CRITIQUE OF THE IDEALIST UNDERSTANDING OF QUANTUM THEORY*

D. I. Blokhintsev

1. THE PHILOSOPHICAL VIEWS OF THE COPENHAGEN SCHOOL AND THE PRINCIPLE OF COMPLEMENTARITY

The Copenhagen school of physics, from its very inception, associated itself with Machism and subsequently contributed greatly to the development of subjectivist views on the essence of quantum mechanics. Thus, Heisenberg places at the foundation of his methodology the so-called “principle of observability,” according to which the subject matter of physical research should be only in principle observable quantities.

It was precisely from this point of view that Heisenberg criticized the concept of Bohr orbits in atoms.

Some foreign physicists are inclined to ascribe to this principle almost the chief role in the creation of quantum mechanics. This is presented in the following way.

Heisenberg, following the “principle of observability,” rejected the concept of electron orbits in atoms as a concept to which nothing observable in experience corresponds; leaving these orbits aside, he turned to directly observable (and measurable) quantities—to the frequencies of radiation and to the intensities of radiation—and on this path created a new, quantum mechanics.

In fact this is not so. The basis for the construction of the new theory was not the principle of observability, but new, previously unknown facts, whose significance had been only partially recognized in Bohr’s primitive theory. These facts consisted in the fact that the measured atomic frequencies $\omega_{mn}$ and radiation intensities $I_{mn}$ were characterized, as experience showed, by two quantum indices $m$ and $n$. One of them (for example, $m$)

* The article was prepared by the author for the collection Philosophical Problems of Modern Physics, published by the Institute of Philosophy of the Academy of Sciences of the USSR.

belonged to the initial state, and the other \((n)\)—to the final state of the atomic system. According to classical mechanics, however, these quantities ought to have been single-indexed: \(I_n,\ \omega_n\), where \(\omega_n\) would denote the frequency of the \(n\)-th overtone, and \(I_n\) the intensity of the radiation of this overtone*). It was precisely the character, revealed by experiment, of the frequencies and intensities emitted by atomic systems that compelled doubt as to the applicability of classical mechanics to them, in particular as to the applicability of the concept of orbits, and led to the construction of a new, quantum mechanics (as it was then called—“matrix” mechanics**).

The circumstance that the orbits predicted by classical mechanics were not found in experiment (incidentally, at that time it had by no means yet been proved that they were in principle impossible to detect) gave only the possibility of supposing that, in the atom, classical mechanics was untenable and had to be replaced by a new one; that is, the unobservability of orbits (factual, not in-principle) was consistent with a structure of frequencies and intensities that contradicted classical mechanics, and nothing more.

Therefore this principle did not play the role in the development of quantum theory that the positivists ascribe to it. Nor could it have played any significant role, since it is flawed in its very essence.

In fact, the adjective “in-principle” (observability) deprives this principle of all meaning, since the question of what is “in principle” observable and what is “in principle” not observable can be decided only on the basis of a completed theory, which can already be regarded as verified by the totality of the data.

But if the theory has already been created, then the “principle of observability” is unnecessary, since it follows from the theory itself what in it has objective significance (is observable in principle), and what does not.

For example, no one has observed velocities of bodies greater than the speed of light. Nevertheless, from the standpoint of Newtonian mechanics such velocities are “in principle” observable. The subsequent development of science showed that this assertion of Newton’s theory goes beyond the limits of its applicability and is in fact incorrect. At high velocities of motion one must turn to the theory of relativity, according to which velocities greater than the speed of light are “in principle” unobservable (since they contradict the laws of nature).

The solution of the question of observability in principle in this case was again based not on that long-known empirical—

*) See, for example, 10.
**) See 11.

in the known fact that the physicist had not encountered, in any phenomena, velocities greater than the speed of light (after all, he might have encountered them in subsequent investigations), but on new facts indicating the unsuitability of classical, Newtonian mechanics—in particular, on facts relating to the propagation of light in moving systems of bodies.

Therefore, to draw a conclusion about the “fundamental” observability of one or another phenomenon solely on the basis that the means at the physicist’s disposal do not allow him to observe it means sliding into idealism. Mach, too, proceeded essentially from the principle of observability, asserting that the molecules, atoms, and electrons of contemporary physicists were the same as “the medieval witches’ Sabbath.” Atoms, in E. Mach’s opinion, were only an invention to which nothing corresponded in the “complex of sensations.”

The later development of physics, on the contrary, confirmed the predictions of the supporters of the kinetic theory of matter, who, on indirect evidence, had long assumed that matter is formed by an aggregate of moving atoms or molecules. The great Lomonosov, as early as the eighteenth century, on the basis of atomic theory developed kinetic views of the nature of heat, anticipating his contemporaries by more than a hundred years. Thus, the development of physics showed that the “principle of fundamental observability,” which ultimately proceeds from Mach’s conception of the “complex of sensations,” is an internally untenable principle and one leading to error.

Meanwhile, many physicists, including some Soviet ones, discern in this “principle” a sound kernel. They say that this principle, supposedly, correctly suggests the necessity of a critical revision of those physical concepts which are contained in a theory but do not find direct confirmation in experiment. Concepts of this sort may, they say, be unnecessary appendages, hindering the development of science (examples: phlogiston, the elastic ether, electron orbits in atoms, etc.).

Such an appraisal of the “principle of fundamental observability” is the result of a narrow craft approach to the methodology of physics, the result of forgetting the foundations of Marxism-Leninism or of ignorance of them. In essence, it represents an attempt to replace the profound and all-embracing doctrine of materialist gnoseology with the anti-scientific principle of the positivists.

In fact, what significance can this principle have in comparison with the teaching of Marxism-Leninism on concepts as reflections of real things, on the necessity of considering phenomena and concepts in their development and in their mutual connection, on the transition of quantity into quality, and so forth.

Mach introduced the fact of the (temporary) unobservability of atoms among the totality of other facts and, together with the energetists, suffered defeat in his predictions.

If Heisenberg had not used the assertion of the unobservability of electron orbits to construct a theory of new facts, but had limited himself only to an assertion about the unobservability of electron orbits, then he, of course, would have created no new mechanics.

“Unobservability” (taken in itself, in isolation) cannot serve as the basis for any predictions, and can lead only to Mach’s idealistic conclusions.

To make still clearer the circumstance that the “beginning of principled observability” did not have the significance in the development of quantum mechanics which is attributed to it by the positivists who advertise it, let us recall that, besides Heisenberg’s matrix mechanics, quantum mechanics also had another source—the wave theories of de Broglie and E. Schrödinger, which developed along a path directly opposed to the ideas of “principled” observability. Meanwhile it was precisely these theories that absorbed Heisenberg’s formal scheme, led to the prediction of diffraction phenomena for particles, and thereby placed the new mechanics on a firm foundation.

Indeed, only after the experimental proof of the diffraction phenomena predicted by the wave theories did quantum mechanics receive a weighty basis in experience. Along the same path the concept of the wave function, the fundamental concept of quantum mechanics, was discovered, and its statistical interpretation was given.

The path of wave (as it was then called) mechanics was the path of seeking a qualitatively new regularity, the path of generalizing regularities found for light to other forms of matter (at first to electrons, and later to all other “particles”). At the very beginning of this path there arose the concept of the wave function, which not only did not belong among the quantities included in the “complex of sensations,” but in general the very method of comparing it with observable, measurable quantities was at first unclear. Its introduction into the theory was based on the conjecture that wave phenomena in nature might have a broader significance than had hitherto been thought. This supposition could be true or false, and only the subsequent development of theory and experiment showed that it was true. It is no accident that E. Schrödinger, in his first article³, notes that he was not sure that “anything reasonable” (i.e., corresponding to reality) would come out of his theory.

It is evident from this that the advertising of the “beginning of principled observability,” imbued with the spirit of subjectivism, is based on the falsification of science, on the artificial emphasizing of some circumstances and the passing over in silence of others, more important ones.

Positivist conceptions of the representatives of the Copenhagen school, in their application to quantum theory, are not, however, limited to the “doctrine” of “in-principle observability.”

As a natural consequence of this principle, N. Bohr laid at the foundation of the understanding of quantum mechanics the so-called “principle of complementarity.” According to this principle, two classes of experimental arrangements are possible: the first class permits the determination of momentum-energy relations, the second—of space-time relations. The simultaneous use of both types of arrangements is excluded.

Thus, the “quantum description” of phenomena breaks down into two observable classes, which are complementary to one another in the sense that their totality in classical physics gives a complete description (see, for example, N. Bohr\(^4\)).

We have set out this principle according to Bohr, leaving aside, for example, Heisenberg’s formulation of complementarity as the complementarity of “acausal description” and “the mathematical scheme outside space and time” (see\(^5\)).

Nevertheless, from the content of the principle of complementarity as presented, it is clear that what is emphasized in it is not the fact of the existence of new objects by their very nature, but the possibilities of macroscopic measuring instruments.

In other words, what is brought to the fore is not the peculiarities of the microworld, the consequence of which is the impossibility of studying them by the methods of classical physics, but the possibilities of the observer operating with macroscopic quantities and concepts. Stated otherwise, “observability” is put in first place, while everything else is regarded as its consequence.

Such a subjectivist orientation of Bohr’s principle of complementarity leads to consequences of a twofold kind.

First, N. Bohr and his followers develop this principle into a special philosophical conception of complementarity, for which the denial of the objectivity of microphenomena is characteristic.

Second, the application of the principle under consideration in physics leads to a subjective interpretation of the wave function and of the concept of state in quantum mechanics. The wave function is regarded not as an objective characteristic of a quantum ensemble, but as an expression of the observer’s information obtained as a result of measurements. The reality of one or another state of microsystems, in this understanding, becomes identical with the observer’s information about the microsystem, i.e. is transformed from an objective category into a subjective one.

We shall first consider the first side of the matter and only later turn to the second.

In his analysis of the problems of quantum theory, N. Bohr constantly moves within the circle of the “observed” and “observability,” concerning himself only with an unambiguous correspondence between words and terms and the description of the experimental situation. Such a striving for precision attracts other physicists, who fail to notice how shaky is the philosophical foundation on which the logic of Bohr’s constructions rests.

In his article—a reply to A. Einstein—N. Bohr writes^4 of the unsuitability of “the ordinary point of view of natural philosophy for describing phenomena of the type with which we are dealing in quantum mechanics. In fact, the finiteness of the interaction between the object and the measuring instrument, conditioned by the very existence of the quantum of action, entails—because of the impossibility of controlling the back action of the object on the measuring instrument (and this impossibility will necessarily occur if only the instrument is to fulfill its purpose)—the necessity of finally renouncing the classical ideal of causality and of radically revising our views on the problem of physical reality.”

In another article of his,^6 N. Bohr characterizes the epistemological problem that has arisen in connection with quantum theory as follows: “...on the one hand, the description of our mental activity requires the opposition of an objectively given content and the studying subject, whereas, on the other hand, as is already evident from such a statement, no strict separation of object from subject can be justified, since this latter concept also belongs to the world of thoughts.”

If from the first quotation it still remains unclear from what standpoint “ordinary natural philosophy” is being criticized, then from the second it is already sufficiently clear that N. Bohr stands on the stereotyped positions of idealism.

In a recent article devoted to causality and the principle of complementarity,^7 N. Bohr in essence merely repeats his former views on the significance of this principle, extending the sphere of its application to psychological, biological, and even social phenomena.

The essence of this article may be briefly reduced to the following: In the atomic domain the situation is such that “any attempt to localize an atomic object in space and time requires an experimental arrangement which leads to an in-principle uncontrollable exchange of energy and momentum between the object and the scales and clocks serving as the system of reference. Conversely, there exists no arrangement suitable for controlling the balance of energy and momentum that permits an exact description of the phenomenon as a chain of events in space and time (cited work, p. 315). The very

the word “phenomenon,” according to Bohr, must include a complete account of the entire experiment as a whole” (ibid., p. 317).

The impossibility of separating the behavior of the object from the measuring instruments requires, according to Bohr, a revision of the problem of “physical explanation.” The place of classical description in the quantum domain must be taken by the concept of complementarity, which represents a “rational generalization of the very classical ideal of causality.”

Thus we see that the entire problem of quantum theory is considered by N. Bohr as a problem of the relation between the instrument and the micro-object, and when he leaves the more definite ground of physics—as a problem of the relation between subject and object.

In this lies the principal methodological defect of the concept of complementarity: in the light of this concept, quantum-mechanical regularities lose their objective character, becoming regularities that follow from the way in which man perceives the phenomena of the microworld. And this is idealism.

Let us explain, with one simple example, where such an “instrumental” approach leads. In foreign physical literature one encounters the so-called “principle of the indistinguishability of particles.” According to this principle, with the aid of instruments it is impossible to distinguish two states of systems that differ by the permutation of particles, one in place of the other. From this principle there follows, in particular, the Pauli principle: in one and the same state there cannot be more than one electron. This regularity has fundamental significance for the structure of atoms and for the bodies formed from them. The following chain of reasoning results: 1) we cannot distinguish particles; 2) therefore the Pauli principle holds, and consequently, 3) physical bodies behave precisely in this way and not otherwise. In other words: only because we cannot observe the difference between particles does there exist the periodic regularity in the structure of atoms discovered by D. I. Mendeleev. It is clear that here everything is turned upside down.

From the standpoint of positivism, such an absurdity is quite natural (what concern is it to positivists that atoms existed long before anyone began to count the electrons in them?); from the standpoint of materialism, however, it is the purest absurdity. It is obvious that the impossibility of distinguishing identical particles is not the initial thing, but a secondary one, flowing from the nature of these particles, from their identity. Indistinguishability is a consequence of identity*), and not the reverse.

The same is true with respect to the principle of complementarity: measuring instruments in reality divide—

*) This term was also adopted by the author in the second edition of his course.^10

3 UFN, vol. XLV, issue 2.

fall into two classes: momentum-energy and space-time; but quantum regularities do not at all follow from this division; on the contrary, this division follows from quantum regularities.

This is also clear from the fact that, starting from the principle of complementarity, it is impossible to construct quantum mechanics. Indeed, let us try from this principle to derive the necessity of the wave function, its physical interpretation, the Schrödinger equation, or anything similar from the concepts and relations that figure in quantum mechanics. This proves completely impossible. From it there would follow only the inapplicability of classical mechanics to the atomic world and nothing more—that is, exactly as much as from Heisenberg’s “principle of principal observability.” Therefore the principle of complementarity cannot serve as the foundation of quantum mechanics. It is only a Machist formulation of one of the consequences of quantum mechanics.

As an opposite example we may cite the theory of relativity. From the principle of relativity and the principle of the independence of the velocity of light from the uniform motion of bodies there follows the Lorentz transformation, and together with it the whole theory of relativity. Proceeding from the principle of complementarity, however, it is impossible to write a single formula.

If N. Bohr still speaks very indistinctly about a certain “natural” philosophy that has become unsuitable, his followers express themselves much more definitely.

P. Jordan, in his book[^8], devotes an entire chapter to the “liquidation of materialism.” It should be borne in mind that P. Jordan did not at all arrive at idealism as a result of analyzing the conclusions of quantum physics. Like the other positivists, he analyzes the problems of both quantum and classical physics from the standpoint of denying the existence of an objective world and objective law-governedness. Following Mach, he believes that the task of science “consists in ordering the results of observation.” In the positivist understanding, both classical and quantum physics are not a reflection of the objective world, but are “mathematical constructions” (cf. P. Jordan[^9]). For the first of these constructions the possibility of separating the concepts of subject and object is characteristic, whereas for the second this separation is impossible, since the subject of measurement “prepares physical reality.” Thus, the question is not of analyzing the relations of the knowing subject and the object as parts of the objective world, but of analyzing these “constructions,” i.e., of an analysis in the sphere of concepts.

From these positions the positivists try to refute materialism in such a way that they first connect it with certain

limited physical or philosophical notions, and then declare it untenable.

V. I. Lenin clearly showed this standard path to idealism¹. Jordan follows this path with an astonishing degree of precision. He foists upon materialism, with striking naïveté and ignorance, the atomism of Democritus and the determinism of Laplace (see p. 144 ff.). As though, since the time of Democritus and Laplace, materialism had undergone no development whatever, as though the teaching of Marx—Engels—Lenin—Stalin did not exist. Such a method of “liquidating materialism” is not new, and from repetition it gains no force.

Jordan writes (cited work, p. 145): “Indeed, comparing the new physics with the materialist picture of the world, one can now establish that what has become obsolete are precisely those features of the materialist conception of nature which expressed the contradiction between materialist theories and other ideas.”

And further: “The atoms of Democritus are indestructible and immutable; contemporary ‘elementary particles,’ by contrast, are capable of unlimited transformations.”

Thus, materialism is identical with Democritus’ doctrine of immutable atoms.

Incidentally, let us note that Jordan has not forgotten Mach’s mistake with his unsuccessful prediction concerning atomic theory, and tries to justify the founder of positivism by saying that the modern conception of the atom differs radically from the conceptions of the atom in the nineteenth century. “The atom,” says Jordan, “is only a framework for classifying experimental facts.”

Such an understanding of the atom is, of course, wholly in the spirit of positivism; but nevertheless, however one classifies the facts, it is impossible to refute the circumstance that Mach, proceeding from observability, rejected that physical view without the further development of which modern quantum physics would have been impossible. The question is legitimate: why was this “framework” invented by positivism not in Mach’s time, but considerably later?

The reason for this is not hard to see. Since Mach’s time, thanks to the development of experimental technique and physical theory, the existence of atoms had been proved; and that is precisely why Jordan hastens to replace Mach’s assertion that the atom is equal to a witches’ sabbath with the more “delicate” assertion that the atom is equal to a framework for arranging facts.

Emphasizing once again (p. 148) that the atoms of Democritus possess clearly perceptible properties, while the atom of “today” is only a “system of formulas,” Jordan asserts that, with the establishment of this distinction, “the most important feature of the materialist picture of the world has been definitively liquidated; at the same time the positivist theory of knowledge has been confirmed and verified.” Thus,

at first positivist epistemology asserted that there are no atoms in the “complex of sensations,” that atoms are only “templates,” etc. But when atoms were nevertheless discovered and their real existence confirmed in a thousand ways, positivist theory came to the conclusion that all this atomic physics is a rather good “framework” for formulas.

Thus, positivist predictions in the field of atomism proved in fact to be in contradiction with the facts.

But it is precisely this philosophy of positivism that lies at the basis of the physical conceptions of the Copenhagen school: the principle of complementarity is a direct product of idealistic positivist epistemology.

2. THE PROBLEM OF QUANTUM STATISTICS

Consistently carrying out the conception of complementarity, N. Bohr assumes that the statistical character of quantum phenomena observed in experiment is the result of the uncontrollable action of the apparatus on the object. It is precisely the uncontrollability of this action that leads, according to Bohr, to the impossibility of constructing a causal description of microphenomena in terms of mechanical determinism. N. Bohr emphasizes that the statistics with which we are dealing here is of an entirely different nature from that known from classical statistical mechanics (see, for example, his article in Dialectica^7).

Classical statistics admits of refinement: by means of a more precise measurement of the coordinates and momenta of particles, one can reduce the spread of all quantities and in principle eliminate statistics altogether to any desired degree of accuracy.

In the quantum domain, the “uncontrollable” intervention of the apparatus does not allow such refinement.

Heisenberg, wishing to bring this aspect of the matter to the fore, in The Physical Principles of the Quantum Theory^5 formulates the principle of complementarity as follows:

Either:

Description in space and time

Uncertainty relation

Either:

Mathematical scheme outside space and time

Causality

By a mathematical scheme outside space and time is meant here a description by means of the wave function (without the intervention of the apparatus). Since Heisenberg considers that the wave function \(\psi\) is, on the one hand, a characteristic of the state of a single particle, and on the other hand, it cannot be measured on this one particle, the wave function is simply-

a merely mathematical symbol existing outside space and time.

Description in space and time (with the aid of trajectories), however, requires the intervention of an apparatus for measuring \(x\) or \(p\) and leads to statistical judgments about the motion of the particle.

Proceeding from these conceptions, Jordan\(^{8}\) arrives at the conclusion that in the quantum domain the Laplacean posing of the problem is impossible: from given initial momenta and coordinates of all particles to predict the future of the whole world; and he draws the conclusion of the collapse of materialism (let us recall that Jordan holds that materialism is the atoms of Democritus plus Laplacean “fatalism”).

Meanwhile dialectical materialism considers the interrelation between the phenomena of the objective world from a far broader and deeper point of view than Laplacean determinism.

At the basis of a truly materialist understanding of nature lies the idea of universal interaction, of the interconnection of phenomena; whereas mechanical determinism represents only a very particular aspect of interaction and is realized in nature only approximately.

However, pre-quantum physics overestimated the significance and force of mechanical determinism and underestimated the category of interaction. Let us dwell on this aspect of the matter in more detail.

As is known, according to classical mechanics, for a system of particles it is sufficient to specify a finite number of parameters (for example, the initial momenta and coordinates of the particles) in order to determine uniquely the motion of the system and, consequently, its state in the future. The corresponding problem may be formulated in the language of Newton’s equations or by the methods of analytical mechanics. Such an “exemplary” solution of the question of the future behavior of particles is, however, an abstraction which only approximately reflects what takes place in nature. The point is that the description of any real system by a finite number of parameters is incomplete. It is tacitly assumed that: 1) the remaining parameters of the system itself are “inessential,” and 2) the system can be isolated from the rest of nature for the entire time for which the prediction is being made. Let us explain this thought by examples. Let us first take the simplest problem of mechanics: the motion of a rigid body under the action of a force. In this case, the coordinates of the center of gravity, the Euler angles, and the corresponding velocities (12 quantities in all) may be taken as parameters. It is thereby assumed that deformations of the body in the course of motion are inessential. Thus it is already presupposed in advance that the body, for example, will not break apart. Further, we must know the external forces \(F\) acting on the body for all those points of space-time where the body will be located.

But since mechanics itself is not in a position to predict forces, they must be specified in advance, i.e. the prediction of the future actually made by mechanics requires the observance of certain conditions: if the forces are known not only in the present, but also in the future, then in the future the body will be located somewhere and will move in some way. The notion of absolutely exact predictions by mechanics is created for the reason that, in many cases, it is indeed possible, from indirect (non-mechanical) data, to predict the force with considerable accuracy. An example of this is the celestial mechanics of the planets. The planetary system is sufficiently isolated from the rest of the universe, and changes in the internal state of the planets and the Sun occur very slowly.

Thus, prediction in mechanics is possible only to the extent that several of the most important interactions determining the phenomenon remain unchanged.

Let us consider another example. The instantaneous transmission of force, allowed by Newtonian mechanics, in fact also does not occur in nature: forces propagate with a velocity not exceeding the speed of light. Therefore, at large velocities of motion of bodies or at large distances between them, it is necessary, in addition to the bodies or particles themselves, also to consider the field.

It turns out that even when the field is taken into account, physical phenomena can be considered with the aid of differential equations.

Indeed, from the fact that the velocity of propagation of any field does not exceed the speed of light, it follows that all processes occurring in a system of bodies and a field can be described by differential equations. This assertion may be explained by Fig. 1. In this figure two spatial surfaces \(\sigma\) and \(\sigma'\) are shown, separated from one another by a small interval of time \(dt\). From the point \(P\) light rays \(PA\) and \(PB\) are drawn (the inclinations \(AP\) and \(BP\) are equal to the speed of light). Then everything that influences the events at \(P\) lies on the segment \(AB\). For small \(dt\), \(AB\) is also small, i.e. events at \(P\) are influenced only by the neighborhood of this point; consequently, this influence can be expressed in the language of differential equations.

Fig. 1.

A field is a material system characterized by an infinitely large number of parameters. Therefore no finite number of operations can specify the initial state of a system consisting of a field and bodies.

In practice, however, one can make use of the fact that the field is sufficiently homogeneous in small, but still finite, regions of space. The more detailed the measurement of the field that has been carried out, the longer the prediction concerning the motion of the system of bodies and field remains valid. Here again it is necessary to specify the boundary conditions in advance, and in this case the prediction has not an absolute, but a relative, character. In Fig. 2 what has been said is explained by the example of a one-dimensional world. \(OX\) is the axis of space, and \(Ot\) is the axis of time. The system under consideration is contained at the moment \(t=0\) in the segment \(AB\). If it remains within this segment thereafter as well, then it is necessary to know the field on the lines \(AA'\) and \(BB'\), i.e., for the entire future of points \(A\) and \(B\). In the real world, instead of lines we shall have three-dimensional surfaces.

Fig. 2.

In the same figure, below, the graph of the field at the moment \(t=0\) is shown by a continuous curve, while the stepped curve is the result of measuring it by a finite number of operations. Because of this replacement of an infinite number of operations by a finite number, prediction into the future is not completely exact. Thus, if the difference between the measured force field and its actual value is \(\Delta F\), then for a body of mass \(M\) one should expect errors in the position of the body

\[ \Delta x \sim \frac{\Delta F}{M} t^2, \]

where \(t\) is time, i.e. the error is proportional to \(t^2\). This example illustrates the general situation: all real physical phenomena obey a given law only with a certain degree of accuracy; there is always dispersion (“scatter”), due to the fact that not one of the laws is capable of exhausting the whole diversity of interactions realized in reality*).

The formulation of the question just outlined—concerning an unambiguous prediction of the future, based on: 1) differential equations, 2) the specification of initial data for the field and bodies, and 3) the specification in advance of boundary conditions—is characteristic of the theory of relativity.

A. Einstein (see his article, ch. II, in Dialectics, p. 320) believes that this formulation of the question, which may be characterized as the “principle of action at short range,” is obligatory. He writes (p. 324): “A complete rejection of this principle would make

*) Here by the word “law” we mean the formulation of a regularity existing in nature. A law only approximately reflects a regularity.

the idea of the existence of (quasi-)closed systems and, at the same time, the establishment of empirically verifiable laws, in the sense familiar to us (emphasis mine.—D. B.), impossible.”

We have allowed ourselves to dwell in such detail on the analysis of causality in the form in which it appears in classical physics (including the theory of relativity) in order to show that the seemingly self-evident possibility of an unambiguous prediction of the future in fact presupposes the observance of a number of essential conditions which may, in reality, not be realized in full.

Therefore causality, too, as formulated in classical physical theory in the form of unambiguous necessity, is an abstraction, an approximation.

This circumstance has long been well known to the founders of materialist dialectics. Therefore the Jordanian method of refuting materialism on the grounds that Laplacian determinism is not realized in nature is, at the very least, naive.

Let us recall what F. Engels wrote concerning determinism in Dialectics of Nature:

“The opposite position is occupied by determinism, which has passed from French materialism into natural science and tries to dispose of chance by denying it altogether” (examples follow).

“With necessity of this kind we likewise do not get beyond the theological view of nature. For science it is almost immaterial whether we call this, with Augustine and Calvin, an eternal decree of God, or, with the Turks, kismet, or necessity. In none of these cases is there any question of tracing the causal chain. Therefore in the one case as in the other we are not a whit the wiser. So-called necessity remains an empty phrase, and with it chance also remains what it was.” “...with this one pod,” Engels continues, “we should have to trace more causal connections than all the botanists in the world could study. Thus chance is not explained here by necessity; rather, on the contrary, necessity is degraded to the production of mere chance. If the fact that a particular pea-pod contains six peas and not five or seven is a phenomenon of the same order as the law of motion of the solar system or the law of the transformation of energy, then in fact chance is not raised to the level of necessity, but necessity is reduced to the level of chance.”

We see that F. Engels ridicules Laplacian determinism, comparing it with divine providence, with kismet. For Engels, necessity and chance are not mutually exclusive

one another’s categories. The accidental has grounds, and the necessary is expressed in the accidental. There is no impassable boundary between the accidental and the necessary.

Thus, if the closure of our planetary system were violated by the intrusion of some cosmic body, this phenomenon would be accidental for the solar system itself, but this very accident could be the expression of a certain more general necessity (for example, of statistical laws of motion of bodies in interstellar space).

Hence it is clear that the form of expression of the necessary and the accidental depends on the nature of the system under consideration and on its connections with the surrounding world, and is not given once and for all for all cases. Therefore, when A. Einstein points out that, in the case of the non-closure of systems, the “customary to us” posing of the question of the laws of nature would become impossible, it does not follow from this that the non-closure of systems must not occur in nature; it follows only that one would have to seek these laws in a “form uncustomary to us.”

After these preliminary remarks of a general character, let us return to quantum mechanics.

As we have seen, N. Bohr and his followers deny the possibility of an objective description of the phenomena of the microworld, and regard statistics as the result of the uncontrolled action of the instrument upon the object.

In this connection we would like first of all to note that quantum statistics has an objective significance in the sense that it is in no way connected with the activity of the observer.

In fact, for example, a radioactive atom decays according to the same statistical laws regardless of whether it is observed or not, whether any observer at all exists or does not exist at all.

Moreover, a very energetic action on the atomic nucleus is necessary in order to change the course of radioactive decay.

This process of decay takes place in a statistically law-governed manner, i.e. different individual nuclei decay at different moments of time, but the mean decay time is one and the same. We may say that here we are dealing with a certain statistical ensemble of radioactive atoms objectively existing in nature.

Another example of a quantum ensemble is cosmic rays. Here again we are dealing with phenomena proceeding according to statistical laws, and these regularities, naturally, in no way depend on the observer.

These are only two examples, but they have a completely general significance: everywhere in the quantum domain we encounter statistical ensembles of this kind.

Meanwhile, the Copenhagen school relegates to the background the fact that quantum mechanics is applicable only to statistical ensembles, and concentrates on analyzing the interrelation of an individual phenomenon and the apparatus. This is a substantial methodological error: in such an interpretation all of quantum mechanics acquires an “instrumental” character, and the objective side of the matter is obscured.

What, then, is in fact the nature of a quantum ensemble? The essence of the matter is that in nature there is no absolute division into the micro- and macroworld. The phenomena of the microworld occur within the macroworld, and if we mentally single out some microphenomenon, it nevertheless in reality remains connected with the macroworld, one may say, with macroscopic bodies. The isolation of microsystems, which from the point of view of classical conceptions seemed possible in principle, in reality, owing to the finiteness of interactions, proves unrealizable.

This finiteness of interaction is connected with the atomism of action, expressed in the existence of Planck’s constant \(h = 1.05 \cdot 10^{-27}\) erg·sec.

Atomism is by no means exhausted by the discreteness of action. It is also manifested in the discreteness of charge, mass, and other quantities. At present we do not know exactly those limitations of classical conceptions and those new concepts and representations that must follow from the atomism of charge and mass—this is a matter for the future theory of particles. But the effect of the atomism of action is precisely what has been thoroughly studied by modern quantum mechanics. Thanks to the atomism of action, closed, isolated microsystems do not exist, and every quantum ensemble includes within itself the connection of microsystems with macrosystems.

The same thought may also be expressed as follows: a quantum ensemble is defined in relation (i.e. in connection) to macrobodies.

Thus, modern atomic physics confirms and develops that well-known thesis of dialectical materialism that things and phenomena should be studied in their interconnection: “...not a single phenomenon in nature can be understood if taken in isolation, outside its connection with surrounding phenomena, for any phenomenon in any domain of nature can be turned into nonsense if it is considered outside its connection with surrounding conditions, in isolation from them, and, conversely, any phenomenon can be understood and substantiated if it is considered in its inseparable connection with surrounding phenomena, in its conditionedness by the phenomena surrounding it.”

Thus, quantum statisticality has as its basis the interconnection of micro- and macrophenomena.

Whether particular macrobodies and macrophenomena are used for constructing measuring instruments or not is a secondary matter. The observer (or, better, the technician, the experimenter) can create measuring instruments only in accordance with the laws of nature.

The wave function \(\psi\), by means of which in quantum mechanics, as is commonly said, “the state of the microsystem is described,” in fact characterizes the quantum ensemble and, consequently, presupposes a definite macroscopic setting. Therefore the wave function is not a characteristic of a microparticle “in itself,” but is a characteristic of its belonging to one or another ensemble (see \({}^{10}\), §§ 14, 29).

If the macroscopic setting changes, then the quantum ensemble also changes. This is usually interpreted as “the intervention of the instrument in the state of the system.” In fact, the instrument is only a very special case of a macroscopic setting. Namely, the instrument is a spectral analyzer of a quantum ensemble\(^*\) (D. I. Blokhintsev, cited work, § 17). Let us explain what this means. If the ensembles in which some quantity of interest to us, for example momentum, has the values \(p_1, p_2, \ldots, p_n\), are characterized by the wave functions \(\psi_1, \psi_2, \ldots, \psi_n,\ldots\), respectively, then the wave function \(\psi\), representing any other ensemble, can be expressed in the form

\[ \psi=\sum C_n\psi_n, \tag{1} \]

i.e., as they say, expanded into a spectrum with respect to the attribute \(p\) (momentum expansion).

A measuring instrument that determines the momentum \(p\) is such a macroscopic device which in practice carries out a spectral decomposition. In the present example such an instrument may be a diffraction grating, since decomposition by momenta is entirely analogous to the decomposition of light into a spectrum.

It is clear that such a decomposition may occur both by itself in nature and be produced artificially by an experimenter.

Already from this example it is evident that the analyzer-instrument changes the ensemble. From an ensemble characterized by the wave function \(\psi\), there arises an ensemble characterized by a set of wave functions \(\psi_1, \psi_2, \ldots, \psi_n,\ldots\) (such an ensemble is called “mixed”). This indeed is “intervention” in the system. Hence the Copenhagen school

\(^*\) The Copenhagen school builds its entire understanding of quantum mechanics precisely on this special case.

draws the conclusion that an objective study of microphenomena is impossible. However, this is profoundly wrong. The point is that, in order to study the nature of an ensemble, it is sufficient to study a small part of it. This part will indeed change in the process of measurement, but on the whole the entire ensemble remains unchanged.

For example, in studying cosmic rays, counters or other instruments are used. These instruments change the state of individual particles detected in them, transferring them into a new ensemble, but they do not change as a whole that quantum ensemble which may be called the ensemble of cosmic rays.

The disturbance introduced by these instruments into the course of the phenomenon of cosmic rays as a whole is, of course, negligible, and therefore nothing prevents the elucidation of the objective regularities inherent in cosmic rays.

This may be expressed as follows: the degree of isolation of the ensemble as a whole is almost not disturbed by measurements. In other words, with respect to the ensemble as a whole there is preserved a situation known from classical physics, where actions may be arbitrarily small.

By virtue of this, the wave function \(\psi\), characterizing the ensemble, obeys Schrödinger’s equation:

\[ ih \frac{d\psi}{dt} = H\psi, \tag{2} \]

which makes it possible to determine this function for any instant of time, if it is known at the initial instant of time.

Thus, for an ensemble the simplest form of causal connection is preserved—classical determinism. This is the result of the fact that the ensemble as a whole (microsystem + macro-setting) can be (of course, approximately) isolated from the rest of the world.

On the contrary, classical determinism, as a form of causal connection, is not realized in application to single microphenomena, and the reason for this inadequacy of “Laplacean” determinism lies in the impossibility of isolating a single microsystem from its environment. An isolated, closed single microsystem does not exist in nature. For this reason, instead of “Laplacean” determinism, in the world of microphenomena the foremost place is occupied by statistical regularity, which reflects the influence of the environment on a single microphenomenon. This latter regularity is not the result of an absence of lawful connections within the world of individual phenomena, as the positivists assert; on the contrary, statistical regularity is precisely the expression of the general law-governed character in individual phenomena.

Therefore, quantum mechanics studies the properties of an individual microphenomenon by studying the statistical regularities of a collective of such phenomena.

It is hardly necessary to speak in detail about the practical success and power of this method. It was precisely along this path that the finest details in the structure of molecules, atoms, and atomic nuclei were uncovered.

It remains for us to consider the question of the possibility of eliminating quantum statistics. This question is known in the literature as the problem of “hidden parameters.”

The problem may be formulated as follows: an individual microphenomenon may be realized in various ways. Quantum mechanics gives only the probability that one or another possibility will be realized. Is it not possible to find such quantities (“hidden parameters”) whose knowledge would make it possible to predict unambiguously each individual microphenomenon?

Is it not possible, for example, to predict the point at which an electron will strike in a diffraction experiment, or the orientation of the spin of an atom in the Stern and Gerlach experiment?

This possibility was investigated by Neumann and later by Reichenbach. Both arrive at a negative answer. Neumann rejects the possibility of hidden parameters in the following way.

Let us have a “pure” ensemble (i.e. an ensemble determined by a single wave function). By the definition of a pure ensemble it cannot be decomposed into parts; that is, if we take a collection of a large number of any identical measurements in this ensemble, \(N\), and arbitrarily divide this collection into two, \(N_1\) and \(N_2\), \((N_1 + N_2 = N)\), then the mathematical expectation of any quantity \(L\) is the same in all subcollections:

\[ \langle L\rangle_N = \langle L\rangle_{N_1} = \langle L\rangle_{N_2}. \]

In other words, by dividing the ensemble into subensembles, one cannot reduce the dispersion, the scatter of the quantity. To verify this, it is enough, instead of \(L\), to substitute in (3) \(\Delta L^2 = (L-\overline{L})^2\); then formula (3) will mean that the mean square deviation \(\Delta L^2\) is the same in any subensemble.

Consequently, according to Neumann, there can exist no parameters by which one could make a selection of subensembles in such a way as to reduce the statistical scatter of the quantities.

Nothing can be objected to in this proof: it simply expresses the internal consistency of quantum mechanics. But the method of reasoning chosen by Neumann still cannot be considered satisfactory, since it is based on quantum mechanics. Meanwhile, if “hidden parameters”

exist, then their analysis lies outside the competence of quantum mechanics.

We shall not reproduce here Reichenbach’s arguments, since they likewise proceed from the fact that the “hidden parameters” obey the laws of quantum mechanics, and then, naturally, we remain within the circle of those ideas which we ought to leave behind.

Therefore we shall consider the problem of “hidden” (better to say, “as yet unknown”) parameters from another point of view, without subordinating these parameters in advance to the laws of quantum mechanics.

Then one can immediately say that these parameters cannot be mutually uniquely related to the quantities that occur in modern experiment and in quantum mechanics.

Indeed, in that case they could be expressed through quantum quantities, and, consequently, would themselves have to obey the laws of quantum mechanics.

Consider, for example, a quantum quantity \(s\) having only two possible values \(s_1\) and \(s_2\). (Such a quantity may be the projection of an atom’s spin onto a magnetic field.) Let there be given some state \(\psi=a_1\psi_1+a_2\psi_2\), where in \(\psi_1\), \(s=s_1\), and in \(\psi_2\), \(s=s_2\). Then, according to quantum mechanics, \(|a_1|^2\) is the probability of finding \(s=s_1\), and \(|a_2|^2\) is the probability of finding \(s=s_2\). In the event that there exist certain hidden parameters \(\lambda\) such that knowledge of them makes it possible to predict unambiguously whether, in the given individual case, \(s\) will be equal to \(s_1\) or \(s_2\), the connection between \(\lambda\) and \(s\) must be such that \(\lambda\) determines the value of \(s\), while the value of \(s\) does not determine \(\lambda\), i.e. the equality

\[ s=F(\lambda) \tag{4} \]

must not be solvable with respect to \(\lambda\)*).

Further, the aggregate of parameters \(\lambda\) cannot be preserved when the arrangement of the experiment is changed. Indeed, suppose that initially an ensemble with \(s=s_1\) (\(a_2=0\)) is given. Let \(s_1\) be the projection of the spin onto a magnetic field directed along the \(OZ\) axis. Direct a beam of such particles into a field directed along the \(OX\) axis and sorting the particles according to the sign of the projection of spin onto the \(OX\) axis. Then, if the initial ensemble corresponds to a region of parameters \(G_1(\lambda)\), this region will split into two

\[ G_1=G'_1+G''_2, \]

where \(G'_1\) is the region of \(\lambda\) corresponding to the projection of spin onto the \(OX\) axis \(s_x=s_1\), and \(G''_2\) is the region with the projection of spin onto the \(OX\) axis equal to \(s_x=s_2\). Let us separate off the beam, say, with \(s_x=s_1\). Let us further split it by a field again directed along the \(OZ\) axis, into a beam with \(s_z=s_1\) and \(s_z=s_2\). If

*) \(F(\lambda)\) is not necessarily a function; it may be a functional.

in the change of the external arrangement that has taken place (separation according to the attribute \(s_X\)) the parameters \(\lambda\) have not changed, then the region \(G'_1\) contains parameters \(\lambda\) corresponding to both values of the projection on the axis \(OZ\): \(s_1\) and \(s_2\). But the region \(G'_1\) is part of the region \(G_1\), in which, by assumption, all \(\lambda\) belonged to \(s_Z=s_1\). Thus we arrive at a contradiction.

It remains to suppose that, when the particles are sorted according to the attribute \(s_X\), new parameters (or new values of parameters) enter into play, so that \(G'_1\) again contains \(\lambda\) referring both to \(s_Z=s_1\) and to \(s_Z=s_2\), i.e., if hidden parameters exist at all, then for each macroscopic apparatus analyzing according to some attribute (for example, \(s_Z\) or \(s_X\)), they must be its own.

What we have considered here is the transformation of pure ensembles into mixed ones. Therefore the result may be formulated as follows: to each transformation of a pure ensemble into a mixed one there correspond its own hidden parameters (if they exist at all). The parameters satisfying this condition evidently do not contradict the laws of quantum mechanics.

Their possible physical meaning is a numerical characteristic of the influence of the macro-arrangement on an individual microphenomenon.

Whether such parameters actually exist in nature is a question that can be resolved only by the further development of theory and experiment. Physically, this is a question of the possibility of isolating, from the surrounding world, some part of it in terms of quantities other than those with which quantum mechanics operates \((s)\). A priori one can neither insist on this possibility nor reject it.

3. ON THE SUBJECTIVE INTERPRETATION OF THE WAVE FUNCTION

Let us now turn to a consideration of the position of the Copenhagen school in its understanding of the physical meaning of the wave function. With the greatest clarity this position can be elucidated in connection with the discussion between A. Einstein and N. Bohr\(^4\). In this discussion the following example was considered.

There are two particles 1 and 2 which undergo a collision. Let their state before the collision, at the initial moment of time, be characterized by the wave function:

\[ \psi^0(x_1,x_2)=\psi^0(x_1)\,\varphi^0(x_2). \tag{1} \]

We shall denote the wave function of these particles after the collision, after a sufficiently long time has elapsed, by \(\psi(x_1,x_2)\). This function will no longer be a product of functions depending on \(x_1\) and \(x_2\) separately.

Let us now measure some quantity pertaining only to the first particle, for definiteness, say the momentum of this particle \(p_1\). After this measurement the wave function of the first particle will be \(\psi_{p_1}(x_1)\). Let us expand \(\psi(x_1x_2)\) in terms of \(\psi_p(x_1)\):

\[ \psi(x_1x_2)=\int \varphi_p(x_2)\psi_p(x_1)\,dp, \tag{2} \]

where \(\varphi_p(x_2)\) is the amplitude in the expansion of \(\psi(x_1x_2)\) in terms of \(\psi_p(x_1)\).

If the measurement of the momentum of the first particle gives \(p_1\), then the wave function \(\psi(x_1x_2)\) reduces to a single term of the expansion:

\[ \psi(x_1x_2)\longrightarrow \varphi_{p_1}\psi_{p_1}(x_1). \tag{3} \]

Thus the state of the second particle also changes, although no measurements were made on it and it had long since ceased to interact with the first, i.e. nothing acted on it, yet its state changed.

Consequently, they say, the “information” (of the observer) about this particle has changed, and hence its state has also changed, i.e. the concept of state in such an interpretation turns out to be equivalent to the concept of “information about the state.”

This is the subjective interpretation of the wave function. This interpretation is connected with the fact that the Copenhagen school generally pushes into the background the statistical character of quantum mechanics.

In quantum mechanics the state of a particle is indeed characterized not “in itself,” but by the particle’s belonging to one or another ensemble (mixed or pure*). This belonging has an entirely objective character and does not depend on the observer’s information. If this information does not correspond to the nature of the ensemble, then no new information—apart, perhaps, from absurdities—can be obtained from it.

As we have already explained, measuring instruments are spectral analyzers. They decompose the initial ensemble into subensembles, the character of which depends on the nature of the analyzer instrument and on the nature of the initial ensemble.

In the example under consideration, the analysis is performed with respect to a feature pertaining to the first particle. But since in the initial ensemble \(\psi(x_1x_2)\) there existed a correlation between both particles, due to their interaction, the decomposition with respect to the feature \(p\) at the same time selects a subensemble for the second particle; that is, after the measurement it proves to belong to another subensemble, characterized by the wave function \(\varphi_{p_1}(x_2)\).

* This point of view on the \(\psi\)-function is consistently carried through by the author in the new edition of his course \({}^{10}\).

Therefore the change in the state of the second particle is caused not by a change in “information” about it, but by the interaction of the first and second particles before the measurement.

If there were no such interaction, then changes in the state of the first particle would have no influence on the state of the second: \(\psi(x_1, x_2)\) would remain equal to the product \(\psi^0(x_1)\varphi^0(x_2)\), and under any measurement on the first particle the state of the second would invariably be \(\varphi^0(x_2)\).

In our example the essence of the correlation due to interaction is especially clear. Indeed, let the momentum of the first particle before the collision be \(p_1^0\), and that of the second \(p_2^0\). Then, if after the collision the momentum of the first particle is equal to \(p_1\), by virtue of the law of conservation of momentum the momentum of the second particle must be equal to \(p_2 = p_1^0 + p_2^0 - p_1\). Consequently, \(\varphi_{p_1}(x_2)\) is a de Broglie wave with momentum \(p_2 = p_1^0 + p_2^0 - p_1\). Therefore sorting particles 1 according to their momenta \((p_1)\) is at the same time a sorting, by momentum \((p_2)\), of particles 2.

We see that the subjective interpretation of the wave function rests on forgetting its statistical essence.

In this same discussion A. Einstein and his coauthors (see the cited work\(^{4}\)) expressed a conviction that quantum mechanics is incomplete. Namely, they showed that it is impossible simultaneously to determine the momentum \(p\) and the coordinate of a particle \(x\), despite the fact that each of these quantities can be measured without directly affecting the particle itself. Einstein and his coauthors consider the following example.

Let the wave function of a system of two particles have the form

\[ \psi(x_1,x_2)=\int_{-\infty}^{+\infty} e^{\frac{i}{h}p(x_1-x_2+a)}\,dp =2\pi\delta(x_1-x_2+a), \tag{4} \]

where \(a\) is a certain constant. Suppose we first measure the momentum of the first particle \(p_1\). From the expansion (4) it is seen that if this momentum is equal to \(p\), then the momentum of the second particle \(p_2=-p\). The coordinate \(x_2\), however, remains completely undetermined. Instead of the momentum, we could likewise measure the coordinate of the first particle \(x_1\). Let \(x_1=x\); then from (4) it follows that \(x_2=x+a\), i.e., by this measurement the coordinate of the second particle is determined. The momentum \(p\), however, will be undetermined.

A. Einstein and his coauthors draw from this the conclusion that quantum mechanics is incomplete, since it does not make it possible simultaneously to determine \(p\) and \(x\) for a particle even in the case when \(p\) and \(x\) are determined separately indirectly, without intervention of the apparatus in the state of the particle (in our example the second …)

particle, whereas the measurement with intervention is carried out on the first particle).

In his reply to A. Einstein, N. Bohr refutes this point of view concerning the completeness of quantum mechanics. In doing so N. Bohr proceeds from the principle of complementarity. He asserts that measuring instruments are in principle always arranged in such a way that one can measure only \(p\) or only \(x\). Therefore quantum mechanics is complete, since it fully corresponds to the possibilities of measuring macroscopic instruments. This reply of N. Bohr’s is only a half-truth.

Taking the principle of complementarity as the basis of his reply, N. Bohr, naturally, brings to the fore the possibilities of measuring instruments, whereas the essence of the matter lies in the new nature of the objects of measurement—microparticles, to which the classical concept of motion along a trajectory is inapplicable.

N. Bohr leaves aside the statistical interpretation of the wave function. L. I. Mandelstam\(^{13}\) showed that in the example adduced by A. Einstein the matter concerns the decomposition of the initial ensemble \(\psi(x_1, x_2)\) into different mutually exclusive subensembles (once according to the property \(p\), another time according to the property \(x\)).

The change in the state of the second particle, as we explained above, is connected not with the action of the apparatus on this particle (which in the example under consideration is absent), but with the correlation of the states of both particles, due to their interaction that took place before the measurement.

Thus, A. Einstein and his coauthors, in criticizing quantum mechanics in connection with the impossibility of measuring \(p\) and \(x\) simultaneously, even in the absence of direct intervention of the apparatus, overlook the fundamentally different nature of macroparticles; they unlawfully suppose that microparticles do not differ from classical particles and that only the indelicate intervention of the apparatus is the cause of the uncertainty relation.

In his later article\(^{7}\), devoted to this same question, A. Einstein repeats his error. A. Einstein considers a dilemma: either a) the particle in fact has \(p\) and \(x\), but the intervention of the apparatus does not allow their simultaneous measurement (then quantum mechanics is incomplete, since it gives no way of measuring what exists in nature); or b) particles are in fact described by the \(\psi\)-function and in reality have neither \(p\) nor \(x\). The latter arise only in a definite setting, for example in measurement.

A. Einstein rejects possibility (b) and inclines toward the first (a). Namely, A. Einstein finds a contradiction between possibility (b) and the principle of “local action,” about which we wrote in Chapter III. The essence of this contradiction consists in the fact that if the \(\psi\)-function describes the state of a particle, then this \(\psi\) can

can be changed throughout all space by a measurement carried out in a local region of space.

Figure 3 shows the \(\psi\)-function of a certain state of a particle. Suppose that the coordinate of the particle is measured and it is found that \(x=x'\). Then to such a particle there corresponds a new wave function \(\psi_{x'}(x)=\delta(x-x')\), shown in the same figure, i.e. the wave function is reduced to a sharp peak. This reduction (“reduction”) means that the process at the point \(x'\) (localization of the particle) “acts” on the wave \(\psi(x)\) throughout all space, i.e. violates the law of action by contact.

Fig. 3.

Having arrived at this contradiction, A. Einstein rules out this possibility on the grounds that “the idea of the existence of closed systems and, together with it, the establishment of empirically verifiable laws, in the customary sense” (emphasis ours) become “impossible.” Above we have already considered this aspect of the matter. The development of science consists precisely in the fact that the “customary” has to be replaced by the “unaccustomed.”

It is easy to see wherein the error in A. Einstein’s reasoning lies. Discussing the alternative possibility (b), A. Einstein, following the Copenhagen school, assumes that the wave function is a characteristic of an individual particle, of the particle itself as such. Proceeding from this incorrect view, he tries to give it physical meaning by regarding the wave function as a quantity characterizing a certain physical field upon which some action or other is possible.

In reality, however, the wave function is a statistical characteristic of a particle’s belonging to one or another ensemble. Therefore in nature there occurs no contraction of the wave function as a real physical process. What occurs pertains to the particle, which possesses material being.*)

In the example given above, there occurs localization of the particle, which makes the particle a member of another statistical collective (a collective with a definite value of \(x\), equal to \(x'\)). In the language of wave functions this means that the particle is now described by a new wave function. But there is not even a trace of a physical action on the wave function.

*) As we shall see below, the nature of a quantum particle is indeed closely connected with the concept of field, but only not with the concept of a “field” of a wave function giving a statistical characteristic of the state of the particle.

Meanwhile, A. Einstein precisely regards the process of contraction of the wave function as a process of the flow of some physical entity, a “fluid,” from one part of space to another. Therefore A. Einstein incorrectly interprets alternative (b) and rejects it without justification. The same incorrect interpretation of alternative (b) is given by H. Reichenbach[^7]. H. Reichenbach also believes that in the contraction of the wave function a violation of ordinary causality is revealed. But the difference between the points of view of H. Reichenbach and A. Einstein consists in the fact that A. Einstein rejects possibility (b) as contradicting, in his opinion, the principle of action by proximity, whereas H. Reichenbach accepts this possibility and, instead of revealing the physical essence of this formulation of the question, declares the existence of an anomaly of causal connection with which one must agree de facto. In this connection H. Reichenbach proclaims the existence of a new principle—the “principle of causal anomaly” (what a passion bourgeois philosophers have for “anomalies”).

Discussing the reduction of the wave function and noting the possibility of a statistical interpretation of the \(\psi\)-function, he writes (cited work, p. 345): “Then the discontinuous transition from \(\psi\) to \(\psi'\) presents no greater difficulty than, say, the discontinuous transition from the probability of death of a twenty-year-old man to the probability of death of the same man under the condition that it is additionally known that he is ill with tuberculosis.” (Evidently, in contemporary Marshallized society it is difficult to invent a more popular example than death at the age of twenty from tuberculosis1.) If, however, says H. Reichenbach, \(\psi\) is regarded as a physical state, then this path (the interpretation of the information \(\psi\) to \(\psi'\)) is closed.

The incorrectness of this argument again follows from the attempt to ascribe the \(\psi\)-function to a single electron and to consider it as a characteristic of that electron, rather than of the ensemble to which it belongs.

Further, H. Reichenbach notes (ibid.): “The probabilistic interpretation of the \(\psi\)-function, of course, also has its anomalies; they are expressed not in the temporal development of the \(\psi\)-function, but in the behavior of particles, since this interpretation in its exhaustive form is identical with the corpuscular interpretation.”

In this assertion H. Reichenbach, in essence, repeats the probabilistic, statistical understanding of the \(\psi\)-function in the spirit of Einstein’s alternative (a) (the particles of the microworld are understood as corpuscles with \(\rho\) and \(x\), which, however, cannot be measured).

Meanwhile, the distinctive feature of quantum mechanics lies precisely in the fact that it directly, in the language of statistics, expresses laws inherent in objects of a different nature, distinct from the material points of classical theory.

In fact, instruments, both in classical physics and in modern atomic physics, are macroscopic devices. It is therefore clear that the essence of the difference between classical and quantum phenomena is rooted not in the instruments as such, but in the new nature of quantum objects.

A. Einstein could be right in the sense of alternative (a) if the situation were indeed such that the modern physical experiment were insufficiently accurate for measuring the “true,” simultaneous values of \(p\) and \(\dot{x}\).

In fact, the modern physical experiment is accurate enough to prove that this pair of quantities is not realized in nature simultaneously.

Fig. 4.

Thus, from the scattering of X-rays or electrons by atoms one can find the distribution of electrons inside the atom, i.e. \(|\psi(r)|^2\), where \(r\) is the distance of the electron from the nucleus. This distribution is shown in Fig. 4. Such an experiment means determining the coordinates of electrons inside atoms. At the same time, the energy of the initial state of the atom is equal to \(E_0=-\dfrac{e^2}{2a}\), where \(a\) is the radius of the Bohr orbit.

As experiment and theory, which is in agreement with experiment, show (see, for example, the author’s book\(^{10}\)), a significant part of the electrons proves to be located at distances \(r\) from the center such that the potential energy of the electron \(U(r)\) is greater than its total energy \(E\) (this part of the electrons is shown in Fig. 4 by the shaded area). Therefore, if we assume that in this state the electron has, in addition to a coordinate, also a momentum \(p\), then, since the total energy is equal to \(E=\dfrac{p^2}{2m}+U(r)\), we obtain that for \(r \gg 2a\), \(U=-\dfrac{e^2}{r}\) and \(\dfrac{p^2}{2m}<0\), i.e. the momentum of the particle is imaginary. If, however, one admits that \(E\) is not ...

not the exact value of the energy, but only a certain mean of the “true” energy values of the individual atoms, then

\[ \overline{\Delta E^2}=\overline{(E-E_0)^2}>0, \]

i.e., there exists a spread of the energy around the mean value \(E_0\). It is not difficult to determine that this spread \(\Delta E\) is, in order of magnitude, equal to \(E_0\). But this conclusion is in complete contradiction with any of the experiments for determining the energy of an electron in the lower state of an atom (the ionization energy), which show that in fact there is no such spread in the value of the ionization energy.

Consequently, our assumption that an electron in an atom, possessing the energy \(E=E_0\), has any simultaneous values \(x(r)\) and \(\rho\) contradicts the experimental data. Microparticles are not objects to which the concept of motion along a trajectory is applicable.

Such motion, from the quantum point of view, is only a special case of motion, realized approximately only under certain definite conditions.

4. ON THE NATURE OF QUANTUM PARTICLES*)

The entire body of known facts indicates that the particles of the microworld differ from the corpuscles of classical theory to a far greater extent than can be seen from nonrelativistic quantum mechanics. One may assert that this latter mechanics, in essence, only poses the problem of particles and prepares the ground for a new conception, but by itself still does not solve this problem.

Since nonrelativistic quantum mechanics deals with ensembles of particles with a finite and specified number of degrees of freedom (a system of spinless particles has, for example, \(3N\) degrees of freedom, while particles with spin have \(4N\) degrees of freedom), it still remains very similar to the mechanics of a system of material points, and therefore many are inclined to think that the object of consideration in quantum mechanics is the same as in classical mechanics, and that, supposedly, quantum mechanics merely gives a new law of motion of particles. This, however, is a profoundly erroneous view. To be convinced of this, it is enough to turn to the region of large energies (the region of relativistic theory).

Experience shows that systems with a specified number of particles exist only so long as the kinetic energy of the particles \(T\) is small in comparison with their own internal energy \(m_0c^2\) (\(m_0\) is the rest mass, \(c\) is the speed of light).

*) This question is treated in greater detail in the author’s papers \(^{14}\); see also \(^{15}\).

If the energy exceeds this value, then the very number of particles becomes variable.

A typical example of such a process is the cascade process in the soft component of cosmic rays. A primary photon of high energy gives rise to a pair (a positron and an electron). The particles of this pair, in their subsequent motion, give rise to new photons (bremsstrahlung); these photons are transformed into new pairs, and so on. As a result, the number of particles is multiplied.

If one considers the whole process as a whole, then here we are dealing with a system possessing an indefinite, in principle arbitrarily large, number of degrees of freedom (i.e., the number of particles produced may be unboundedly large).

Such material systems are called a field. Formerly the field was regarded as something through which the interaction of particles is effected, and it was opposed to particles. This had its legitimate grounds in the fact that particles were considered immutable. Physics considered only changes of motion in a system consisting of a given number of particles.

Since phenomena have now been discovered in which the very number of particles changes (they are born and annihilated, being transformed into others), the classical division into field and particles, preserved also in nonrelativistic quantum mechanics, becomes untenable.

For example, it was formerly assumed that the electromagnetic field (photons) determines the interaction of electrons, but that electrons, in turn, do not give rise to any interaction of photons. At present we know that photons can interact with one another through electrons; i.e., if formerly photons were opposed to electrons as a field to “true” particles, then now such an opposition is illegitimate: electrons, in their turn, determine the interaction of photons and, consequently, the “true” particles—electrons—play the role of a field for the “ephemeral” particles—photons.

Another example: mesons may be regarded as true particles in relation to photons, which determine the electromagnetic interaction of mesons (the field), but the mesons themselves determine the interaction of heavier particles—nucleons—and are, for this interaction, the field (meson field).

Therefore the division into particles and field becomes unfounded and has only relative significance.

The concept of a field as a material entity with an unboundedly large number of degrees of freedom apparently has a more fundamental significance than particles, which from this point of view are only a particular manifestation of the field.

A field can transmit to another field its charge, its mass, energy, momentum, etc., only in certain discrete por—

tions, which we call particles. Particles appear in this aspect as definite manifestations of field interactions. If we say that we are dealing with one, two, three, etc. particles, then from the “field” point of view these are only different degrees of excitation of the field.

This excitation may be localized in a small region or, conversely, occupy a large region of space. In the first case we say: the particle has a coordinate; in the second we say: there is no coordinate, it has no meaning.

The physical content of this statement may be illustrated by the example of photons (quanta of light), which have a rest mass equal to zero and therefore under all conditions are relativistic particles. The concept of spatial localization is not applicable to these particles in general (but it is applicable to the excitation of the field).

If the excitation of the field \(F(x)\) is localized in some region of space \((\simeq \Delta x)\), then such a field can be represented in the form of a superposition of waves

\[ F(x)=\int_{k_0-\Delta k}^{k_0+\Delta k} A(k)e^{ikx}\,dk . \]

By a well-known theorem, the interval of waves \(\Delta k\) essentially participating in the superposition of waves is connected with the localization region \(\Delta x\) by the relation*):

\[ \Delta k\cdot \Delta x \geqslant 2\pi . \]

It follows from this that the smaller the region \(\Delta k\) in which the excited field is localized, the broader the set of waves creating this excitation. But to each wave with a definite \(k\) there corresponds a photon of one kind (the kind “\(k\)”). Therefore a localized field cannot be realized by one photon, but only by a set of photons. Thus an excitation corresponding to a single photon is necessarily nonlocalized (occupies a large region of space).

The same must also apply to particles with rest mass different from zero. If the region of localization of an excitation, for example of the electron field, is small in comparison with \(\Delta x=\dfrac{h}{m_0 c}\) (here \(h\) is Planck’s constant, \(m\) is the rest mass of the electron, \(c\) is the speed of light), then such an excitation cannot be realized by one electron or one positron: the number of participating positrons and electrons becomes indefinitely large.

This effect is known under the name “vacuum polarization.”

*) This relation is applicable to any waves.

Modern particle theory is still at a very embryonic stage of its development. The considerations set out above concerning the nature of the particle are based partly on new, experimentally established facts, and partly on provisional theories. It is therefore quite possible that the concept of the particle as an atomistic, quantum manifestation of the field is not fully adequate to the actual essence of micro-objects. Nevertheless, one may think that many features of the conception of particles outlined above will be preserved also in future theories, more fundamental than the existing ones.

This thought finds its confirmation in the circumstance that modern quantum-field theory, despite its sufficiently well-known imperfections, has in recent years received new, unexpected confirmations precisely on those points which until now had been considered doubtful.

Let us give here one of them, which is of special interest for the range of questions we are considering.

It is known that the theory of the quantized electromagnetic field leads to the existence of the so-called zero field, i.e. a field without photons. Photons-particles appear only as the result of excitation of the field[^14][^15]. This theoretical conclusion points to an analogy between the properties of the field and of a solid body. In a solid body at absolute zero temperature there exist oscillations of matter—zero oscillations. When the solid body is excited (for example, by heating), waves arise in it, whose energy changes in portions $\varepsilon = h\omega$, and momentum—in portions $\mathbf{p}=h\mathbf{k}$ (here $\omega$ is the frequency of the wave, $\mathbf{k}$ is the wave vector), with

\[ k=\frac{\omega}{v}, \]

where $v$ is the speed of sound, i.e. when the body is excited, corpuscular properties appear in it, particles—“phonons”—appear, having energy $\varepsilon$ and momentum $\mathbf{p}$. No one has ever doubted that even in the absence of phonons, i.e. in its lowest energy state, the solid body continues to exist. Meanwhile, with respect to the electromagnetic field, it was believed that it exists only insofar as photons exist. The existence of the zero field and its energy were inclined to be regarded as an unnecessary conclusion of the theory, not corresponding to reality.

It was shown by the author of this article as early as 1938 that zero oscillations of the field, if they exist, must lead to a displacement of levels in atoms. At that time a qualitatively correct formula for this displacement was also obtained. More exactly and consistently, this question was considered in recent years by a number of foreign scientists, and the predictions of the theory proved to be in complete agreement with experiment[^14].

The question of the real existence of the background also belongs to the number of questions concerning which there is no complete unanimity

among physicists. Such a background is capable, by analogy with a dielectric, of becoming polarized. Polarization should have led to the scattering of light by light. Experimentally, no one has succeeded in proving this phenomenon, because of its smallness.

However, it has been shown (see \(^{13}\)) that the polarization of the background (or, as it is sometimes said, of the “vacuum”) makes a definite contribution to the shift of levels in atoms. More convincing is the circumstance that recently it has been shown experimentally that the ratio of the magnetic moment of the electron to the mechanical one does not quite correspond to that predicted by Dirac’s theory, but differs from it by

\[ \frac{1}{2\pi}\cdot\frac{e^3}{hc}. \]

If this correction is calculated on the basis of the Brownian motion of the electron in the zero field, the sign of the correction obtained is wrong, and only taking into account the fact that, in addition to the zero field, there also exist fluctuations of the polarization of the background leads to the correct value of the correction. This result indicates the real existence of fluctuations of the polarization of the background, and therefore, it would seem, of the background itself. Modern theory does not describe quite correctly either the field of zero oscillations or the background: the energy of both turns out to be infinitely large. The connection between the various fields also remains unclear.

However, the theoretical results cited above, confirmed by experiment, indicate that both the zero electromagnetic field and the background nevertheless do exist in reality. And this means that what we call emptiness is in fact a certain medium. Whether we call it, in the old manner, the “ether,” or by the more modern word “vacuum,” the essence of the matter is not changed by this. This medium has certain properties in common with a solid dielectric. It is understood that this commonality of properties has only the significance of an analogy: the “vacuum,” of course, is neither a solid, nor a liquid, nor a gaseous body, but possesses its own special nature.

Welton gave a very clear interpretation of the phenomenon under consideration. (See also the later work of the author \(^{13}\).) It was he who showed that the entire effect of the displacement of levels is caused by the fact that the electron, under the action of the zero oscillations of the electromagnetic field, executes Brownian motion, as a result of which the mean value of the electron’s potential energy changes in comparison with what it would have been in the absence of zero oscillations. Referring the reader, for a more complete acquaintance with this question, to the survey by Ya. A. Smorodinskii \(^{16}\), we draw attention to the fact that, as a result of the works mentioned, it may be considered established that the zero oscillations of the electromagnetic field do indeed exist, and at the same time there also exists an electromagnetic field without photons, just as a solid body exists without phonons. Thus it has also been proved that the field is primary and general, while particles, in the present case photons, are secondary and particular.

The analogy with a solid body continues further as well, if one turns to the positron-electron field. It is well known that the energy spectrum of a dielectric consists of a number of bands (“zones”) of allowed and forbidden energies. In the normal, unexcited state the lower band is completely filled with electrons. If one extracts from it one electron and transfers it to another, upper, band, then a pair arises: an electron in the upper band and a “hole” in the lower band. The electrons located in the upper band give rise to normal electronic conductivity, while the “holes” in the lower band give rise to “hole” conductivity, corresponding to the positive sign of the charge of the current carriers. In a completely analogous way, in the theory of the electron-positron field there exists a lower band \((E < -mc^2)\), filled with electrons (the so-called “background”), and an upper one \((E > +mc^2)\), normally free. Upon excitation of such a field an electron from the background passes into the upper band, and in the lower band, in the background, a “hole” is formed, which is a positron.

We have undertaken this excursion into a region in which there is as yet no beaten road, but only barely marked paths vanishing into the unknown, in order to show that contemporary physics cannot go backward, but, on the contrary, is compelled to seek new paths, breaking the framework even of quantum mechanics itself.

Quantum mechanics regards the purely corpuscular representations and concepts of classical atomism as approximate. But contemporary theory must take new steps along the path of the further development of the concept of the particle, leading us still farther away from classical atomistics. On this path the physicist will probably encounter new “strangenesses” and “unusual features.” But V. I. Lenin is profoundly right in asserting that “however marvelous from the standpoint of ‘common sense’ the transformation of imponderable ether into ponderable matter and vice versa may be, however ‘strange’ the absence in the electron of any other mass except electromagnetic mass, however unusual the limitation of the mechanical laws of motion to only one domain of natural phenomena and their subordination to the deeper laws of electromagnetic phenomena, etc., all this is but a further confirmation of dialectical materialism.”2

From all that has been said, what is important for us is the basic conclusion concerning particles: particles are only excitations of the “vacuum,” or, one may say, of a new quantum “ether,” which continues to live even when there are no particles at all; in it the electromagnetic field and electric polarization fluctuate. Here there is no rest, but eternal motion, like a swell on the surface of the sea.

From this point of view it also seems clear that no isolated, self-contained “free” particles (as they say) exist. Even in the case of a considerable separation of particles from one another, they nevertheless continue to belong to the generating

their medium, which is in a state of continuous, chaotic, “turbulent” motion.

In this connection between the particle and the medium lies the nature of that impossibility of isolating the particle which manifests itself in the apparatus of quantum mechanics.

Cited Literature

1. V. I. Lenin, Materialism and Empirio-Criticism.
2. History of the CPSU(b), Short Course, p. 101.
3. E. Schrödinger, Abhandlungen über die Wellenmechanik.
4. N. Bohr, UFN, 16, 446 (1936).
5. V. Heisenberg, The Physical Principles of the Quantum Theory, GTTI, 1932.
6. N. Bohr, Atomtheorie und Naturbeschreibung, J. Springer, Berlin (1931), p. 62.
7. Dialectics, No. 7/8, 1948, p. 312.
8. P. Jordan, Physics of 20th Century, Phil. Lib., New York, 1944, p. 160.
9. P. Jordan, Anschauliche Quantentheorie, J. Springer, Berlin, 1936, p. 303.
10. D. I. Blokhintsev, Foundations of Quantum Mechanics, Gostekhizdat (1949).
11. A. Ioffe, Collection, The Foundation of the New Mechanics, GIZ, 1927, article by A. Grinberg.
12. E. Schrödinger, “What Is Life?”, IL, 1947, p. 123.
13. D. I. Blokhintsev, ZhETF, 16, 965 (1946).
14. D. I. Blokhintsev, UFN, 42, 78 (1950); 44, 104 (1951).
15. Ya. I. Frenkel, UFN, 42, 69 (1950).
16. Ya. A. Smorodinsky, UFN, 39, 323 (1950).

1. There is a well-known example of E. Schrödinger, in which, with the aid of atomic physics, a signal is sent to some cat. Big-eared and mangy young people, mangy cats, and the like now make up the arsenal of examples of bourgeois philosophers. ↩

2. V. I. Lenin, Materialism and Empirio-Criticism. ↩