Foundations of Physics, Vol. 11, Nos. 7/8, 1981

Nonlocality in Quantum Theory Understood in Terms of Einstein's Nonlinear Field Approach 1 D. Bohm 2 and B. J. Hiley 2 Received September 24, 1980 We discuss Einstein's ideas on the need for a theory that is both objective and local and also his suggestion for realizing such a theory through nonlinear field equations. We go on to analyze the nonIocality implied by the quantum theory, especially in terms of the experiment of Einstein, Podolsky, and Rosen. We then suggest an objective local fietd model along Einstein's lines, which might explain quantum nonlocatity as a coordination of the properties of pulse-like solutions of the nonlinear equations that would represent particles. Finally, we discuss the implications of our model for Bell's inequality.

1. I N T R O D U C T I O N It is well known that Einstein did not accept the fundamental and irreducible indeterminism of the usual interpretation of the quantum theory, e.g., as revealed in his statement(2): "God does not play dice." However what is much more important is his rejection of another fundamental and irreducible feature of the quantum theory, i.e., nonlocality. Indeed it was implicit in his entire world view that connections between any elements whatsoever had to be local. That is, they could take place either when such elements were in contact at the same point in space-time, or else, they could be propagated continuously, across infinitesimal distances by the actions of fields. He regarded the failure of quantum mechanics to fit in with this notion of locality as a fundamental criticism of the whole structure of quantum mechanics, which indicated the need for developing a basically new kind of theory, making possible a deeper concept of reality that would ultimately This article is an extension and modification of a previouslypublished article. (See Ref. 1.) z Department of Physics, Birkbeck College, University of London, London, England. 529 0015-9018/81/0800-0529503.00/0 © I98I Plenum Publishing Corporation

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permit us to properly understand this feature of nonlocality in terms of an underlying local field structure. We feel that Einstein was right to ask for a deeper concept of reality, though we do not agree with his insistence that the basic elements of such a theory could only be local. Instead, we regard this question as open and indeed, we have elsewhere ~3~ explored specific possibilities for nonlocal theories. However, it is not our intention to discuss our views on the subject in this paper. Instead, we shall indicate an alternative set of ideas that may open the possibility of explaining quantum-mechanical nonlocality along Einstein's lines, i.e., in terms of a local field with local connections, obeying nonlinear equations. As we shall see such ideas may point to some radically new concepts that could help us towards a better understanding of the universe.

2. EINSTEIN'S VIEWS ON LOCALITY Let us begin with a brief discussion of why Einstein's notion of tocality is essential to the theory of relativity. Now, by contrast, the previous nonrelativistic theory did not require locality. Indeed, one of the basic notions of nonrelativistic mechanics was that of the extended rigid body, which would instantly respond as a whole (i.e., nonlocally) if a force were applied to any part. We illustrate this feature in the space-time diagram in Fig. 1. The world lines of the boundaries of an object at rest are given by A'A and B'B. A force, F, is applied at A, and the entire object accelerates as a single rigid body, so that the subsequent world lines of its boundaries are AA" and BB". The force, F, is thus transmitted instantaneously from A to B. But, as is well known, relativity theory implies that such an instantaneous transmission of force across a finite distance would be inconsistent with the

t

/A"

A[

F ....

B"

B

/

A"

Fig.

1.

Instantaneous

x response

body to an impulse F.

of a rigid

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x

Fig. 2. World line of an extensionless point object. principle of causality, It follows that the concept of the extended rigid body cannot be used in relativity theory. Nor can we regard such a body as made of smaller objects (e.g., atoms) that are rigid, since the same problem of infinite speed of transmission of force would arise with these. Ultimately, we would have to suppose that the particle was a mathematical point, P, of no extension at all (i.e., entirely local). Its world line is illustrated in Fig. 2. However, it is necessary, in general, that particles be considered as sources of fields. As soon as we do this we discover that the field at such a point is infinite. This leads to various inconsistencies, such as an infinite mass for the particle. As Dirac ~4) has shown (in a classical model of the electron) this infinity can be consistently "subtracted off," but then this leads to other incorrect features for the motion of the particle, such as instability (i.e., self-acceleration). So it appears that relativity denies the extended rigid body, and yet must also reject the point particle which appears to be a natural alternative to this? Einstein hoped that a way out of this dilemma could be found through the general theory of relativity. This theory begins, indeed, with a further extension of the concept of locality. Thus, the principle of equivalence of inertial and gravitational masses is interpreted through a curvilinear transformation of space-time coordinates, which can "transform away" the gravitational field in a given infinitesimal region, thus demonstrating a connection between the properties of inertia and the local gravitational field. Einstein then deduced a set of nonlinear differential equations for these fields which were, of course, also local. It was Einstein's further new idea that he could explain the extended object (e.g., a particle) as a feature of such a field. In particular he supposed 3 Quantum mechanically, similar infinities arise, and these can be also "subtracted off" to give what is called a renormalized theory. But it is not clear what this means with regard to the local and nonlocal character of the electron, especially since the very process of renormalization is, in itself, a nonlocal transformation of the "quantum state of the universe."

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Z_

p,

×

Fig. 3. Einstein's view of an extended object in a fieLd-theoretic view.

that there could exist a singularity of this field, or at least a field function ~, which was large only in a small region, as illustrated in Fig. 3. The maximum field would be at A, and the field would quickly fall to small values at appreciable distances from A. If the field equations had been linear, such a field would have been unstable, and indeed would have spread out very quickly. But it was well known that nonlinear equations could in principle admit of such pulse-like solutions that were stable, even allowing the field pulse to remain together as it moved as a whole through space, with a certain overall momentum. Einstein took such a pulse as a model of a particle. In this theory, the basic concept is that of a field, with entirely local properties. A particle is regarded as a nonlocal form in this field. But this nonlocal form plays no fundamental part in the theory. It is an abstraction, a shadow, so to speak, that has to move according to the laws of the local field on which this form lies. Nevertheless, it is important to note that highly nonlocal features of particles may thus be comprehended in terms of local field entities. Consider, for example, two particles as shown in Fig. 4. Actually, the fields of these bodies never fall to zero, so that in some sense, the two particles merge, in one continuous field that has two maxima.

A

B

×

Fig. 4. Representation of two particles illustrating their connection through the continuous field.

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The separate "particles," A and B, are thus abstractions, that may be generally useful, when the pulses are sufficiently separated. But in a fundamental sense there is no localizable particle at all. Instead, Einstein describes the universe as an undivided whole. Nevertheless, he constructs it as a set of extended forms lying on entirely local field elements, obeying laws based only on local connections of these elements. It has to be added, however, that Einstein was well aware that the actual stability of matter (e.g., stationary states of atoms) must be closely related to the laws of quantum theory. Therefore, it would be of no use to postulate nonlinear equations that gave structures whose size and general behavior were unrelated to these laws. Indeed he hoped that by including some sort of random fluctuations in his field quantities, he could simultaneously comprehend relativistic and quantum laws. In this connection, one should note that the present quantum theory implies that even in empty space, any field undergoes certain irreducible "zero-point" fluctuations, which do not directly show up in the movement of matter or light. Current calculations lead to an infinite value for this energy, but one may surmise that some factor not clearly evident in current theories will limit this to a finite, but rather large value. Einstein's idea might, thus, take the form of supposing that empty space contains a random background of fluctuating fields obeying nonlinear equations, and that with the aid of this, the statistical feature of the quantum theory (as well as other features) could ultimately be explained. During the latter part of his life, Einstein made a serious and sustained attempt to realize such a program. In doing this he was hampered by the fact that his nonlinear field equations were too difficult to solve by any known method. On the whole, what he was able to achieve was not very promising particularly with regard to the ultimate goal of incorporating quantum laws in his nonlinear field equations. At best, one can say that his results were inconclusive, but most physicists who work in the field seem to regard such a goal as unrealizable. To sum up Einstein's views, then, we repeat that Einstein took locality as an absolutely inevitable requirement for any reasonable physical theory. Thus, in commenting on the possibility of nonlocality in quantum theory he said: "Quantum theory cannot be reconciled with the idea that physics should represent a reality in time and space, free from spooky action at a distance. ''(5) This way of thinking was certainly required for consistency with relativity theory. But Einstein did not primarily regard the need for locality as an inference from relativity theory. Instead, he felt that locality was so self-evident that he would regard the feature of nonlocality in a theory as evidence that it was either incorrect, or at the very best, a fragmentary

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abstraction from a more nearly correct theory that would be local (e.g., as his proposed nonlinear field theory, nonlocal features of extended particles are explained as forms on a deeper local field).

3. N O N L O C A L I T Y AS I M P L I E D BY Q U A N T U M T H E O R Y Let us now go on to consider the quantum theory. Here one encounters nonlocal features even at a very elementary level. Thus, if electrons are diffracted by a pair of slits, one finds that when two slits are open, electrons fail to arrive at certain points where they actually can arrive when only one slit is open. If the electron is thought of as a particle which either goes through one slit or the other, this experiment indicates that its behavior depends on conditions at the second slit, which is far away. This already is at least some form of nonlocality. On the other hand, the electron might be thought of as an extended entity that goes through both slits together. But this would be a more extreme form of nonlocality, in the sense that it would deny that the electron occupies a definite region of space, even when the region is defined only within microscopic orders of magnitude. A much more deeply penetrating kind of nonlocality in the quantum theory is demonstrated by the well-known experiment of Einstein, Podolsky, and Rosen. (6) We shall discuss this experiment briefly in terms of the measurement of spin variables as indicated in Fig. 5. Let us begin with a molecule, M, of total spin zero, consisting of two atoms, A and B, each having spin, h/2. Let this atom be disintegrated by some means (e.g., purely electrostatic forces) that do not affect the spin variables in any significant way. The atoms start to separate, after which they cease to interact. To make the implications more clearly, let us imagine that the experiment is done in interstellar space, leaving room to allow the atoms to separate by many miles, before they are measured, so that a considerable amount of time (perhaps a minute or an hour) might elapse, between the disintegration of the

A B

Fig. 5. Schematic representation of the measurement of spin variables in the Einstein, Podolsky, and Rosen experiment.

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molecule, and the detection of the atoms. The experiment then consists in measuring the spins of atoms A and B. Let us begin by measuring the Z component of the spins of both atoms. The quantum theory predicts that these two results will always come out opposite (as indicated by the arrows in Fig. 5). We then go on to measure the X component or, indeed, any other component of the spin of each atom. The theory states that these also will come out opposite. It is easy to see that a similar result would be obtained in classical mechanics. Here a molecule of total spin zero would have the respective spin vectors of atom A and atom B oppositely directed. As the atoms fly apart, the spin vectors would remain opposite (since they would not be affected by the force that disintegrated the atom). And so, no matter which components were measured they would evidently have to come out opposite. This is explained with the aid of the simple, classical model, which always attributes to each atom a well-defined spin vector, opposite to that of the other. In quantum mechanics, however, this kind of simple model can no longer be used. As is well known, the operators for the spin components in different directions do not commute, from which it follows that they all cannot be determined together in a measurement. Indeed only one component can be measured at a given time, though this may be any component. One may illustrate the meaning of this by considering an atom whose spin is well defined in the Z direction (say +h/2). It follows then that the X and Y directions are not defined. In a rough picture we may imagine that the spin vector lies in a cone, whose axis is in the Z direction, as illustrated in Fig. 6. Its direction, normal to Z however, fluctuates at random so that the X

spin vector

Fig. 6. Illustration of a particular way of imagining the spin vector to account for the fluctuations in the X and Y components when the Z-component is well defined.

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and Y components are not detected. If one now measures the spin in the X direction then, after the measurement is over, this cone will point in the X direction, while the Y and Z components will fluctuate at random. More generally, if the spin is measured in the direction of any unit vector fi, the components perpendicular to fi will fluctuate at random. It seems clear from the above description that a measurement of the component of the spin in the direction, fi, must do more than merely establish the value of that component. It must also somehow introduce a random fluctuation in the two perpendicular components. This evidently implies that a measurement disturbs or modifies the observed system in a certain unpredictable and uncontrollable way. Indeed, the notion of such a disturbance has long been familiar to us, having first been suggested by Heisenberg, in connection with how, in his hypothetical gamma-ray microscope, the particle momentum is unpredictably altered in its interaction with the quantum of electromagnetic energy used to make it visible. Let us now return to the measurement of spin variables for the two atoms, A and B. It is clear that in measuring the spin of atom A in any direction, fi, we may suppose that the measuring apparatus disturbs the two components of the spin perpendicular to fi in an unpredictable and uncontrollable way. But what about atom B? Quantum theory predicts not only that the spin component in the direction of fi will be opposite to that of B, but that the two perpendicular spin components will also fluctuate at random. But recall that after atoms A and B separate, they do not interact in any way at all. How then can a random disturbance of the spin components of atom A normal to fi bring about a corresponding random disturbance of these components of the spin of atom B? The difficulty can be made even more evident, by considering that while the atoms are still in flight, the apparatus measuring atom A may be orientated in a different direction, fi'. Atom B must then respond with an opposite spin in this direction and with a corresponding random fluctuation in the two perpendicular directions. All of this must happen without any interaction between these two atoms. Surely this would seem to be an example of the "ghostly" nonlocal connection of distant events that Einstein regarded as absurd. Certainly, at least we can say that any simple model resembling the classical type which attributes independent spin properties to each of two noninteracting atoms separated in space, would not be able to fit in with the implications of the quantum theory (as we shall indeed bring out in more detail later). As we have already indicated, Einstein regarded such an idea as so unacceptable that he took the analysis of this experiment as a devastating criticism of the quantum theory. In particular, his proposal was that the quantum theory, while probably yielding correct predictions for this

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experiment, did not provide a complete or even adequate description of the reality which underlies this peculiar connection of distant events. Instead, he considered it to be little more than a fragmentary abstraction that is useful for calculations. In doing this he was, of course, going against the view of most physicists, who regarded (and still generally regard) the quantum theory as capable of yielding the most complete description of reality that is, in principle, possible. In further orienting our thoughts about this question, it is evidently important to consider experimental tests for whether or not this nonlocal feature is actually present. One of the earliest such tests ~7) was made possible by observations carried out on a pair of gamma-ray photons, with mutually perpendicular plane polarizations, arising in the annihilation of positronium. Mathematically, this leads to the same sort of situation as that arising in the example treated in this paper (i.e., the molecule of total spin zero). In an analysis of the two-photon experiment (8~ the predictions of the quantum theory were compared with those of a model originally suggested by Furry. °) In this model one supposed that, after the two quanta separated, each went into a definite polarization state that was opposite to that of the other, but with a random distribution of orientations of this state. The actual experimental results agreed with the quantum theory and clearly did not agree with the predictions arising from Furry's model. Since the essential assumption of this model is that after the two particles separate, their spin states are independent, it is clear that the experiment constitutes a confirmation of quantum mechanical nonlocality. Later, an important refinement of the criteria distinguishing locality and nonlocality was developed by Bell, (1°) in the form of an inequality that has to be satisfied, if the states of two separated particles are to be functionally independent. A considerable number of experiments has been done, aimed at testing this inequality, (I1) and it may be said that they generally confirm the quantum theory with its feature of nonloeality. Other experiments ~12) which are refinements of the test of the Furry model provide a similar confirmation of up to distances of the particle which are as great as 5 m.

4. A SUGGESTED WAY OF UNDERSTANDING NONLOCALITY IN EINSTEIN'S A P P R O A C H It seems clear from both the theoretical analysis and from the experiments that some kind of nontocality is a fact and that we have to understand what it means. If we accept the fact that quantum theory provides the most complete possible description of reality, then nonlocality means simply a further extension of the many ways in which this theory

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defies understanding in terms of any ordinary conception of reality. So we have merely to agree that to obtain a successful algorithm for computations, correctly predicting the results of experiment is the only genuine goal of physics. On the other hand, Einstein could not accept this approach. Instead, as we indicated earlier, he felt that the theory must yield an objective description of the universe, whose elements are moreover local, as well as locally connected. As we have already pointed out, an experimental proof of nonlocality as implied by quantum theory, was not contrary to Einstein's expectations. What Einstein insisted on was the need for some new and broader theoretical framework, within which the question of how nonlocality of this kind comes about could be discussed on the basis of a more fundamental notion of a reality that is objective and local. To indicate how this question may be approached in a new way, we first call attention to a well-known property of nonlinear equations, i.e., the existence of stable orbits or limit cycles, which are such that small deviations lead either to rapid oscillatory motion around the stable movement, or to motions that die away exponentially. Work on these lines has recently been extended by Thom (~3) who has systematically expressed the conditions in which whole families of orbits bifurcate into two groups, one of which goes toward one stable state of motion while the other goes to a second such state, This sort of behavior might well be considered to be a possible model of the transition between quantum states, in which a given atom may jump from any particular state into one of a range of possible states. If we consider that in Einstein's nonlinear field theory the vacuum would contain an intense excitation of field energy, a kind of random background, we can see that even if the basic equations of this field are locat, their nonlinearity might bring about stable movements involving a coordination of fields at separated parts of space and time (e.g., as coupled nonlinear oscillators that are separated can tend to fall into synchronism). So the universal field may get into a state of motion, near a stable set of "orbits" in which the distant and separate particle-like pulses move together in coordination (more like what happens in a ballet dance, rather than in a crowd of people who jostle each other at random). We must recall here that this applies not only to the observed system, but also to the "particles" that constitute the observing apparatus. Of course, such coordinated sets of movements would correspond only to certain states of movement of the whole (e.g., those containing a molecule of total spin zero) while there would be other states (e.g., those containing two atoms, each with well-defined spins) in which there would not be this kind of coordination. Let us now consider the experiment in which positronium decays into a pair of gamma-rays with mutually perpendicular planes of polarization (see Fig. 7).

Quantum Theory Understood in Terms of Einstein's Nonlinear Field Approach Detector B

0

positronmm

-~

.~

539

Detector A

~

0

Fig. 7. Schematic representation of the EPR experiment using annihilation photons from positronium decay.

We are supposing that through nonlinearity the movements of the positronium and of the "particles" constituting the detectors, A and B, are coordinated in a certain way. The nature of the coordination is such that a positronium atom decays into two photons with suitably related, but welldefined polarizations, only when the detectors are in a condition to absorb these photons. Once the photon has started to move toward the detector, it simply "carries" its own state of polarization. So there is no direct nonlocal connection in the movements of the photons. Instead, the nonlocality is of a different kind, arising in a new quality of coordinated movement. This coordination is not the result of "preestablished harmony" (e.g., on the lines of the monads of Leibniz). Nor is it like a "conspiracy" (any more than the common phase of two coupled synchronous motors would be a conspiracy). Instead, it is a stable form of overall movement, carried in a continuous and local way by the "vacuum" fields in the background. Small deviations from this stable form of the whole movement either oscillate so rapidly that they produce negligible effects, or else die away exponentially. The results of measurements carried out in any one place may thus not be totally independent of what measurements are being carried out at another far away place, and this makes possible an explanation of the experiment of Einstein, Podolsky, and Rosen. And yet one may still be able to understand all of this by means of the underlying local field. The above discussion deals with those photons that are actually detected and measured. But one may ask what happens to those that go off in some other direction, and which do not go through any detectors, to be measured? To deal with these, one generalizes the idea of coordinated movement and says that any material system which absorbs a quantum is similarly coordinated with the source of that quantum. (Thus, the measuring apparatus now plays no special rote in our theory, but it is treated as just a special case of the properties of the universal field.) Thus, it is essentially an extension of Wheeler and Feynman's absorber theory of radiation (14~ in which no photon is emitted unless there is matter somewhere which will absorb it. As Wheeler and Feynman showed, this theory can give just as consistent a description of the process of radiation as is given by the current theory, in which an atom radiates independently of whether an absorber is available or not. More generally, no transitions from one state to another take place

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unless the rest of the universe, in its coordinated movements, is ready to absorb (or emit) the energy which has to be exchanged in this transition. To return to the experiment with the gamma-rays from positronium decay, one sees that it is crucial to understand the question of relative timing of the operation of detectors A and B. Thus, in one of the experiments that has actually been done, (11) the detectors are arranged to work simultaneously within 10 -9 sec. The distance from source to detector is so large that a light signal would take about 10 -8 sec so that there evidently can be no immediate connections of the two events through the fluctuations of the "vacuum" field. Nevertheless, all of the parts of the system have had a long time (many days at least) to come to equilibrium through the forces carried in these vacuum fluctuations. So the positronium will be ready to decay into photons of a certain polarization only when the detectors are ready to receive these photons. This experiment will, therefore, still be comprehended within this "local" explanation of quantum-mechanical nonlocality. If we recall that the coordinated movements are only the stable limit of fluctuations around them, the idea arises that one might destroy the coordination by suddenly disturbing one of the detectors in a way that is not related to what happens to the rest of the system. Thus, we might hope to obtain a result, in which the predictions of the quantum theory break down. The extreme difficulty of doing this becomes evident, however, when we recall that all matter is constituted of field pulses, on top of the vacuum fluctuations, and that these pulses have generally been in existence tbr at least some thousands of millions of years. So it seems tikely that even the apparatus that is used to "disturb" the detector will be in "equilibrium" and will, thus, simply combine with the rest, to form a larger whole, that is still coordinated according to the laws of the quantum theory. Of course, one cannot say a priori how far these coordinations will actually go. It is always in principle possible that when the detectors are sufficiently separated, and when they are "disturbed" in a sufficiently rapid and "random" way, the coordination will break down. Thus, experiments such as those of Aspect, "9) which are designed to test this point, are always worth doing. Yet one must remember that at present there is no known reason to suppose that they will be successful in finding a set of conditions in which quantum theory will yield wrong predictions. We see, then, that the nonlocality of quantum theory points to the need to develop a new principle of coordinated movements, which may well extend over the entire universe. Even though the fields are propagated locally, the forms of matter will have to be understood as an undivided whole not only because they merge and unite through the background of vacuum fluctuations, but also because through nonlinearity they may work together even over long distances. This indicates that though the universe

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may be immense, the various parts are not basically isolated from each other. Such a universe could perhaps be contained within Einstein's vision of a fundamental reality, ordered and lawful, and yet rich and subtle enough to comprehend all the complexities of the world as we know it. 5. IMPLICATIONS OF O U R M O D E L FOR BELL'S I N E Q U A L I T Y

Bell's inequality (1°) has commonly been taken to imply that the quantum theory is not compatible with local hidden variables. Since we have proposed a model based on determinate local connections of field quantities that would explain the quantum theory, we have evidently to ask whether our proposal is ruled out by Bell's theorem. We shall see that it is indeed not ruled out, but that to demonstrate this we have to go more carefully into the meaning of the Bell inequality. To formulate this question more precisely we shall begin by briefly reviewing Bell's arguments, which are applied to the case of measurements of the spins of two atoms, by means of two instruments, 1 and 2, which are separated by a large distance. It is supposed that each experimental result is determined completely by a set of hidden variables, ,t. (whose meaning will be discussed in more detail later). However, along with Bell, (9) we also assume that the result, A, of measurement of spin in the direction, d, depends only on it and on d, while the result B, of measurement of spin in the direction 6, depends only on )~ and 6. In other words, while nothing is said about the general dynamical laws of the hidden variables, ~, which may be as nonlocally connected as we please, we are requiring that the response of each particular observing instrument to the set, ~, depends only on its own state and not on the state of any other piece of apparatus that is far away. Thus, we write

a =A(a, ~) B = B(6, it)

while we specifically exclude possibilities, such as A =A(c~, 6, it) and B = B(d, 6, ~). Finally, we state Bell's additional basic assumption which is that the statistical distribution of hidden variables is a function p(2) that is independent of d and 6. The correlation function for the observations A and B is then given by P(d, 6) = .f p(~) A (,~, ,~) B(6, ~t) d~t For these assumptions, one readily obtains the well-known Bell inequality for four measured directions, d, ~, & d.

[P(d, ~) -- P(d, a)t + [P(~, d) + P(8,/~)1 ~ 2

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As already emphasized in the work of Stapp (~6) this result depends on the more general supposition that whatever may be the deeper structure of the universe, whether this be local or nonlocal, a given large scale system will respond to this structure in a way that is not affected by the states of other such systems that are far away. However, what we wish to emphasize further here is that it also depends crucially on the assumption that the probability distribution of hidden variables does not depend on the states of any large scale systems (such as the various pieces of observing apparatus). In our model, which works in terms of a rather different set of concepts, this latter assumption is not valid. The main similarity between our model and the set of theories that are compatible with Bell's assumptions is that we are indeed regarding the underlying fields as equivalent to a set of hidden variables, ~.. But the key difference is that we do not begin by assuming the independent existence of various large scale systems (e.g., pieces of observing apparatus). Instead, these are, like "elementary particles" mere abstractions or empty forms, which are (as emphasized in Section 2) "shadows" of the activities of the underlying fields. What further distinguishes these abstractions is, of course, that they have a certain persistence, relative autonomy, and stability (much like a pattern of stable vortices in a stream). In this context, we evidently cannot discuss consistently in terms of direct connections between large scale systems and the hidden variables, nor indeed can we properly talk about direct connections of large scale systems with each other. This means that the ordinary description of an experiment (such as "the observing instruments and the observed system interact with each other") is just a picturesque way of implying an infinitely complex set of movements in the underlying field. In these movements, there is no way of separating the observed system from the observing apparatus, nor can one even properly regard the various large scale forms observable in common experience (e.g., "pieces of apparatus" and "the experimenter who assembles and operates this apparatus") as sharply divided from each other. All such forms merely reveal some abstract properties of the whole set of underlying field variables. Actually, this sort of situation is fairly common in physics in other contexts. For example, the temperature and pressure of an aggregate of atoms is not a separate property, but just an abstract form on the underlying atomic variables a. This is revealed by the functional relationship between these variables and the parameters concerned. Thus, the probability distribution over a takes the form

p =p(a, ~r; p,...) and it is clear that this does not mean that T and P "interact" with a or with each other.

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In our model of the quantum theory we are, thus, led to write p

=p(a,6,2)

implying that there is a functional relationship between the distribution of hidden variables, and the abstract tbrms corresponding to the orientations, d and #. We emphasize once again that, as with T and P, we are not implying that the two pieces of apparatus interact directly, nor are we implying that the response of A to the hidden variables directly depends "nonlocally" on B. Rather the distribution of the total set of hidden variables is restricted in such a way that whenever it gives rise to the emergence of given orientations of the two pieces of apparatus, it gives rise also to the emergence of properly correlated pairs of photons. To go back to the proof of Bell's theorem, we may still assume that the laws of the underlying field are such that a given result A is determined as a specified function of the hidden variables A(d, k). But when we compute the correlation function we must write

P(d, #) = f p(d, 6, )t) A (d, 2) B(#, 2) d2 and from this, it is easy to show that Bell's inequality no longer follows. (The failure of this inequality to follow is clearly a consequence of the very different set of physical concepts from which we are proceeding.) At this point, we can throw some light on an earlier model of hidden variables proposed by one of us (17) in 1952. In this model, the hidden variables are the actual positions )~, of all the particles constituting the total system, consisting of "observed object" plus all the pieces of observing apparatus. These positions are conceived as well defined, and follow continuous and well defined trajectories. ~18) In addition to these hidden variables, we also introduce the wave function of the entire system, ~()~i, t), as a basic concept, and suppose that this satisfies the Schr6dinger equation for the corresponding many-body problem. For the case of spin measurements, the total wave function will be functionally related to the orientations of both pieces of observing apparatus, so that we can write 7:'= ~'(a, 6 , £ , t)-- ~'(a, 6,2, 0 (where we have denoted the set of hidden variables, Xi, by the symbol 2). We now assume that in addition to the classical potentials operating o n the hidden variables, there is a further quantum potential

Q=

2, t))

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where Q is a certain specified function of the overall wave function. We then show that on the basis of this potential it is consistent to define a probability distribution over 2, which is

p=

6, 4, 0 )

In this model, the observed result for each piece of apparatus corresponds to a set of hidden variables associated with that piece of apparatus alone, so that Bell's first condition A =A(d, 2) and B =B(6,)~) is satisfied. However, as indicated above, Bell's second condition, p = pot) is not satisfied, so that Bell's inequality is violated. 4 In this case, the dynamical laws of the system are nonlocal in the sense that the quantum potential, Q, permits strong instantaneous interactions of particles at indefinitely large distance. The model now under discussion, thus, differs significantly from the one suggested earlier in this paper, which has only locally connected hidden variables. But as we have already pointed out, dynamical laws of the hidden variables play no direct part in the proof of Bell's theories. All that is important in this proof is whether, for any reason whatsoever, the probability distribution of the hidden variables is functionally related to the states of the various pieces of observing apparatus, even when these are indefinitely far apart. In the model based on locally connected fields, this functional relationship arises because the observing apparatus and the observed system emerge together from the underlying field, in a way that requires certain restrictions on the distribution of hidden variables. In the model proposed in 1952, such a relationship arises because of the quantum potential which implies a nonlocal connection in the total set of hidden variables, that depends on the state of the observing apparatus. To return to the model based on nonlinear, but locally connected fields, we recall that, as pointed out in Section 4, in the process in which the "observed system" and "the apparatus" emerge from the underlying universal background, there has been plenty of time for correlations between these to arise through stable limit cycles in the total field. Thus, as we have already indicated it is quite possible that even experiments such as those of Aspect ~15~will involve processes too slow to go beyond the limits of stability of these cycles. In this connection, it must be remembered that the "particles" constituting the entire apparatus including the "switching device" are also generally assumed to obey the laws of the quantum theory. In terms of our model, this means that they too are merely abstract forms of the underlying field, and that their relationships depend on the limit cycles in 4This point has already been noted by Selleri and Tarozzi.~9~ However,they do not take into account that p depends on d and 5, and so, they are led to the wrong conclusion that this satisfies Bell's inequality.

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question. For example, during the long period while these systems were being assembled the laws of quantum mechanics would have to apply. Thus, although the "switching time" in this experiment may be very short compared with the flight time of the photons, the precise details of how the switch operates may have emerged from the field background in such a manner that whenever the latter gives rise to a certain switching process, it also gives rise to photons that emerge in a corresponding way. Of course, in terms of our usual mode of thinking, it is hard to believe that this sort of extremely extensive correlation could ever happen. Most physicists are, however, ready to accept that it may happen at a sufficiently short distance (of a few interatomic spacings). But logically the distance has nothing to do with the question. It is only a matter of fact, to be tested experimentally, as to how far such correlation of emergent forms will actually go. It is interesting to ask whether this sort of chain of correlations would include even the human observer. Of course, we are not at present in a position to test this experimentally because human actions are even slower than those which we have been discussing. Nevertheless, one may wonder whether all the actions of the human being, including the choices that he makes, can also be understood as emerging out of the same general field from which matter emerges. Here, it must be pointed out that the sense of free will that one commonly experiences is not relevant. For as Schopenhauer has pointed out, we may perhaps be free to do as we will, but we are not able to will the content of this will. One possibility is that all of this content emerges, as suggested above, from the general field, and another is that at least some of it does not. Whether this question can ever be resolved in a physical experiment is something that we can, for the present, say nothing about. Indeed, even if, as Wigner (2°) suggests, there is a connection between mental and physical events, one cannot exclude the possibility that both of these are abstract forms emerging from the underlying field. So this question must remain open.

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