ON THE INTERPRETATION OF QUANTUM MECHANICS*

V. A. Fock

A characteristic feature of modern physics is that it penetrates ever more deeply into the regularities of the world of atoms and of other smallest particles of matter. These regularities require not only new experimental techniques for their discovery, but also new concepts for their formulation. To describe atomic phenomena, new methods are needed, different from those used in the study of larger objects of the external world. The new formulation of the problem of describing atomic objects has great fundamental significance, since the concepts associated with it may find application in other areas of natural science as well. Thus, on the ground of modern physics, there arise certain questions of a philosophical character, to which we should like to draw attention here.

As has more than once happened in the history of physics, the mathematical part of the theory, together with certain formal recipes connecting the theory with experiment, was constructed earlier than the corresponding physical concepts had been worked out. The apparatus of nonrelativistic quantum mechanics, containing no internal contradictions, was successfully applied to the solution of concrete problems of atomic physics, but its physical interpretation long remained unclear. The need for a proper physical interpretation of the ready-made apparatus of quantum mechanics came to be felt more and more acutely.

§ 1. ATTEMPTS AT A CLASSICAL INTERPRETATION OF THE WAVE FUNCTION AND THE REASONS FOR THEIR FAILURE

The point of view originally advanced by de Broglie and Schrödinger consisted in the claim that, in quantum mechanics, the wave function represents a certain field distributed in space, similar to the electromagnetic field and other previously known fields; according to Schrödinger, the stationary states of atoms correspond to the natural oscillations of this field. Somewhat later de Broglie put forward a somewhat different point of view, according to which the field distributed in space is the carrier of particles and determines their motion in the classical sense (pilot wave, or, more precisely, guiding wave). This point of view was soon abandoned by de Broglie, but subsequently, after 25 years, he returned to it. The works of Bohm are also closely connected with it; he attempted to preserve the concept of trajectory and to reconcile it with the formulas of ordinary quantum mechanics by introducing, in each separate case, specially selected

* The English text of the present article is being published in the Czechoslovak Journal of Physics.

case of a “quantum potential.” A certain modification of the “pilot-wave” point of view is represented by de Broglie’s attempts to apply to the present case Einstein’s idea of a particle as a singular point of a field; however, these attempts are not supported by any convincing mathematical arguments.

If in the first years of the development of wave mechanics it was natural to try to interpret it in a classical spirit, the same cannot be said of the attempts of de Broglie and his followers undertaken by them in recent years. The common feature of these attempts is their extreme artificiality and the complete absence of any heuristic value: the authors of these attempts have not tried to solve a single new problem. On the contrary, their reasoning was adjusted (and not convincingly at that) to a result already known from quantum mechanics. Thus the criterion of practice speaks decisively against this scientific direction.

What, then, are those features of quantum mechanics which do not allow it to be interpreted in a classical spirit and to see in the wave function a field, distributed in space, similar to a classical one? Leaving aside for the time being deeper foundations of a gnoseological order, one may indicate a number of formal reasons that prevent such an interpretation. First, in the case of a complex system consisting of many particles, the wave function depends not on three coordinates, but on all the degrees of freedom of the system. It is a function of a point in a multidimensional configuration space, and not in real physical space. Second, in quantum mechanics canonical transformations of the wave function of the Fourier-transform type are admissible, and all wave functions transformed in this way describe one and the same state and are entirely equivalent to the initial wave function expressed through coordinates; physical meaning is possessed not only by the square of the modulus of the initial wave function, but also by the squares of the moduli of the transformed functions. Third, the many-body problem (in particular, the problem of many identical particles) has in quantum mechanics features which do not permit reducing it to a problem of individual particles, still less formulating it as a problem of a field in ordinary three-dimensional space. Thus, if a complex system has a wave function common to all particles, no special wave functions can be ascribed to the individual particles; moreover, in the case of identical particles obeying the Pauli principle, there exists a special kind of quantum interaction between particles, not reducible to force interaction in ordinary space. Another kind of interaction, likewise not reducible to the classical one, exists between identical particles if they are described by symmetric wave functions. Finally, not only in the case of many particles but also for an individual particle the wave function does not always exist, and it does not always change according to Schrödinger’s equation; under certain conditions it is simply crossed out and replaced by another (the so-called reduction of the wave packet, see § 11). It is obvious that this kind of “instantaneous change” is not consistent with the concept of a field.

The indicated features of quantum mechanics doom in advance to failure all attempts to interpret the wave function in a classical spirit.

The connection of quantum mechanics with classical mechanics lies elsewhere, namely in the correspondence principle, according to which there exists a limiting case when those formulas of quantum mechanics which are directly compared with experiment pass over into classical ones. In this limiting case, the quantities characteristic of the given mechanical system, possess-

dimension of action may be regarded as large in comparison with the “quantum of action” \(h\)*). The correspondence principle was established by Bohr at the very beginning of the development of quantum mechanics and played a major role in its development.

§ 2. NIELS BOHR’S IDEAS AND HIS TERMINOLOGY

The true meaning of the wave function and of other concepts of quantum mechanics began gradually to become clear, starting with Max Born’s works on the statistical interpretation of quantum mechanics. The fundamental significance of the concept of probability became clear, although initially it was not clear probabilities of what exactly were meant. An essential role in clarifying this question, as in the interpretation of quantum mechanics in general, was played by Niels Bohr’s ideas that the quantum-mechanical description of the properties of an atomic object must be combined with the classical description of the means of observation (the experimental apparatus).

In his works devoted to fundamental questions of quantum mechanics, N. Bohr especially emphasizes the necessity of considering the whole experiment as a whole and of carrying the description of the experiment through to the readings of instruments. In itself this idea is correct, in the sense that in principle it must be possible to carry the description through to the readings of instruments. But excessive emphasis on the role of instruments gives grounds for reproaching Bohr with underestimating the necessity of abstraction and, as it were, forgetting that the subject of study is the properties of the micro-object, and not the readings of instruments. The properties of atomic objects, such as charge, mass, spin, the form of the energy operator, and the law of interaction of particles with an external field, are, on the one hand, completely objective and can be abstracted from the means of observation; on the other hand, these properties require for their formulation new, quantum-mechanical concepts. This applies in particular to the formulation of the many-body problem.

A source of misunderstandings is also the terminology used by Bohr. Thus, he speaks of “uncontrollable interaction,” although an interaction considered as a physical process is always controllable. Bohr has to speak of “uncontrollability” only in order to cover the inconsistency arising from the use of classical concepts outside their domain of applicability. Further, one may point to Bohr’s opposition of the “principle of complementarity” to the “principle of causality.” If these terms are understood literally, such an opposition is, of course, incorrect. But by the “principle of complementarity” Bohr understands not only Heisenberg’s relations, but in general all the characteristic differences of quantum mechanics from classical mechanics. By the principle of causality Bohr understands causality in the narrow mechanical sense—in the sense of determinism of the Laplace type. Thus, in fact Bohr has in mind the incompatibility of quantum mechanics with determinism of the Laplace type, but not with the principle of causality in the more general sense. And in that he can be agreed with.

The principle of causality in the general sense should be understood as the assertion of the existence of laws of nature and, in particular, those which

*) For a material point in a time-independent force field one may take as such a characteristic quantity

\[ \frac{mv^3}{w}, \]

where \(m\) is the mass of the particle, \(v\) is its velocity, and \(w\) is its acceleration. Since this is a question of estimating an order of magnitude, instead of \(v\) and \(w\) one may take their mean values (in one sense or another). If by \(h\) one understands Planck’s constant divided by \(2\pi\), then the criterion of applicability of classical mechanics to the motion of a material point is \(mv^3 \gg hw\) (see V. Fock, Uch. zap. LGU, ser. phys., vol. 3, pp. 5–9, 1937).

are connected with the general properties of space and time (the finite speed of propagation of interactions, the impossibility of acting on the past). In this understanding, quantum mechanics not only does not contradict the principle of causality, but gives it a new expression and extends its application to probabilistic laws.

As I have had occasion to convince myself from personal conversations with Niels Bohr, in fact his position is much closer to a materialist one than might appear from reading his works on the fundamental questions of quantum mechanics. Above all, Bohr believes that nature must be taken as it is. He resolutely expresses his disagreement with the positivist point of view and fully recognizes the objectivity of the properties of atomic objects. As for terminology, Bohr is prepared to abandon the use of the term “uncontrolled interaction,” which he regards as unfortunate. Bohr also agrees that the general principle of causality should be distinguished from determinism of the Laplacian type, and that only such determinism contradicts the regularities of atomic physics.

§ 3. THE DENIAL OF NEW IDEAS AS A REACTION TO THEIR POSITIVIST INTERPRETATION

The novelty of Bohr’s ideas and their difficult-to-understand exposition, employing terminology that was not always successful, gave rise to many misunderstandings and incorrect interpretations of them in a positivist spirit. (It should be noted that precisely this positivist interpretation of the new ideas is usually what is meant by the term “Copenhagen school.”) The most extreme positivist position is occupied by P. Jordan; other, more serious physicists, such as M. Born, W. Heisenberg, and others, were at one time strongly attracted by positivist views, but are now gradually moving away from them. Thus, in one of his recent works, printed in a volume dedicated to the seventieth birthday of Niels Bohr, W. Heisenberg already recognizes the objectivity of the concept of a quantum state.

The interpretation of Bohr’s ideas in the spirit of positivism, carried out by some of his followers, naturally gave rise to a reaction which, in the name of materialism, denied the new ideas (de Broglie, Bohm, Vigier, and others). The principal motive of the named scientists, compelling them to adopt the position of refusing to recognize the usual probabilistic interpretation of quantum mechanics, is the false conviction that the probabilistic interpretation means a rejection of the objectivity of the microworld and its laws, i.e. a rejection of the basic tenet of materialism. In the opinion of the followers of de Broglie’s school, only determinism of the classical type is compatible with materialism. They therefore call their point of view deterministic.

The narrowness, and therefore the incorrectness, of such an understanding of materialism appears to us undoubted. To impose on nature precisely a deterministic form of regularities, while refusing, contrary to the evidence, to recognize their more general probabilistic form, means proceeding from certain dogmas, and not from the properties of nature itself. Such a position is philosophically incorrect. Therefore one should not be surprised at the failure of all attempts at a “deterministic” interpretation of quantum mechanics—the failure whose formal causes we analyzed in § 1. At the same time, the persistent renewal of attempts of this kind and the interest shown in them by non-specialists, conditioned by the fact that they are undertaken in the name of materialism, make a more profound analysis of the new ideas of quantum mechanics from the point of view of materialism urgently necessary.

tion of materialist philosophy. Such an analysis must undoubtedly lead to the conclusion that the new ideas considerably expand the range of concepts with which materialist philosophy operates, but in no way contradict its spirit.

§ 4. RELATIVITY TO THE MEANS OF OBSERVATION

Let us try to indicate the basic features of quantum mechanics that distinguish it from classical mechanics.

The properties of objects are always manifested in their interaction with other objects, in particular with the means of observation (instruments). This is true both in classical and in quantum physics. But in classical physics it was possible to abstract from the means of observation to a far greater degree than is possible in quantum physics. This becomes understandable if we recall that the means of observation are always on a “human” scale, whereas the scales of the objects with which classical physics, on the one hand, and quantum physics, on the other, operate are quite different: classical objects, generally speaking, are not smaller than the means of observation, whereas quantum objects are immeasurably smaller.

In classical physics it is assumed that, with sufficiently careful use of the means of observation, they cannot noticeably influence the object being studied; and if they do exert an influence, then a correction can be introduced for this influence. Therefore one may reason there as if the means of observation played no role at all and speak, for example, of the state of motion of an object irrespective of the means of observation, thereby absolutizing the very concept of the “state of motion.” True, an element of relativity remains here as well, since even in classical physics reference to a definite reference frame is necessary; from the point of view developed here this may be interpreted as taking account of the motion of the means of observation. But in quantum physics it is necessary to take into account not only the motion of the means of observation, but also, in some schematized form, their internal structure.

Thus, by a classical description one may understand a description that is independent of the means of observation (if one does not count taking account of their motion). The accuracy of such a description is limited by Heisenberg’s uncertainty relations. Such accuracy is sufficient for describing the mechanical properties of larger bodies, but it becomes insufficient for describing objects of the microworld. Not only accuracy in the quantitative sense, but also the formulation of qualitatively new properties of micro-objects requires new methods of description; and above all it is necessary to introduce into their description a new element of relativity—relativity to the means of observation.

It is perfectly clear that relativity does not hinder objectivity. Even in classical physics, such very simple concepts as the trajectory of a material point, while being entirely objective, are at the same time relative, since they acquire a definite meaning only in a definite reference frame. Similarly, in quantum physics relativity to the means of observation only makes physical concepts more precise and allows new ones to be introduced, and by no means deprives them of objectivity. The objects of the microworld are just as real and their properties just as objective as the properties of the objects studied by classical physics.

§ 5. THE CONCEPT OF AN INSTRUMENT

In the preceding section we indicated certain general conclusions from the simple but fundamental fact that the study of the world of atoms is possible only through larger objects, which

and serve as means of observation (instruments). Since the concept of an instrument plays a large role in our reasoning, we must clarify it. We may call an “instrument” such a device which, on the one hand, can interact with a micro-object and respond to its actions, and, on the other hand, admits, with an accuracy sufficient for the given purpose, a classical description (and, consequently, does not require further “means of observation”). It should be noted at once that in this definition of an instrument it is completely immaterial whether the “instrument” is made by human hands or represents a natural combination of external conditions, convenient for observation, in which the micro-object is placed. What is important is only that these conditions, like the actual means of observation, must be describable classically.

Understanding the term “instrument” in this sense, we can formulate the problem of the quantum-mechanical description of a micro-object as follows.

All the properties of a micro-object, including the specifically quantum ones, i.e. those for the description of which classical mechanics is insufficient, must be characterized by the capacity of the micro-object to act upon instruments that admit a classical description.

§ 6. THE ESSENCE OF WAVE–PARTICLE DUALISM

Different external conditions are needed for the manifestation of different properties of an atomic object. It may turn out that different types of external conditions are incompatible with one another. Consider, for example, the scattering of electrons by a crystal. The regular arrangement of the scattering centers is necessary in order to obtain a distinct diffraction pattern, and thereby also to reveal the wave properties of the electron. But this same regularity of arrangement is an obstacle to the precise spatial localization of the electron undergoing diffraction: if the regularity of the arrangement of the centers is not disturbed, it is impossible to establish from which of them the electron was reflected. In the literature, especially in the works of N. Bohr and W. Heisenberg, many other examples have been analyzed which may be regarded as illustrating the incompatibility of the external conditions necessary for revealing the wave and corpuscular properties of the electron.

There are also possible conditions under which the wave and corpuscular properties of the electron manifest themselves simultaneously, but then these properties are not expressed sharply. For example, for an electron bound in an atom, the wave function has the character of a standing wave with an amplitude that rapidly decreases with distance from the center of the atom. This means precisely that the electron is approximately localized (a corpuscular property), but at the same time it partly exhibits wave properties.

Thus, for atomic objects, under some conditions wave properties come to the fore, and under others corpuscular properties do; there are also possible conditions under which both kinds of properties appear, albeit not sharply, but simultaneously. One may say that for an atomic object there exists a potential possibility of manifesting itself, depending on the external conditions, either as a wave, or as a particle, or in an intermediate manner. It is precisely in this potential possibility of different manifestations of properties inherent in an atomic object that wave–particle dualism consists. Any other, more literal understanding of this dualism in the form of some model is incorrect. In particular, the model proposed by de Broglie and his school of a particle carried by a wave, or the model of a particle as a special point of a field, are absolutely unsuitable, as was already indicated in § 1.

Moreover, it must be remembered that the peculiarities of quantum mechanics (even nonrelativistic quantum mechanics) are not exhausted by wave–particle dualism. Such properties of electrons as spin and quantum statistics (the Pauli principle) are not reducible to this dualism, but they too can be formulated with the aid of the apparatus of quantum mechanics. The fundamental character of these properties is evident at least from the fact that it is precisely they that determine the structure of the electron shells of atoms, and thereby their optical and chemical properties.

The apparatus of quantum mechanics, correctly reflecting a number of fundamental properties of atomic objects, finds a rational interpretation only on the basis of an extended formulation of the problem of describing micro-objects—namely, one in which their behavior is not separated from their interaction with the means of observation.

§ 7. PROBABILISTIC DESCRIPTION OF THE INTERACTION BETWEEN OBJECT AND APPARATUS

Speaking of the interaction between a micro-object and an apparatus, we must distinguish two aspects of the matter: first, interaction as a physical process, and, second, interaction as the junction between a part of the system described quantum-mechanically (the micro-object) and a part described classically. In the first case the word “interaction” is used in a more literal sense, and in the second case in a more conventional sense. Here we are interested chiefly in the second aspect of the matter, since the very concept of a quantum-mechanical description must be based on an analysis of interaction in this second sense.

What, then, are the peculiarities of the interaction of an atomic object with a classical apparatus?

In answering this question it should be remembered that both the external conditions in which the object is found and the result of its interaction with the apparatus must be described in the language of classical physics. From these classical data one must draw a conclusion about the quantum characteristics of the atomic object.

Even if an atomic object is in fixed external conditions, the result of its interaction with an apparatus is, in the general case, not unique. This result cannot be predicted with certainty on the basis of preceding observations, however exact the latter may have been. Only the probability of a given result is determinate. The fullest expression of the results of a series of measurements will be not an exact value of the measured quantity, but a probability distribution for it.

Of course, it may turn out that the probability of one value of the measured quantity (or of one narrow interval of values) so predominates over the probabilities of the remaining values that, in practice, precisely that value may be assigned to the quantity. In this case an exact or nearly exact prediction of the result of the measurement is possible. However, this case is only an exception, or rather a special case. Characteristic of the state of affairs that obtains in quantum mechanics is the general case, in which measurements lead to some probability distribution.

The fact that, in the general case, no refinement of previous observations leads to an unambiguous prediction of the result of a measurement has great fundamental significance. This fact should be regarded as the expression of a certain law of nature connected with the properties of atomic objects, in particular with the wave–particle dualism inherent in them. Recognition of this fact means the renunciation of the clas-

cal determinism and requires new forms of expression of the principle of causality.

In itself, the expression of the results of a series of measurements in the form of a probability distribution is not alien to classical physics either. But there probabilities were regarded as a “foreign element,” as the result of a failure to take into account certain unknown circumstances and of averaging over unknown data. In classical physics it was always assumed that it is in principle possible to sort the observed objects in advance in such a way that subsequently, as a result of observation, for the measured quantity only a single value would be obtained, and not an entire probability distribution. On the contrary, in quantum physics such sorting of atomic objects is impossible, since, by the very nature of atomic objects, the measured quantities may, under the given conditions, have no definite values. In quantum physics the concept of probability is a primary concept, and it plays a fundamental role there. The quantum-mechanical concept of the state of an object is also connected with it.

§ 8. PROBABILISTIC CHARACTERISTIC OF THE STATE OF AN OBJECT

For the study of the properties of atomic objects, the most important arrangement of an experiment is one in which three stages can be distinguished: the preparation of the object, the behavior of the object under fixed external conditions, and the measurement proper. Correspondingly, in the apparatus one may distinguish the preparing part, the working part, and the registering part. For example, when observing the diffraction of electrons by a crystal, the preparing part is the source of a monochromatic beam of electrons, as well as diaphragms and other devices placed before the crystal; the working part is the crystal itself; and the registering part is a photographic plate or a counter.

With such an arrangement of the experiment, it is possible to vary the final stage (the measurement) while leaving the first two stages unchanged. The physical interpretation of the apparatus of quantum mechanics is most conveniently traced precisely for such an arrangement of the experiment.

By varying the final stage of the experiment, one may carry out measurements of different quantities (for example, the energy of the particle, or its velocity, or its position in space), starting from one and the same initial state of the object. To each quantity there corresponds its own series of measurements, the results of which are expressed in the form of a probability distribution for that quantity.

All the indicated probability distributions may be expressed parametrically through one and the same wave function, which does not depend on the final stage of the experiment and thereby is an objective characteristic of the state of the object immediately before the final stage.

The state of the object described by the wave function is objective in the sense that it represents an objective (observer-independent) characteristic of the potential possibilities of one or another result of the interaction of an atomic object with an apparatus. In this same sense it pertains precisely to the given, individual object. But this objective state is not yet actual in the sense that, for the object in the given state, the indicated potential possibilities have not yet been realized. The transition from the potentially possible to the realized, to the actual, occurs in the final stage of the experiment. For the statistical characterization of this transition, i.e., for the experimental obtaining of the corresponding probability distribution, a series of measurements is already necessary; the probability distribution is obt—

is obtained after statistical processing of this series of measurements. This experimental probability distribution can then be compared with the theoretical one obtained from the wave function.

It should be noted that, although the immediate result of the final stage of the experiment is formulated classically, on the basis of the theory one can also derive from it the values of those quantities that are specifically quantum—such as the spin of a particle, the energy levels of an atomic system, and so on. Thus, from the statistical processing of series of measurements one can obtain probability distributions not only for quantities analogous to classical ones, but also for specifically quantum quantities.

§ 9. THE CONCEPTS OF THE POTENTIALLY POSSIBLE AND THE ACTUALIZED IN CLASSICAL PHYSICS

In classical deterministic physics the question of the transition from the potentially possible to the actualized does not arise at all, since there an unambiguous predetermination of the course of events is postulated, by virtue of which everything potentially possible is also actualized, so that there is no need to distinguish one from the other. The practical impossibility of predicting all events is there attributed to the incompleteness of the initial data.

Such a deterministic point of view is by no means a logical necessity, but is due to historical causes, above all the successes of celestial mechanics in the eighteenth and nineteenth centuries. The high precision achieved by astronomers in predicting the motion of celestial bodies gave rise to an enthusiasm for mechanistic determinism (Laplacean determinism). As a result of this enthusiasm, the deterministic worldview spread to all of physics (with the exception, perhaps, of thermodynamics) and began to claim to be the only scientific one. The successes of the electromagnetic theory of light, which introduced the concept of the field as a physical reality, although they showed the limitations of the mechanistic point of view, did not undermine faith in determinism.

True, the experience of everyday life, in which one has to distinguish strictly between a possibility and its actualization, spoke against it; but this experience was rejected as “unscientific.” In the domain of physics, thermodynamics remained “unreliable” in the sense of determinism, and it never proved possible to reconcile it with determinism. But the real collapse of determinism occurred with the development of quantum mechanics, beginning with A. Einstein’s work on the theory of radiation (1916), where he first introduced a priori probabilities into physics.* The correct interpretation of the quantum-mechanical description of the properties of atomic objects completely excludes the deterministic point of view. Quantum mechanics restores, in its full rights, the distinction dictated by lived experience between potential possibility and its actualization.

§ 10. PROBABILITY AND STATISTICS IN QUANTUM MECHANICS

The probabilistic character of quantum mechanics is not subject to doubt; indeed, it is almost never disputed by anyone. But the questions of what exactly these probabilities refer to, in what statistical ensemble they are taken, and whether quantum mechanics represents a theory of individual

*) It is curious that Einstein, who did much for quantum theory in the initial period of its development and was the first to introduce a priori probabilities into physics, subsequently became an opponent of quantum mechanics and a supporter of determinism; as he put it, half-jokingly, he could not believe that the Lord God plays dice (dass der liebe Gott würfelt).

atomic objects, or only the theory of ensembles of such objects—these questions continue to be discussed, although at the present time a quite unambiguous answer can be given to them.

In the first years of the development of quantum mechanics, in the early attempts at its statistical interpretation, physicists had not yet freed themselves from the conception of the electron as a classical material point. The electron was spoken of as if it were a particle with definite values of coordinate and velocity, but it was unknown exactly which ones. Heisenberg’s relations were interpreted as relations of inaccuracies, and not as uncertainty relations. The square of the modulus of the wave function was interpreted as the probability density for a particle to have given coordinates (as if the coordinates were always definite). The square of the modulus of the wave function in momentum space was interpreted analogously; moreover, both probabilities (in coordinate space and in momentum space) were considered simultaneously, as if the values of coordinates and momenta were compatible. The factual impossibility, expressed by Heisenberg’s relations, of measuring them jointly was represented, under such a view, as some paradox or caprice of nature, by virtue of which, supposedly, not everything that exists is knowable.

All these difficulties disappear if one fully accepts the dual corpuscular-wave nature of the electron, clarifies the essence of this dualism, and understands to what the probabilities considered in quantum mechanics refer. In order not to repeat what has already been explained above, let us recall only that the probabilities obtained from the wave function for various quantities refer to different experimental arrangements and that they characterize not the behavior of the particle “in itself,” but its action on an apparatus of a definite type.

The question of the statistical ensemble in which the probabilities are taken was also a subject of discussion. One of the first to pose this question was Academician L. I. Mandelstam (Works, vol. 5, p. 356), but he gave an incorrect answer to it. Mandelstam speaks of a “micromechanical ensemble to which the wave function refers,” and also calls it an “electron ensemble,” thereby emphasizing that he has in mind an aggregate of micro-objects prepared in a definite manner. In these initial propositions of Mandelstam there are inaccuracies connected with an insufficiently clear definition of a statistical ensemble. Let us try to correct them and to give a clearer definition of the concept of an ensemble.

Let us imagine an unlimited series of elements possessing various attributes, according to which these elements can be sorted, and let us observe the frequency with which an element with a given attribute appears. If, for the appearance of an element with each given attribute, there exists a definite probability*), then the series of elements under consideration represents a statistical ensemble.

What statistical ensemble, then, can be considered in quantum mechanics? Obviously, only an ensemble of elements described classically, since only to such elements can one always ascribe definite values of the parameters by which the sorting is performed. For this reason a quantum object cannot be an element of a statistical ensemble, even if it is under such conditions that

*) The existence of a definite probability represents a hypothesis which is introduced either a priori (for example, from considerations of symmetry), or on the basis of the constancy of those external conditions under which the physical realization of the series of elements under consideration takes place. The hypothesis of the existence of a probability is equivalent to the hypothesis that the given series of elements represents a statistical ensemble.

one may associate a wave function with it. Thus, one cannot speak of a “micromechanical” and an “electronic” ensemble in Mandelstam’s sense.

The elements of the statistical ensembles considered in quantum mechanics are not the micro-objects themselves, but the results of experiments on them; moreover, a definite experimental arrangement corresponds to one definite ensemble. Since the probability distributions for different quantities obtained from the wave function correspond to different experimental arrangements, they also correspond to different ensembles. Thus, the wave function cannot belong to any definite statistical ensemble.

What has been said may be illustrated by the following scheme:

Each cell of this scheme corresponds to a definite statistical ensemble with its own probability distribution. In one row are placed ensembles obtained by measuring different quantities $E$, $p$, $x$, starting from one and the same initial state. In one column are placed ensembles obtained by measuring a given quantity, starting from different states $\psi_1$, $\psi_2$, $\psi_3\ldots$

The deeper reason why no statistical ensemble can be associated with the wave function is that the concept of a wave function refers to the potentially possible (to experiments not yet performed), whereas the concept of a statistical ensemble refers to what has been realized (to the results of experiments already performed).

Attempts to assign the wave function to a collection of micro-objects have been made by various authors; moreover, the opinion has been expressed that all of quantum mechanics is a theory of such collections of micro-objects (ensembles), while a theory of separate, individual micro-objects supposedly does not yet exist. Such an opinion is based above all on a misunderstanding of what probability is. The probability of a given behavior of an object under given external conditions is determined by the internal properties of the given individual object and by these external conditions; it is a numerical estimate of the potential possibilities of one or another behavior of the object. This probability manifests itself in the relative number of realized cases of the given behavior of the object; this number is its measure. Thus, probability pertains to an individual object and characterizes its potential possibilities; at the same time, for the experimental determination of its numerical value, statistics of the realization of these possibilities is necessary, i.e., repeated repetition of the experiment. Hence it is clear that the probabilistic character of a theory does not preclude its pertaining to an individual object. This is also true for quantum mechanics.

§ 11. FORMS OF EXPRESSION OF THE PRINCIPLE OF CAUSALITY IN QUANTUM MECHANICS

The quantum-mechanical concept of state makes it possible to formulate the principle of causality as applied to atomic phenomena. According to quantum mechanics, the wave function of an atomic system satisfies a wave equation which uniquely determines it from its initial value (the Schrödinger equation). Thereby the law of change of the probabilities expressed through the wave function is also determined. The wave equation makes it possible to solve nonstationary problems of quantum mechanics corresponding to experiments whose various stages are separated in time. A typical example of such problems is the problem of the decay of an almost stationary state of an atomic system, in particular the problem of the ionization of an atom by an electric field; in principle, the problem of the radioactive decay of an atomic nucleus also belongs here.

In modern physics the principle of causality is associated not only with the impossibility of acting upon the past, but also with the existence of a limiting velocity of propagation of actions, equal to the velocity of light in free space. Both these requirements are also satisfied in quantum mechanics. It is true that in its nonrelativistic form (in Schrödinger’s theory) the existence of a limiting velocity is taken into account only indirectly, in the form of an additional requirement that the velocities under consideration be small in comparison with the limiting one. But in all relativistic generalizations of quantum mechanics the existence of a limiting velocity is taken into account automatically. The relations following from the principle of causality, in particular the relations for scattering amplitudes, play a large role in quantum field theory.

In connection with the existence of a limiting velocity of propagation of actions, one should consider the question of the so-called “reduction of the wave packet.” By this is meant the following. If one assumes that the final stage of one experiment is at the same time the initial stage of another, then the wave function which gave the probability distribution of the results of this experiment must be replaced by another, corresponding to the result actually obtained. Such a replacement occurs suddenly; the change of the wave function does not take place according to the Schrödinger equation. It may seem (and this question has indeed been debated) that a sudden change of the wave function is in contradiction with the finite velocity of propagation of actions. But it is easy to see that here we are dealing not with the propagation of any action, but with a change in the formulation of the question about probabilities. In the experiment performed, one of the potentially possible results provided for by the initial wave function has been realized. The change in the formulation of the question about probabilities consists precisely in taking account of the result that has occurred, i.e. in taking account of new data. And to the new data there corresponds a new wave function as well.

These considerations show how important it is, in interpreting quantum mechanics, to distinguish the potentially possible from the realized. They also show with complete clarity that the wave function is not some real field, and that its sudden change is not any physical process analogous to a change of field. A physical process is indeed connected with the performance of the experiment, but it is reflected in the wave function not directly, but by virtue of the necessity caused by it to reformulate the problem of probabilities.

The quantum-mechanical understanding of causality differs significantly from the classical one, although it represents its natural generaliza—

Classical (Laplacean) determinism, of which we have already spoken in § 9, may be defined as the point of view according to which the improvement of methods of observation, together with the refinement of the formulation of the laws of nature and of their mathematical treatment, can in principle make possible an unambiguous prediction of the entire subsequent course of events. The study of the atomic world shows that classical determinism not only does not correspond to the laws of nature, but does not even make it possible to formulate them with sufficient precision. This discrepancy occurs even in the case of the simplest elementary processes (quantum transitions), so that the issue here is not at all the complexity of the phenomenon, but the unsuitability of the old methods for describing it. The essential features of the new methods consist, as we have seen, in the probabilistic character of the description, owing to which it is necessary to distinguish the potentially possible from what has been realized, in taking account of relativity with respect to the means of observation, and, finally, in a new understanding of the principle of causality, according to which this principle refers directly to probabilities, i.e., to the potentially possible, and not to events actually taking place.

§ 12. PHILOSOPHICAL QUESTIONS POSED BY QUANTUM MECHANICS

The development of the new ideas introduced by atomic physics requires the elaboration of a number of philosophical questions, especially questions connected with the analysis of the act of cognition. These questions arise in connection with the impossibility, noted above, of abstracting in the study of atomic objects from the means of observation, and also in connection with the necessity of considering probability as a fundamental concept and of distinguishing the potentially possible from what has been realized, with which, in turn, the new formulation of the principle of causality is connected. Here one cannot get by with the study of the classical heritage and the selection of quotations from the classics; it is necessary to approach the solution of the philosophical questions of natural science creatively. It is necessary creatively to develop dialectical materialism. At the same time it must be remembered that the ideas of atomic physics are indeed radically new, and that it is in no way possible to dismiss them by trying to reduce the matter to those ideas about which we have ready-made judgments of the classics.

Nor can one appeal to the fact that the concepts of ordinary quantum mechanics are not the final word of science, or to the fact that a satisfactory quantum theory of the field has not yet been constructed. Every theory, including quantum mechanics, represents only a relative truth, but this gives no grounds for refusing to acknowledge the new ideas and concepts it has introduced.

Physical concepts will undoubtedly develop, but it is already clear now that this development will proceed in the direction of a further departure from classical notions, and in no way in the direction of a return to them. In particular, the hopes expressed by some physicists of the de Broglie school for a return, in some new form, to classical determinism have no foundation whatever. Whoever, in the name of materialism, tries to deny the new ideas and to restore the old ones is rendering materialism a poor service.

The philosophical generalization of the new ideas that first arose in atomic physics may also prove useful for the development of other areas of science, in which questions analogous to those already resolved in quantum mechanics may arise.

The resolution achieved in quantum mechanics of the contradictions between the wave and corpuscular nature of the electron, between probability and causality, between the quantum description of an atomic object and the classical description of the apparatus, and, finally, between the properties of an individual object and their statistical manifestations, provides a number of striking examples of the practical application of dialectics to questions of natural science. This remains a fact regardless of whether the dialectical method was applied consciously or not. The achievements of quantum mechanics must serve as a powerful stimulus for the development of dialectical materialism. The incorporation of new ideas into its treasury is a primary task of materialist philosophy.