Can It Be Considered That the Quantum-Mechanical Description of Physical Reality Is Complete?

V. A. Fock, A. Einstein, B. Podolsky and N. Rosen, N. Bohr

I. Introductory Article by V. A. Fock, Leningrad

In quantum mechanics we encounter new physical ideas so different from the customary notions of classical theory that their assimilation presents considerable difficulty, especially for minds brought up on classical physics. How difficult it is for such minds to “accept” these new ideas is shown by the fact that even the creator of the theory of relativity—Einstein—who enriched human thought with no less profound physical ideas, and, paradoxical as it may be, one of the creators of quantum mechanics—Schrödinger—still cannot reconcile themselves to those consequences that follow from the discovery of quantum mechanics.

During 1935, two articles appeared in the American journal Physical Review under the same title, which we have also adopted as the title of the present combined article. The first article was written by Einstein, Podolsky, and Rosen. In it the authors, proceeding from a certain “criterion of physical reality,” come to the conclusion that the question posed in the title must be answered in the negative. In response to this article, in one of the following issues of the same journal, there appeared the second of the articles mentioned, written by Bohr. In it the author, developing ideas he had repeatedly expressed earlier, gives a profound and brilliant analysis of the concept of physical reality in the light of the new quantum physics. Using the simplest means, Bohr gives an exhaustive clarification of the paradox indicated by Einstein, Podolsky, and Rosen, and shows that the question posed by these authors should be answered in the affirmative.

Although in the dispute between the authors of the two named works no essentially new idea was expressed, and although the answer to the question posed in the first article, coinciding with Bohr’s answer, follows directly from considerations that can be found even in courses on quantum mechanics[^1], nevertheless this dispute is of great fundamental interest, since in it the points of view of both sides were formulated with great clarity, as a result of which it greatly contributed

to the clarification, for many physicists, of those new physical concepts that are connected with quantum mechanics.

In view of the interest represented by the two indicated articles, we print here their complete translation*. We do not include Schrödinger’s article², devoted to the same question, since in it only the paradoxes arising from the incorrect point of view of Einstein, Podolsky, and Rosen are developed, and no answer to them is given. Likewise we do not include other works that appeared in connection with the aforementioned dispute, and in particular Furry’s work³, which contains valuable mathematical explanations of Bohr’s arguments. We can confine ourselves to the two principal articles because the essence of the dispute is exhausted by them: in the article by Einstein, Podolsky, and Rosen the natural, though incorrect, point of view is formulated very clearly and the paradoxes to which it leads are indicated, while in Bohr’s article an exhaustive clarification of these paradoxes is given on the basis of the concept of “complementarity” introduced by him.

The polemic between Einstein, Podolsky, and Rosen, on the one hand, and Bohr, on the other, may, if one wishes, be regarded as a dispute about the physical meaning of the wave function. Proceeding to the exposition of the mathematical part of his arguments, Einstein** says that the fundamental concept of the theory is the concept of a state described by a wave function. Einstein understands the word “state” in the sense usually assigned to it in classical physics, i.e. in the sense of something entirely objective and completely independent of any information about it. From this all the paradoxes arise. Quantum mechanics indeed deals with the study of the objective properties of nature in the sense that its laws are dictated by nature itself and not by human fantasy. But the concept of a state in the quantum sense does not belong among objective concepts. In quantum mechanics the concept of a state merges with the concept of “information about the state obtained as the result of a definite maximally exact experiment.” In it the wave function describes not the state in the ordinary sense, but rather this “information about the state.” Einstein shows that, without touching the system, one can give its wave function one form or another. If, together with Einstein, one considers that the wave function describes an objective state, then, of course, his result will have the character of a paradox. For it is impossible to imagine that the objective state of a system (whatever we may mean by this) should change as a result of any operations performed not on it, but on another system with which it does not at all inter—

* The article by Einstein, Podolsky, and Rosen was translated by A. G. Lyubina under the editorship of V. A. Fock; Bohr’s article was translated by V. A. Fock.

** In what follows, for brevity, we shall omit the surnames of the two other coauthors.

acts. But although, as a result of such operations, the “objective state” of the system cannot change, the “information about the state,” i.e. the state in the quantum sense, can change. Therefore all the paradoxes disappear as soon as we abandon Einstein’s incorrect “objective” interpretation of the wave function and adopt its correct interpretation, i.e. regard it as describing the “state in the quantum sense,” or “information about the state, obtained as the result of a certain maximally precise experiment.”

By a maximally precise experiment we mean one that makes it possible to find all those quantities that can in general be known simultaneously. This definition is applicable both to classical and to quantum mechanics. But in classical mechanics the maximally precise experiment was essentially unique, namely the one that gave the values of all mechanical quantities in general, in particular the coordinates and the components of momentum. Precisely because in classical mechanics any two maximally precise experiments give the very same information about the system, it was possible there to speak of the state of the system as something objective, without specifying by which particular experiment the information about it had been obtained. Matters are otherwise in quantum mechanics. There such a qualification is essentially necessary. Indeed, the Heisenberg relations show that different experiments may interfere with one another. Therefore there may be an infinite multitude of maximally precise experiments: some of them will give maximally precise information about the coordinates, others about the momentum, and so on. In quantum mechanics a wave function is associated with each result of a certain maximally precise experiment. It thus represents a record of the information obtained as the result of such an experiment. From this point of view it is also easy to understand the physical meaning of the dependence of the wave function on time, i.e. the physical meaning of the Schrödinger equation: it makes it possible to use information pertaining to the initial instant of time for predictions pertaining to later instants of time.

We have already said above that in quantum mechanics the boundary is erased between the concepts of “state” and “maximally precise information about the state.” This assertion is only another formulation of the thought that Bohr has in mind when he speaks of the necessity of a “radical revision of our views on the problem of physical reality.” This thought is brilliantly substantiated by Bohr in his article. Therefore we need not dwell on its substantiation, but may proceed to an analysis of some consequences that follow from the interpretation of the wave function as a record of information.

First of all, the information about a system need not be maximally precise. This means that the system need not have a definite (even if unknown) wave function. It may happen, for example, that we know about the momentum of a particle only

that it is equal either to \(p_1\) (with probability \(w_1\)), or to \(p_2\) (with probability \(w_2\)), and know nothing about its coordinates. From this information we cannot construct any wave function; we can only construct (following Neumann\(^4\)) a certain “statistical operator,” which makes it possible to compute the probabilities and mathematical expectations for all mechanical quantities corresponding to the available information.

Thus, to maximally precise information about a system we can associate a wave function (this will be the so-called case of a pure state, reiner Fall), whereas to less precise information we cannot associate a wave function (this will be the so-called case of a mixture of states, Gemisch).

Connected with this circumstance is a characteristic feature of the quantum-mechanical description of a composite system consisting of two or several subsystems (the case considered by Einstein).

Suppose that our system consists of two subsystems. Let the coordinates (or other variables) of the first subsystem be denoted by \(x_1\), and the coordinates of the second subsystem by \(x_2\). Suppose that on the complete system a maximally precise experiment has been performed, which has given us its wave function \(\Psi(x_1, x_2)\). In this case, although the information about the complete system will be maximally precise, the information about the subsystems, generally speaking\(^*\), will not be so. Therefore, even if both subsystems have ceased to interact, we are not entitled to ascribe to them separately any wave functions whatever, even unknown ones (Einstein, apparently, thought the contrary\(^ {**}\)). We can only ascribe to them statistical operators, which are easily expressed in terms of \(\Psi(x_1, x_2)^{***}\). In order to obtain maximally precise information about the individual subsystems, i.e. in order to determine their wave functions, we must perform one more experiment. The knowledge of the wave function \(\Psi(x_1, x_2)\) here introduces the simplification that it is sufficient to perform this experiment on one of the subsystems.

\(^*\) That is, in the general case, when \(\Psi(x_1,x_2)\) does not decompose into a product of the form \(\varphi(x_1)\psi(x_2)\).

\(^ {**}\) On p. 443 he says: “We cannot, however, compute the state in which each of the two systems will remain after the interaction.”

\(^ {***}\) Namely, if \(\Psi(x_1,x_2)\) is normalized to unity, then the matrix element of the statistical operator \(U_1\) of the first subsystem will be

\[ (x_1 \mid U_1 \mid x_1') = \int \overline{\Psi}(x_1', x_2)\,\Psi(x_1, x_2)\,dx_2 \]

and, correspondingly, for the second subsystem

\[ (x_2 \mid U_2 \mid x_2') = \int \overline{\Psi}(x_1', x_2')\,\Psi(x_1, x_2)\,dx_1. \]

This process, too, is considered by Einstein. From what has been said here, it is sufficiently clear that only an incorrect interpretation of the physical meaning of the wave function led him to the conclusion that the quantum-mechanical description is incomplete. The incorrectness of this conclusion follows with still greater clarity from Bohr’s profound analysis. We therefore hope that the attentive reader will have no doubt as to who is right in the dispute to which this article is devoted.

II. Article by A. Einstein, B. Podolsky and N. Rosen*, Princeton

1. In analyzing a physical theory it is necessary to distinguish between objective reality, which is independent of any theory, and those physical concepts with which the theory operates. These concepts are introduced as elements that must correspond to objective reality, and by means of these concepts we also form a picture of this reality.

In judging the success of a physical theory we may ask ourselves two questions: 1) Is the theory correct? and 2) Is the description given by the theory complete? Only if both of these questions can be answered in the affirmative can the concepts of the theory be recognized as satisfactory. The first question—concerning the correctness of the theory—is decided according to the degree of agreement between the conclusions of the theory and human experience. This experience, which alone permits us to draw conclusions about reality, takes in physics the form of experiment and measurement. We wish to consider here, with quantum mechanics in view, the second of the questions posed above.

Whatever meaning may be attached to the term complete, it seems to us that of any complete theory the following must be required: every element of physical reality must have a counterpart in the physical theory. We shall call this the condition of completeness. Thus the second question can easily be answered if we can determine what the elements of physical reality are.

The elements of physical reality cannot be defined by means of a priori philosophical reasoning; they must be found on the basis of the results of experiments and measurements. However, for our purposes there is no need to give an exhaustive definition of reality. We shall be satisfied with the following criterion, which we regard as reasonable. If we can, without in any way disturbing the system, predict with certainty (i.e. with probability equal to unity) the value of some physical quantity, then there exists an element of physical reality corresponding

* A. Einstein, B. Podolsky and N. Rosen, Phys. Rev., 47, 777, 1935. Translated by A. G. Lyubina under the editorship of V. A. Fock.

corresponding to this physical quantity. It seems to us that this criterion, although it by no means exhausts all possible ways of recognizing physical reality, at least gives us one such way, provided the conditions formulated in it are fulfilled. This criterion, regarded not as a necessary but only as a sufficient condition of reality, is in agreement with both the classical and the quantum-mechanical conception of reality.

To illustrate our thought, let us consider the quantum-mechanical description of the behavior of a particle having one degree of freedom. The basic concept of the theory is the concept of a state, which by assumption is completely characterized by the wave function \(\psi\). The latter is a function of the variables chosen to describe the behavior of the particle. To every physically observable quantity \(A\) there is made to correspond an operator, which may be denoted by the same letter.

If \(\psi\) is an eigenfunction of the operator \(A\), i.e., if, where \(a\) is a number,

\[ \psi' \equiv A\psi = a\psi, \tag{1} \]

then the physical quantity \(A\) has with certainty the value \(a\), provided the particle is in the state \(\psi\). If \(\psi\) satisfies equation (1), then, according to our criterion of reality, for a particle in the state \(\psi\) there exists an element of physical reality corresponding to the physical quantity \(A\). Let, for example,

\[ \psi = e^{\frac{2\pi i}{h}p_0 x}, \tag{2} \]

where \(h\) is Planck’s constant, \(p_0\) is some constant number, and \(x\) is an independent variable. Since the operator corresponding to the momentum of the particle has the form

\[ p = \frac{h}{2\pi i}\frac{\partial}{\partial x}, \tag{3} \]

we obtain:

\[ \psi' \equiv p\psi = \frac{h}{2\pi i}\frac{\partial\psi}{\partial x} = p_0\psi, \tag{4} \]

Thus, in the state determined by equation (2), the momentum has with certainty the value \(p_0\). Hence, in this case it makes sense to say that the momentum of the particle in the state \(\psi\) is real.

On the other hand, if equation (1) is not satisfied, we can no longer say that the physical quantity \(A\) has a definite value; such is the case, for example, with the

ordinate of the particle. The operator \(q\), corresponding to the coordinate, is the operator of multiplication by the independent variable. Thus

\[ q\psi = x\psi \ne a\psi . \tag{5} \]

According to quantum mechanics, we can only say that the relative probability that a measurement of the coordinate gives a result lying between \(a\) and \(b\) is equal to

\[ P(a,b)=\int_a^b \bar{\psi}\psi\,dx=\int_a^b dx=b-a. \tag{6} \]

Since this probability does not depend on \(a\), but depends only on the difference \(b-a\), we see that all values of the coordinate are equally probable.

Thus, for a particle in the state \(\psi\) [formula (2)] it is impossible to predict a definite value of the coordinate, and it can be obtained only by direct measurement. Such a measurement, however, will perturb the particle and, in this way, change its state. After the coordinate has been determined, the particle will no longer be in the state given by formula (2). Usually in quantum mechanics the following conclusion is drawn from this: if the momentum of a particle is known, then its coordinate has no physical reality.

In quantum mechanics a more general proposition is also proved: if the operators corresponding to two physical quantities, say \(A\) and \(B\), do not commute, i.e., if \(AB \ne BA\), then exact knowledge of one of these quantities excludes exact knowledge of the other. Moreover, every attempt experimentally to determine the second quantity changes the state in such a way as to destroy the knowledge of the first.

It follows from this that either 1) the quantum-mechanical description of reality by means of the wave function is incomplete, or 2) when the operators corresponding to two physical quantities do not commute, these two quantities cannot simultaneously be real. For, if both of them were simultaneously real and, consequently, had definite values, then these values would, by the condition of completeness, have to be contained in the complete description. Hence, if the wave function provided a complete description of reality, it would have had to include these values, and they could have been predicted. Since this is not the case, we are left with the alternative formulated above.

In quantum mechanics it is usually assumed that the wave function does indeed give a complete description of physical reality for the system in the state to which it corresponds. At first glance this assumption is quite acceptable, since

information that can be derived from knowledge of the wave function seems to correspond exactly to that which can be obtained by means of measurements without changing the state of the system. We shall show, however, that this assumption, together with the criterion of reality given above, leads to a contradiction.

1. For this purpose let us imagine two systems I and II, which we allow to interact from the moment \(t=0\) to \(t=T\), after which there is no longer any interaction between the two parts. Moreover, let us suppose that the states of both systems before \(t=0\) were known. We can then calculate, by means of the Schrödinger equation, the state of the combined system I + II at any subsequent moment of time, in particular for any \(t>T\). Let us denote the corresponding wave function by \(\Psi\). We cannot, however, calculate the state in which each of the two systems will remain after the interaction. According to quantum mechanics, this state can be found only with the aid of subsequent measurements, by means of the process known as “reduction of the wave packet.” Let us consider the essence of this process.

Let \(a_1, a_2, a_3\ldots\) be the eigenvalues of some physical quantity \(A\) belonging to system I, and \(u_1(x_1), u_2(x_1), u_3(x_1)\ldots\) the corresponding eigenfunctions, where \(x_1\) denotes the set of variables that serve to describe the first system. Then \(\Psi\), regarded as a function of \(x_1\), can be expressed in the form of the series

\[ \Psi(x_1,x_2)=\sum_{n=1}^{\infty}\psi_n(x_2)u_n(x_1), \tag{7} \]

where \(x_2\) denotes the variables that serve to describe the second system. Here the quantities \(\psi_n(x_2)\) are to be regarded simply as the coefficients of the expansion of \(\Psi\) in a series in the orthogonal functions \(u_n(x_1)\). Suppose now that the quantity \(A\) is measured, and that it is found to have the value \(a_k\). From this it is concluded that after the measurement the first system remains in the state described by the wave function \(u_k(x_1)\), while the second system remains in the state with wave function \(\psi_k(x_2)\). This is precisely the process of reduction, or collapse, of the wave packet: the wave packet given by the infinite series (7) is reduced to a single term \(\psi_k(x_2)u_k(x_1)\).

The sequence of functions \(u_n(x_1)\) is determined by the choice of the physical quantity \(A\). If instead of it we were to choose another quantity, say \(B\), having eigenvalues \(b_1, b_2, b_3,\ldots\) and eigenfunctions \(v_1(x_1), v_2(x_1), v_3(x_1),\ldots\), we would obtain, instead of equation (7), the expansion

\[ \Psi(x_1,x_2)=\sum_{s=1}^{\infty}\varphi_s(x_2)v_s(x_1), \tag{8} \]

where the quantities \(\varphi_s(x_2)\) are the new coefficients. If now the quantity \(B\) is measured and it turns out to be equal to \(b_r\), then we ...

we conclude that after the measurement the first system remains in a state described by the function \(v_r(x_1)\), while the second system remains in a state described by the function \(\varphi_2(x_2)\).

We see, therefore, that as a result of two different measurements performed on the first system, the second system may turn out to be in two different states, described by different wave functions. On the other hand, since during the measurement these two systems no longer interact, no real changes whatever can result in the second system from any operations on the first system. This, of course, is merely another formulation of what is meant by the absence of interaction between the two systems. Thus, to one and the same reality (the second system after interaction with the first) one can assign two different wave functions (in our example \(\psi_k\) and \(\varphi_r\)).

But it may happen that the two wave functions \(\psi_k\) and \(\varphi_r\) are eigenfunctions of two noncommuting operators corresponding to certain physical quantities \(P\) and \(Q\). That such a case is indeed possible is best shown by an example. Suppose that the two systems are two particles and that the function \(\Psi(x_1,x_2)\) is equal to

\[ \Psi(x_1,x_2)=\int_{-\infty}^{+\infty} e^{\frac{2\pi i}{h}(x_1-x_2+x_0)p}\,dp, \tag{9} \]

where \(x_0\) is some constant. Let the quantity \(A\) be the momentum of the first particle; then, as we know from equation (4), its eigenfunctions corresponding to the eigenvalue \(p\) will be

\[ u_p(x_1)=e^{\frac{2\pi i}{h}px_1}. \tag{10} \]

Since we have here a case of a continuous spectrum, equation (7) is rewritten in the form

\[ \Psi(x_1,x_2)=\int_{-\infty}^{+\infty}\psi_p(x_2)u_p(x_1)\,dp, \tag{11} \]

where

\[ \psi_p(x_2)=e^{-\frac{2\pi i}{h}(x_2-x_0)p}. \tag{12} \]

But this \(\psi_p\) is an eigenfunction of the operator

\[ P=\frac{h}{2\pi i}\frac{\partial}{\partial x_2}, \tag{13} \]

corresponding to the eigenvalue \(-p\) of the momentum of the second particle. On the other hand, if the quantity \(B\) is the coordi-

to the coordinate of the first particle, then its eigenfunction, corresponding to the eigenvalue \(x\), will be

\[ v_x(x_1)=\delta(x_1-x), \tag{14} \]

where \(\delta(x_1-x)\) is the well-known Dirac delta function. Equation (8) in this case will take the form

\[ \Psi(x_1,x_2)=\int_{-\infty}^{+\infty}\varphi_x(x_2)v_x(x_1)\,dx, \tag{15} \]

where

\[ \varphi_x(x_2)=\int_{-\infty}^{+\infty} e^{\frac{2\pi i}{h}(x-x_2+x_0)p}\,dp = h\delta(x-x_2+x_0). \tag{16} \]

But this \(\varphi_x\) is an eigenfunction of the operator

\[ Q=x_2, \tag{17} \]

corresponding to the eigenvalue \(x+x_0\) of the coordinate of the second particle. Since

\[ PQ-QP=\frac{h}{2\pi i}, \tag{18} \]

we have shown that, generally speaking, a case is possible in which \(\psi_k\) and \(\varphi_n\) are eigenfunctions of two noncommuting operators corresponding to two physical quantities.

Let us now return to the general case, to which equations (7) and (8) refer. We shall suppose that \(\psi_k\) and \(\varphi_r\) are indeed eigenfunctions of certain noncommuting operators \(P\) and \(Q\), with \(\psi_k\) corresponding to the eigenvalue \(p_k\), and \(\varphi_r\) corresponding to the eigenvalue \(q_r\). In such a case, having measured \(A\) and \(B\), we shall be able to predict with certainty and without any disturbance whatsoever of the second system either the value of the quantity \(P\) (i.e. \(p_k\)) or the value of the quantity \(Q\) (i.e. \(q_r\)). According to our criterion of reality, in the first case we must regard the quantity \(P\) as an element of reality, and in the second case the element of reality will be the quantity \(Q\). But, as we have seen, both wave functions \(\psi_k\) and \(\varphi_r\) refer to one and the same reality.

Above we proved that either 1) the quantum-mechanical description of reality by means of the wave function is not complete, or 2) if the operators corresponding to two physical quantities do not commute, these two quantities cannot simultaneously possess reality. Proceeding then from the assumption that the wave function really does give a complete description of physical reality, we

came to the conclusion that two physical quantities with noncommuting operators can be real simultaneously. Thus the negation of 1 leads to the negation of the only remaining assumption 2. Consequently, we are forced to conclude that the quantum-mechanical description of physical reality by means of wave functions is not complete.

One might object to this conclusion on the grounds that our criterion of reality is not sufficiently restrictive. Indeed, we would not have arrived at our conclusion if we had insisted that two or more physical quantities may simultaneously be regarded as elements of reality only in the case where they can be simultaneously measured or predicted. From this point of view the quantities \(P\) and \(Q\) do not simultaneously possess reality, since one can predict either \(P\) or \(Q\), but not \(P\) and \(Q\) simultaneously. Here the reality of \(P\) and \(Q\) is made dependent on the process of measurement performed on the first system, although this process in no way affects the second system. No reasonable definition of reality, it would seem, should allow this.

Although we have shown that the wave function does not give a complete description of physical reality, we have left open the question whether such a description exists or not. We think, however, that such a theory is possible.

III. Niels Bohr’s Reply*; Copenhagen

In their recent article under the same title A. Einstein, B. Podolsky, and N. Rosen present arguments which led them to answer the question posed in the title in the negative sense. However, it seems to me that the general course of their reasoning does not fully correspond to the state of affairs with which we meet in atomic physics. I shall therefore gladly take the opportunity thus presented to explain in somewhat greater detail a general point of view which it is convenient to call “complementarity.” From this point of view, to which I have repeatedly already referred\(^5\), quantum mechanics, within the limits of its field of applicability, appears as a wholly rational description of those physical phenomena which we encounter in the study of atomic processes.

The question of within what limits an unambiguous meaning can be ascribed to such an expression as “physical reality” cannot, of course, be decided on the basis of a priori philosophical considerations. As the authors of the article in question themselves emphasize, in order to decide this question one must turn directly to experiments and measurements. To this end they propose a certain “criterion of reality,” formulated by them as follows: “If we can, without in any way disturbing the system, predict with certainty the value of some physical

* Niels Bohr, Phys. Rev., 48, 696, 1935. Translated by V. A. Fock.

magnitudes, then there exists an element of physical reality corresponding to this physical magnitude.” On an interesting example, to which we shall return, they then show the following. In quantum mechanics, just as in classical mechanics, the value of any variable may, under known conditions, be predicted on the basis of measurements made entirely on other systems that have previously interacted with the given system. Relying on their criterion, the authors therefore seek to ascribe an element of reality to each of the quantities represented by these variables. But, on the other hand, a characteristic feature of the existing mathematical formulation of quantum mechanics is, as is well known, that if we have two canonically conjugate quantities, then in describing the state of a mechanical system it is impossible to ascribe definite values to both of them. In view of this they regard the existing mathematical formulation as incomplete and express the conviction that a more satisfactory theory can be constructed.

However, argumentation of this kind is hardly suitable for undermining the reliability of the quantum-mechanical description based on a coherent mathematical theory, which automatically covers all cases of measurement similar to the one indicated.* The apparent contradiction in fact reveals only the essential inadequacy of the ordinary point of view of natural philosophy for describing physical phenomena of the type with which we have to deal in quantum mechanics. Indeed, 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 reaction

* In this respect the conclusions of the article cited may be regarded as direct consequences of the theorems on canonical transformations in quantum mechanics. These theorems ensure its mathematical completeness and rational correspondence with classical mechanics, perhaps to a greater extent than any other feature of this theory. Indeed, let us have a mechanical system consisting of two subsystems (1) and (2), which may interact with each other, but may also not interact. In describing a system of this kind it is always possible to replace any two pairs of canonically conjugate variables, referring respectively to (1) and to (2) and satisfying the usual commutation relations

\[ \left. \begin{aligned} [q_1 p_1] &= [q_2 p_2] = \frac{ih}{2\pi},\\ [q_1 q_2] &= [p_1 p_2] = [q_1 p_2] = [q_2 p_1] = 0, \end{aligned} \right\} \]

by two pairs of new canonically conjugate variables \((Q_1,P_1)\), \((Q_2,P_2)\), connected with the original ones by a simple orthogonal transformation corresponding to a rotation through an angle \(\theta\) in the planes \((q_1 q_2)\) and \((p_1 p_2)\), namely

\[ \left. \begin{aligned} q_1 &= Q_1 \cos\theta - Q_2 \sin\theta &\qquad p_1 &= P_1 \cos\theta - P_2 \sin\theta\\ q_2 &= Q_1 \sin\theta + Q_2 \cos\theta &\qquad p_2 &= P_1 \sin\theta + P_2 \cos\theta \end{aligned} \right\} \]

of the object on the measuring instrument (and this impossibility will necessarily occur if only the instrument fulfills its purpose)—the necessity of a final renunciation of the classical ideal of causality and a radical revision of our views on the problem of physical reality. As we shall see below, any criterion of reality similar to that proposed by the authors mentioned will—however cautious its formulation may seem—contain an essential ambiguity if we begin to apply it to the actual problems that interest us here. In order to give the arguments that we shall present in support of this proposition the greatest possible clarity, I shall first consider in some detail several simple examples of measuring arrangements.

Let us begin with the simple case of a particle passing through a slit in a diaphragm, which may form part of a more or less complicated experimental arrangement. Even if the momentum of this particle before its incidence on the diaphragm were completely known, the diffraction of a plane wave (which gives a symbolic representation of the state of the particle) by the edges of the slit will entail an uncertainty in the momentum of the particle after its passage through the diaphragm; moreover, this uncertainty will be the greater, the narrower the slit. But the width of the slit (at least if it is still large in comparison with the wavelength) may be taken as a measure of the uncertainty \(\Delta q\) in the position of the particle relative to the diaphragm in the direction perpendicular to the slit. Further, from the de Broglie relation between momentum and wavelength it is easy to see that the uncertainty \(\Delta p\) in the momentum of the particle in this direction is connected with \(\Delta q\) by Heisenberg’s relation

\[ \Delta p \Delta q \sim h, \]

which, as can be shown by using the mathematical apparatus of quantum mechanics, is an immediate consequence of the commutation relations for any pair of canonically conjugate

In view of the fact that these variables satisfy analogous commutation relations, in particular

\[ [Q_1 P_1] = \frac{ih}{2\pi}, \qquad [Q_1 P_2] = 0, \]

it is obvious that, in describing the state of a composite system, one cannot assign definite values to the quantities \(Q_1\) and \(P_1\) at the same time, but that they can be assigned to the quantities \(Q_1\) and \(P_2\). In this case, from the expressions for these variables in terms of \((q_1p_1)\) and \((q_2p_2)\), namely from

\[ Q_1 = q_1 \cos \theta + q_2 \sin \theta, \qquad P_2 = -p_1 \sin \theta + p_2 \cos \theta \]

it follows, further, that a subsequent measurement of one of the quantities \(q_2\) or \(p_2\) will allow us to predict in advance the value of \(q_1\) or respectively \(p_1\) (in the preceding formulas the square brackets \([AB]\) denote, as is customary, the expression \([AB]=AB-BA\). Ed. note).

variables. It is obvious that the uncertainty \(\Delta p\) is inseparably connected with the exchange of momentum between the particle and the diaphragm. In connection with this, a question of fundamental importance for our reasoning arises: to what extent can the momentum thus transferred be taken into account, to what extent can it be included in the description of the phenomenon that we study by means of the given experimental arrangement, the first stage of which may be regarded as the passage of the particle through the diaphragm.

In accordance with the usual arrangement of experiments on the remarkable phenomenon of electron diffraction, let us first suppose that our diaphragm, as well as the other parts of the apparatus—for example, a second diaphragm with several slits parallel to the first, and the photographic plate—are rigidly connected with the support that defines the spatial frame of reference. Then the momentum transferred by the particle to the diaphragm, and also to the other parts of the apparatus, will pass into their common support. Thus, in this case we consciously renounce any possibility of taking into account the reaction of the particle upon the separate parts of the apparatus and of including these reactions in our predictions concerning the final result of the experiment—for example, concerning the position of the spot that the particle produces on the photographic plate. The impossibility of a more detailed analysis of the interactions occurring between the particle and the measuring apparatus is, obviously, not a peculiarity of precisely this experimental arrangement, but constitutes an essential property of any arrangement suitable for the study of phenomena of the type under consideration, in which we encounter a distinctive feature of individuality entirely foreign to classical physics. Indeed, if we had any possibility of taking into account the momentum transferred by the particle to the separate parts of the apparatus, this would at once enable us to draw conclusions concerning the “course” of phenomena of this kind. For example, we could then indicate through exactly which slit in the second diaphragm the particle passed on its way to the photographic plate—and this cannot in any way be reconciled with the fact that the probability of the particle’s arriving at a given region of the surface of the plate is determined not by the presence of one or another slit taken separately, but by the arrangement of all the slits in the second diaphragm that can be reached by the wave associated with the particle after diffraction by the slit in the first diaphragm.

But we could make use of another experimental arrangement, in which the first diaphragm would no longer be rigidly connected with the remaining parts of the apparatus. In such an arrangement we would have, at least in principle,* the possibility of meas—

* The obvious impossibility of actually carrying out, with the experimental technique available, measuring procedures similar to those analyzed here and below, of course in no way undermines

determine with any desired accuracy the momentum of the diaphragm before and after the passage of the particle, and hence also to specify in advance the momentum of the latter after its passage through the slit. In fact, measurements of this kind presuppose only the possibility of an unambiguous application of the classical law of conservation of momentum; moreover, it must be applied, for example, to the collision process between the diaphragm and some test body, whose momentum is suitably controlled before and after the collision. True, such control will depend essentially on the study of the course, in space and time, of some such process to which the concepts of classical mechanics would be applicable; however, if all spatial dimensions and time intervals are taken sufficiently large, then this, obviously, is not connected with any limitations of accuracy in determining the momentum of the test bodies, but is connected only with a renunciation of exact control of their localization in space and time. The latter circumstance presents a complete analogy with that renunciation of taking into account the momentum of the fixed diaphragm which we encountered above in discussing the original arrangement. Such a renunciation is ultimately conditioned by the requirement of a purely classical description of the measuring apparatus; this requirement entails the necessity of introducing into the description of the action of the apparatus certain allowances corresponding to the uncertainty relations of quantum mechanics.

But the most essential difference between the two experimental arrangements we have considered consists in the following. In the arrangement which is suitable for measuring the momentum of the first diaphragm, we can no longer use this diaphragm as a measuring instrument and employ it for the same purpose as in the original arrangement. Since we are interested in the position of the diaphragm relative to the rest of the apparatus, we must already regard it, like the particle passing through the slit, as an object of investigation; this means that we must explicitly take into account the quantum-mechanical uncertainty relations for its position and momentum. In fact, even if we knew the position (relative to the spatial frame of reference, i.e. the stand) occupied by the diaphragm before the first measurement of its momentum, and even if we precisely established its position after the second measurement, nevertheless, in using the second arrangement, we lose the possibility of judging the position of the diaphragm at the moment when the particle passed through the slit; this will be because in every collision process of the diaphragm with a test—

validity of our theoretical reasoning. For these methods are essentially equivalent to the application of atomic processes, similar to the Compton phenomenon, for which the applicability of the law of conservation of momentum is well established.

bodies it undergoes a displacement that cannot be controlled. Therefore our entire arrangement in its second variant is obviously unsuitable for the study of those phenomena which were studied by means of its first variant. In particular, one may point out the following. Suppose that the momentum of the first diaphragm has been measured with an accuracy sufficient to judge whether or not the particle passed through some definite slit in the second diaphragm. In that case even the minimal uncertainty in the position of the first diaphragm, combined with the presence of this kind of information about its momentum, will erase the entire interference pattern that determines the disposition of those regions on the photographic plate where the particle may arrive. Meanwhile, the presence of several slits in the second diaphragm would necessarily have led to an interference effect of this kind if the relative disposition of all parts of the apparatus had been fixed.

Suppose that we are using an apparatus suitable for measuring the momentum of the first diaphragm. It is clear that even if we have measured this momentum before the particle passes through the slit, after this passage we have a free choice between two possibilities: namely, we may set ourselves the aim of finding out either the momentum of the particle, or its initial position with respect to the remaining part of the apparatus. In the first case it is enough for us to make one more determination of the momentum of the diaphragm, thereby depriving ourselves forever of the possibility of knowing its exact position at the time when the particle was passing through it. In the second case it is enough for us to determine the position of the diaphragm relative to the frame of reference, with which is associated the loss of the possibility of taking into account the momentum transmitted to the diaphragm by the particle. If the diaphragm has a sufficiently large mass in comparison with the mass of the particle, we may even arrange matters so that after the first determination of the diaphragm’s momentum it remains at rest in some unknown position relative to the other parts of the apparatus; then the subsequent fixation of the position may consist simply in establishing a rigid connection between the diaphragm and the support.

If I have repeated here these simple and essentially well-known considerations, I have been guided by the wish to emphasize the following. In the phenomena under consideration we are dealing not at all with some incomplete description, with an arbitrary snatching out of different elements of physical reality at the expense of other such elements, but with a rational distinction between essentially different experimental arrangements and measurement processes, of which some allow an unambiguous application of the concept of spatial localization, while others allow a legitimate application of the theorem on the conservation of momentum. If any arbitrariness remains, it relates only to our freedom of choice and use of various-

…measuring instruments, which is characteristic of the very concept of experiment. With each arrangement of an experiment there is associated the renunciation of one of the two sides in the description of physical phenomena; these two sides will here be, as it were, complementary to one another, whereas their combination characterizes the methods of classical physics. This renunciation is essentially due to the fact that, in the domain of quantum phenomena, it is impossible to take exact account of the reaction of the object on the measuring instruments, i.e. to take account of the transfer of momentum in the case of a measurement of position, and to take account of displacement in the case of a measurement of momentum. In this connection no comparisons and analogies between quantum mechanics and ordinary statistical mechanics will ever be able to convey the essence of the matter—however useful such analogies may be for the formal exposition of the theory. For in every experimental arrangement suitable for the study of properly quantum phenomena, we encounter not only ignorance of the values of certain physical quantities, but also the impossibility of giving these quantities an unambiguous definition.

The last remarks apply equally to the special problem mentioned above, which was considered by Einstein, Podolsky, and Rosen. This problem requires no more complicated reasoning than the simple examples that we have considered above. The special case of the quantum-mechanical state of two free particles, for which these authors give an explicit analytical expression, can be reproduced, at least in principle, by means of a simple experimental arrangement: this arrangement consists of a rigid diaphragm with two parallel slits, very narrow in comparison with the distance between them, and through each of these slits there passes, independently of the other, one particle with a previously measured momentum. If the momentum of this diaphragm is measured before and after the passage of the particles, then we shall indeed know, first, the sum of the components of the momenta of both particles in the direction perpendicular to the slits and, second, the difference of their initial coordinates measured in the same direction. At the same time the canonically conjugate quantities, i.e. the difference of the components of their momenta and the sum of their coordinates, will of course remain completely unknown.* With such an arrangement of the experiment it is clear that if one then makes a single measurement either of the position or of

* This description will evidently correspond, up to an inessential normalization factor, precisely to that transformation of variables which was given in one of the preceding remarks, where \((q_1\ p_1)\), \((q_2\ p_2)\) are to denote the coordinates and components of momentum of the two particles and the angle \(\theta\) is to be equal to \(-\frac{\pi}{4}\). Let us also note that the wave function given in formula (3) of the article cited above corresponds to the special case \(P_2 = 0\) and to the limiting case of two infinitely narrow slits.

of the momentum of one of the particles, then thereby the position, or respectively the momentum, of the other particle will automatically be determined with any desired accuracy; this will be so at least in the case where the wavelength corresponding to the free motion of each of the particles is sufficiently small in comparison with the width of the slits. As the authors mentioned have indicated, at this stage of the experiment we have the full possibility of freely choosing one or another variant of the experiment, depending on which of the quantities named we wish to determine, and in neither variant do we directly touch the particle in which we are interested.

That “freedom of choice” which this arrangement of the experiment affords us means precisely that we must stop at one of two different experimental manipulations, permitting the unambiguous application of one of two additional classical concepts—this is exactly the same as in the simple case considered above of a single particle that has passed through the slit of a diaphragm, where we could choose between the manipulations needed for predicting its position and its momentum. In fact, to measure the position of one of the particles means nothing other than to establish how it will behave with respect to some apparatus rigidly fastened to the support defining the spatial reference system. Under the experimental conditions described above, such a measurement also gives us knowledge of the position which our diaphragm occupied relative to this reference system after the particles had passed through the slits, whereas without such a measurement the position of the diaphragm remains completely unknown. It is obvious that only in this way do we obtain data allowing conclusions to be drawn about the initial position of the other particle with respect to the rest of the apparatus. But, having thereby allowed an essentially indeterminate transfer of momentum from the first particle to the aforementioned support, we have thereby deprived ourselves of any future possibility of applying the law of conservation of momentum to the system consisting of the diaphragm and both particles, and hence have lost the only basis that could have allowed us unambiguously to apply the concept of momentum to predictions concerning the behavior of the second particle. Conversely, if we wished to measure the momentum of one of the particles, we would lose, owing to the inevitable and in such a measurement unaccountable displacement, any possibility of judging from the behavior of this particle the position of the diaphragm relative to the rest of the apparatus, and would deprive ourselves of any basis for predictions concerning the localization of the other particle.

From our point of view we now see that the formulation of the above-mentioned criterion of physical reality, proposed by Einstein, Podolsky, and Rosen, contains an ambiguity

in the expression “without any disturbance whatever of the system.” Of course, in a case such as the one just considered, there is no question of the system under study being subjected, during the last critical stage of the measuring process, to any mechanical disturbance. But even at this stage what is involved is essentially a disturbance in the sense of an influence on the very conditions that determine the possible types of predictions of the future behavior of the system. Since these conditions constitute an essential element in the description of any phenomenon to which the term “physical reality” can be applied, we see that the argument of the authors mentioned does not justify their conclusion that the quantum-mechanical description is essentially incomplete. On the contrary, as follows from our preceding considerations, this description may be characterized as a rational use of all the possibilities of unambiguous interpretation of measurements compatible with the finite and uncontrollable interaction, characteristic of quantum phenomena, between the object and the measuring instruments. Indeed, it is only the mutual exclusion of any two experimental manipulations that would make it possible to give an unambiguous determination of two mutually complementary physical quantities—only this mutual exclusion that makes room for new physical laws, whose joint existence might at first sight seem to contradict the fundamental principles of the construction of science. It is precisely this completely new situation in the description of physical phenomena that we have tried to characterize by the term complementarity.

The experimental arrangements we have investigated so far are distinguished by a special simplicity in that, in the description of the phenomena considered, the concept of time plays a secondary role. True, we have repeatedly made use of such expressions as “before” and “after,” implying a connection in time; but in every such case one must keep in mind the corresponding inaccuracy. This inaccuracy will, however, be inessential so long as the intervals of time with which we are dealing are sufficiently large in comparison with those intrinsic periods that are connected with the given phenomenon and are revealed in a more detailed analysis of it. But as soon as we proceed to a more precise description of the course of quantum phenomena in time, we encounter the well-known new paradoxes, for whose clarification it is necessary to take into account further features of the interaction between objects and measuring instruments. Indeed, in phenomena of this kind we are dealing not with experimental arrangements in which all essential parts of the apparatus are at rest with respect to one another, but with arrangements containing moving parts, such as shutters opening and closing the slits of diaphragms, these parts being controlled by mechanisms playing the role of clocks. In addition to the transfer, already considered above, of momentum between the object and the bodies defining the spatial sy-

system of reference, when studying experimental arrangements of this kind we shall now have to investigate the possible exchange of energy between the object and these “clock” mechanisms.

An essential point in the reasoning concerning measurements of time in quantum mechanics is quite analogous to the argument that applies to measurements of position. Just as the transfer of momentum to individual parts of the apparatus, whose relative position must be known in order to describe the phenomenon, turns out, as we have seen, to be wholly beyond control, so too it is impossible to analyze the exchange of energy between the object and the various bodies whose relative motion must be known for the desired use of the apparatus. Indeed, the possibility of controlling the energy transmitted to the clock without disturbing its action as an indicator of time is excluded in principle. In fact, the use of clocks as indicators of time is entirely based on the presumed possibility of applying the methods of classical physics to the description of the action of each clock and of the means of checking them against other clocks. In this description we must obviously introduce into the energy balance an allowance corresponding to the quantum-mechanical uncertainty relations between the canonically conjugate variables—energy and time. In the final analysis, it is precisely this circumstance that entails the relation of complementarity between any detailed description of the course of atomic processes in time, on the one hand, and those properties of the internal stability of atoms, alien to classical mechanics, which were revealed in the study of the transfer of energy in atomic reactions, on the other. The situation here is exactly the same as in the question considered above concerning the mutually exclusive character of any unambiguous application to quantum phenomena of the concepts of position and momentum.

As we have seen, in every experimental arrangement it is necessary to draw a boundary between those parts of the physical system under consideration which we classify as measuring instruments and those which are the objects to be investigated. It may be said that the necessity of such a distinction constitutes the chief difference between the classical and the quantum-mechanical description of physical phenomena. True, within each measurement process we may draw this boundary, as we wish, at one place or another; the choice of its place is determined, both in the classical and in the quantum case, mainly by considerations of convenience. However, whereas in classical physics the choice of one place or another for the boundary between object and measuring instrument is not connected with any changes in the character of the description of the physical phenomena under study, in quantum theory it entails changes in this description. The fundamental importance of the distinction between object and apparatus in quantum theory is due, as we have seen, to the fact that, for the interpretation of all measurements in the proper sense, it is necessary

make use of classical representations, despite the fact that classical theory cannot by itself explain the new regularities with which we are dealing in atomic physics.

In view of this state of affairs, there can be no question of any other unambiguous interpretation of the symbols of quantum mechanics than that which is embodied in the well-known rules concerning the prediction of results obtained with the aid of a given experimental arrangement described in a purely classical way; these rules find their general expression in the theorems on canonical transformations mentioned above. By ensuring the proper correspondence of quantum theory with classical theory, these theorems exclude, in particular, any internal contradiction in the quantum-mechanical description which might arise in connection with changing the place where the boundary is drawn between the object and the measuring instrument. Indeed, an obvious consequence of the considerations adduced above is the following: for any arrangement of an experiment and any measuring manipulations, the choice of the place for this boundary is possible only within that region where the quantum-mechanical description of the process in question is essentially equivalent to the classical description.

In conclusion, I should like to note the enormous significance, for the question of physical reality in the domain of quantum theory, of the lesson taught by the general theory of relativity. Indeed, despite all the characteristic differences, there is a striking analogy between the state of affairs in both generalizations of classical theory, an analogy that has been noted more than once. In particular, the special position, just discussed by us, occupied in the description of quantum phenomena by measuring instruments, presents a close analogy with the necessity, in the theory of relativity as well, of using the ordinary description of all measuring processes, including the sharp separation into space and time, although this necessity obtains despite the fact that the very essence of the theory of relativity is the establishment of new physical laws of such a kind that, in order to understand them, we must abandon the customary separation of the concepts of space and time*. The characteristic

* It is precisely this circumstance, together with the relativistic invariance of the quantum-mechanical uncertainty relations, that guarantees for us the compatibility of the arguments set forth in the present article with all the requirements of the theory of relativity. This question will be considered in greater detail in a work now being prepared for publication, where the author, in particular, will examine a very interesting paradox put forward by Einstein and relating to the application of the theory of gravitation to measurements of energy; the resolution of this paradox provides a particularly instructive illustration of the generality of arguments based on the concept of complementarity. In the same work, space-time measurements in quantum theory will be discussed in greater detail, and all the necessary mathematical calculations and diagrams of experimental arrangements will be given—in short, everything that has been omitted in the present article, where our chief attention was directed to the dialectical side of the question posed in the title.

for the theory of relativity, the dependence of all readings of scales and clocks on the adopted reference system may, furthermore, be compared with the uncontrollable exchange of momentum and energy between the objects measured and all the instruments that define the space-time reference system, which in quantum theory brings us to a state of affairs characterized by the concept of complementarity. Indeed, this new feature of natural philosophy signifies a radical revision of our views of physical reality, which may be set in parallel with that fundamental alteration of all notions of the absolute character of physical phenomena which was brought about by the general theory of relativity.

References

1. See, for example, W. Pauli, Die allgemeinen Prinzipien der Wellenmechanik, Handb. d. Physik. Geiger u. Scheel, Berlin 1933.

2. E. Schrödinger, Proc. Cambr. Phil. Soc., 31, 555, 1935.

3. W. Furry, Phys. Rev., 31, 393, 1935.

4. See, for example, J. v. Neumann, Mathematische Grundlagen der Quantenmechanik, Springer, Berlin, 1932.

5. Cf. N. Bohr, Atomic Theory and Description of Nature, 1 (Cambridge, 1934).