical experiment (Fig. 3). In source O, the systemof two neutrons is prepared in such a way that its spin is zero. Subsequently, the

32

system spontaneously breaks apart, so that the neutrons fly in opposite directionstoward observers A and B who have Stern-Gerlach-type detectors at their disposal.For brevity, the detectors and neutrons are denoted by the letters corresponding tothe observers. The source and the observers are considered in a common inertialframe of reference; the distances OA and OB are assumed to be equal. (Theconclusions do not change fundamentally if, say, OB is slightly larger than OA,so that the measurement in B is made slightly later than that in A.) If thedetectors are far apart or are separated by an impermeable baffle, then the neutronsno longer interact, in the classical understanding of this word, with one anothershortly before their recording. Since the result of measuring the spin projectionfor one particle is random, it is clear that the result of each individual experimentto measure the spins of two neutrons with the corresponding detectors will also berandom. The question is how correlated the counts of detectors A and B are.

First, we will attempt to predict the results of the experiment using a semiclas-sical model that makes it possible to independently consider the remote neutronsbut takes into account the quantum character of the spin measurement expressedby Eqs. (16) firmly established in experiments. Subsequently, we will give theconclusions of an analytical study of the same problem in terms of the quantumtheory and, finally, will present the corresponding experimental data.

10.2. Semiclassical model

Taking into account spin conservation in a closed system, we should consider thetotal spin of a system of two neutrons flying apart to be always zero. In theclassical approximation, if this was the case for the spin, it could be assumed thatthe neutrons are independent and their spins are directed oppositely along somestraight line whose orientation can change arbitrarily in successive experiments.We will consider the direction from the south pole of the detector to its north poleto be positive. The subsequent explanations will be simplified if we equip eachdetector with two lamps, green and red, and adopt the following condition: thegreen (G) and red (R) lamps turn on if the recorded neutron spin projection isoriented, respectively, in the positive and negative directions of a given detector.Thus, one of the four combinations of turned-on lamp colors can be realized in aseparate experiment: GG, GR, RG, or RR. Let us find the probabilities of theseevents by assuming, for simplicity, that the detectors are oriented in the samedirection, say, in direction 1 (Fig. 3).

Denote the angle between the positive direction 1 and the direction of thespin vector for neutron A realized in a given experiment by Θ ∈ [0, π]; the anglebetween the spin of neutron B and the same direction 1+ will be π − Θ. As wassaid, the spin projection measured in A will be randomly oriented in the positive or

33

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Θ

g(Θ

)

Figure 4: Recording probability of oppositely directed neutron spin projec-tions versus angle between the detector and spin orientations for the semi-classical model.

negative direction of vector 1. Denote the possible results of the measurement forneutron A by A1+ and A1−, respectively; for neutron B, the possible results willbe B1+ and B1−. Taking into account the independence of the neutron recordingevents and Eq. (16), we obtain for the sought probabilities of the events:

Pr(GG) = Pr(A1+ ·B1+) = cos2(

Θ2

)

cos2(

π−Θ2

)

= cos2(Θ/2) sin2(Θ/2),

P r(GR) = Pr(A1+ ·B1−) = cos2(

Θ2

)

sin2(

π−Θ2

)

= cos4(Θ/2),

P r(RG) = Pr(A1− ·B1+) = sin2(

Θ2

)

cos2(

π−Θ2

)

= sin4(Θ/2),

P r(RR) = Pr(A1− · B1−) = sin2(

Θ2

)

sin2(

π−Θ2

)

= sin2(Θ/2) cos2(Θ/2).

(17)It is easy to verify that the sum of all probabilities (17) is equal to 1.

We are particularly interested in the probability that oppositely directed spinprojections are recorded, i.e., the lamps of different colors will turn on:

Pr(GR) + Pr(RG) ≡ g(Θ) = 1− 1

2sin2 Θ. (18)

The function g(Θ) has the meaning of a conditional probability of occurrence anevent given angle Θ. As Fig. 4 shows, only at Θ = 0 and Θ = π, i.e., when the‘spin-line’ is oriented in the same way as the detectors, are oppositely directed spinprojections recorded with confidence; for noncoincident orientations, the values ofg(Θ) lie between 0.5 and 1.

Obviously, the model with fixed Θ is insufficient to explain real experiments;it is more appropriate to assume that this angle changes in some way in successive

34

experiments. This means that Θ should be considered as a random variable. Inparticular, for an isotropic orientation of the straight line along which the neutronspins are directed, the probability that this straight line will fall within the solidangle dω is dω/4π = sin θdθdϕ/4π, where the factor (1/2) sin θ is the distributiondensity of the polar angle Θ and 1/2π is the distribution density of the azimuthangle Φ. Averaging p+ from (16) over all angles yields the probability that detectorA in a given experiment will record the positive spin direction, i.e., the green lampwill turn on:

1

4π

∫

2π

0

dϕ

∫ π

0

p+(θ) sin θdθ =1

4π

∫

2π

0

dϕ

∫ π

0

cos2(θ/2) sin θdθ = 1/2. (19)

As would be expected, each of the detectors records one or the other spin pro-jection with a probability of 1/2, so in a long series of experiments, the lamps ofdifferent colors turn on equally frequently. However, we are more interested inthe correlation between the results, which requires calculating the unconditionalprobability pd of recording oppositely directed spins. This is achieved by averagingEq. (18) for the function g(Θ) over all angles:

pd =1

4π

∫

2π

0

dϕ

∫ π

0

(

1− 1

2sin2 θ

)

sin θdθ = 2/3. (20)

Thus, the semiclassical model predicts that for an isotropic distribution of theneutron spin direction, on average, 2/3 and 1/3 of the experiments will lead to therecording of oppositely and identically directed spin projections, respectively.

The isotropic model we considered cannot be reckoned to be mandatory; it isonly an example of one of the possible spin direction distributions for the particlesflying apart. Only the fact that follows from the form of the function g(Θ) in (18) isactually important: for any spin direction distribution that admits a deviation fromthe orientation of the detectors, the probability of obtaining oppositely directedspin projections in the case of independent recording events is less than 1.

10.3. Results of quantum mechanical calculations

We will now briefly present the conclusions that follow from rigorous quantummechanical calculations. David Bohm [1952] performed such calculations for asituation where the detectors were oriented in the same direction and the recordingevents occurred simultaneously12. His conclusions are:

12The basically analogous process of positronium annihilation is considered in Feynman’slectures on physics [1965b], Sect. 18.3. This solution is briefly reproduced in the Appendixto this chapter.

35

1. In the initial state of the system, when its total spin is fixed, it is admis-sible to talk about the spins of individual neutrons only provisionally. Thepossible spin states of a pair of neutrons depicted as ↑↓ and ↓↑ interferebetween themselves and just this interference provides both the fixed totalspin during the experiment and the conjugacy of the results of measuringthe spins for remote neutrons.

2. In individual experiments, the possible values of the spin projection for eachof the neutrons, positive and negative, occur randomly with a probability of1/2; detectors A and B always record the opposite spin directions.

3. The unequivocal connection between the results of measuring the particlespins does not point to any influence of one of them on the other after thetermination of the interaction understood in the classical sense.

Thus, the predictions of the two theories are significantly different: accordingto quantum mechanics, oppositely directed particle spin projections are alwaysrecorded, i.e., pd = 1, while for all nontrivial semiclassical models there is a nonzeroprobability of identically directed projections and pd < 1.

Why does not the assumption of the independence of the neutron states holdwhile the quantum theory confirms that there is no interaction between the neu-trons? The reason is that both neutrons always constitute a single quantum systemthat turns to one of its two eigenstates as a result of the measurement. In eachof the system’s eigenstates, the neutron spin projections are opposite. “If thereare two particles in nature which are interacting, there is no way of describingwhat happens to one of the particles by trying to write down a wave function forit alone. The famous paradoxes... where the measurements made on one particlewere claimed to be able to tell what was going to happen to another particle, orwere able to destroy an interference have caused people all sorts of trouble becausethey have tried to think of the wave function of one particle alone, rather than thecorrect wave function in the coordinates of both particles. The complete descrip-tion can be given correctly only in terms of functions of the coordinates of bothparticles.” (Feynman 1965b, vol. 3, p. 231).

10.4. Verification of Bell’s inequalities

The EPR thought experiment remained as such until the appearance of the papersby Bell [1964, 1966], who pointed out the possibility of its experimental verifica-tion. Imagine that a series of experiments of the type described above is madein the scheme shown in Fig. 3, but now the detectors are not fixed in the samedirection – the observers randomly choose one of the three possible orientations of

36

their instruments independently of one another. As above, the observer records anevent of one of the two types in an individual experiment: (G) the direction of themeasured spin projection coincides with the arbitrarily chosen positive orientationof the magnet poles, and (R) the measured spin projection is directed oppositely.Bell proved that considering the neutrons as independent particles in the classicalsense (in other words, the existence of hidden variables) entailed the fulfillmentof the rigorous inequality for the probabilities of the combinations of events of acertain type.

Bell’s theorem, especially in the form that was imparted to it by Clauser etal. [1969], allows the conclusions of the classical and quantum models to be reallycompared (one can find the clear formal discussion in a lecture course by Kiselev2009). The fact that in almost all experiments photons were used instead ofneutrons and the photon polarization rather than the spin projection was measureddoes not change the essence of the problem. Pairs of photons with correlatedlinear polarizations were produced under two-photon transitions of exited atoms;the polarization direction of each photon was ascertained using calcite crystals orother analyzers (Shimony 1988).

Almost all experiments, starting from the first of them performed by Freedmanand Clauser [1972], suggest that Bell’s inequalities break down; in the cases wherethe opposite result was obtained subsequent verification revealed shortcomings ofthe experiment. As was said above, Bell’s inequalities must hold if the classical un-derstanding of independence is valid, therefore, their breakdown suggests that theexistence of hidden parameters is incompatible with the behavior of the microworld.The experiments performed by Aspect et al. [1982] became widely known (see alsoAspect 1999; Mermin 1981; Brida et al. 2000). To exclude the possibility of mu-tual influence of the recording events in a given experiment, the orientation of thepolarization analyzers was chosen during the flight of photons. Experiments onthe correlation of photons in the parts of the setup the separation between whichreached 18 km have been performed recently (Salar et al. 2008); these experimentshave again demonstrated the unavoidability of the quantum behavior of light.

Although the described experiments are spectacular, it should be recognizedthat fundamentally they have added little since the debate between Einstein andBohr regarding the meaning of the EPR effect and, especially, after the formalclarification of the problem by Bohm. However, the work of Bell is of great impor-tance in more general relation, namely, substantiation of quantum mechanics. Thepoint is that in his article of 1966 Bell discovered that the known von Neumann’s(1932) proof of impossibility to incorporate hidden parameters into quantum me-chanics relied on an erroneous assumption (see also an excellent discussion byRudolf Peierls 1979). The Bell’s argument has given a new, this time a correctproof that the theory of classical realism with hidden variables can reproduce the

37

experimentally confirmed predictions of quantum mechanics only by violation anessential physical requirement, the condition of locality.

10.5. Illusory superluminal speed

The classical interpretation of the above-mentioned experimental data inevitablyleads to the supposition that the signals between remote observers can be transmit-ted with a superluminal speed, which is in conflict with the experimentally testedpostulate of the special theory of relativity. In numerous publications, not onlypopular ones, fairly vague explanations are given on this subject. In particular,one alludes to the mysterious ‘collapse of the wave function’ (wave packet) thatwe have already commented on in Section 9 devoted to the laws of quantum me-chanics. Meanwhile, the transmission of a signal (information) between observersis not required at all within the framework of the latter for both identical andindependent orientations of the detectors. In both cases, the lamps of differentcolors on each of the detectors turn on equally frequently and if the observers areisolated from each other, then the strong correlation between the measurementresults remains unknown to them. The existence of a correlation will be revealedonly after the completion of a fairly long series of experiments, when all data willbe collected in one place. The same is also true for the situation where one of theobservers will change the orientation of his detector, thereby attempting to codethe message to his colleague: the lamps of different colors on each detector turnon equally frequently, while the correlation between the results remain hidden.

It is also worth noting that knowledge of the quantum laws by the observersdoes not promote the transmission of information between them either. Supposethat the detectors are oriented identically and observer A is slightly closer to thesource than B, so the former will perform a measurement slightly earlier than thelatter. If observer A sees the turn-on of the green lamp, then he immediatelyreceives one bit of information relative to the result of observer B: the latter willsee the turn-on of the red lamp. However, this information is provided by the set ofall stages of the experiment: preparing the initial two-particle system, providingthe corresponding information to the observers, and, finally, the flight of particlesfrom the source to the detectors.

Let us explain this by a simple example. Suppose that an inhabitant of Siberiasends messages to two friends living in Paris and Tokyo. All three agreed in ad-vance that a piece of green paper be randomly (e.g., equiprobably) enclosed inone of the envelopes and a piece of red paper be enclosed in the other. Obvi-ously, having opened the envelope, the Parisian will immediately learn the colorof the enclosed paper in the envelope of Tokyo’s resident. In this case, no infor-mation is transmitted between the remote points – the connectivity of the events

38

is attributable to their common past history.Basically, we see in the EPR effect the same manifestation of the specific quan-

tum behavior as in the experiment on the diffraction of electrons by two widelyseparated slits. This is quite sufficient to once again, as Bohr said, experience ashock when one familiarizes oneself with quantum mechanics, but it is absolutelyunnecessary to invoke the reasoning about the influence on remote objects with asuperluminal speed. Like most other paradoxes of quantum mechanics, this rea-soning is attributable to the improper use of the concept of a state of a quantumsystem (see Section 9.2 above). Changing the detector orientation, the researchercarries out a new experiment whose description enters as a component into the newstate of the entire system. It is quite natural that changing the experimental con-ditions can affect the result obtained. By choosing the orientation of his detector,the observer selects the possible alternatives of the experiment, thereby creatingthe illusion of influence on a remote object. For the subsequent ‘explanation’ ofthe effect, it remains only to repeat what was said in Section 10.3 and to refer tothe analysis given in the Appendix.

Einstein rightly believed the direct action at a distance to be inadmissiblein physical theories, but there is no need to resort to this concept in quantummechanics.

11. Schrodinger’s cat could play dice

An impressive illustration of the concept of entanglement of quantum states wasproposed by Erwin Schrodinger in his 1935 papers initiated by the work of Einstein,Podolsky, and Rosen. The case in point is a thought experiment, in which a catis in a superposition of partially alive and partially dead states. Consider theprobabilistic aspect of this construction.

Imagine that the following objects were placed in a closed box: a cat, a grainof radioactive material, a Geiger counter, an ampoule with a quickly acting poison,and some actuating device that breaks the ampoule when the counter is triggered.The decay half-life of the radioactive atoms is 1 hour; poison release leads to im-mediate death of the cat. The critical question is: How must an external observerdescribe the cat’s state after several hours?

One usually reasons as follows. There are two eigenstates of the entire (closed)system: the first corresponds to the situation where a β-particle has not yet trig-gered the counter, the ampoule is intact, and the cat is alive; the second statecorresponds to the case where the poison spilled, causing the death of the cat.The complete description of the system is given by a wave function that is a su-perposition of eigenstates with weight coefficients determined by the details of the

39

experiment. As has already been noted above, one of the possible states of anyclosed system is realized only when the measurement is done. Therefore, the catis partially alive and, at the same time, partially dead until the observer opens thebox and, thus, ‘measures’ its state.

The classical experiment tells us that the object is in one of the possible statesirrespective of whether the measurement is made. (Einstein: “The moon existseven when I don’t look at it.”) Accordingly, the ‘classical cat’ is either alive ordead but we do not know precisely in which state it is. In contrast, the abovedescription leads to the conclusion that the cat is in a strange superposition ofeigenstates.

Let us now consider the experiment more carefully. First of all, it should beemphasized that a subsystem of a closed system, in our case, the cat, cannotbe characterized by a wave function. The state of the subsystem is describedusing the density matrix introduced by von Neumann, which corresponds to amixture of several pure states taken with fixed weights. In the probability theory,this corresponds to the so-called randomization of the distribution function overthe possible values of some parameter. Landau and Lifshitz [1963] pointed out,“For the states that have only the density matrix, there is no complete systemof measurements that would lead to unequivocally predictable results”. Note alsothat the evolution of the subsystem depends on all details of its interaction withthe remaining part of the system.

Thus, there is no way of ascertaining the cat’s state until the observer opensthe box, hence, the question about how it feels is vacuous. Physics answers onlyreasonable questions13. The present-day experiments with macroscopic systemsthat have two eigenstates are properly described using the apparatus of quantummechanics (Blatter 2000).

In Schrodinger’s experiment, the situation is dramatized by the fact that ahabitual breather is involved in it. Nothing will change essentially if we replacethe cat, say, with a pendulum clock and the poison with a stopper. We can go evenfarther and place a microlaser or even a vibrating diatomic molecule in a closedsystem instead of the clock.

The paradox with Schrodinger’s cat can also be resolved without using suchformal concepts as the density matrix: the terminology itself invoking the con-cept of a system that is partially in different states is provisional. For example,analyzing the experiment with the passage of light through a tourmaline crystaldescribed in Section 5, Dirac [1958] wrote: “Some further description is necessaryin order to correlate the results of this experiment with the results of other exper-iments that might be performed with photons and to fit them all into a general

13However, as science in general, physics is able to answer not all reasonable questions.

40

scheme. Such further description should be regarded, not as an attempt to answerquestions outside the domain of science, but as an aid to the formulation of rulesfor expressing concisely the results of large numbers of experiments. The furtherdescription provided by quantum mechanics runs as follows. It is supposed thata photon polarized obliquely to the optic axis may be regarded as being partlyin the state of polarization parallel to the axis and partly in the state of polar-ization perpendicular to the axis.” Obviously, the cat in two states is no moresurprising than the analogous behavior of a photon. The ‘simultaneous’ passageof an electron along all possible trajectories in Feynman’s approach should also beunderstood in the same sense (see Section 9.1).

Let us also touch on the following curious question: Will the description ofthe system change from the standpoint of an external observer if the radioactivematerial is replaced with an ordinary die? Say, if the clock hand (or the cat itself)throws the die from a height very large compared to its sizes after a given time. Ifan odd number occurs, the actuating device (or the cat) breaks the ampoule withpoison.

The question touches on the widespread conviction that “...without a β-particle,nobody could even think about the admission of such a strange superposition”(Kadomtsev 1999). Only atomic phenomena, in particular, radioactive decay, arewidely believed to provide ‘true’ randomness, unlike the behavior of macroscopicbodies governed by the laws of classical mechanics. However, in the experimentwith Schrodinger’s cat, if it is considered from the viewpoint of quantum mechan-ics, it does not matter precisely how the necessity of a probabilistic descriptionof the subsystem from the standpoint of an external observer is provided. As forquality of the device that implements stochasticity, analysis of the probabilisticmodels of classical physics suggest the following: when the simple conditions thatprovide instability of motion are met, the behavior of the die is as unpredictableas radioactive decay. After all, the die can be reduced to the sizes of a Brownianparticle without violating the fundamental aspect of the experiment.

12. Conclusion

The above examples strongly suggest that God does not play dice; in the quantumworld, as in classical mechanics, the probabilistic concepts are inherent only inmodels of real phenomena but not in Nature itself. The word ‘model’ is a key onehere – as soon as the stochastic behavior is ascribed to some real objects, such assea waves, billiard balls, molecules, or photons, the appearance of contradictionsbecomes inevitable. This assertion seems so obvious that it remains only to won-der how widely the opposite viewpoint is covered in literature, including classical

41

works. The probabilistic concepts, as a branch of mathematics, were created forseveral centuries in order to formalize various life experiences and scientific data.It would be strange to believe that these concepts were ‘built’ in objective real-ity from the outset; the probability theory is only an efficient tool for modelingNature.

Another conclusion is that standard quantum mechanics in terms of which ourdiscussion was conducted gives a strange but logically consistent explanation for allof the experiments performed to date. This is enough to leave aside the widespread,in recent years, attempts to associate the interpretation of experiments with theconsciousness of an observer, the birth of universes at each act of observation,the transmission of information with a superluminal speed, and other mysticalphenomena. The aforesaid by no means rules out the quest for models differentfrom the theories of Schrodinger, Heisenberg, and Feynman; they may give a deeperand clearer picture of quantum processes (in fact, Feynman’s approach servesas such an example). New experiments may also require a fundamentally newapproach, but it is quite unrealistic to expect the appearance of a deterministicbasic model.

Note also that the probabilistic formulation of models for real processes de-termines the statistical nature of the inverse problem that consists in ascertainingthe true properties of the processes from their observed manifestations (Terebizh1995, 2005).

It has been repeatedly pointed out that the difficulties in understanding thequantum theory primarily stem from the fact that the behavior of quantum sys-tems is unusual: our everyday experience concerns the properties of surroundingus massive bodies and waves, while the microparticles are neither the former northe latter. This is true, but still it is hard to avoid the feeling that the rules ofquantum mechanics are a set of strange procedures justified by nothing but theirunconditional practical efficiency. If there was no statistical physics, similar feel-ings would also be aroused by the laws of thermodynamics – while being importantin engineering. In a similar context, Richard Feynman described the algorithm forpredicting solar eclipses developed for centuries by South American Indians. Fromgeneration to generation, priests handed ropes with many tied knots to their dis-ciples; it took many years to memorize the rules for handling the knots, but themeaning of these rules remained completely mysterious for Indians. We will addthat the system worked like the model proposed by Thales of Milet in the 6thcentury B.C.; a significant difference between the approaches is that, without ex-plaining the nature of the motion of celestial bodies, Thales’s theory neverthelessproceeded from a simple model of the Solar system. Critics of quantum mechanicsalso would like to have a simpler underlying model of microworld phenomena fromwhich the set of quantum rules would follow.

42

In response to reasoning of this kind, an advocate of the orthodox interpre-tation of quantum mechanics can recall the origin of the classical equations ofelectrodynamics. James Clerk Maxwell, who wrote a complete system of equa-tions, made much effort to create a mechanistic model of phenomena that led tothe equations of electrodynamics. This proved to be an unsolvable problem, whileMaxwell’s equations per se became habitual in time to an extent that physicistsleft aside the question about their origin. A similar situation also arose with regardto the basic principles of classical mechanics, in particular, the law of inertia.

Eight centuries ago, the King of Castile Alfonso X named ‘The Wise’ hadreasons to note, “Had I been present at the creation of the World, I should haverecommended something simpler.” We can only guess what Alfonso X would sayafter familiarizing himself with quantum mechanics. It probably would not besimpler.

Acknowledgements

I am deeply grateful to V.V. Biryukov (Moscow State University), Yu.A. Kravtsov(Space Research Institute, Moscow), and M.A. Mnatsakanian (California Instituteof Technology) for a stimulating discussion of the issues under consideration andconstructive suggestions. In particular, Yu.A. Kravtsov pointed to the importanceof allowance for the low-frequency radiation in collisions of molecules and thenecessity of discussing the problem of quantum chaos.

Appendix. Positronium annihilation

In our description of the experiment on the recording of neutrons, we gave onlya reference to the formal solution of the problem in terms of quantum mechanicsperformed by Bohm [1952]. Below, we reproduce with minor changes the analyti-cal consideration of a basically similar process – the annihilation of a positroniumatom – contained in Sect. 18.3 of the lectures by Feynman [1965b]. This solutionis all the more instructive, because it is given with clarity characteristic of RichardFeynman and is performed in the context of a modern approach to quantum me-chanics.

A positronium atom is composed of an electron e− and a particle with oppositecharge sign, a positron e+. The spin of each of these particles is 1/2 (in units of h);initially, we consider an atom at rest with antiparallel spins of its components anda zero total spin. The characteristic lifetime of this system is 10−10 s, followingwhich the electron and the positron annihilate with the emission of two γ-rayquanta. The latter fly apart in opposite directions with equal speeds; the direction

43

of the flight is distributed isotropically. We are interested in the polarization ofthe produced photons.

In this case, to describe the photons, it is convenient to choose a state ofcircular polarization. Recall that we arbitrarily attribute a right-hand circularpolarization to the monochromatic light wave if the electric field vector rotatescounterclockwise as it travels toward the observer; the photons constituting thewave, accordingly, are assumed to be right-hand circularly polarized (state |R〉).In a beam of left-hand circularly polarized photons (state |L〉), the electric fieldvector rotates clockwise if we look at the approaching wave.

Obviously, two alternate modes of decay that conserve the system’s zero totalspin are admissible (Fig. 5). Two right-hand circularly polarized photons areproduced in the first mode; each of them has an angular momentum of +1 relativeto its momentum direction, while the angular momenta relative to the z axis are+1 and −1. Denote this state of the system by |R1R2〉. Two left-hand circularlypolarized photons are produced in the second mode denoted by |L1L2〉. We willassume that |R1R2〉 and |L1L2〉 constitute an orthonormal basis.

According to the superposition principle, the final state of the system afterannihilation |F 〉 is a linear combination of the alternate states |R1R2〉 and |L1L2〉.To find the corresponding coefficients, we should take into account two conditions:(1) the initial state – a positronium atom with zero spin – is characterized by oddparity and the parity of |F 〉 must be the same; (2) the normalization of |F 〉 mustprovide a unit sum of the probabilities of all possible realizations. The only linearcombination of the alternative states that satisfies both conditions is

|F 〉 = (|R1R2〉 − |L1L2〉) /√2. (A1)

Indeed, the operator of spatial inversion P changes both the direction of pho-ton motion and the direction of its polarization; therefore, P |R1R2〉 = |L1L2〉,P |L1L2〉 = |R1R2〉. As a result, P |F 〉 = −|F 〉, suggesting that parity is conservedunder positronium annihilation.

Having the final state of the system, we can calculate the amplitudes andprobabilities of events of various kinds. In particular, the amplitudes of the twoalternative types of decay that we mentioned above are

〈R1R2|F 〉 = 1/√2, 〈L1L2|F 〉 = −1/

√2, (A2)

so the probabilities of both modes are 1/2. Physically, the representation (A1)means that the detectors placed in the positive and negative directions of the z-axis will always record equiprobably either a pair of right-hand photons or a pairof left-hand photons.

The scheme of the experiment shown in Fig. 6 is of interest in the context of thequestion about the possibility of action at a distance discussed in Section 10. The

44

Figure 5: Alternate states of the pair of photons that resulted from positro-nium annihilation.

photons that fly apart after positronium annihilation pass through calcite crystals,as a result they become linearly polarized either in the x- or in the y- direction.Each of the four possible channels of photon propagation is equipped with a lightdetector. It is required to ascertain the way in which this scheme operates whenmany annihilation processes are observed successively. More specifically, since thetriggering of one of the four pairs of counters is admissible at each annihilation:D1x and D2x, D1x and D2y, D1y and D2x or, finally, D1y and D2y, the probabilitiesof the corresponding processes should be found.

As an example, let us calculate the amplitude 〈y1x2|F 〉 of the event that con-sists in the triggering of counters D1y and D2x. Taking into account the represen-tation (A1), we find

〈y1x2|F 〉 ·√2 = 〈y1x2|R1R2〉 − 〈y1x2|L1L2〉. (A3)

Since the photon recordings by different counters are independent events, we as-sume that

〈y1x2|R1R2〉 = 〈y1|R1〉〈x2|R2〉 (A4)

and similarly for 〈y1x2|L1L2〉. As a result, Eq. (A3) takes the form

〈y1x2|F 〉 ·√2 = 〈y1|R1〉〈x2|R2〉 − 〈y1|L1〉〈x2|L2〉. (A5)

Next, we should take into account the fact that the states of right-hand and left-hand circular polarization are related to the states of linear polarization along thex- and y- directions by the relations

{

|R〉 = (|x〉+ i |y〉) /√2,

|L〉 = (|x〉 − i |y〉) /√2.

(A6)

45

Figure 6: Scheme for recording the photons produced by positronium anni-hilation. C1 and C2 are the calcite crystals; x1, y1, x2 and y2 are the paths ofthe photons linearly polarized in the x and y directions; D1x, D1y, D2x andD2y are the photon counters.

Taking into account the orthonormality of the system |x〉 and |y〉, we find from(A6):

{

〈y1|R1〉 = +i/√2, 〈x2|R2〉 = 1/

√2,

〈y1|L1〉 = −i/√2, 〈x2|L2〉 = 1/

√2.

(A7)

Substituting these expressions into (A5) finally yields

〈y1x2|F 〉 = i/√2, (A8)

so that the probability of the corresponding process is 1/2.The amplitudes of the three remaining processes are calculated similarly. As

a result, we obtain:{

〈x1x2|F 〉 = 〈y1y2|F 〉 = 0,

〈x1y2|F 〉 = 〈y1x2|F 〉 = i/√2.

(A9)

These expressions show that one of the pairs of counters in the channels withmutually orthogonal polarizations of photons is triggered always, and with equalprobabilities: eitherD1x andD2y orD1y andD2x; the triggering probabilities of thepairs with identically directed polarizations of photons are zero. Thus, Feynman’sanalysis of the positronium annihilation process leads to the same conclusions asthose listed in Section 10.3 from Bohm’s calculations concerning the recording ofparticles with a spin of 1/2.

Such are the predictions of the quantum theory that are fully consistent withexperimental data. Where is then the paradox in the described situation? Supposethat observer 2 is slightly farther from the positronium atom than observer 1. Asa consequence, the triggering of a particular counter of observer 1 will allow him topredict with certainty precisely which counter of observer 2 will be triggered. Onthe other hand, the photon flying toward observer 2 is in a superposition of states|x2〉 and |y2〉 with linear polarization, therefore, this photon could seemingly reachone of the detectors of observer 2 with some nonzero probabilities irrespective of the

46

result obtained by observer 1. Why the recording of a photon by a remote observercompletely determines the result of an experiment that has not yet happened?Does this mean that there is some interaction that propagates with a speed higherthan the speed of light in a vacuum?

In his lectures, Feynman gives a detailed interpretation of the experiment withwhich, of course, one should familiarize oneself carefully. In our view, even thestructure of representation (A1) gives an answer to the above questions: The pair ofphotons produced by annihilation was initially prepared in such a way that the twocorresponding detectors always recorded a circular polarization of one type. This‘preparedness’ of the system is retained after the passage of photons through calcitecrystals and determines the fact of triggering the pairs of detectors with mutuallyorthogonal polarizations of photons. No influence, not to mention superluminalone, on the remote process is required. The situation is similar to the examplewith the sending of letters to remote points considered in Section 10.5 with the onlysignificant (but now familiar) difference that the quantum processes are nonlocal.

“Do you still think there is a ‘paradox’? Make sure that it is, in fact, a paradoxabout the behavior of Nature, by setting up an imaginary experiment for which thetheory of quantum mechanics would predict inconsistent results via two differentarguments. Otherwise the ‘paradox’ is only a conflict between reality and yourfeeling of what reality ought to be” (R. Feynman 1965b, p. 261).

References

[1] Alimov, Yu.I.; Kravtsov, Yu.A. Physics-Uspechi 1992, 162, 149.

[2] Aspect, A. Nature 1999, 398, 189.

[3] Aspect, A.; Dalibard, J.; Roger, G. Phys. Rev. Lett. 1982, 49, 91,1804.

[4] Ballentine, L.E. Quantum Mechanics: A Modern Development ;World Scientific Publishing, 1998.

[5] Bell, J.S. Physics 1964, 1, 195.

[6] Bell, J.S. Rev. Mod. Phys. 1966, 38, 447.

[7] Bell, J.S. Speakable and Unspeakable in Quantum Mechanics; Cam-bridge Univ. Press, Cambridge, 1987.

[8] Blatter, G. Nature 2000, 406, 25.

[9] Bohm, D. Quantum Theory ; Prentice-Hall, NY, 1952.

47

[10] Bohr, N. Phys. Rev. 1935, 48, 696.

[11] Born, M., Atoq mic Physics; Blackie and Son, London, 1963.

[12] Brewer, R.G.; Hahn, E.L. Sci. Am. 1984, 251, No. 6.

[13] Brida, G.; Genovese, M.; Novero, C.; Predazzi, E. Phys. Lett. A 2000,268, 12.

[14] Byrne, P. Sci. Am. 2007, Dec 2007, 98.

[15] Clauser, I.F.; Horne, M.A.; Shimony A.; Holt, R.A. Phys. Rev. Lett.1969, 23, 880.

[16] Chandrasekhar, S. Rev. Mod. Phys. 1943, 15, 1.

[17] Chirikov, B.V. Preprint IYaF SO AN USSR (In Russian), 1978,No. 66, Novosibirsk.

[18] Dirac, P.A.M. The Principles of Quantum Mechanics; ClarendonPress, Oxford, 1958.

[19] Einstein, A. Phys. Zs. 1909, 10, 817.

[20] Einstein, A.; Podolsky, B.; Rosen, N. Phys. Rev. 1935, 47, 777.

[21] Faddeev, L.D.; Yakubovsky, O.A. Lectures in Quantum Mechanics forthe Students-Mathematicians (In Russian); LGU Press, Leningrad,1980.

[22] Feynman, R.P.; Leighton, R.B.; Sands, M. The Feynman Lectures onPhysics; Addison-Wesley, Reading, 1964, Vol. 2.

[23] Feynman, R.P. The Character of Physical Laws; Cox and Wyman,London, 1965a.

[24] Feynman, R.P.; Leighton, R.B.; Sands, M. The Feynman Lectures onPhysics; Addison-Wesley, Reading, 1965b, Vol. 3.

[25] Feynman, R.P. QED: The Strange Theory of Light and Matter ;Princeton Univ. Press, Princeton, 1985.

[26] Feynman, R.P.; Morinigo, F.B.; Wagner, W.G. Feynman Lectures onGravitation; Addison-Wesley, Reading, 1995.

[27] Fok, V.A. Physics-Uspechi 1936, 16, 436.

48

[28] Fok, V.A. Physics-Uspechi 1957, 62, 461.

[29] Fok, V.A. Foundations of Quantum Mechanics 2007, Forth ed., LKI,Moscow.

[30] Freedman, S.I.; Clauser, I.F. Phys. Rev. Lett. 1972, 28, 938.

[31] Fridman, A.A. Universe as Space and Time; Sec. ed., Nauka,Moscow, 1965, 54.

[32] Gertsenstein, M.E.; Kravtsov, Yu.A. JETP 2000, 59, 658.

[33] Huang, K. Statistical Mechanics; Wiley and Sons, NY, 1963.

[34] Kac, M. Probabilities and Related Topics in Physical Sciences; Inter-science Publ., London, 1957.

[35] Kadomtsev, B.B. Dynamics and Information (In Russian); PhysicsUspechi Publ., Moscow, 1999.

[36] Kiselev, V.V. Quantum Mechanics (In Russian); Moscow, 2009.

[37] Kravtsov, Yu.A. Physics-Uspechi 1989, 158, 93.

[38] Krylov, N.S.Works on Foundation of Statistical Physics (In Russian);USSR Academy Publ., Moscow, 1950.

[39] Landau, L.D.; Lifshitz, E.M. Quantum Mechanics (In Russian); Fiz-matlit, Moscow, 1963, §14.

[40] Lichtenberg, A.J.; Lieberman, M.A. Regular and Stochastic Motion;Springer, NY, 1983.

[41] Mandelstam, L.I. Complete Works (In Russian); USSR AcademyPubl., Moscow, 1950, Vol. 5, pp 385-389.

[42] Mermin, N.D. Am. J. of Physics 1981, 49, 940.

[43] Mermin, N.D. ArXiv:quant-ph/9801057v2 2 Sep 1998.

[44] Pais, A. The Science and the Life of Albert Einstein; Oxford Univ.Press., Oxford, 1982.

[45] Pauli, W. Dialectica 1954, 8, 283.

[46] Peierls, R. Surprises in Theoretical Physics; Princeton Univ. Press,Princeton, 1979.

49

[47] Poulin, D. A rough guide to quantum chaos; Univ. of Waterloo, 2002.

[48] Salart, D.; Baas, A.; Branciard, C.; Gisin, N.; Zbinden, H.arXiv:0808.3316v1 [quant-ph] 25 Aug 2008.

[49] Schuster, H.G. Deterministic Chaos; Physik-Verlag, Weinheim, 1984.

[50] Shimony, A. Sci. Am. 258, No. 1, Jan 1988.

[51] Shpol’skiy, E.V. Atomic Physics (In Russian); Nauka, Moscow, 1974.

[52] Sinai, Ya.G. Mathematics-Uspekhi 1970, 25, 141.

[53] St. Augustine Confessions; 398, 11.23.30.

[54] Terebizh, V.Yu. Physics-Uspechi 1995, 165, 143.

[55] Terebizh, V.Yu. Introduction to Statistical Theory of Inverse Prob-lems (In Russian); Fizmatlit, Moscow, 2005.

[56] Terebizh, V.Yu. Proc. Crimean Astrophys. Obs. 2010, 106, 103-126.

[57] Uhlenbeck, G.E.; Ford, G.W. Lectures in Statistical Mechanics; Am.Math. Soc., Providence, 1963.

[58] Weinberg, S. Dreams of a Final Theory ; Random House, NY, 1992.

[59] Wigner, E.P. Phys. Rev. 1932, 40, 749-759.

[60] Wigner, E.P. Comm. Pure and Appl. Math. 1960, 131, 1.

[61] Zaslavsky, G.M. Stochasticity of Dynamical Systems (In Russian);Nauka, Moscow, 1984.

[62] Zaslavsky, G.M. Physics Today Aug 1999, 39.

50