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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 classical mechanics, statistical physics, and quantum theory are discussed. The viewpoint according to which there is no room for probabilities in Nature, as such, is consistently substantiated; the researcher is forced to introduce probabilistic concepts and the corresponding models in conditions that provide only partial predictability of the phenomena being studied. This approach allows one to achieve a consistent interpretation of some important physical phenomena, in particular, the relationship between instability of processes and their irreversibility in time, the stochastic evolution of systems in the theory of deterministic chaos, Boltzmann’s H-theorem, and paradoxes of quantum mechanics. ∗ “Advances in Quantum Systems Research”, Chapter 8. Nova Publishers, Zoheir Ezziane (Ed.), ISBN: 978-1-62948-645-1. † 98409 Nauchny, Crimea, Ukraine; E-mail: [email protected] 1 A year or so ago, while Philip Candelas (of the physics department at Texas) and I were waiting for an elevator, our conversation turned to a young theorist who had been quite promising as a graduate student and who had then dropped out of sight. I asked Phil what had interfered with the ex-student’s research. Phil shook his head sadly and 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 unlikely remain after several years of specific research. In the same book, Weinberg notes, “Most physicists use quantum mechanics every day in their working lives without needing to worry about the fundamental problem of its interpretation... But I admit to some discomfort in working all my life in a theoretical framework that no one fully understands.” Another outstanding physicist, Richard Feynman [1985], conveyed his attitude toward the orthodox quantum theory even more emotionally, “I can’t explain why Nature behaves in this peculiar way. . . The theory of quantum electrodynamics describes Nature as absurd from the point of view of common sense. And it agrees fully with experiment. So I hope you can accept Nature as She is – absurd.” The negative attitude of Albert Einstein and Erwin Schrödinger in this respect is well known. What did not suit the scientists to whom physics owes the introduction of the concept of photons, the fundamental equation for the wave function, the overcoming of serious difficulties of quantum electrodynamics, and the creation of a unified theory of electroweak interactions? (Nobel Prizes were awarded for all four contributions.) It is often said that Einstein did not accept the probabilistic interpretation of the reality that quantum mechanics brought in. However, it should be remembered that even in his first works, Einstein gave a theory of Brownian motion, independently built a basic framework of statistical mechanics, anticipated an important probabilistic expansion that many years later was associated with names of Karhunen and Loéve, introduced the probabilities of transitions in atoms, and performed a number of other studies clearly showing that he subtly understood the probabilistic problems and had a good command of the corresponding technique. Of course, that’s not the point; what matters is the understanding of the 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 simplest model for the range of phenomena being studied that is consistent with experimental data and that has a forecasting power. Previously, the term theory was commonly used instead of the term model; the latter term reflects better the incomplete, transient character of our understanding of Nature. Why on earth were the 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 analysis of real and thought experiments, is devoted to the analysis of probabilistic foundations of physics and the corresponding interpretation of quantum theory (see, in particular, Bohr 1935; Mandelstam 1950; Fok 1957; Wigner 1960; Born 1963; Chirikov 1978; Kravtsov 1989; Alimov and Kravtsov 1992; Mermin 1998). The approach suggested by quantum mechanics will become clearer if we initially trace the emergence of probabilistic models in classical physics. Even an examination of a simple experiment with die gives clear evidence of this kind. The objective of this chapter is, as far as possible, to facilitate familiarization with the probabilistic problems of physics to avoid situations like that described in the epigraph2 . We consistently substantiate the viewpoint according to which there is no room for probabilities in Nature, as such; the researcher is forced to introduce probabilistic concepts and the corresponding models in conditions that provide only partial predictability of the phenomena being studied in both classical and quantum mechanics. Light was diffracted by holes, atoms retained stability, and dice and systems consisting of a large number of molecules demonstrated certain regularities long before Christiaan Huygens, Blaise Pascal, Pierre Fermat, and Jakob Bernoulli laid the foundations of the probability theory. For coherence, we had to cursorily touch on some of the facts that are covered well in textbooks. From the probability theory, to understand the subsequent material, it will suffice to know that a discrete random variable ξ is specified by a set of values x1 , x2 , . . . , xN that it can take in an experiment and by the corresponding probabilities of these values p1 , p2 , . . . , pN . The number of values N can 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 its possible values. In applied research, such confusion often leads to misunderstandings; experts on the probability theory occasionally agree to that when there is a shortage of symbols. 1 Not always unconditionally, though; it will suffice to recall the long-term debate concerning Boltzmann’s H-theorem. 2 See also the preliminary paper by Terebizh [2010]. 3 2. Throws of a die and other unstable processes Consider once again the old problem of the throws of a die by focusing attention on the physical aspect of the situation. Suppose that such throws are made in the following conditions: (1) the die has the shape of a cube whose edges and angles are identical with a microscopic accuracy; (2) the inscriptions 1, 2, . . . , 6 that distinguish the faces from one another are made in such a way that their mass is smaller than that of the die by several orders of magnitude; (3) the die and the table on which it falls are made of an elastic material; (4) the initial orientation of the die is always the same, say, face ‘6’ is directed upward and face ‘4’ is oriented northward3 ; (5) the die is released from fixation in some delicate way that is now not concretized; (6) the number written on the upper face of the stopped die is taken as the result of a single throw. The problem consists in predicting the result of 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 edge length a. Clearly, at such a small height, ‘6’ will prevail in a series of throws; the faces that have been initially sideways will occur rarely, while many throws would have 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 need to 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 of one of the side faces as a consequence of inexact adherence to the experimental conditions, and consider the occurrence of ‘1’ as an exceptional event that requires an additional study. At h/a ≃ 1 − 2, the results of experiments will become more varied. For their interpretation, we can invoke sophisticated means of recording the initial position of the die and then calculate its motion on a supercomputer using the corresponding equations of aerodynamics and elasticity theory. Obviously, in this way, we will be able to achieve such a good accuracy of predictions that the dynamical model of the process will be considered acceptable. However, a much simpler, probabilistic, model turns out to be also useful even here. It does not require a high accuracy of the information about the initial position of the die and laborious calculations. This model postulates that the number written on the upper face is a random variable ξ that takes one of the values 1, 2, . . . , 6 with probabilities p1 , p2 , . . . , p6 . Analysis of the results of a series 3 The die is assumed to be numbered in a standard way, so that the sums of the numbers on opposite faces are equal to 7. 4 of throws of the die from a fixed initial height shows that for the subsequent results to be predicted successfully, we must assign the highest probability, p6 , to the upper face, slightly lower probabilities, p2 = p3 = p4 = p5 , to the four side faces, and the minimum probability, p1 , to the lower face. Of course, all of the introduced probabilities depend on the initial height of the die. As the ratio h/a increases, the dynamical model leads to errors increasingly often, while the probabilistic model retains its efficiency, when the probability distribution {pk (h)}6k=1 is properly redefined. In particular, if the observed frequencies of occurrence of the faces at h/a ≫ 1 do not differ from 1/6 within the boundaries prescribed by mathematical statistics4 , we have no reason to suspect asymmetry of the die. Otherwise, both models require a scrupulous study of the degree of homogeneity of the die material and the experimental conditions. Why are we forced to abandon the deterministic model at a significant initial height of the die? The reason is that as h/a increases, the factors that we disregarded play an increasingly important role: a small spread in initial positions of the die, nonidentical operations of the die release mechanism, the influence of the environment, etc. In those cases where the result of the experiment changes significantly with small variations in initial conditions and behavior of the process, the latter is called unstable. In classical mechanics, instability had not been considered as a fundamental difficulty for a long time: the problem seemed to consist only 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 is often exponential, so that extremely small deviations rapidly reach a significant value (see, e.g., Lichtenberg and Lieberman 1983; Zaslavsky 1984, 1999; Schuster 1984). Since some uncertainty in the initial and boundary conditions is unavoidable in view of the atomic structure of the material, many processes of the world that surrounds us are fundamentally unstable (these issues are discussed in more detail in the next section). The applicability of probabilistic models to the description of processes similar to the throwing of a die caused no particular disagreement. The stochasticity of the quantum behavior, about which we will talk below, apart, it could be assumed that as all details of the process are refined (at least mentally), its result becomes increasingly definite. The Brownian motion of particles ∼ 1 µm in size in a liquid can serve as an example. By invoking the simplest model of a random walk to describe the motion of an individual particle, we draw useful probabilistic conclusions about the behavior of an ensemble of many such particles. At the same time, analysis of the long-term observations of one specific particle using a refined theory of the motion of small particles in a liquid allows us to take into account 4 This discipline ascertains the results of imaginary and, hence, completely ‘pure’ experiments for samples of an arbitrarily large size. 5 the particle inertia and the viscosity effects and, hence, to predict its behavior with a greater certainty than that of the prediction provided by the model of a purely random walk (Chandrasekhar 1943). Similarly, it is impossible to create a final, complete theory of turbulence; various probabilistic models of this complex phenomenon reflect to some degree the properties of viscous fluid flow. The traditional description of traffic in big cities may be considered as a good example of natural transition to a probabilistic model. Any driver will undoubtedly reject the assumption that his trips on Friday are random: he initially drove to work, subsequently visited several predetermined shops, and, finally, taking his family, drove to the country cottage. In principle, municipal authorities could collect information about the plans of each car owner beforehand, but such an extensive picture on the scales of the city is not required at all – it will suffice to introduce a stochastic model of traffic along major highways, correcting the parameters of the corresponding probability distribution depending on the time of the day. 3. Statistical mechanics 3.1. Instability of motion and irreversibility The irreversibility of the evolution of large ensembles of particles is usually illustrated 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 are separated by a baffle. Experiment shows that after the removal of the baffle, both types mix between themselves, so that the contents of the vessel appear ‘cyan’ after the relaxation time. Why has the inverse process, when the initial cyan mixture is separated with time and the white molecules are in one part of the vessel, while the 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 simple thought experiment. Suppose that small identical balls move in a closed box in the shape of a rectangular parallelepiped. At the initial time, all balls touch one of the walls, their velocities are directed exactly perpendicular to this wall, and the 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 be different in accordance with some law. We assume the wall surfaces, along with the ball surfaces, to be perfectly smooth and the collisions to be absolutely elastic; there is no mutual attraction. Classical mechanics easily predicts the behavior of this system for an arbitrarily long time interval: each of the balls independently of others vibrates along a straight line segment perpendicular to the two walls. If the 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 or less uniformly between the two walls of the box. At the same time, if all velocities are reversed at an arbitrary instant of time, then the set of balls will return exactly to 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 they turn back by the signal of the referee and each will retain his speed. Let us complicate the experimental conditions only slightly: suppose that there is a small asymmetric convexity on the box wall at the point of collision of one of the balls. At the very first impact on the wall, the ball under consideration will be reflected slightly sideways and, some finite time later, will hit one of the side walls. Generally, the subsequent trajectory of the ball is very complex; specially chosen initial conditions apart, it can be asserted that, after a lapse of time, the balls will begin to collide not only with the box walls but also between themselves. Nevertheless, classical mechanics insists on the possibility of predicting the state of 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 with an infinitely high accuracy (Krylov 1950; Sinai 1970). The fully developed state of such models is called deterministic chaos (Schuster 1984). Determinacy is attributable to the complete absence of random factors (noise), so that the evolution is described by a system of differential or difference equations, while the allusion to chaos is attributable to extreme entanglement of the trajectories even in the case of systems with only a few degrees of freedom. We emphasize: there is no chaos here in the true sense of this word, which implies dominant stochasticity of the behavior. It should be kept in mind that states appropriately called illusory chaos can be realized 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 Brewer and Hahn [1984]: “A viscous fluid is placed in the ring-shaped space between two concentric plastic cylinders. Whereas the outer cylinder is stationary, the inner one is free to rotate about its axis. A streak of colored dye, representing an initial alignment 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 between the cylinders to a thermodynamicist, he or she would say that the dye is completely disordered... Reversal of the rotation of the inner cylinder reverses the mixing process; after an equal number of reverse rotations the dye streak reappears.” The cited paper includes impressive photographs. Obviously, a situation similar to the above example with runners is realized here. The phenomenon of nuclear spin echo discovered by E.L. Hahn in 1950 also demonstrates the possibility of a long-lived memory in systems with stable motion but 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 a memory of the pulse sequence, and at time 2t seconds after the first radio-frequency pulse the sample itself emitted a third pulse, an echo” (Brewer and Hahn 1984). The experiment can be explained as follows. Glycerin is initially prepared by orienting the proton spins parallel to the external magnetic field. The first pulse triggers a complex precession of the proton spins and the second pulse turns the spins through 180◦ , so that all spins are again oriented identically after time t. At this 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 a significant change in its state after a long time interval. The very possibility of a regular evolution of macroscopic systems is of fundamental importance for the understanding of statistical mechanics: Generally speaking, we do not always have to refer to probabilistic models. On the other hand, even the early works by Henri Poincare (1892) and Jacques Hadamard (1898) and, presently, many of the studies that followed the paper by E.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 strictly classical systems with a small number of degrees of freedom in the absence of noise exhibit instability of motion (generally, behavior): negligibly small variations in initial conditions lead to a radical difference of the final pictures. In contrast to regular evolution, where the divergence of the phase elements with time is no higher than the linear one, the divergence in unstable systems is very fast – it is exponential. Any finite accuracy of specifying the initial conditions guarantees the possibility of keeping track of the evolution of an unstable system only for a short time; its subsequent behavior is indistinguishable from the evolution of a system with a different initial state. The questions of what the relaxation time is, whether a stationary, on average, density distribution will be established, and whether this distribution will be uniform, or the regions of avoidance will remain, as well as many other problems concerning deterministic chaos, have been solved only partially. We are forced to conclude that there are no tools at the disposal of classical mechanics that would allow one to keep track of the evolution of an even ideal system 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 experiment we consider, not only the very complex pattern of roughness of the walls and ball surfaces but also – since their constituent atoms move – the variability of these characteristics with time, the inelasticity of the impacts, and many other phenomena 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 closed vessel has been assessed, the interaction of molecules with the radiation field. It can still be imagined how to isolate the gas from thermal radiation from the vessel walls, but the collisions between molecules even at moderate temperatures are inevitably accompanied by low-frequency electromagnetic radiation. Gertsenstein and Kravtsov [2000] showed that this mechanism leads to a significant deviation of the trajectory of a molecule from the results of purely Newtonian calculations in an astonishingly short time, which only a few times longer the mean free-flight time of the molecules. So, with the aid of classical mechanics, only some general features of the evolution of a many-particle system can be established and other models should be invoked to create a real picture. Two factors necessarily considered jointly – the character of the initial state and the instability of evolution – allow us to answer the question of why we see so many irreversible processes in the world that surrounds us, say, the sea waves breaking against the shore do not recover their shape and do not go back. Many systems were initially prepared – ordered – by Nature or man and then the instability of evolution intervened. The same roughly ordered sequence of sea waves is generated in a natural way – by a strong wind, whereas the recovery of a regular system of waves requires enormous purposeful work. As Richard Feynman [1965a] believed, the hypothesis that the Universe was more ordered in the past should be added to the known physical laws. Now, we are ready to return to the experiment with two types of molecules in a closed vessel. Clearly, in conditions that approach the real ones at least in part, the motion of an arbitrarily chosen molecule will be unstable. For example, if we launch several realizations of the mixing process from the same – within the theoretical possibilities of the experiment – initial state, with the external conditions being retained as scrupulously as possible, then this molecule will be in completely different regions of the vessel after a finite time in different realizations of the process. So far, when discussing the thought experiments concerning the evolution of many-particle systems, we have said no word about probabilities. Their introduction is inevitable, because the evolution of typical systems is unstable; for this reason, real processes evolve in a way unpredictable for the researcher. (Einstein said, “God does not care about our mathematical difficulties. He integrates empirically.”) In these conditions, which are much more complicated than those in the experiment with the throw of a die, the researcher is forced to invoke a particular probabilistic model by specifying an appropriate stochastic apparatus. Only afterward and only within the framework of the adopted stochastic model do we have the right to say that the initially separated state will eventually become uniformly colored 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 of all molecules will not return the system to the separated state because arbitrarily weak stochasticity admitted by the researcher will not allow this to happen. This is all the more true for the models that represent real systems with a distinct instability of motion. As an example of a successful probabilistic model, we will mention the ‘dogand-flea model’ suggested by Boltzmann and considered in detail by Paul and Tatyana Ehrenfest in 1907 (see Kac 1957). The model illustrates the transition to statistical equilibrium of the gas that nonuniformly fills a closed volume. Initially, 2N numbered balls are distributed among two boxes in some specified way. Subsequently, a random number generator creates an integer that is uniformly distributed on the set 1, 2, . . . , 2N ; the ball with this number is moved from the box where it lies to the other box. The procedure is repeated many times. This simple model admits an exhaustive analytical study; it is now easy to perform a number of corresponding computer realizations of the process as well. In particular, it is curious to trace its evolution from the state when all balls were in the same box. Above, we deliberately emphasized that the distribution of the random ball number is uniform: in general, some different discrete probability distribution can also be specified. In this case, a new probabilistic model of the process will be introduced which may describe better the actual behavior of the specific gas sample. Obviously, at N ≫ 1, the system that reached statistical equilibrium in the adopted probabilistic model will return to the original state only with a very low probability. It is easy to continue examples similar to those considered above, by successively passing from simple situations to more complicated ones, from Boltzmann’s ideas to Gibbs’ ensembles (see, in particular, the lectures by Uhlenbeck and Ford 1963). Analysis convincingly indicates that the deterministic model proposed by classical mechanics is unproductive when systems with an unstable behavior are studied. The main objective of statistical physics is to construct adequate probabilistic models of such phenomena. This is also suggested by the name of this field of physics itself5 . 5 However, the word ‘statistical’ is illegitimately used in the literature instead of ‘probabilistic’ or ‘stochastic’. The latter terms emphasize the presence of random factors in the model and must be contrasted with the term ‘deterministic’. In contrast, mathematical statistics deals with the problem that is inverse to the main range of problems of the probability theory, namely, the reconstruction of information about the probabilistic laws from a specified random realization. Therefore, one cannot say that the behavior of the system has a ‘statistical’ character. 10 3.2. Boltzmann’s H-theorem The best known case of prolonged debates in classical physics spawned by the probabilistic treatment of the phenomenon is related to the H-theorem. In 1872, Ludwig Boltzmann concluded that some function of time H(t) that characterizes the state of a rarefied gas either decreases with time or retains its value6 . In this form, the H-theorem is in conflict with the symmetry of the laws of mechanics relative to time reversal and with the theorem proven by Henri Poincare according to which a closed mechanical system returns to an arbitrarily small vicinity of almost any initial state after a fairly long time interval. These contradictions called the Loschmidt and Zermelo paradoxes forced Boltzmann to turn to the probabilistic treatment of the process of approaching equilibrium: the change in H(t) describes only the most probable evolution of the system. The present-day formulation (Huang 1963) of the H-theorem also includes an important refinement of the initial state: “If at a given instant t the state of the gas satisfies the assumption of molecular chaos, then with an overwhelming probability at the instant t + ǫ (ǫ → 0) dH/dt ≤ 0.” The aforesaid clearly shows that the H-theorem says not about the behavior of an ensemble of classical particles but only about some probabilistic model intended to sufficiently describe the approach of the system being studied to equilibrium. The physical mechanism that destroys the order is the instability of motion, while the 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 the adopted 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, the arguments of Loschmidt and Zermelo remain valid. In contrast, the ‘gas’ introduced in the appropriate stochastic model need not obey all the laws of classical mechanics. A good model developed for a specific situation has the right to disregard in full extent the classical reversibility in time, thus, not to obey Poincare’s recurrent theorem (especially since the corresponding cycle is excessively long!). For example, the H-theorem invokes a specific model of a rarefied gas that sufficiently describes its evolution from the hypothetical state of molecular chaos being realized in practice only with a limited accuracy. Precisely “This statistical assumption [about molecular chaos] introduces irreversibility in time” (Uhlenbeck and Ford 1963). Present-day models of statistical physics, in particular, the 6 For a rarefied gas, H(t) coincides with entropy taken with the opposite sign, so the H-theorem may be considered as a special case of the law of entropy increase in a closed system. 11 well-known Bogolyubov-Born-Green-Kirkwood-Yvon approach, introduce similar probabilistic assumptions but on a deeper level than that of Boltzmann’s theory. Clearly, being forced to choose some probabilistic model, we can no longer purport to describe in detail the process, which we hoped to do within the framework of classical dynamics. 3.3. Arrow of time The widely discussed question of the difference between the past and the future is usually assigned to the same range of problems of statistical physics. In view of aforesaid it seems obvious that it is illegitimate to condition the directionality of time (an ‘Arrow of time’, according to Arthur Eddington) by the behavior of large ensembles of particles described by statistical mechanics. Let us add that otherwise we would have to observe not only fluctuations of the characteristics of many-particle systems, but time also. On the contrary, just the flow of time allows us to observe successive states of an unstable system and to describe its evolution using probabilistic models. As was noted by St. Augustine [398], “Let no man then tell me that the motions of the heavenly bodies are times, because, when at the prayer of one the sun stood still in order that he might achieve his victorious battle, the sun stood still, but time went on.” Clarifying the nature of time and, in particular, its directionality is a much deeper problem of physics. 4. Characteristic features of classical stochastic models Let us summarize the conclusions that follow from the analysis of probabilistic models of classical physics in order to subsequently trace the corresponding changes in quantum theory7 . It is well known that any model of the phenomenon under study is neither unique nor exhaustive. Even a very successful model has boundaries within which it is more preferable than other models, but a more perfect model usually replaces it when the range of studies is extended. For example, Eintein’s general theory of relativity replaced Newton’s theory of gravitation; the latter retained its efficiency in the case of low velocities and weak gravitational fields. The choice of one of the many possible models is determined by the principle formulated by William 7 The author hopes that he will not be rebuked for “playing with words specially made up for this”. 12 Occam in the 14th century: The researcher must prefer the simplest model from a number of alternatives that give an explanation for the experimental results. Obviously, the simplicity of a model should be understood in a comparative context. Quite often, one has to use sophisticated constructions even within the framework of the simplest possible models. Say, when the distribution of people in height is studied, the histogram is approximated by a Gaussian function that contains the seemingly irrelevant constant π (see Wigner’s 1960 well-known essay). Present-day models for the structure of matter invoke incomparably more complicated tools; as a rule, abstract mathematical constructions are used in them. Technically, the general theory of relativity is much more complex than Newtonian theory, but the former invokes fewer a priory assumptions, which justifies its comparative simplicity. If the model is simple and productive and especially when it works for decades or even centuries (as is now commonly said, became a paradigm), people begin to assign the concepts with which their theories operate to Nature itself. For example, in the past, a wide debate was caused by the gravitational paradox – the divergence of the Newtonian potential in an unbounded homogeneous Universe. The divergence of the potential was believed to reject the infinite model. As usual, the paradox only emphasized the conditionality of the tool we choose; if we talk about the gravitational force at an arbitrary point of an unbounded and, on average, homogeneous Universe, then it is finite and obeys Holtzmark’s probability distribution. To be more precise, if the spatial distribution of stars is described by a Poisson model, then the probability that the value of the force is larger than a given 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 worth noting once again, because it plays an important role in the picture of the world to which quantum mechanics leads (see Section 9). Strange as it may seem, many debates were caused by the confusion of the concepts of ‘randomness’ and ‘unpredictability’. Unless you know the schedule of a bus, its arrival is unpredictable for you, but you are free either to choose some probabilistic waiting model or to call the dispatcher. The fact that we are unable to predict the position of a Brownian particle cannot serve as a reason for considering its motion to be objectively random; only the economical model we chose is such. Similarly, the fact that we can not predict the sequence of digits in the infinite decimal expansion of the number π = 3.14 . . . does not mean that they appear randomly. They are completely determined by Nature, but for a non-specialist in number theory (such as the author) their seemingly random and plausible equiprobable appearance is a fairly good probabilistic model. 13 The above discussion of the procedure for throwing die, the models for the Brownian motion of small particles, the traffic in a big city, and typical models of statistical mechanics clearly illustrate the consistent viewpoint of classical physics on the fundamental question of how probabilities appear in the theory: (A) There is no room for probabilities in Nature itself. The researcher is forced to invoke probabilistic models in describing unstable processes, more specifically, when the initial data and external conditions for the experiment are known only approximately, while its results depend on these circumstances so significantly that they become partly unpredictable. Einstein may have had something of this kind in mind in his aphorism about God playing dice: It seemed that, in contrast to classical physics, quantum mechanics prescribes a probabilistic behavior to the objects of investigation themselves. The incentives for such a radical assertion can be understood by considering an experiment with the passage of light through two slits, but we will defer this discussion until Section 6. In the next section, we consider an intermediate situation where the 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 recording of light, its discreteness, was found. If the brightness of a light source is gradually reduced, then the picture being recorded becomes increasingly ‘grainy’. It is easy to make the light source so weak that individual flashes, photoevents, separated by long time intervals, say, more than an hour, are recorded. Significantly, when monochromatic light is used, the same portion of energy is recorded each time. This phenomenon can be explained by invoking the hypothesis put forward by Einstein in 1905: light consists of spatially localized quanta, photons, that have energy and momentum. Einstein obtained further evidence for the discrete model of light 4 years later, when he analytically found an expression describing the fluctuations of radiation energy in a closed volume. The above explanations were needed in connection with the description of an experiment on the passage of polarized light through a crystal of tourmaline given by Paul Dirac [1958]: “It is known experimentally that when plane-polarized light is used for ejecting photo-electrons, there is a preferential direction for the electron emission. Thus the 14 polarization properties of light are closely connected with its corpuscular properties and 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 perpendicular to its optic axis. Classical electrodynamics tells us what will happen for any given polarization of the incident beam. If this beam is polarized perpendicular to the optical axis, it will all go through the crystal; if parallel to the axis, none of it will go through; while if polarized at an angle α to the axis, a fraction sin2 α will go through. 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 made up 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 photons is then obliquely polarized and it is not clear what will happen to such a photon when it reaches the tourmaline. A question about what will happen to a particular photon under certain conditions is not really very precise. To make it precise one must imagine some experiment performed having a bearing on the question and inquire what will be the result of the experiment. Only questions about the results of experiments have a real significance and it is only such questions that theoretical physics has to consider. In our present example the obvious experiment is to use an incident beam consisting of only a single photon and to observe what appears on the back side of the crystal. According to quantum mechanics the result of this experiment will be that sometimes one will find a whole photon, of energy equal to the energy of 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 optic axis. One will never find only a part of a photon on the back side. If one repeats the experiment a large number of times, one will find the photon on the back side in a fraction sin2 α of the total number of times. Thus we may say that the photon has a probability sin2 α of passing through the tourmaline and appearing on the back side polarized perpendicular to the axis and a probability cos2 α of being absorbed. These values for the probabilities lead to the correct classical results for an incident beam containing a large number of photons. In this way we preserve the individuality of the photon in all cases. We are able to do this, however, only because we abandon the determinacy of the classical theory. The result of an experiment is not determined, as it would be according to classical ideas, by the conditions under the control of the experimenter. The most that can be predicted is a set of possible results, with a probability of occurrence for 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 of a probabilistic model in the experiment considered stems from the fact that “the result of an experiment is not determined... by the conditions under the control of the experimenter.” However, there is also a fundamental difference between the situations. Whereas in classical physics we can still hope for a refinement of the conditions in which the experiment is carried out, in the microworld the experimenter influences the process under study so significantly that turning to a probabilistic model becomes inevitable. Having stepped on an anthill, one should not be surprised by the fussiness of its inhabitants. Nevertheless, if this was the only peculiarity of quantum phenomena, then the difficulties with the perception of quantum mechanics that were mentioned in the Introduction would remain incomprehensible. The problem consists not in the rejection of the probabilistic description of the results of measurements; the point is that the probabilistic model suggested by quantum mechanics is considered not as an approximate description of some deeper picture of microworld phenomena but is given as a fundamental property of Nature. The situation is often characterized by asserting that there are no ‘hidden parameters’ in quantum theory that could give 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 explain the peculiar features of the quantum behavior and, first of all, the seeming nonlocality of quantum interaction. These features have been repeatedly illustrated using thought and real experiments on the interference of light, to the description of which we will now turn. For the subsequent comparison with the quantum procedure, we will briefly repeat an elementary derivation of the expression for the intensity of light in the interference pattern. 6. Interference of light according to the classical wave theory Let us first assume (Fig. 1) that there is only one narrow slit in an opaque screen on which the light from a bright source O is incident; the radiation passed through the slit is recorded by the detector located on the other side of the screen. If the light is bright and the detector resolution is low, then a continuous flux distribution with one maximum at point C lying on the axial line is recorded. The experiment with one narrow slit imposes no serious constraints on the model invoked for its interpretation. In contrast, Young’s experiment with two narrow slits performed in 1801 (Fig. 2) required significant concretization of the model, namely, the wave theory of light that includes the concept of wave interference. 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 increasing distance from the axial line. The classical wave theory of light proposed by Thomas Young, Christiaan Huygens, and Augustin Fresnel excellently coped with this. According to the wave model in the form that it attained by the late 19th century, light is a set of harmonic ether oscillations with various temporal periods T and spatial wavelengths λ. Outside an ordinary material, the oscillation frequency ν ≡ 1/T is related to the wavelength by the relation νλ = c, where c is the speed of light. In the experiment considered in Fig. 2, the primary wave from source O is incident on the screen and generates electron oscillations in it; this is equivalent to the fact that each of the slits serves as a source of coherent secondary waves. The oscillation amplitude at some point Q of the detector is determined by the phase difference between the waves arriving at this point from both slits. The reaction of the detector at this point depends on the local light flux, which is proportional to the wave amplitude squared. Consider, for simplicity, a monochromatic light source located at equal distances from slits A and B. Near the slits, the source generates harmonic oscillations a sin ωt with amplitude a and angular frequency ω = 2πν = 2πc/λ. Let D be the separation between the slits and z be the distance of the detector from the screen; we are interested in the oscillation amplitude at point Q located at distance x from the axis. If we neglect the difference in the degree of attenuation of the secondary waves on their way from the screen to the detector due to the distances 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 /c can 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) δφ(x) = 2π(ℓ2 − ℓ1 )/λ (3) where is the phase difference between the secondary waves. At x ≪ z, we can assume that ℓ2 − ℓ1 ≃ Dx/z, so that Dx F (x) ≃ 2 1 + cos 2π λz . (4) Formula (4) describes a periodic flux distribution on the detector. The spatial period of the pattern that specifies the linear resolution when the structure of the light 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, it should take into account the increase in the distance of the point of observation from the slits and the finiteness of their width. Allowance for the latter factor leads to the expression Dx F (x) ≃ 2 sinc (bx/λz) 1 + cos 2π λz 2 , (6) where the function sinc(t) ≡ sin(πt)/(πt) and b denotes the width of each slit. The additional – compared to Eq. (4) – factor sinc2 (bx/λz) predicts the decrease in brightness with increasing distance from the axis to zero at x0 = λz/b, 18 (7) and, since we assume that b ≪ D, we have x0 ≫ ∆x and the resolution of the pattern is still specified by Eq. (5). A consistent allowance for other features of the experiment within the framework of a classical model leads to a detailed interpretation of the observed pattern, with the exception of one significant fact – discreteness of the detector counts. We will discuss this phenomenon in the next section. Let us give an example. Assume that the light wavelength be λ = 0.5 µm, the slit width be 10 wavelengths, i.e., b = 5 µm, the separation between the slits be D = 150 µm, 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 slit width is x0 = 100 mm. Thus, the classical wave theory of light gives a satisfactory description for the pattern in Young’s experiment averaged over a long time interval. This theory was also 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 that take into account its quantum nature. First, let us turn to the passage of light through a single slit (Fig. 1). As has already been said, the observed distribution of photo-counts in the shape of a single-humped curve increasingly exhibits granularity due to the absorption of individual photons with decreasing brightness of the light source. These photons are simply like the classical particles (micropellets), because both the discreteness of counts and the overall shape of their distribution on the light detector can be explained in this case. In Young’s experiment with two slits (Fig. 2), the decrease in the brightness of the light source also reveals the discreteness of photo-counts, but the classical view of photons as micropellets turns out to be untenable. Indeed, the micropellets must come to the detector either through slit A or through slit B, so that the expected image is a superposition of two single-humped distributions shifted relative to the axis – basically, the projections of slits A and B onto the screen from point O. This pattern clearly differs from the observed distribution, which is characterized by sharp intensity variations. The real distribution is formed by identical photo-events, but the arrival of a photon at a given point of the detector depends 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 model operating with photons-micropellets can reduce the frequency of their falling to some place of the detector! Thus, we get the impression that the propagation of light obeys the laws of the wave theory, while it interacts with the detector as if it consisted of localized particles. It is this situation that is usually characterized by the concept of wave-particle duality. The brightness of the light source can be made so low that independent photon recording events will be recorded with a certainty. The distribution of photo-events accumulated over a long time closely coincides with the interference pattern that is observed in the case of a bright source. To explain this 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 quantum behavior. Indeed, the slits can be separated very far from each other on the scale of the light wavelength and, nevertheless, both the interference pattern and the discreteness of counts are retained. In particular, for the example described at 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. The telescopes constituting stellar interferometers are tens and hundreds of meters apart. Albert Einstein was the first to realize the inevitability of wave-particle duality in describing light in 1908: “I already attempted earlier to show that our current foundations of the radiation theory have to be abandoned... It is my opinion that the next phase in the development of theoretical physics will bring us a theory of light which can be interpreted as a kind of fusion of the wave and the emission theory.” Recall that the ‘emission theory’ that considered light as a flux of very unusual 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 of electromagnetic waves, which can naturally reach simultaneously two slits spaced far apart, helps us in the case of light. However, a similar diffraction pattern is observed if light is replaced by a flux of electrons or other particles with a nonzero rest mass (for brevity, such particles are often called ‘material’ ones), or even whole atoms! This is evidenced by various experiments, the first of which was carried out 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. A satisfactory 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 wavelength is associated with any particle whose rest mass m0 is nonzero: λ= h q 1 − v 2 /c2 , m0 v (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 accelerated in 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 Å. If such a beam of electrons is directed to a nickel single crystal, for which the lattice spacing is about 2 Å, then the appearance of interference extrema in the distribution of scattered electrons should be expected. This pattern was first observed in 1927 by C. Davisson and L. Germer. Thus, the experimental data suggest the wave nature of not only photons but also material particles. Since the electron can be likened neither to a micropellet nor to a wave, we have to abandon the seemingly only possible alternatives when considering the diffraction of electrons by two slits: (1) the electron passes either through one slit or through the other; (2) it passes through both slits simultaneously. The point is that we unjustifiably transfer the concept of body ‘trajectory’ worked out by macroscopic experience to the quantum world. This is discussed in more detail in the next section. The wave-particle duality has been discussed for about a century. Excellent explanations in this connection were given by E.V. Shpol’skiy [1974] in his course on atomic physics: “Since the properties of particles and waves are not only too different but also, in many respects, exclude each other and the electrons undoubtedly have a single nature, we have to conclude that the electrons are actually neither the former nor the latter, so that the pictures of waves and particles are suitable in some cases and unsuitable in other cases. The properties of microparticles are so peculiar and their behavior is so different from that of the macroscopic bodies that surround us in everyday life that we have no suitable images for them. However, it is clear that since we are forced to use both wave and particle pictures to describe the same objects, we can no longer ascribe all properties of particles and all properties of waves to these objects.” New names have been repeatedly proposed for microworld objects. For example, Feynman once used the term wavicles, a derivative of the words waves and particles. Unfortunately, none of the names was so apt to be widely used in the literature. 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 in the mid-1920th to the creation of quantum mechanics – the theory of phenomena in describing which Planck’s constant plays a crucial role. Quantum mechanics and its development including the special theory of relativity – quantum field theory – give a satisfactory description of the entire set of phenomena in the world that surrounds us, with the exception of gravity. Some of the aspects of this description achieved remarkable agreement with experimental data, ∼ 10−10 , which is indicative of a high efficiency of the model created by Erwin Schrödinger, Werner Heisenberg, Max Born, and Paul Dirac. At the same time, the words of outstanding physicists given at the beginning of this chapter suggest that, while providing a consistent formal procedure for calculating the results of experiments in the domain of atomic phenomena, quantum mechanics raises difficult conceptual questions. John Bell [1987], who proposed well-known experiments to test quantum mechanics, reached a bitter conclusion: “When I look at quantum mechanics I see that it’s a dirty theory: You have a theory which is fundamentally ambiguous.” 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 description can also be equally extended to the experiment with photons discussed in Section 6). Analysis of this experiment allows the formal aspect of the quantum-theory calculations to be perceived. For simplicity, consider a stationary process where source 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 from the axis. In the case of two open slits, the calculation is as follows. With the aid of the procedure briefly described below two complex numbers should be formed: ϕ1 (x) and ϕ2 (x) – the amplitudes of the probability of electron passage through slits A and B, respectively, followed by their detecting at Q(x). By definition, the total amplitude of the probability of detecting at Q that we denote by 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 arbitrary particle emerged from O will fall into an infinitely small interval of width dx near point Q: f (x)dx = |ϕ(x)|2 dx. (10) 22 The electron flux near point Q and the number of photo-events per unit time are proportional to the probability density f (x). It follows from the two previous formulas that f (x) = |ϕ1 (x)|2 + |ϕ2 (x)|2 + 2 ℜ[ϕ∗1 (x)ϕ2 (x)], (11) where the symbols ℜ and ∗ correspond to the separation of the real part of the complex number and complex conjugation, respectively. The classical description would be restricted to the first two terms in Eq. (11), which define the probability of passage either through the first slit or through the second one. Quantum mechanics introduces the third term dependent on the phases; it is the relationship between the phases ϕ1 (x) and ϕ2 (x) that allows the interference of microparticles to be properly described. All of this resembles the operations that we performed in Section 6 when analyzing the diffraction of light by two slits, but the probability amplitudes cannot be treated as waves in ordinary space. Obviously, before we turn to Eq. (11), we should specify how the probability amplitudes are calculated and what the rules for handling these quantities are. We will only cursorily touch on these questions here within the framework of nonrelativistic quantum mechanics; for a detailed description, see the lectures by Feynman [1965b] and textbooks on quantum mechanics. Feynman showed that the probability amplitude ϕ for some path is defined by the action S, i.e., the time integral of the difference between the kinetic and potential energies along this path: ϕ ∝ eiS/h̄ , (12) where i is the imaginary unit and h̄ is Planck’s constant divided by 2π. The transition amplitude, considered as a function of the final state, is the famous ‘psifunction’ ψ(x, t), which forms the basis for the adopted description of quantum phenomena. To find ψ(x, t), one should either solve Schrödinger’s equation or calculate the action S and use Eq. (12). Note, incidentally, that representation (12) elucidates the nature of the wellknown principle of least action. Since Planck’s constant is small, paths with greatly differing S are characterized by an enormous phase difference and, hence, their contributions cancel each other out; only for the paths near the S extremum is the phase 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 finite state is attainable along several independent paths, then the total probability amplitude of the process is the sum of the amplitudes for all paths considered separately. We emphasize that the linearity of the system holds for the probability amplitudes, while the probabilities themselves are related to the amplitudes in a quadratic 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 from O to Q enclosed by the slits, but, as a first approximation, when the slit width is small, each of the beams can be replaced by one path. It is this approximation that 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 amplitude can be represented as the product of the amplitudes corresponding to the motion along separate parts of this route. For example, in our problem, the amplitude ϕ1 (x) is the product of ϕ(O → A), the transition amplitude from O to slit A, and ϕ(A → Q), the transition amplitude from slit A to point Q(x). Solving Schrödinger’s equation or performing calculations according to Eq. (12) for separate paths of two possible transition ways from the source to the detector (point of interest), and then multiplying the corresponding partial amplitudes, we can find the distribution of the mean particle flux along the detector with the aid of Eq. (11). 9.2. Interpretation The above brief description is intended only to emphasize the most important features of the quantum mechanical approach: (B) The basic picture of phenomena is described in the language of probability amplitudes – new concepts that have no classical analogue. The complex probability amplitudes are calculated according to the specified set of rules. It is possible then to find the probabilities of various events that are defined as the squares of the absolute values of the corresponding amplitudes. Thus, according to Max Born (1926), quantum mechanics initially dealt with the probabilistic picture of physical phenomena. This inference, per se, is consistent with conclusion A in Section 4: If probabilistic models are needed even for the description of classical unstable models, they are all the more inevitable in describing microworld phenomena, where the predictability of measurements is restricted by several fundamental circumstances, in particular, by Heisenberg’s uncertainty relations. For example, when the experiment in which the position x of a particle and the conjugate momentum p are measured simultaneously is carried out, the standard deviations σx and σp of the measured quantities obey the inequality σx σp ≥ h̄/2. (13) Restrictions of this type are formally included in the apparatus of quantum mechanics; 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 classical models is attributable not to the probabilistic way of reasoning but primarily to the peculiarity of the quantum behavior itself (above all, nonlocal – in the classical sense – interaction), the abandonment of searches for a deeper deterministic underlying picture, and the clarification of the key role of experimental conditions for the possibilities of describing the phenomenon under study. Collectively, these features required introducing new concepts, including the concept of wave function. Classical physics invoked a probabilistic model for some phenomenon in a situation where the impossibility of describing it exhaustively was obvious either in view of the instability of its behavior, or due to the extreme complexity of the accompanying processes. Such is the origin of the models related to the throws of die, the theory of Brownian motion, and the ensembles of statistical physics. The reality of the deterministic, in principle, behavior of the system under study has always been implied, even if it was not possible to give its detailed description8 . Quantum mechanics considers a probabilistic model not as an approximate description of some deeper picture of microworld phenomena but as a fundamental property of Nature. This statement has repeatedly appeared in various forms throughout the history of development of quantum theory; it will suffice to cite the opinion of Wolfgang Pauli [1954]: “It was wave or quantum mechanics that was 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 auxiliary hypotheses, as do for example the thermodynamic probabilities of classical physics. This revolutionary consequence is regarded as irrevocable by the great majority of modern theoretical physicists – primarily by M. Born, W. Heisenberg and N. Bohr, with whom I also associate myself”. In the early 1950s, when this authoritative statement was made, the crucial role 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 statistical physics and thermodynamics to deterministic laws seems untenable. As for the “primary probabilities” of quantum mechanics, Pauli most likely had in mind only the inevitability of the probabilistic description of the microworld but not God playing dice. The latter would become inevitable if we were dealing only with Nature, as such. In reality, however, we are always forced to give a joint description of the phenomenon and the experimental setup. In this connection, 8 Einstein also had this in mind in 1916 in his theory of interaction of the radiation field with atoms. However, the Einsteinian transition probabilities were the proclaimers of a new quantum theory... 25 Max Born [1963] noted that before the creation of the relativity theory, the concept of ‘simultaneity’ of two spatially separated events was also considered self-evident, and only the analysis of the experimental foundations of this concept performed by Einstein showed its dependence on the frame of reference. V.A. Fok characterized the situation in the microworld as “relativity to the means of observations”. The following two opinions separated by seven decades clarify well the essence of the matter. In his introductory article to the debate between Einstein and Bohr on completeness of the quantum mechanical description of reality, Fok [1936] gave the following clarifications9 : “Quantum mechanics actually studies the objective properties of Nature in the sense that its laws are dictated by Nature itself, not by human imagination. However, the concept of state in the quantum sense is not among the objective concepts. In quantum mechanics, the concept of state merges with the concept of ‘information about the state obtained through a certain maximally accurate experiment’. The wave function in it describes not the state in an ordinary sense but this ‘information about the state’... By the maximally accurate experiment we mean such an experiment that allows all of the quantities that can be known simultaneously to be found. This definition is applicable to both classical and quantum mechanics. However, in classical mechanics, there was basically one maximally accurate experiment, namely, the experiment that gave the values of all mechanical quantities, in particular, the positions and momentum components. It is because any two maximally accurate experiments in classical mechanics give the same information about the system that one could talk about the state of the system there as about something objective, without specifying through which experiment the information was obtained”. The second extract allows the viewpoint of a modern researcher, David Mermin, to be judged: “A wave function is a human construction. Its purpose is to enable us to make sense of our macroscopic observations. My point of view is exactly the opposite of the many-worlds interpretation. Quantum mechanics is a device for enabling us to make our observations coherent, and to say that we are inside of quantum mechanics and that quantum mechanics must apply to our perceptions is inconsistent.” (a quotation from the paper by Byrne 2007). The many-worlds interpretation mentioned by Mermin concerns the interpretation of the measurement process proposed by H. Everett in the mid-1950s (for the history and references, see Byrne 2007). Everett’s dissertation was discussed before its publication in Copenhagen by leading physicists; the reaction was negative. Fok’s clarifications and Mermin’s remark clearly reveal the reason why Everett’s idea cannot be considered acceptable (see also comments by Feynman regarding 9 Now, 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 example may prove to be useful. Suppose that we have a set of dice at our disposal, each of which is made asymmetric in some known way. Say, one of the dice is made so that number ‘6’ occurs very rarely. Obviously, the result of a throw is determined both by the (subjective) choice of a die from the set and by the (objective) structure of this die. Similarly, the result of a quantum mechanical experiment depends both on the ‘objective state’ of the system unknown to us and on the character of the question asked by the experimenter by choosing the experimental conditions. As John Wheeler noted, “Schrödinger’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 framework of the orthodox interpretation of quantum mechanics, allows some of the known paradoxes to be avoided10 . For example, putting the wave function in the list 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. However, we should then also say that the probability distribution {pk } of the possible results of throwing a die collapses similarly when the die stops. This did not happen; paradoxes appear when the concepts introduced by us are ascribed to Nature itself. A reasonable interpretation of the experiment on the diffraction of electrons by two slits should be sought in the same direction. The experimental data strongly suggest that separate electrons are diffracted as if each of them came to the detector through both slits at the same time. Emphasizing the inevitability of a joint description of the physical process under study and the recording, necessarily macroscopic, instrument, quantum mechanics requires particular thoroughness in choosing the words and concepts used. In the case under consideration, the posed question is formulated as follows: Can the interference pattern and the passage of electrons through a particular slit be observed simultaneously in some experiment? A comprehensive analysis shows that using any means that allows the electron trajectory to be established immediately destroys the interference (see, in particular, the lectures on quantum mechanics by Feynman 1965b). Therefore, the question of whether an electron passes through both slits in the experiment on interference is empty to the same extent as the question about the number of devils fit at the needle tip widely debated in the Middle Ages. No physical experiment can answer questions of this kind, hence, physics is forced to abandon the classical 10 We everywhere understand a ‘paradox’ as an ‘imaginary contradiction’. Feynman emphasized: “In physics there are never any real paradoxes because there is only one correct 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 various paths to which markedly differing probability amplitudes correspond. Reality can be successfully described using the rule of addition of the probability amplitudes and the fact that the laws of quantum mechanics seem absurd to us only says about the degree of discrepancy between our everyday experience and the microworld laws. It should be said that, apart from the underestimation of the macroscopic experimental conditions specified by an observer, the viewpoint that clearly overestimates the role of an observer in studying quantum phenomena is also fairly popular. More specifically, it is believed that Nature is real only to the extent to which it appears before the observer. The feelings of an acting physicist that continuously ponders new experimental data was vividly expressed by Richard Feynman [1995]: “This is a horrible viewpoint. Do you seriously entertain the thought that without the observer there is no reality? Which observer? Any observer? Is a fly an observer? Is a star an observer? Was there no reality in the universe before 109 B.C. when life began? Or are you the observer? Then there is no reality to the world after you are dead?” 9.3. Some probabilistic aspects of the uncertainty relation Let us note, referring to the physical meaning of the inequality (13), that quite often met ambiguity in interpretation of this relation caused largely by use of vague concept of ‘measurements error’. Meanwhile, the inequality concerns only the relationship between the characteristic widths σX and σP of the probability distributions of X and its conjugate momentum P , which are interpreted as random variables. For definiteness, we will continue discussion in a frame of experiment on diffraction of weak flux of electrons on a narrow slit. Obviously, at registration of any electron, 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 deviation of impact point from an axial line. We can make width of the very massive slit arbitrarily small, and the measurements with the detector arbitrarily detailed, so the accuracy of measurement of realizations (x1 , p1 ), (x2 , p2 ),. . ., (xN , pN ) in N consecutive passages of electrons through the slit is defined only by devices which are used in experiment. The actually important feature of quantum phenomena is that narrowing a slit leads to increase of average width of a whole set of impact points on the detector. Thus, the true uncertainty of experiment is limited not by accuracy of individual measurements, but the wave nature of microparticles generating the diffraction phenomenon. 28 From the formal point of view, the inequality (13) is a consequence of the fact that the coordinate density distribution f (x) and the conjugate momentum density distribution g(p) are interdependent. Really, for any state |ψi the probability amplitude in coordinate representation, hx|ψi ≡ ψ(x), and the probability amplitude in momentum representation, hp|ψi ≡ η(p), are connected by Fourier transform: η(p) = √ 1 2πh̄ Z e−ipx/h̄ ψ(x) dx, 1 ψ(x) = √ 2πh̄ Z eipx/h̄ η(p) dp. (14) The mentioned interrelation of probability densities is caused by their quadratic definition 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 consequence of equations (14) and (15); corresponding derivation can be found in the book by Fok [2007], P. II, Ch. I, §7. Said above suggests that conjugate random variables X and P possess not only the particular probability densities, but also a joint probability density f (x, p). Nevertheless, searches for a physically sensible joint density, the beginning to which has put Wigner [1932], were unsuccessful. This circumstance deservs to be 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 his recent monograph. 9.4. Quantum chaos In classical mechanics, the concept of ‘chaos’ is associated with a distinct instability of particle trajectories under small variations in initial data and ambient conditions. As we saw, it is the instability that impels us to introduce probabilistic models of classical phenomena. Therefore, the following question is quite legitimate: Can the probabilistic nature of the quantum theory be attributable to a similar instability? Obviously, the classical concept of chaos cannot be extended directly to quantum mechanics, which not only insists on the fundamental impossibility of a simultaneous exact measurement of the conjugate coordinates and momenta of particles but also dispenses with the word ‘trajectory’. If the complete description of reality is 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 wave function defined by Schrödinger’s equation. However, the latter is a first-order 29 equation that always specifies a stable time evolution: two close, in some appropriate sense, states remain so during the entire subsequent evolution11 . As David Poulin [2002] points out, this conclusion follows even from the requirement that the system’s energy be real. In contrast, the description of chaotic systems in classical physics is accompanied by differential equations of the second or higher orders (of course, invoking such equations is not a sufficient condition for the described phenomenon being chaotic). Thus, a productive definition of quantum chaos, if there 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 small difference, but now of the Hamiltonians rather than the initial states, can lead to an exponential divergence of initially close systems. Another important direction is the development of decoherence of quantum systems due to the influence of the environment. In view of the peculiar behavior of chaotic systems, we will also mention that it is desirable to refine Ehrenfest’s theorem concerning the passage to the classical limit. All these studies are of interest in their own right, but the question about the origins of the probabilistic interpretation of quantum mechanics should be associated not with them. We have in mind the “relativity to the means of observations” and the peculiarity of the concept of ‘state’ in quantum mechanics that were discussed in the previous section. Schrödinger’s equation describes not the evolution of the system’s state that exists independently of the experimentation but, speaking somewhat simplified, the time behavior of the potentialities of a specific experiment with regard to the system under study. Therefore, it is generally illegitimate to expect that the variation in wave function will resemble the behavior of classical particles. On the other hand, the instability of evolution is important not in itself; in classical physics, it determines the fact that serves as a real basis for turning to probabilistic models, namely, it produces situations where, as Dirac said, “The result of an experiment is not determined... by the conditions under the control of the experimenter.” But the latter, for a number of reasons, is also highly characteristic of our experience in microphysics. It is not only a matter of the influence of a classical observer on quantum processes. The inappropriateness of the concept of particle trajectory to the results of quantum experiments reflects only one aspect of a more general concept – the microparticle identity principle. Indistinguishability of two particles of a given class (say, electrons) themselves and their trajectories serves as a basis for the amplitude addition rule, which determines the characteristic features of the quantum theory. So far we have restricted ourselves to the discussion of nonrelativistic quantum mechanics, because the inevitability of introducing a probabilistic model is clear 11 The 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 wide range of experimental data is possible without invoking the special theory of relativity. The picture of the world painted by the corresponding model – quantum field theory – is much richer in colors. For example, it appears that the smaller the interaction scale, the more intensely the physical vacuum ‘boils’: new particles incessantly ‘evaporate’ from the vacuum and annihilate almost immediately. Nevertheless, virtual particles often have time to interact with real particles and atoms, which determines the observed effects (in particular, Einsteinian spontaneous transition coefficient). This picture, which more deserves the name quantum chaos, clearly suggests that the probabilistic approach in describing the microworld is 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 see striking corollaries of the quantum theory. In classical physics, we got accustomed to the fact that the interrelation between events is attributable either to the action of one of them on the other or to their common past history. In the opinion of Einstein, Podolsky, and Rosen [1935], the orthodox quantum theory introduces a new type of interaction that can manifest itself in the parts of the system that have already ceased to influence each other in an ordinary sense. This paper is so often referred to that the abbreviation EPR became common for it and for the corresponding effect. In Schrödinger’s papers that followed the EPR work in the same year, the term ‘entanglement’ of the properties of noninteracting systems was used. Over the last quarter of the century, this phenomenon was subjected to thorough experimental testing; we will consider an idealized version of the real experiment that retains the 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 will deal 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 specific example. Let it be required to measure the spin of a neutron – its intrinsic mechanical moment and the related magnetic moment. This can be done using the Stern-Gerlach setup, in which the microparticles fly between the poles of a magnet that produces a strong and, what is important, nonuniform magnetic field. Since 31 Figure 3: Scheme of the thought experiment on measuring the spins of two neutrons emerging from source O. Detectors A and B are separated by an impermeable baffle (hatched), numbers 1, 2, and 3 mark the directions in which the spin is measured. the spin of a free neutron is an arbitrarily directed vector, one could expect that when measuring its projection in the direction specified by the arrangement of the magnet poles, we would obtain a certain value from a continuous range of values. However, experiment shows that the neutron spin projection vector is always equal in magnitude to h̄/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 a purely quantum property; the angular momentum of a body about an axis passing through the center of inertia represents its indirect analogy in classical mechanics. According to the interpretation adopted in quantum mechanics, the choice of one of the two possible neutron spin projection directions is random. In the experiment under 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 is measured. As we see from Eqs. (16), if the spin was initially directed to one of the poles, then its measured projection would retain its direction. These peculiarities of 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 eigenvalues corresponding to this quantity will be obtained (Dirac 1958, Sect. 10). In our case, the eigenvalues of the spin are ±h̄/2. Let us now turn to the critical experiment (Fig. 3). In source O, the system of 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 directions toward 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 to the observers. The source and the observers are considered in a common inertial frame of reference; the distances OA and OB are assumed to be equal. (The conclusions 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 the detectors are far apart or are separated by an impermeable baffle, then the neutrons no longer interact, in the classical understanding of this word, with one another shortly before their recording. Since the result of measuring the spin projection for one particle is random, it is clear that the result of each individual experiment to measure the spins of two neutrons with the corresponding detectors will also be random. 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 semiclassical model that makes it possible to independently consider the remote neutrons but takes into account the quantum character of the spin measurement expressed by Eqs. (16) firmly established in experiments. Subsequently, we will give the conclusions of an analytical study of the same problem in terms of the quantum theory and, finally, will present the corresponding experimental data. 10.2. Semiclassical model Taking into account spin conservation in a closed system, we should consider the total spin of a system of two neutrons flying apart to be always zero. In the classical approximation, if this was the case for the spin, it could be assumed that the neutrons are independent and their spins are directed oppositely along some straight line whose orientation can change arbitrarily in successive experiments. We will consider the direction from the south pole of the detector to its north pole to be positive. The subsequent explanations will be simplified if we equip each detector with two lamps, green and red, and adopt the following condition: the green (G) and red (R) lamps turn on if the recorded neutron spin projection is oriented, 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 a separate experiment: GG, GR, RG, or RR. Let us find the probabilities of these events by assuming, for simplicity, that the detectors are oriented in the same direction, say, in direction 1 (Fig. 3). Denote the angle between the positive direction 1 and the direction of the spin vector for neutron A realized in a given experiment by Θ ∈ [0, π]; the angle between the spin of neutron B and the same direction 1+ will be π − Θ. As was said, the spin projection measured in A will be randomly oriented in the positive or 33 1 0.9 0.8 0.7 g(Θ) 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.5 1 1.5 Θ 2 2.5 3 Figure 4: Recording probability of oppositely directed neutron spin projections versus angle between the detector and spin orientations for the semiclassical model. negative direction of vector 1. Denote the possible results of the measurement for neutron A by A1+ and A1− , respectively; for neutron B, the possible results will be B1+ and B1− . Taking into account the independence of the neutron recording events and Eq. (16), we obtain for the sought probabilities of the events: + · B1+ ) = cos2 Θ cos2 π−Θ = cos2 (Θ/2) sin2 (Θ/2), P r(GG) = P r(A1 2 2 P r(GR) = P r(A1+ · B1− ) = cos2 Θ sin2 π−Θ = cos4 (Θ/2), 2 2 π−Θ 2 Θ 4 − + 2 P r(RG) = P r(A1 · B1 ) = sin 2 cos 2 = sin (Θ/2), P r(RR) = P r(A1− · B1− ) = sin2 Θ sin2 π−Θ = sin2 (Θ/2) cos2 (Θ/2). 2 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 spin projections are recorded, i.e., the lamps of different colors will turn on: P r(GR) + P r(RG) ≡ g(Θ) = 1 − 1 sin2 Θ. 2 (18) The function g(Θ) has the meaning of a conditional probability of occurrence an event 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 spin projections recorded with confidence; for noncoincident orientations, the values of g(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. In particular, for an isotropic orientation of the straight line along which the neutron spins are directed, the probability that this straight line will fall within the solid angle dω is dω/4π = sin θdθdϕ/4π, where the factor (1/2) sin θ is the distribution density of the polar angle Θ and 1/2π is the distribution density of the azimuth angle Φ. Averaging p+ from (16) over all angles yields the probability that detector A in a given experiment will record the positive spin direction, i.e., the green lamp will turn on: 1 4π Z 0 2π dϕ Z π 0 1 p+ (θ) sin θdθ = 4π Z 2π dϕ 0 Z π cos2 (θ/2) sin θdθ = 1/2. (19) 0 As would be expected, each of the detectors records one or the other spin projection with a probability of 1/2, so in a long series of experiments, the lamps of different colors turn on equally frequently. However, we are more interested in the correlation between the results, which requires calculating the unconditional probability pd of recording oppositely directed spins. This is achieved by averaging Eq. (18) for the function g(Θ) over all angles: pd = 1 4π Z 0 2π dϕ Z π 0 1− 1 sin2 θ sin θdθ = 2/3. 2 (20) Thus, the semiclassical model predicts that for an isotropic distribution of the neutron spin direction, on average, 2/3 and 1/3 of the experiments will lead to the recording of oppositely and identically directed spin projections, respectively. The isotropic model we considered cannot be reckoned to be mandatory; it is only an example of one of the possible spin direction distributions for the particles flying apart. Only the fact that follows from the form of the function g(Θ) in (18) is actually important: for any spin direction distribution that admits a deviation from the orientation of the detectors, the probability of obtaining oppositely directed spin 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 quantum mechanical calculations. David Bohm [1952] performed such calculations for a situation where the detectors were oriented in the same direction and the recording events occurred simultaneously12 . His conclusions are: 12 The basically analogous process of positronium annihilation is considered in Feynman’s lectures on physics [1965b], Sect. 18.3. This solution is briefly reproduced in the Appendix to this chapter. 35 1. In the initial state of the system, when its total spin is fixed, it is admissible to talk about the spins of individual neutrons only provisionally. The possible spin states of a pair of neutrons depicted as ↑↓ and ↓↑ interfere between themselves and just this interference provides both the fixed total spin during the experiment and the conjugacy of the results of measuring the spins for remote neutrons. 2. In individual experiments, the possible values of the spin projection for each of the neutrons, positive and negative, occur randomly with a probability of 1/2; detectors A and B always record the opposite spin directions. 3. The unequivocal connection between the results of measuring the particle spins does not point to any influence of one of them on the other after the termination of the interaction understood in the classical sense. Thus, the predictions of the two theories are significantly different: according to quantum mechanics, oppositely directed particle spin projections are always recorded, i.e., pd = 1, while for all nontrivial semiclassical models there is a nonzero probability of identically directed projections and pd < 1. Why does not the assumption of the independence of the neutron states hold while the quantum theory confirms that there is no interaction between the neutrons? The reason is that both neutrons always constitute a single quantum system that turns to one of its two eigenstates as a result of the measurement. In each of the system’s eigenstates, the neutron spin projections are opposite. “If there are two particles in nature which are interacting, there is no way of describing what happens to one of the particles by trying to write down a wave function for it alone. The famous paradoxes... where the measurements made on one particle were claimed to be able to tell what was going to happen to another particle, or were able to destroy an interference have caused people all sorts of trouble because they have tried to think of the wave function of one particle alone, rather than the correct wave function in the coordinates of both particles. The complete description can be given correctly only in terms of functions of the coordinates of both particles.” (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 papers by Bell [1964, 1966], who pointed out the possibility of its experimental verification. Imagine that a series of experiments of the type described above is made in the scheme shown in Fig. 3, but now the detectors are not fixed in the same direction – the observers randomly choose one of the three possible orientations of 36 their instruments independently of one another. As above, the observer records an event of one of the two types in an individual experiment: (G) the direction of the measured spin projection coincides with the arbitrarily chosen positive orientation of the magnet poles, and (R) the measured spin projection is directed oppositely. Bell proved that considering the neutrons as independent particles in the classical sense (in other words, the existence of hidden variables) entailed the fulfillment of the rigorous inequality for the probabilities of the combinations of events of a certain type. Bell’s theorem, especially in the form that was imparted to it by Clauser et al. [1969], allows the conclusions of the classical and quantum models to be really compared (one can find the clear formal discussion in a lecture course by Kiselev 2009). The fact that in almost all experiments photons were used instead of neutrons and the photon polarization rather than the spin projection was measured does not change the essence of the problem. Pairs of photons with correlated linear polarizations were produced under two-photon transitions of exited atoms; the polarization direction of each photon was ascertained using calcite crystals or other analyzers (Shimony 1988). Almost all experiments, starting from the first of them performed by Freedman and Clauser [1972], suggest that Bell’s inequalities break down; in the cases where the opposite result was obtained subsequent verification revealed shortcomings of the experiment. As was said above, Bell’s inequalities must hold if the classical understanding of independence is valid, therefore, their breakdown suggests that the existence of hidden parameters is incompatible with the behavior of the microworld. The experiments performed by Aspect et al. [1982] became widely known (see also Aspect 1999; Mermin 1981; Brida et al. 2000). To exclude the possibility of mutual influence of the recording events in a given experiment, the orientation of the polarization analyzers was chosen during the flight of photons. Experiments on the correlation of photons in the parts of the setup the separation between which reached 18 km have been performed recently (Salar et al. 2008); these experiments have again demonstrated the unavoidability of the quantum behavior of light. Although the described experiments are spectacular, it should be recognized that fundamentally they have added little since the debate between Einstein and Bohr regarding the meaning of the EPR effect and, especially, after the formal clarification of the problem by Bohm. However, the work of Bell is of great importance in more general relation, namely, substantiation of quantum mechanics. The point 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 mechanics relied on an erroneous assumption (see also an excellent discussion by Rudolf Peierls 1979). The Bell’s argument has given a new, this time a correct proof that the theory of classical realism with hidden variables can reproduce the 37 experimentally confirmed predictions of quantum mechanics only by violation an essential physical requirement, the condition of locality. 10.5. Illusory superluminal speed The classical interpretation of the above-mentioned experimental data inevitably leads to the supposition that the signals between remote observers can be transmitted with a superluminal speed, which is in conflict with the experimentally tested postulate of the special theory of relativity. In numerous publications, not only popular ones, fairly vague explanations are given on this subject. In particular, one alludes to the mysterious ‘collapse of the wave function’ (wave packet) that we have already commented on in Section 9 devoted to the laws of quantum mechanics. Meanwhile, the transmission of a signal (information) between observers is not required at all within the framework of the latter for both identical and independent orientations of the detectors. In both cases, the lamps of different colors on each of the detectors turn on equally frequently and if the observers are isolated from each other, then the strong correlation between the measurement results remains unknown to them. The existence of a correlation will be revealed only after the completion of a fairly long series of experiments, when all data will be collected in one place. The same is also true for the situation where one of the observers will change the orientation of his detector, thereby attempting to code the message to his colleague: the lamps of different colors on each detector turn on equally frequently, while the correlation between the results remain hidden. It is also worth noting that knowledge of the quantum laws by the observers does not promote the transmission of information between them either. Suppose that the detectors are oriented identically and observer A is slightly closer to the source than B, so the former will perform a measurement slightly earlier than the latter. If observer A sees the turn-on of the green lamp, then he immediately receives one bit of information relative to the result of observer B: the latter will see the turn-on of the red lamp. However, this information is provided by the set of all stages of the experiment: preparing the initial two-particle system, providing the corresponding information to the observers, and, finally, the flight of particles from the source to the detectors. Let us explain this by a simple example. Suppose that an inhabitant of Siberia sends messages to two friends living in Paris and Tokyo. All three agreed in advance that a piece of green paper be randomly (e.g., equiprobably) enclosed in one of the envelopes and a piece of red paper be enclosed in the other. Obviously, having opened the envelope, the Parisian will immediately learn the color of the enclosed paper in the envelope of Tokyo’s resident. In this case, no information 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 quantum behavior as in the experiment on the diffraction of electrons by two widely separated slits. This is quite sufficient to once again, as Bohr said, experience a shock when one familiarizes oneself with quantum mechanics, but it is absolutely unnecessary to invoke the reasoning about the influence on remote objects with a superluminal speed. Like most other paradoxes of quantum mechanics, this reasoning is attributable to the improper use of the concept of a state of a quantum system (see Section 9.2 above). Changing the detector orientation, the researcher carries out a new experiment whose description enters as a component into the new state of the entire system. It is quite natural that changing the experimental conditions can affect the result obtained. By choosing the orientation of his detector, the observer selects the possible alternatives of the experiment, thereby creating the illusion of influence on a remote object. For the subsequent ‘explanation’ of the effect, it remains only to repeat what was said in Section 10.3 and to refer to the analysis given in the Appendix. Einstein rightly believed the direct action at a distance to be inadmissible in physical theories, but there is no need to resort to this concept in quantum mechanics. 11. Schrödinger’s cat could play dice An impressive illustration of the concept of entanglement of quantum states was proposed by Erwin Schrödinger in his 1935 papers initiated by the work of Einstein, Podolsky, and Rosen. The case in point is a thought experiment, in which a cat is in a superposition of partially alive and partially dead states. Consider the probabilistic aspect of this construction. Imagine that the following objects were placed in a closed box: a cat, a grain of 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 immediate death of the cat. The critical question is: How must an external observer describe 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 triggered the counter, the ampoule is intact, and the cat is alive; the second state corresponds 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 superposition 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 any closed system is realized only when the measurement is done. Therefore, the cat is partially alive and, at the same time, partially dead until the observer opens the box and, thus, ‘measures’ its state. The classical experiment tells us that the object is in one of the possible states irrespective of whether the measurement is made. (Einstein: “The moon exists even when I don’t look at it.”) Accordingly, the ‘classical cat’ is either alive or dead but we do not know precisely in which state it is. In contrast, the above description leads to the conclusion that the cat is in a strange superposition of eigenstates. Let us now consider the experiment more carefully. First of all, it should be emphasized that a subsystem of a closed system, in our case, the cat, cannot be characterized by a wave function. The state of the subsystem is described using the density matrix introduced by von Neumann, which corresponds to a mixture of several pure states taken with fixed weights. In the probability theory, this corresponds to the so-called randomization of the distribution function over the possible values of some parameter. Landau and Lifshitz [1963] pointed out, “For the states that have only the density matrix, there is no complete system of measurements that would lead to unequivocally predictable results”. Note also that the evolution of the subsystem depends on all details of its interaction with the remaining part of the system. Thus, there is no way of ascertaining the cat’s state until the observer opens the box, hence, the question about how it feels is vacuous. Physics answers only reasonable questions13 . The present-day experiments with macroscopic systems that have two eigenstates are properly described using the apparatus of quantum mechanics (Blatter 2000). In Schrödinger’s experiment, the situation is dramatized by the fact that a habitual breather is involved in it. Nothing will change essentially if we replace the cat, say, with a pendulum clock and the poison with a stopper. We can go even farther and place a microlaser or even a vibrating diatomic molecule in a closed system instead of the clock. The paradox with Schrödinger’s cat can also be resolved without using such formal concepts as the density matrix: the terminology itself invoking the concept of a system that is partially in different states is provisional. For example, analyzing the experiment with the passage of light through a tourmaline crystal described in Section 5, Dirac [1958] wrote: “Some further description is necessary in order to correlate the results of this experiment with the results of other experiments that might be performed with photons and to fit them all into a general 13 However, 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 answer questions outside the domain of science, but as an aid to the formulation of rules for expressing concisely the results of large numbers of experiments. The further description provided by quantum mechanics runs as follows. It is supposed that a photon polarized obliquely to the optic axis may be regarded as being partly in the state of polarization parallel to the axis and partly in the state of polarization perpendicular to the axis.” Obviously, the cat in two states is no more surprising than the analogous behavior of a photon. The ‘simultaneous’ passage of an electron along all possible trajectories in Feynman’s approach should also be understood in the same sense (see Section 9.1). Let us also touch on the following curious question: Will the description of the system change from the standpoint of an external observer if the radioactive material 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. If an odd number occurs, the actuating device (or the cat) breaks the ampoule with poison. 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, are widely believed to provide ‘true’ randomness, unlike the behavior of macroscopic bodies governed by the laws of classical mechanics. However, in the experiment with Schrödinger’s cat, if it is considered from the viewpoint of quantum mechanics, it does not matter precisely how the necessity of a probabilistic description of the subsystem from the standpoint of an external observer is provided. As for quality of the device that implements stochasticity, analysis of the probabilistic models of classical physics suggest the following: when the simple conditions that provide instability of motion are met, the behavior of the die is as unpredictable as radioactive decay. After all, the die can be reduced to the sizes of a Brownian particle without violating the fundamental aspect of the experiment. 12. Conclusion The above examples strongly suggest that God does not play dice; in the quantum world, as in classical mechanics, the probabilistic concepts are inherent only in models of real phenomena but not in Nature itself. The word ‘model’ is a key one here – as soon as the stochastic behavior is ascribed to some real objects, such as sea waves, billiard balls, molecules, or photons, the appearance of contradictions becomes inevitable. This assertion seems so obvious that it remains only to wonder how widely the opposite viewpoint is covered in literature, including classical 41 works. The probabilistic concepts, as a branch of mathematics, were created for several centuries in order to formalize various life experiences and scientific data. It would be strange to believe that these concepts were ‘built’ in objective reality from the outset; the probability theory is only an efficient tool for modeling Nature. Another conclusion is that standard quantum mechanics in terms of which our discussion was conducted gives a strange but logically consistent explanation for all of the experiments performed to date. This is enough to leave aside the widespread, in recent years, attempts to associate the interpretation of experiments with the consciousness of an observer, the birth of universes at each act of observation, the transmission of information with a superluminal speed, and other mystical phenomena. The aforesaid by no means rules out the quest for models different from the theories of Schrödinger, Heisenberg, and Feynman; they may give a deeper and clearer picture of quantum processes (in fact, Feynman’s approach serves as such an example). New experiments may also require a fundamentally new approach, but it is quite unrealistic to expect the appearance of a deterministic basic model. Note also that the probabilistic formulation of models for real processes determines the statistical nature of the inverse problem that consists in ascertaining the true properties of the processes from their observed manifestations (Terebizh 1995, 2005). It has been repeatedly pointed out that the difficulties in understanding the quantum theory primarily stem from the fact that the behavior of quantum systems is unusual: our everyday experience concerns the properties of surrounding us massive bodies and waves, while the microparticles are neither the former nor the latter. This is true, but still it is hard to avoid the feeling that the rules of quantum mechanics are a set of strange procedures justified by nothing but their unconditional practical efficiency. If there was no statistical physics, similar feelings would also be aroused by the laws of thermodynamics – while being important in engineering. In a similar context, Richard Feynman described the algorithm for predicting solar eclipses developed for centuries by South American Indians. From generation to generation, priests handed ropes with many tied knots to their disciples; it took many years to memorize the rules for handling the knots, but the meaning of these rules remained completely mysterious for Indians. We will add that the system worked like the model proposed by Thales of Milet in the 6th century B.C.; a significant difference between the approaches is that, without explaining the nature of the motion of celestial bodies, Thales’s theory nevertheless proceeded from a simple model of the Solar system. Critics of quantum mechanics also would like to have a simpler underlying model of microworld phenomena from which the set of quantum rules would follow. 42 In response to reasoning of this kind, an advocate of the orthodox interpretation of quantum mechanics can recall the origin of the classical equations of electrodynamics. James Clerk Maxwell, who wrote a complete system of equations, made much effort to create a mechanistic model of phenomena that led to the equations of electrodynamics. This proved to be an unsolvable problem, while Maxwell’s equations per se became habitual in time to an extent that physicists left aside the question about their origin. A similar situation also arose with regard to the basic principles of classical mechanics, in particular, the law of inertia. Eight centuries ago, the King of Castile Alfonso X named ‘The Wise’ had reasons to note, “Had I been present at the creation of the World, I should have recommended something simpler.” We can only guess what Alfonso X would say after familiarizing himself with quantum mechanics. It probably would not be simpler. Acknowledgements I am deeply grateful to V.V. Biryukov (Moscow State University), Yu.A. Kravtsov (Space Research Institute, Moscow), and M.A. Mnatsakanian (California Institute of Technology) for a stimulating discussion of the issues under consideration and constructive suggestions. In particular, Yu.A. Kravtsov pointed to the importance of allowance for the low-frequency radiation in collisions of molecules and the necessity of discussing the problem of quantum chaos. Appendix. Positronium annihilation In our description of the experiment on the recording of neutrons, we gave only a reference to the formal solution of the problem in terms of quantum mechanics performed by Bohm [1952]. Below, we reproduce with minor changes the analytical consideration of a basically similar process – the annihilation of a positronium atom – contained in Sect. 18.3 of the lectures by Feynman [1965b]. This solution is all the more instructive, because it is given with clarity characteristic of Richard Feynman and is performed in the context of a modern approach to quantum mechanics. A positronium atom is composed of an electron e− and a particle with opposite charge sign, a positron e+ . The spin of each of these particles is 1/2 (in units of h̄); initially, we consider an atom at rest with antiparallel spins of its components and a zero total spin. The characteristic lifetime of this system is 10−10 s, following which the electron and the positron annihilate with the emission of two γ-ray quanta. 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 of the produced photons. In this case, to describe the photons, it is convenient to choose a state of circular polarization. Recall that we arbitrarily attribute a right-hand circular polarization to the monochromatic light wave if the electric field vector rotates counterclockwise as it travels toward the observer; the photons constituting the wave, accordingly, are assumed to be right-hand circularly polarized (state |Ri). In a beam of left-hand circularly polarized photons (state |Li), the electric field vector rotates clockwise if we look at the approaching wave. Obviously, two alternate modes of decay that conserve the system’s zero total spin are admissible (Fig. 5). Two right-hand circularly polarized photons are produced in the first mode; each of them has an angular momentum of +1 relative to its momentum direction, while the angular momenta relative to the z axis are +1 and −1. Denote this state of the system by |R1 R2 i. Two left-hand circularly polarized photons are produced in the second mode denoted by |L1 L2 i. We will assume that |R1 R2 i and |L1 L2 i constitute an orthonormal basis. According to the superposition principle, the final state of the system after annihilation |F i is a linear combination of the alternate states |R1 R2 i and |L1 L2 i. 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 odd parity and the parity of |F i must be the same; (2) the normalization of |F i must provide a unit sum of the probabilities of all possible realizations. The only linear combination of the alternative states that satisfies both conditions is √ (A1) |F i = (|R1 R2 i − |L1 L2 i) / 2. Indeed, the operator of spatial inversion P̂ changes both the direction of photon motion and the direction of its polarization; therefore, P̂ |R1 R2 i = |L1 L2 i, P̂ |L1 L2 i = |R1 R2 i. As a result, P̂ |F i = −|F i, suggesting that parity is conserved under positronium annihilation. Having the final state of the system, we can calculate the amplitudes and probabilities of events of various kinds. In particular, the amplitudes of the two alternative types of decay that we mentioned above are √ √ hR1 R2 |F i = 1/ 2, hL1 L2 |F i = −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 zaxis will always record equiprobably either a pair of right-hand photons or a pair of left-hand photons. The scheme of the experiment shown in Fig. 6 is of interest in the context of the question 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 positronium 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 light detector. It is required to ascertain the way in which this scheme operates when many annihilation processes are observed successively. More specifically, since the triggering 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 probabilities of the corresponding processes should be found. As an example, let us calculate the amplitude hy1 x2 |F i of the event that consists in the triggering of counters D1y and D2x . Taking into account the representation (A1), we find √ (A3) hy1 x2 |F i · 2 = hy1 x2 |R1 R2 i − hy1 x2 |L1 L2 i. Since the photon recordings by different counters are independent events, we assume that (A4) hy1 x2 |R1 R2 i = hy1 |R1 ihx2 |R2 i and similarly for hy1 x2 |L1 L2 i. As a result, Eq. (A3) takes the form √ hy1 x2 |F i · 2 = hy1 |R1 ihx2 |R2 i − hy1 |L1 ihx2 |L2 i. (A5) Next, we should take into account the fact that the states of right-hand and lefthand circular polarization are related to the states of linear polarization along the x- and y- directions by the relations ( √ |Ri = (|xi + i |yi) /√ 2, (A6) |Li = (|xi − i |yi) / 2. 45 Figure 6: Scheme for recording the photons produced by positronium annihilation. C1 and C2 are the calcite crystals; x1 , y1 , x2 and y2 are the paths of the photons linearly polarized in the x and y directions; D1x , D1y , D2x and D2y are the photon counters. Taking into account the orthonormality of the system |xi and |yi, we find from (A6): ( √ √ hy1 |R1 i = +i/√ 2, hx2 |R2 i = 1/√ 2, (A7) hy1 |L1 i = −i/ 2, hx2 |L2 i = 1/ 2. Substituting these expressions into (A5) finally yields √ hy1 x2 |F i = 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: ( hx1 x2 |F i = hy1 y2 |F i = 0, √ (A9) hx1 y2 |F i = hy1 x2 |F i = i/ 2. These expressions show that one of the pairs of counters in the channels with mutually orthogonal polarizations of photons is triggered always, and with equal probabilities: either D1x and D2y or D1y and D2x ; the triggering probabilities of the pairs with identically directed polarizations of photons are zero. Thus, Feynman’s analysis of the positronium annihilation process leads to the same conclusions as those listed in Section 10.3 from Bohm’s calculations concerning the recording of particles with a spin of 1/2. Such are the predictions of the quantum theory that are fully consistent with experimental data. Where is then the paradox in the described situation? Suppose that observer 2 is slightly farther from the positronium atom than observer 1. As a consequence, the triggering of a particular counter of observer 1 will allow him to predict with certainty precisely which counter of observer 2 will be triggered. On the other hand, the photon flying toward observer 2 is in a superposition of states |x2 i and |y2 i with linear polarization, therefore, this photon could seemingly reach one 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 observer completely determines the result of an experiment that has not yet happened? Does this mean that there is some interaction that propagates with a speed higher than the speed of light in a vacuum? In his lectures, Feynman gives a detailed interpretation of the experiment with which, of course, one should familiarize oneself carefully. In our view, even the structure of representation (A1) gives an answer to the above questions: The pair of photons produced by annihilation was initially prepared in such a way that the two corresponding detectors always recorded a circular polarization of one type. This ‘preparedness’ of the system is retained after the passage of photons through calcite crystals and determines the fact of triggering the pairs of detectors with mutually orthogonal polarizations of photons. No influence, not to mention superluminal one, on the remote process is required. 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