57

Can QIantum-Mechanical Description of Physical Reality Be Considered Complete?

Nathan Rosen

Introduction

In 1935 there appeared a paper with the above title (Einstein, Podolsky and Rosen 1935) in The Physical Review. This paper was the outgrowth of a number of discussions held by Albert Einstein, Boris Podolsky and myself at the Institute for Advanced Study in Princeton. The purpose of the discus­sions was to help us understand the concepts and principles of quantum mechanics, and what we thought we understood troubled us. For the con­clusion we reached in these discussions was that the answer to the above question is, "No'.

The paper aroused considerable controversy among physicists at the time. Now, more than forty years later, discussion is still going on. It seems therefore appropriate on the occasion of the centennial of the birth of Albert Einstein to go back to this paper and re-examine it from the perspective of the present time.

The next section presents a detailed review of the above paper (to be referred to hereafter simply as "the paper") together with some critical remarks. This is followed by a section presenting a contrasting point of view, that of Bohr, and then a section of discussion.

II The Paper

The paper begins with the statement:

Any serious consideration of a physical theory must take into account the distinc­tion between the objective reality, which is independent of any theory, and the physical concepts with which the theory operates. These concepts are intended to correspond with the objective reality, and by means of these concepts we picture this reality to ourselves.

Obviously, it is tacitly assumed, as most physicists believe, that there exists an objective reality, a physical world independent of the human ob­server, and that a physical theory describes some aspects of this reality, thereby enabling us to form some sort of picture of it.

P. C. Aichelburg et al. (eds.), Albert Einstein© Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig 1979

58 Nathan Rosen

In attempting to judge 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?" It is only in the case in which positive answers may be given to both of these questions, that the concepts of the theory may be said to be satisfactory. The correctness of the theory is judged by the degree of agreement between the conclusions of the theory and human experience. This experience, which alone enables us to make inferences about reality, in physics takes the form of experiment and measurement. It is the second ques­tion that we wish to consider here, as applied to quantum mechanics.

Nowadays one sometimes hears it said that all that one wants of a theory is that it should be correct, i. e., that it should enable one to carry out calcula­tions so as to obtain numbers that agree with the results of experiments, and that it is not necessary for the theory to provide us with any picture of the reality. However, it seems to me that most physicists want such a picture, and for them the second question, that of completeness, is important.

Whatever the meaning assigned to the term complete, the following requirement for a complete theory seems to be a necessary one: every element of the physical reality must have a counterpart in the physical theory. We shall call this the condition of com­pleteness. The second question is thus easily answered, as soon as we are able to decide what are the elements of the physical reality.

It might be remarked that, in addition to deciding what is an element of the physical reality, one also has to decide what is its counterpart in the theory. It is taken as self-evident here that the corresponding concept and the numerical value associated with it should appear in the theory.

The elements of the physical reality cannot be determined by a priori philosophical considerations, but must be found by an appeal to results of experiments and measure­ments. A comprehensive definition of reality is, however, unnecessary for our purpose. We shall be satisfied with the following criterion, which we regard as reasonable. If, without in any way disturbing a system, we can predict with certainty (i. e., with prob­ability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity. It seems to us that this criterion, while far from exhausting all possible ways of recognizing a physical reality, at least provides us with one such way, whenever the conditions set down in it occur. Regarded not as a necessary, but merely as a sufficient, condition of reality, this criterion is in agreement with classical as well as quantum-mechanical ideas of reality.

This criterion is of crucial importance to the discussion. The key point in it is: "without in any way disturbing the system". This will be considered below.

To illustrate the ideas involved the paper considers the quantum­mechanical description of the behavior of a particle having one degree of free­dom. Reference is made to the concept of state, completely characterized by the wave function l/J, to the correspondence between each physically observ­able quantity A and an operator (also denoted by A), and to the eigenfunc-

Quantum-Mechanical Description of Physical Reality 59

tions and eigenvalues of this operator. It is recalled that, if l/J is an eigenfunc­tion of the operator A, 1. e.,

A l/J = a '1/1, (1)

where a is a number, then the observable A has with certainty the value a whenever the particle is in the state given by '1/1.

In accordance with our criterion of reality, for a particle in the state given by t/J for which Eq. (1) holds, there is an element of physical reality corresponding to the physical quantity A.

As an example, the state is considered for which '1/1 is given by

'1/1 = e(2rrilh)pox, (2)

where Po is a constant and x the independent variable. With the momentum operator P = (h/21f i) a/ax, one sees that l/J is a momentum eigenfunction with eigenvalue Po.

Thus, in the state given by Eq. (2), the momentum has certainly the value Po. It thus has meaning to say that the momentum of the particle in the state given by Eq. (2)

is real.

On the other hand, if one considers the particle coordinate, for which the operator q is that of multiplication by x, Eq. (1) does not hold in this case, and one cannot say that the coordinate has a particular value. In accord­ance with quantum mechanics one can talk only about probabilities of various values, and in the case of the state given by Eq. (2), it is shown that all values of the coordinate are equally probable.

A definite value of the coordinate, for a particle in the state given by Eq. (2), is thus not predictable, but may be obtained only by a direct measurement. Such a measure­ment however disturbs the particle and thus alters its state. After the coordinate is determined, the particle will no longer be in the state given by Eq. (2). The usual conclu­sion from this in quantum mechanics is that when the momentum of a particle is known, its coordinate has no physical reality.

More generally, it is shown in quantum mechanics that, if the operators correspond­ing to two physical quantities, say A and B, do not commute, that is, if AB f. BA, then the precise knowledge of one of them precludes such a knowledge of the other. Further­more, any attempt to determine the latter experimentally will alter the state of the system in such a way as to destroy the knowledge of the first.

From this follows that either (1) the quantum-mechanical description of reality given by the wave functions is not complete or (2) when the operators corresponding to two physical quantities do not commute the two quantities cannot have simultaneous reality. For if both of them had simultaneous reality - and thus definite values - these values would enter into the complete description, according to the condition of com­pleteness. If then the wave function provided such a complete description of reality, it

60 Nathan Rosen

would contain these values; these would then be predictable. This not being the case, we are left with the alternatives stated.

In quantum mechanics it is usually assumed that we wave function does contain a complete description of the physical reality of the system in the state to which it cor­responds. At first sight this assumption is entirely reasonable, for the information obtain­able from a wave function seems to correspond exactly to what can be measured without altering the state of the system. We shall show, however, that this assumption, together with the criterion of reality given above, leads to a contradiction.

For this purpose the paper considers the case of two systems I and II, which interact from the time t = 0 to t = T, after which there is no longer any interaction between them. From the states of the systems for t < 0 one can calculate with the help of the Schrodinger equation the wave function t/J of the combined system I + II at a time t > T. This does not enable one to calculate the states of the individual systems after the interaction. That can be done only with the help of further measurements by a process known as the reduction of the wave packet, as follows:

Let A be some physical quantity pertaining to system I with eigen­functions Un (Xl) and corresponding eigenvalues an, where Xl stands for the variables describing this system. (It is tacitly assumed that there is no degener­acy.) Then t/J can be expanded in a series of the orthogonal functions Un (Xl),

00

t/J(X I ,X2)= L t/Jn(X 2)Un(X I ), (3) n=l

where X2 stands for the variables describing system II and t/Jn (x2) are merely the coefficients of the expansion. If the quantity A is measured and is found to have the value ak, it is concluded that I is left in the state given by the wave function Uk (Xl) and II in the state given by t/Jk (X2)' Thus the wave packet given by the infinite series (3) is reduced to a single term t/Jk (X2) Uk (Xl)' This is the process of the reduction of the wave packet.

Some additional remarks are appropriate at this point. The assumption that the measurement results in the reduction of the wave packet is some­times referred to as the projection postulate. The validity of this assumption depends on the nature of the measurement process. Suppose that we carry out on a system a measurement of a physical quantity A and obtain the value a (which must be one of its eigenvalues, according to quantum mechan­ics). It may be that we are dealing here with a reproducible measurement, i. e., such that an immediate repetition of the measurement is certain to give the same result as before. In that case we can conclude that, after the first measurement, the system is left in a state given by t/J satisfying Eq. (1) (in order to account for the certainty of obtaining a by another measurement). The change from the original state of the system to that given by the eigen­function t/J represents a reduction of the wave packet. On the other hand, one may have a measurement that is not reproducible, in which case one

Quantum-Mechanical Description of Physical Reality 61

cannot draw this conclusion. (We can think of such a measurement as one which leaves the physical quantity with a value different from that given by the measurement.)

One can raise the question: is it always possible to choose a repro­ducible measurement in order to determine the value of a given observable? For our purpose, in view of the discussion below, it is enough to consider measurements of the position and momentum of a particle. In these cases it seems clear that reproducible measurements are possible, in principle.

In the paper it is tacitly assumed that the measurements are repro­ducible and hence the reductions of the wave packets take place. For the state given by Eq. (3) this means that, if the measurement of A gives the value ak, then system I is left in the state with the wave function Uk(XI)' Since nothing was done to system II, the coefficient 1/!k(X2) is unaltered (except possibly for a normalization factor arising from probability consider­ations).

If, instead of A, we had taken another quantity B having eigenfunctions Us (Xl) and eigenvalues bs, we should have obtained, instead of Eq. (3), the expansIOn

00

(4)

with new coefficients ¢S(X2)' If B is now measured and found to have the value br , one concludes that system I is left in the state given by Ur(XI) and II in that given by ¢r (X2)'

We see therefore that, as a consequence of two different measurements performed upon the first system, the second system may be left in states with two different wave functions. On the other hand, since at the time of measurement the two systems no longer interact, no real change can take place in the second system in consequence of anything that may be done to the first system. This is, of course, merely a statement of what is meant by the absence of an interaction between the two systems. Thus, it is possible to assign two different wave functions (in our example IPk and if>r) to the same

reality (the second system after the interaction with the first).

Now it may happen that the two wave functions of system II, 1/lk and ¢r, are eigenfunctions of noncommuting operators corresponding to some physical quantities P and Q, respectively, as can be seen from the following example:

Suppose that the two systems are two particles and that

00

(5)

- 00

62 Nathan Rosen

where x 0 is a constant. Let A be the momentum of particle I so that, accord­ing to Eq. (2), its eigenfunctions are

up (Xl) = e(2nilh)px l, (6)

corresponding to the eigenvalue p. Since we now have a continuous spectrum, let us write Eq. (3) in the form

00

-00

with

l/Ip(X2) = e(2nilh) (XO-X2)P.

This however is the eigenfunction of

P = (bl2ni) a/ax2,

corresponding to the eigenvalue - p of the momentum of particle II. Now let B be the coordinate of particle I, with eigenfunctions

VX(XI)=O(XI-X),

(7)

(8)

(9)

(10)

corresponding to the eigenvalue x, where 0 (Xl - X) is the Dirac delta-func­tion. Then Eq. (4) becomes

00

-00

where

00

-00

This however is the eigenfunction of the operator

Q = X2,

(11)

(12)

(13)

corresponding to the eigenvalue X + Xo of the coordinate of particle II. Since

PQ - QP = hl2ni, (14)

we see that it is possible for l/I k and ¢r to be eigenfunctions of two non­commuting operators, corresponding to physical quantities.

Another example, and a very interesting one, was later given by Bohm and Aharonov (1957). They considered the case in which the system I and II are two particles, each having a spin ~ (in units of bI27T), while the state of the combined system given by l/I corresponds to a total spin o. In this

Quantum-Mechanical Description of Physical Reality 63

case, if one measures the component of the spin of I in any direction, the corresponding component of II must be equal to the measured value, but of opposite sign. Hence, by measuring the spin of I in the x or the y direction, one can determine the spin of II in the x or y direction. However, according to quantum mechanics, the spins in these directions do not commute. Hence one has a situation analogous to that in the previous example.

Returning now to the general case contemplated in Eqs. (3) and (4), we assume that Wk and rfJr are indeed eigenfunctions of some noncommuting operators P and Q, corresponding to the eigenvalue Pk and qr, respectively. Thus, by measuring either A or B we are in a position to predict with certainty, and without in any way disturbing the second system, either the value of the quantity P (that is Pk) or the value of the quantity Q (that IS qrl. In accordance with our criterion of reality, in the first case we must consider the quantity P as being an element of reality, in the second case the quan­tity Q is an element of reality. But, as we have seen, both wave functions Wk and rfJr

belong to the same reality. Previously we proved that either (1) the quantum-mechanical description of reality

given by the wave function is not complete or (2) when the operators corresponding to two physical quantities do not commute the two quantities cannot have simultaneous reality. Starting then with the assumption that the wave function does give a complete description of the physical reality, we arrived at the conclusion that two physical quan­tities, with noncommuting operators, can have simultaneous reality. Thus the negation of (1) leads to the negation of the only other alternative (2). We are thus forced to

conclude that the quantum-mechanical description of physical reality given by wave functions is not complete.

One could object to this conclusion on the grounds that our criterion of reality is not sufficiently restrictive. Indeed, one would not arrive at our conclusion if one insisted that two or more physical quantities can be regarded as simultaneous elements of reality only when they can be simultaneously measured or predicted. On this point of view, since either one of the other, but not both simultaneously, of the quantities P and Q can be predicted, they are not simultaneously real. This makes the reality of P and Q depend upon the process of measurement carried out on the first system, which does not disturb the second system in any way. No reasonable definition of reality could be expected to permit this.

The paper ends with the remark:

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

III Bohr's Reply

The appearance of the paper was followed by a spate of replies attempt­ing to refute its conclusions. Of all of them one will be singled out for con­sideration here, that by Niels Bohr (1936). The justification for this choice

64 Nathan Rosen

lies in the fact that Bohr presented the "orthodox" Copenhagen interpreta­tion of quantum mechanics, which he, more than anyone else, had developed and which is accepted today by most of the workers in this field. The juxta­position of the ideas of the two articles should help to understand the differ­ences in the Weltanschauung of these two great scientists, Einstein and Bohr.

Bohr disagreed strongly with the paper and, in particular, with the criterion of reality. According to him, the conclusion indicates the "inade­quacy of the customary viewpoint of natural philosophy for a rational account of physical phenomena of the type with which we are concerned in quantum mechanics", and the criterion of reality contains an essential ambiguity when applied to the problems under discussion.

Let us look briefly at that part of Bohr's work dealing directly with the paper. He begins by considering the simple case of a particle passing through a slit in a diaphragm. On the one hand, one can choose an experimental arrangement in which the diaphragm is rigidly attached to the support so that the slit is at a fixed position. On the other hand, one can choose an experimental arrangement in which the diaphragm moves freely and it is possible to determine the transfer of momentum from the particle to the diaphragm. In the first case the slit defines the position of the particle just after it has passed through, but there is an uncontrollable transfer of momen­tum between particle and diaphragm so that one is ignorant of the particle's momentum. In the second case, if one knew the particle's momentum pre­viously, one knows its momentum after it has passed through, but its posi­tion is unkown since one does not know the position of the slit on the (mov­ing) diaphragm at the moment the particle passes through. The two experi­mental arrangements, which can be thought of as suitable for predicting the position or the momentum of the particle which has just passed through the slit, are mutually exclusive and allow the use of complementary classical concepts (e.g., a coordinate or a momentum) which are mutually exclusive according to Bohr's complementarity principle.

Bohr's purpose in discussing this situation is to emphasize that one is not dealing with an incomplete description in which one is ignorant of cer­tain quantities, but rather that in each experimental arrangement one is faced with the impossibility of defining certain quantities in an unambiguous way.

Bohr then considers the problem of the two particles discussed in the paper. According to him the situation is not very different from that of the single particle above. In principle, one can imagine that the two particles go through two slits in a diaphragm with an experimental arrangement for deter­mining the momentum transferred from the particles to the diaphragm. The distance between the slits gives x 2 - X I just after they have passed through. Knowing the momenta of the particles before they reach the slits, one can determine their total momentum PI + P2 after passing through. On the basis of the commutation relations between operators corresponding to conjugate

Quantum-Mechanical Description of Physical Reality 65

variables one sees that these two operators commute, so that we can have a state of the combined system I + II which is an eigenstate of both of them. If we now have an additional experimental arrangement for measuring x I,

then we can determine x 2; if we have one for measuring PI, we can deter­mine P2' This is the situation in the paper.

However, Bohr interprets this situation differently. The measurement process used to determine x 2 prevents one from determining P2 and vice versa. According to Bohr, the criterion of physical reality used in the paper contains an ambiguity as regards the meaning of the expression, "without in any way disturbing a system". In the case considered there is no mechani­cal disturbance of system II during the last critical stage of the measuring procedure (when one measures either XI or PI)' But there is "an influence on the very conditions which define the possible types of predictions regard­ing the future behavior of the system. Since these conditions constitute an inherent element of the description of any phenomenon to which the term 'physical reality' can be properly attached, we see that the argumentation of the mentioned authors does not justify their conclusion that quantum­mechanical description is essentially incomplete". On the contrary, as he sees it, this description "may be characterized as a rational utilization of all possi­bilities of unambiguous interpretation of measurements compatible with the finite and uncontrollable interaction between the objects and the measuring instruments in the field of quantum theory". Therefore, according to Bohr, the quantum-mechanical description of physical reality is complete.

IV Discussion

So what is the answer to the question: can quantum-mechanical descrip­tion of physical reality be considered complete?

It is clear that this is not a question that can be anwered in an opera­tional way, by means of experiments and measurements. The answer depends on how one defines the elements of physical reality or, more generally, how one views physical reality. It appears then that, ultimately, it is a question of what one believes.

Einstein believed in the existence of an objective reality, independent of the observer. With the help of measurements one can get information about this reality, but it exists independently of these measurements (pro­vided the measurements do not distrub or change it), and it would continue to exist in the absence of human observes. Starting from this standpoint and applying the criterion for an element of physical reality, the paper arrives at the conclusion that the quantum-mechanical description is incomplete.

Bohr, on the other hand, viewed reality differently. According to him, the elements of reality in a given system are determined by the experimental arrangements that are set up to investigate the system. For the experimental

66 Nathan Rosen

arrangements determine the possible results of the measurements to be car­ried out and, one might say, they therefore mold the reality into a form that corresponds to these possible results. If one has set up an apparatus to measure the position of a particle, this position is an element of reality while the momentum is not, and vice versa. According to Bohr, the quantum­mechanical description is complete because it corresponds exactly to what it is possible to determine in a given situation, i. e., with a given experimental arrangement. What it seems to amount to is that the description of reality by quantum mechanics is complete because reality is whatever quantum mechanics is capable of describing.

When Bohr considered the influence of the experimental arrangement on the physical reality, he did not distinguish between the case in which a direct measurement is carried out on the system of interest and the case in which one carries out a measurement on one system (I) to get information about another system (II). According to Einstein the two cases are quite different; only the second case can fulfill the condition of the reality criter­ion, "without in any way disturbing a system". To him a disturbance meant a physical interaction with another system, not just the presence of some measuring instrument at a distance.

Let us go back to the paper. If one accepts its conclusion (and there are some who do), one can raise the question: what can be done to get a complete description of reality? One possibility is to retain the present form of quantum mechanics, which has proved to be so successful in giving agree­ment with observation, but to supplement the information provided by the wave function with additional information given by other quantities, often referred to as hidden variables or hidden parameters (Belinfante 1973). These hidden variables, if known, would help to give a complete description of reality. For example, in the case of the two systems considered in the paper, a knowledge of the hidden variables would tell us which term l/Ik (x z) Uk (x 1)

in Eq. (3) and which term CPr (Xz) Vr (x 1) in Eq. (4) would be obtained if A or B were measured. However, these hidden variables are not known, and this is what gives a statistical character to quantum mechanics.

It appears, however, that matters are not so simple. The hidden para­meters, if they exist, give correlations between the results of measurements carried out on system I and II after they have ceased to interact. If one assumes the principle of "locality", i.e., that the result of a measurement on one system does not depend on what is being done to the other system, then it is found that the correlations determined by the hidden parameters lead to certain statistical relations, such as Bell's inequalities (Bell 1971). These relations can be different from those given by quantum mechanics on the basis of correlations described by the wave function t/I in Eqs. (3) and (4). However, experiments carried out in recent years seem to confirm the predictions of quantum mechanics (Clauser and Shimoni 1978). It appe­ars therefore that, if one wants to have hidden parameters, one must also

Quantum-Mechanical Description of Physical Reality 67

have some sort of non-local interaction between the systems, i. e., an inter­action at a distance, which causes one system to be influenced by a measure­ment carried out on the other. Most physicists would find this idea rather unattractive.

Let us now turn to the last paragraph of the paper. What are the prospects of finding a satisfactory theory that will give a complete description of re­ality? One must not be overly optimistic. It appears that such a theory will not be obtained by some simple modification of quantum mechanics, such as the addition of hidden variables. If someday quantum mechanics is re­placed by another theory, this is likely to involve revolutionary changes in concepts and principles - perhaps even changes in our concepts of space and time. In that case it may even turn out that the question posed by the paper - is the description of physical reality complete? - no longer has a mean­ing, or that it has to be given a different interpretation. The consequences of a revolution in physics are hard to foresee.

References

Belinfante, F.]. 1973. A Survey of Hidden-Variables Theories, Pergamon Press, Oxford. Bell,]. S. 1971. Foundations of Quantum Mechanics, Proc. of Int. School of Physics

"Enrico Fermi", B. d'Espagnat, ed., Course 49, p. 171, Academic Press, N.Y. Bohm, D. and Aharonov, Y. 1957. Phys. Rev. 108, 1070. Bohr, N. 1936. Phys. Rev. 48, 696. Clauser,]. F. and Shimoni, A. 1978. Reports on Progress in Physics (to appear). Einstein, A., Podolsky, B. and Rosen, N. 1935. Phys. Rev. 47, 777.