hiddenVarNotes

https://chat.openai.com/g/g-CcNlCDjI1-nash-linter/c/c193d1e3-98f3-4720-8b57-5ce7d877e046

http://www.gci.org.uk/Documents/DavidBohm-WholenessAndTheImplicateOrder.pdf

83 The question of whether there are hidden variables underlying the quantum theory was thought to have been settled definitely in the negative long ago. As a result, the majority of modern physicists no longer regard this question as relevant for physical theory. In the past few years, however, a number of physicists, including the author, have developed a new approach to this problem, which raises the question of hidden variables again.1

In the course of this chapter, we shall show a number of reasons why theories involving hidden variables promise to be significant for the treatment of new physical problems, especially those arising in the domain of very short distances (of the order of 10−13cm or less) and of very high energies (of the order of 109 ev or more).

the difficulties of dealing with the Heisenberg indeterminacy relations, the quantization of action, the paradox of Einstein, Rosen and Podolsky, and von Neumann’s arguments against the possibility of such variables.

This means that not all physically significant observables can be determined together, and even more important, that those which are not determined will fluctuate lawlessly (at random) in a series of measurements on an ensemble represented by the same wave function.

it follows that if we begin with an ensemble of undisintegrated nuclei, represented by the same wave function, each hidden variables in the quantum theory 85 individual nucleus will decay at an unpredictable time.

discovered that there is a random distribution of clicks of the Geiger counter, together with a regular mean distribution that obeys the probability laws implied by the quantum theory.

, it is evident that the indeterministic features of quantum mechanics are in some way a reflection of the real behaviour of matter in the atomic and nuclear domains, but here the question arises as to just how to interpret this indeterminism

some analogous problems. Thus, it is well known that insurance companies operate on the basis of certain statistical laws, which predict to a high degree of approximation the mean number of people in a given class of age, height, weight, etc., that will die of a certain disease in a specified period of time. They can do this even though they cannot predict the precise time of death of an individual policy-holder, and even though such individual deaths are distributed at random in a way having no lawful relationship to the kind of data that the insurance company is able to collect. Nevertheless, the fact that statistical laws of this kind are operating does not prevent the simultaneous operation of individual laws which determine in more detail the precise conditions of death of each policy-holder

when it was discovered that spores and smoke particles suffer a random movement obeying certain statistical laws (the Brownian motion) it was supposed that this was due to impacts from myriads of molecules, obeying deeper individual laws. The statistical laws were then seen to be consistent with the possibility of deeper individual laws, for, as in the case of insurance statistics, the overall behaviour of an individual Brownian particle would be determined by a very large number of essentially independent factors.

to put the case more generally: lawlessness of individual behaviour in the context of a given statistical law is, in general, consistent with the notion of more detailed individual laws applying in a broader context.

consider the hypothesis that results of individual quantum-mechanical measurements are determined by a multitude of new kinds of factors, outside the context of what can enter into the quantum theory.

These factors would be represented mathematically by a further set of variables, describing the states of new kinds of entities existing in a deeper, sub-quantum-mechanical level and obeying qualitatively new types of individual laws

Such entities and their laws would then constitute a new side of nature, a side that is, for the present ‘hidden’.

he atoms, first postulated to explain Brownian motion and large-scale regularities, were also hidden variables in the quantum theory 87 originally ‘hidden’ in a similar way, and were revealed in detail only later by new kinds of experiments (e.g., Geiger counters, cloud chambers, etc.) that were sensitive to the properties of individual atoms.

. Similarly, one may suppose that the variables describing the sub-quantum-mechanical entities will be revealed in detail when we have discovered still other kinds of experiments, which may be as different from those of the current type as the latter are from experiments that are able to reveal the laws of the large-scale level (e.g., measurements of temperature, pressure, etc.).

At this point it must be stated that as is well known – the majority of modern theoretical physicists3 have come to reject any suggestion of the type described above. They do this mainly on the basis of the conclusion that the statistical laws of the quantum theory are incompatible with the possibility of deeper individual laws. In other words, while they would in general admit that some kinds of statistical laws are consistent with the assumption of further individual laws operating in a broader context, they believe that quantum mechanics could never satisfactorily be regarded as a law of this kind. The statistical features of the quantum theory are thus regarded as representing a kind of irreducible lawlessness of individual phenomena in the quantum domain. All individual laws (e.g. classical mechanics) are then regarded as limiting cases of the probability laws of the quantum theory, approximately valid for systems involving large numbers of molecules.

4.1 Heisenberg’s indeterminacy principle We begin with a discussion of Heisenberg’s indeterminacy principle. He showed that even if one supposes that the physically significant variables actually existed with sharply defined values (as is demanded by classical mechanics) then we could never measure all of them simultaneously, for the interaction between the observing apparatus and what is observed always involves an exchange of one or more indivisible and uncontrollably fluctuating quanta. For example, if one tries to measure the coordinate, x, and the associated momentum, p, of a particle, then the particle is disturbed in such a way that the maximum accuracy for the simultaneous determination of both is given by the wellknown relation ∆p∆x h. As a result, even if there were deeper sub-quantum laws determining the precise behaviour of an individual electron, there would be no way for us to verify by any conceivable kind of measurement that these laws were really operating. It is therefore concluded that the notion of a subquantum level would be ‘metaphysical’, or empty of real experimental content. Heisenberg argued that it is desirable to formulate physical laws in terms of the minimum possible number of such notions, for they add nothing to the physical predictions of the theory, while they complicate the expression in an irrelevant way. 4.2 Von Neumann’s arguments against hidden variables It is well known that in such an experiment a statistical interference pattern is still obtained, even if we pass the particles through the apparatus at intervals so far apart that each particle essentially enters separately and independently of all the others. But, if the whole ensemble of such particles were to split into 90 wholeness and the implicate order sub-ensembles, each corresponding to the electron striking the grating at a definite value of x, then the statistical behaviour of every sub-ensemble would be represented by a state corresponding to a delta function of the point in question. As a result, a single sub-ensemble could have no interference that would represent the contributions from different parts of the grating. Because the electrons enter separately and independently no interference between sub-ensembles corresponding to different positions will be possible either. In this way we show that the notion of hidden variables is not compatible with the interference properties of matter, which are both experimentally observed and necessary consequences of the quantum theory. Von Neumann generalized the above argument and made it more precise; but he came to essentially the same result. In other words, he concluded that nothing (not even hypothetical hidden variables) can be consistently supposed to determine beforehand the results of an individual measurement in more detail than is possible according to the quantum theory.

4.3 The paradox of Einstein, Rosen and Podolsky However, in the quantum theory we have the additional fact that only one component of the spin can be sharply defined at one time, while the other two are then subject to random fluctuations. If we wish to interpret the fluctuations as nothing but the result of disturbances due to the measuring apparatus, we can do this for atom A, which is directly observed, but how does atom B, which interacts in no way either with atom A or with the observing apparatus, ‘know’ in what direction it ought to allow its spin to fluctuate at random? The problem is made even more difficult if we consider that, while the atoms are still in flight, we are free to re-orientate the observing apparatus arbitrarily, and in this way to measure the spin of atom A in some other direction. This change is somehow transmitted immediately to atom B, which responds accordingly. Thus, we are led to contradict one of the basic principles of the theory of relativity, which states that no physical influences can be propagated faster than light.

5 BOHR’S RESOLUTION OF THE PARADOX OF EINSTEIN, ROSEN AND PODOLSKY – THE INDIVISIBILITY OF ALL MATERIAL PROCESSES The paradox of Einstein, Rosen and Podolsky was resolved by Niels Bohr in a way that retained the notion of indeterminism in quantum theory as a kind of irreducible lawlessness in nature.7 To do this he used the indivisibility of a quantum as his basis.

To deal with this new property of matter in the quantum domain, Bohr proposed to begin with the classical level, which is immediately accessible to observation. The various events which take place in this level can be adequately described with the aid of our customary general concepts, involving indefinite analysability. It is then found that up to a certain degree of approximation these events are related by a definite set of laws, i.e., Newton’s laws of motion, which would, in principle, determine the future course of these events in terms of their characteristics at a given time.

Now comes the essential point. In order to give the classical laws a real experimental content, we must be able to determine the momenta and positions of all the relevant parts of the system of interest. Such a determination requires that the system of interest be connected to an apparatus which yields some observable large-scale result that is definitely correlated to the state of the system of interest. But in order to satisfy the requirement that we must be able to know the state of the observed system by observing that of the large-scale apparatus, it must be possible, in principle at least, for us to distinguish between the two systems by means of a suitable conceptual analysis, even though they are connected and in some kind of interaction. In the quantum domain, however, such an analysis can no longer be correctly carried out. Consequently, one must regard what has previously been called the ‘combined system’ as a single, indivisible, overall experimental situation. The result of the operation of the whole experimental set-up does not tell us about the system that we wish to observe but, rather, only about itself as a whole

The above discussion of the meaning of a measurement then leads directly to an interpretation of the indeterminacy relationships of Heisenberg. As a simple analysis shows, the impossibil94 wholeness and the implicate order ity of theoretically defining two non-commuting observables by a single wave function is matched exactly, and in full detail, by the impossibility of the operation together of two overall set-ups that would permit the simultaneous experimental determination of these two variables. This suggests that the non-commutativity of two operators is to be interpreted as a mathematical representation of the incompatibility of the arrangements of apparatus needed to define the corresponding quantities experimentally.

In the classical domain it is of course essential that pairs of canonically conjugate variables of the kind described above shall be defined together. Each one of such a pair describes a necessary aspect of the whole system, an aspect which must be combined with the other if the physical state of the system is to be defined uniquely and unambiguously. Nevertheless, in the quantum domain each one of such a pair, as we have seen, can be defined more precisely only in an experimental situation in which the other must become correspondingly less precisely defined. In a certain sense, each of the variables then opposes the other. Nevertheless, they still remain ‘complementary’ because each describes an essential aspect of the system that the other misses. Both variables must therefore still be used together but now they can be defined only within the limits set by Heisenberg’s principle. As a result, such variables can no longer provide us with a definite, unique, and unambiguous concept of matter in the quantum domain. Only in the classical domain is such a concept in adequate approximation.

If there is no definite concept of matter in the quantum domain, what then is the meaning of the quantum theory? In Bohr’s point of view it is just a ‘generalization’ of classical mechanics. Instead of relating to observable classical phenomena by Newton’s equations, which are a completely deterministic and indefinitely analysable set of laws, we relate these same phenomena by the quantum theory, which provides a probabilistic set of laws that does not permit analysis of the phenomena in hidden variables in the quantum theory 95 indefinite detail. The same concepts (e.g., position and momentum) appear in both classical and quantum theories. In both theories, all concepts obtain their experimental content in essentially the same way, i.e., by their being related to a specific experimental set-up involving observable large-scale phenomena. The only difference between classical and quantum theories is that they involve the use of different kinds of laws to relate the concepts.

It is evident that according to Bohr’s interpretation nothing is measured in the quantum domain. Indeed, in his point of view, there can be nothing to measure there, because all ‘unambiguous’ concepts that could be used to describe, define, and think about the meaning of the results of such a measurement belong to the classical domain only. Hence, there can be no talk about the ‘disturbance’ due to a measurement, since there is no meaning to the supposition that there was something there to be disturbed in the first place.

It is now clear that the paradox of Einstein, Rosen and Podolsky will not arise, because the notion of some kind of actually existing molecule, which was originally combined, and which later ‘disintegrated’, and which was ‘disturbed’ by the ‘spinmeasuring’ device, has no meaning either. Such ideas should be regarded as nothing more than picturesque terms that it is convenient to use in describing the whole experimental set-up by which we observe certain correlated pairs of classical events (e.g., two parallel ‘spin-measuring’ devices that are on opposite sides of the ‘molecule’ will always register opposite results).

As long as we restrict ourselves to computing the probabilities of pairs of events in this way, we will not obtain any paradoxes similar to that described above.

In such a computation the wave function should be regarded as just a mathematical symbol, which will help us to calculate the right relationships between classical events, provided that it is manipulated in accordance 96 wholeness and the implicate order with a certain technique, but which has no other significance whatsoever.

It is now clear that Bohr’s point of view necessarily leads us to interpret the indeterministic features of the quantum theory as representing irreducible lawlessness; for, because of the indivisibility of the experimental arrangement as a whole, there is no room in the conceptual scheme for an ascription of causal factors which is more precise and detailed than that permitted by the Heisenberg relations. This characteristic then reveals itself as an irreducible random fluctuation in the detailed properties of the individual large-scale phenomena, a fluctuation, however, that still satisfies the statistical laws of the quantum theory. Bohr’s rejection of hidden variables is therefore based on a very radical revision of the notion of what a physical theory is supposed to mean, a revision that in turn follows from the fundamental role which he assigns to the indivisibility of the quantum

The first systematic suggestions for an interpretation of the quantum theory in terms of hidden variables were made by the author.8 Based at first on an extension and completion of certain hidden variables in the quantum theory 97 ideas originally proposed by de Broglie,9 this new interpretation was then carried further in a later work jointly by the author and Vigier.10 After some additional development, it finally took a form the main points of which will be summarized as follows:11

1 The wave function, ψ, is assumed to represent an objectively real field and not just a mathematical symbol.

2 We suppose that there is, beside the field, a particle represented mathematically by a set of coordinates, which are always well defined and which vary in a definite way.

3 < complex math >

It has been demonstrated13 that the above theory predicts physical results that are identical with those predicted by the usual interpretation of the quantum theory, but it does so with the aid of very different assumptions concerning the existence of a deeper level of individual law.

As we saw in previous sections, the usual point of view does not permit us to analyse this process in detail, even conceptually; nor does it permit us to regard the places at which individual electrons will arrive as determined beforehand by the hidden variables. It is our belief, however, that this process can be analysed with the aid of a new conceptual model.

This model is based, as we have seen, on the supposition that there is a particle following a definite but randomly fluctuating track, the behaviour of which is strongly dependent on an objectively real and randomly fluctuating ψ-field, satisfying Schrödinger’s equation in the mean. When the ψ-field passes through the grating, it diffracts in much the same way as other fields would (e.g., the electromagnetic). As a result, there will be an interference pattern in the later intensity of the ψ-field, an interference pattern that reflects the structure of the grating. But the behaviour of the ψ-field also reflects the hidden variables in the sub-quantum level, which determine the details of its fluctuations around the mean value, obtained by solving Schrödinger’s equation. Thus, the place where each particle will arrive is finally determined in principle by a combination of factors including the initial position of the particle, the initial form of its ψ-field, the systematic changes of the ψ-field due to the grating, and the random changes of this field originating in the sub-quantum level. In a statistical ensemble of cases having the same mean initial wave function, the fluctuations of the ψ-field will, as has been shown,14 produce just the same interference pattern that is predicted in the usual interpretation of the quantum theory

At this point, we must ask how we have been able to come to a result opposite to that deduced by von Neumann (section 4.2). The answer is to be found in a certain unnecessarily restrictive 100 wholeness and the implicate order assumption behind von Neumann’s arguments. This assumption is that the particles arriving at the grating in a given position, x (determined beforehand by the hidden variable), must belong to a sub-ensemble having the same statistical properties as those of an ensemble of particles whose position, x, has actually been measured (and whose functions are therefore all a corresponding delta function of position). Now it is well known that if the position of each electron as it passes through the grating were to be measured, no interference would be obtained (because of the disturbance due to the measurement that causes the system to divide into the non-interfering ensembles represented by delta functions as discussed in section 4.2). Hence, von Neumann’s procedure is equivalent to an implicit assumption, that any factors (such as hidden variables) which determine x beforehand must destroy interference in the same way as it is destroyed in a measurement of the coordinate x

In our model, we go beyond the above implicit assumption by admitting at the outset that the electron has more properties than can be described in terms of the so-called ‘observables’ of the quantum theory.

Thus, as we have seen, it has a position, a momentum, a wave field, ψ, and sub-quantum fluctuations, all of which combine to determine the detailed behaviour of each individual system with the passage of time.

As a result, the theory has room to describe within it the difference between an experiment in which the electrons pass through the grating undisturbed by anything else, and one in which they are disturbed by a position-measuring apparatus

These two sets of experimental conditions would lead to very different ψ-fields, even if in both cases the particles were to strike the grating at the same position.

. The differences in the subsequent behaviour of the electron (i.e., interference in one case and not in the other) will therefore follow from the different ψ-fields which exist in the two cases.

The interpretation of the quantum theory discussed in the previous section is subject to a number of serious criticisms.

First of all, it must be admitted that the notion of the ‘quantum potential’ is not entirely a satisfactory one, for not only is the proposed form, U = − (2 /2m) (2 R/R), rather strange and arbitrary but also (unlike other fields such as the electromagnetic) it has no visible source. This criticism by no means invalidates the theory as a logical self-consistent structure but only attacks its plausibility. Nevertheless, we evidently cannot be satisfied with accepting such a potential in a definitive theory. Rather, we should regard it as at best a schematic representation of some more plausible physical idea to which we hope to advance later, as we develop the theory further.

Second, in the many-body problem, we are led to introduce a 102 wholeness and the implicate order many-dimensional ψ-field [ψ(x1, x2, . . ., xn, . . ., xN)] and a corresponding many-dimensional quantum potential U = — 2 2m N i = 1 ∇2 i R R , with ψ = Reis/ as in the one-body case. The momentum of each particle is then given by Pi = S(x1 ... xn ... xN) xi All of these notions are quite consistent logically. Yet it must be admitted that they are difficult to understand from a physical point of view. As best, they should be regarded, like the quantum potential itself, as schematic or preliminary representations of certain features of some more plausible physical ideas to be obtained later.

Third, the criticism has been levelled against this interpretation that the precise values of the fluctuating ψ-field and of the particle coordinates are empty of real physical content. The theory has been constructed in just such a way that the observable large-scale results of any possible kind of measurements are identical with those predicted by current quantum theory. In other words, from the experimental results, one can find no evidence for the existence of the hidden variables, nor does the theory permit their definition to be ever good enough to predict any result more accurately than the current quantum theory does.

. To show that it was wrong to throw out hidden variables because they could not be imagined, it was therefore sufficient to propose any logically consistent theory that explained the quantum mechanics, through hidden variables, no matter how abstract and hypothetical it might be. Thus, the existence of even a single consistent theory of this kind showed that whatever arguments one might continue to use against hidden variables, one could no longer use the argument that they are inconceivable

… the logical structure of the theory makes room for the possibility of its being changed in such a way that it ceases to be completely identical with the current quantum mechanics in its experimental content. As a result, the details of the hidden variables (e.g. the fluctuations of the ψ-field and of the particle positions) will be able to reveal themselves in new experimental results not predicted by the quantum theory as it is now formulated.

, we first point out that even if there 104 wholeness and the implicate order existed no known experiments that the current quantumtheoretical framework failed to treat satisfactorily, the possibility would always still be open for new experimental results, not fitting into this framework. All experiments are necessarily done only in some limited domain, and even in this domain, only to a limited degree of approximation. Room is therefore always left open, logically speaking, for the possibility that when experiments are done in new domains and to new degrees of approximation, results will be obtained that do not fit completely into the framework of the current theories.

Physics has quite frequently developed in the way described above. Thus, Newtonian mechanics, thought originally to be of completely universal validity, was eventually found to be valid in a limited domain (velocity small compared with that of light) and only to a limited degree of approximation. Newtonian mechanics had to give way to the theory of relativity which utilized basic conceptions concerning space and time which were in many ways not consistent with those of Newtonian mechanics. The new theory was, therefore, in certain essential features qualitatively and fundamentally different from the old one. Nevertheless, within the domain of low velocities, the new theory approached the old one as a limiting case. In a similar way, classical mechanics eventually gave way to the quantum theory, which is very different in its basic structure, but which still contains classical theory as a limiting case, valid approximately in the domain of large quantum numbers. Agreement with experiments in a limited domain and to a limited degree of approximation is evidently no proof, therefore, that the basic concepts of a given theory have a completely universal validity.

From the above discussion we see that the experimental evidence taken by itself will always leave open the possibility of a theory of hidden variables that yields results differing from those of the quantum theory in new domains (and even in the old domains when carried to a sufficiently high degree of hidden variables in the quantum theory 105 approximation).

Now, however, we must have some more definite ideas as to which are the domains in which we expect the results to be new, and as to just what are the ways in which they ought to be new.

There are the difficulties arising in connection with the divergences (infinite results) obtained in calculations of the effects of interactions of various kinds of particles and fields. It is true that for the special case of electromagnetic interactions such divergences can be avoided to a certain extent by means of the so-called ‘renormalization’ techniques. It is by no means clear, however, that these techniques can be placed on a secure logical mathematical basis.16 Moreover, for the problem of mesonic and other interactions, the renormalization method does not work well even when considered as a purely technical manipulation of mathematical symbols, apart from the question of its logical justification. While it has not been proved conclusively, as yet, that the infinities described above are essential characteristics of the theory, there is already a considerable amount of evidence in favour of such a conclusion.17 It is generally agreed that, if as seems rather likely, the theory does not converge, then some fundamental change must be made in its treatment of interactions involving very short distances, from which domain all the difficulties arise (as one sees in a detailed mathematical analysis)

Most of the proponents of the usual interpretation of the quantum theory would not deny that such a fundamental change seems to be needed in the present theory. Indeed, some of them, including Heisenberg, are even ready to go so far as to 106 wholeness and the implicate order give up completely our notions of a definable space and time, in connection with such very short distances, while comparably fundamental changes in other principles, such as those of relativity, have also been considered by a number of physicists (in connection with the theory of non-local fields). But there seems to exist a widespread impression that the principles of quantum mechanics almost certainly will not have to be changed in essence. In other words, it is felt that however radical the changes in physical theories may be they will only build upon the principles of the present quantum theory as a foundation, and perhaps enrich and generalize these principles by supplying them with a newer and broader scope of application.

I have never been able to discover any well-founded reasons as to why there exists so high a degree of confidence in the general principles of the current form of the quantum theory. Several physicists18 have suggested that the trend of the century is away from determinism, and that a step backwards is not very likely.

This, however, is a speculation of a kind that could easily be made in any period concerning theories that have hitherto been successful. (For example, classical physicists of the nineteenth century could have argued with equal justification that the trend of the times was toward more determinism, whereas future events would have proved this speculation wrong. Still others have adduced a psychological preference for indeterministic theories, but this may well be just a result of their having become accustomed to such theories. Classical physicists of the nineteenth century would surely have expressed an equally powerful psychological bias toward determinism.)

Finally, there is a widespread belief that it will not really be possible to carry out our suggested programme of developing a theory of hidden variables which will be genuinely different in experimental content from the quantum theory, and which will still agree with the latter theory in the domain where this theory is already known to be essentially correct. This view is held in hidden variables in the quantum theory 107 particular by Niels Bohr, who expressed especially strong doubts19 that such a theory could treat all significant aspects of the problem of the indivisibility of the quantum of action – but then this argument stands or falls on the question of whether an alternative theory of the kind described above can really be produced, and in the next sections, we shall see that such a position is not a very secure one.

8 STEPS TOWARD A MORE DETAILED THEORY OF HIDDEN VARIABLES

From the discussion given in the previous section, it is clear that our central task is to develop a new theory of hidden variables.

This theory should be quite different from the current quantum theory both in its basic concepts and in its general experimental content, and yet be capable of yielding essentially the same results as those of the current theory in the domain in which this latter theory has thus far been verified, and to the degree of approximation of the measurements that have actually been carried out. The possibility of distinguishing between the two theories experimentally will then arise either in new domains (e.g., very short distances) or in more accurate measurements carried out in the older domains.

Our basic starting point will be to try to provide a more concrete physical theory leading to ideas resembling those discussed in connection with our preliminary interpretation (section 6). In doing this, we must first recall that we have been regarding indeterminism as a real and objective property of matter, but one associated with a given limited context (in this case that of the variables of the quantum-mechanical level). We are supposing that in a deeper sub-quantum level, there are further variables which determine in more detail the fluctuations of the results of individual quantum-mechanical measurements

Does the existing physical theory provide us with any hints as 108 wholeness and the implicate order to the nature of these deeper sub-quantum-mechanical variables? To guide us in our search, we can begin by considering the current quantum theory in its most developed form, namely that of relativistic field theory. According to the principles of the current theory, it is essential that every field operator, µ, be a function of a sharply defined point, x, and that all interactions shall be between fields at the same point. This leads us to formulate our theories in terms of a non-denumerable infinity of field variables. Of course, such a formulation must be made, even classically, but in classical physics one can assume that the fields vary continuously. As a result, one can effectively reduce the number of variables to a denumerable set (e.g., the average values of the fields in very small regions), essentially because the field changes over very short distances are negligibly small. As a simple calculation shows, however, this is not possible in the quantum theory, because the shorter the distances one considers, the more violent are the quantum fluctuations associated with the ‘zero-point energy’ of the vacuum. Indeed, these fluctuations are so large that the assumption that the field operators are continuous functions of positions (and time) is not valid in a strict sense.

Even in the usual quantum theory, the problem of a nondenumerable infinity of field variables presents several as yet unsolved basic mathematical difficulties. Thus, it is customary to deal with field theoretical calculations by starting with certain assumptions concerning the ‘vacuum’ state, and thereafter applying perturbation theory. Yet, in principle, it is possible to start with an infinite variety of very different assumptions for the vacuum state, involving the assignment of definite values to a set of completely discontinuous functions of the field variables, functions which ‘fill’ the space densely and yet leave a dense set of ‘holes’. These new states cannot be reached from the original ‘vacuum’ state by any canonical transformation20

0. Hence they lead to theories that are, in general, different in physical content from those obtained hidden variables in the quantum theory 109 with the original starting point. It is entirely possible that because of the divergences in field theoretical results, even the current renormalization techniques imply such an ‘infinitely different’ vacuum state; but even more important, it is necessary to stress that a reorganization of a non-denumerable infinity of variables usually leads to a different theory, and that the principles of such a reorganization will then be equivalent to basic assumptions about the corresponding new laws of nature.

At this point we ask ourselves whether it would ever be possible to reorganize a classical field theory in such a way that it becomes equivalent (at least in some approximation and within some domain) to the modern quantum field theory. In order to answer this question, we must evidently reproduce from the basic ‘deterministic’ law of our assumed non-denumerable infinity of ‘classical’ field variables, the fluctuations of quantum processes, the indivisibility of the quantum, and other essential quantum-mechanical properties, such as interference and the correlations associated with the paradox of Einstein, Rosen and Podolsky. It is with these problems that we shall concern ourselves in subsequent sections.

9 TREATMENT OF QUANTUM FLUCTUATIONS

Let us begin by assuming some ‘deterministic’ field theory. Its precise characteristics are unimportant for our purposes here. All that is important is to suppose the following properties.

1 There is a set of field equations which completely determines the changes of the field with time.

2 These equations are sufficiently non-linear to guarantee a significant coupling between all wave components, so that (except perhaps in some approximation) solutions cannot be linearly superposed.

3 Even in the ‘vacuum’ the field is so highly excited that the mean field in each region, however small, fluctuates significantly, with a kind of turbulent motion that leads to a high degree of randomness in the fluctuations. This excitation guarantees the discontinuity of the fields in the smallest regions.

4 What we usually call ‘particles’ are relatively stable and conserved excitations on top of this vacuum. Such particles will be registered at the large-scale level, where all apparatus is sensitive only to those features of the field that will last a long time, but not to those features that fluctuate rapidly. Thus, the ‘vacuum’ will produce no visible effects at the large-scale level, since its fields will cancel themselves out on the average, and space will be effectively ‘empty’ for every large-scale process (e.g. as a perfect crystal lattice is effectively ‘empty’ for an electron in the lowest band, even though the space is full of atoms).

It is evident that there would be no way to solve such a set of field equations directly. The only possibility would be to try to deal with some kind of average field quantities (taken over small regions of space and time). In general, we could hope that a group of such average quantities would, at least within some approximation, determine themselves independently of the infinitely complex fluctuations inside the associated regions of space.21 To the extent that this happened, we could obtain approximate field laws, associated with a certain level of size, but these laws cannot be exact because the non-linearity of the equations means fields will necessarily be coupled in some way to the inner fluctuations that have been neglected. As a result, the mean fields will also fluctuate at random about their average behaviour. There will be hidden variables in the quantum theory 111 a typical domain of fluctuation of the mean fields, determined by the character of the deeper field motions that have been left out.

As in the case of the Brownian motion of a particle, this fluctuation will determine a probability distribution dP = P(1, 2, . . ., k . . .) d1 d2 . . . dk . . . (5) which yields the mean fraction of the time in which the variables, 1, 2 . . ., k . . ., representing the mean fields in the regions, 1, 2 . . ., k . . . respectively, will be in the range d1 d2 . . . dk . . . (Note that P is in general a multidimensional function, which can describe statistical correlations in the field distributions.)

To sum up, we are reorganizing the non-denumerable infinity of field variables, and we are treating explicitly only some denumerable sets of these reorganized coordinates. We do this by defining a series of levels by average fields, each associated with a certain dimension, over which the averages are taken. Such a treatment can be justified only in those cases in which the denumerable sets of variables form a totality that, within certain limits, determines its own motions independently of the precise details of the non-denumerable infinity of coordinates that has necessarily been left out of account. Such self-determination is never complete, however, and its basic limits are defined by a certain minimum degree of fluctuation over a domain that depends on the coupling of the field coordinates in question to those that have been neglected. Thus we obtain a real and objective limitation on the degree of self-determination of a certain level, along with a probability function that represents the character of the statistical fluctuations which are responsible for the above described limitations on self-determination.

10 HEISENBERG’S INDETERMINACY PRINCIPLE We are now ready to show how Heisenberg’s indeterminacy principle fits into our general scheme. We shall do this by discussing the degree of determinism associated with a spaceaveraged field coordinate, k, and the corresponding average of the canonically conjugate field momentum, πk.

It is immediately clear that the above result shows a strong analogy to Heisenberg’s principle,22 δp δq h. The constant, ab, appearing in Eq. (9) plays the role that Planck’s constant, h, plays in Heisenberg’s principle. The universality of h therefore implies the universality of ab.

Now a is just a constant relating the field momentum to its time derivative and will evidently be a universal constant. The constant, b, represents the basic intensity of the random fluctuation. To suppose that b is a universal constant is the same as to assume that the random field fluctuations are at all places, at all times, and in all levels of size, essentially the same in character.

With regard to different places and times the assumption of the universality of the constant, b, is not at all implausible. The random field fluctuations (which here play a role similar to that of the ‘zero-point’ vacuum fluctuations in the usual quantum theory) are infinitely large, so that any disturbances that might be 114 wholeness and the implicate order made by further localized excitations or concentrations of energy occurring naturally, or produced in a laboratory experiment, would have a negligible influence on the general magnitudes of the basic random fluctuations. (Thus, the presence of matter as we know it on a large scale would mean the concentration of a non-fluctuating part of the energy, associated with a few extra grams per cubic centimetre on top of the infinite zero-point fluctuations of the ‘vacuum’ field.)

With regard to the problem of different levels of space and time intervals, however, the assumption of the universality of b is not so plausible. Thus, it is quite possible that the quantity b will remain constant for fields averaged over shorter and shorter time intervals only down to some characteristic time interval ∆t0, beyond which the quantity b may change. This is equivalent to the possibility that the degree of self-determination may not be limited by Planck’s constant, h, for very short times (and for correspondingly short distances).

We have thus constructed a theory which contains Heisenberg’s relations as a limiting case, valid approximately for fields averaged over a certain level of intervals of space and time. Nevertheless, fields averaged over smaller intervals are subject to a greater degree of self-determination than is consistent with this principle. From this, it follows that our new theory is able to reproduce, in essence at least, one of the essential features of the quantum theory, i.e. Heisenberg’s principle and yet have a different content in new levels.

…the divergencies of present-day field theories are directly a result of contributions to the energy, charge, etc., coming from quantum fluctuations associated with infinitely short distance and times. Our point of view permits us to assume that while the total fluctuation is still infinite, the fluctuation per degree of freedom ceases to increase without limit as shorter and shorter times are considered. In this way, field-theoretical calculations could be made to give finite results. Thus, it is clear already that divergences of the current quantum field theory may come from the extrapolation of the basic principles of this theory to excessively short intervals of time and space.

11 THE INDIVISIBILITY OF QUANTUM PROCESSES

Our next step is to show how quantization, i.e., the indivisibility of the quantum of action, fits into our notions concerning a sub-quantum-mechanical level. To do this, we begin by considering in more detail the problem of just how to define the field averages that are needed for the treatment of a non-denumerable infinity of variables.

Here, we shall guide ourselves by certain results obtained in the very analogous many-body problem (e.g., the analysis of solids, liquids, plasmas, etc., in terms of their constituent atomic particles). In this problem, we are likewise confronted with the need to treat certain kinds of averages of deeper (atomic) variables. The totality of a set of such averages then determines itself in some approximation, while its details are subject to characteristic domains of random fluctuations arising from the lower-level (atomic) motions, in much the same way that was suggested for the averages of the non-denumerable infinity of field variables discussed in the previous sections

Now, in the many-body problem, one deals with large-scale behaviour by working with collective coordinates, 23 which are an approximately self-determining set of symmetrical functions of the particle variables, representing certain overall aspects of the motions (e.g., oscillations). The collective motions are determined (within their characteristic domains of random fluctuation) by approximate constants of the motion. For the special but very widespread case that the collective coordinates describe nearly harmonic oscillations, the constants of the motion are the amplitudes of the oscillations and their initial phases. More generally, however, they may take the form of more complex functions of the collective coordinates.

It is often very instructive to solve for the collective coordinates by means of a canonical transformation. In classical mechanics,24 this takes the form hidden variables in the quantum theory 117 Pk = S qk (q1 . . . qk . . .; J1 . . . Jn ) Qn = S Jn (q1 . . . qk . . .; J1 . . . Jn . . .) (12) where S is the transformation function, pk and qk are the momenta and one coordinates of the particles, and Jn and Qn are the momenta of the collective degrees of freedom. Here, we suppose the Jn to be constants of the motion. In other words, we assume that the transformation is such that, at least in the domain in which the approximation of collective coordinates is a good one, the Hamiltonian is only a function of the Jn, and not of the Qn. It then follows that the Qn increase linearly with time so that they have the properties of the so-called ‘angle-variables’.25

It is clear that a similar attack can be made on the problem of a non-denumerable infinity of field variables subject to a nonlinear coupling with each other. To do this, we now let qk, pk represent the original canonically conjugate set of field variables, and we assume that there will be a set of overall large-scale motions which we represent by the constants of the motion, Jn and the canonically conjugate angle-variables, Qn. It is clear that if such overall motions exist, they will manifest themselves relatively directly in high-level interactions, for by hypothesis, they are the motions that retain their characteristic features for a long time without being lost in the infinitely rapid random fluctuations, which average out to zero on a higher level.

Our next task is to show that the constants of the motion (which are, for harmonic oscillators, proportional to the energy of a large-scale collective degree of freedom) are quantized by the rule J = nh, where n is an integer, and h is Planck’s constant. Such a demonstration will constitute an explanation of the waveparticle duality, since the collective degrees of freedom are already known to be wavelike motions, with harmonically 118 wholeness and the implicate order oscillating amplitudes. In general, these waves will take the form of fairly localized packets, and if these packets have discrete and well-defined quantities of energy, momentum, and other properties, they will at the higher level, reproduce all the essential characteristics of particles. Yet they will have inner wavelike motions which will reveal themselves only under conditions in which there exist systems that can respond significantly to these finer details.

In order to show the quantization of the constants of the motion as described above, we first return to the preliminary interpretation of the quantum theory, given in sections 6 and 7. Here, we encountered a relation very similar to (12). Pk = S qk (q1 . . . qk . . .). (13) The main difference between (4) and (12) is that the former does not contain any constants of the motion, whereas the latter does. But once the constants of the motion are specified, they are just numbers, which need only be given certain values which they thereafter retain. If this is done, the S of Eq. (12) will also no longer contain the Jn as explicitly represented variables. We can therefore regard the S of our preliminary interpretation, (4), as the actual S function, in which the constants of the motion have already been specified. S is then determined by the wave function, ψ = Reis/ . Thus, when we give the wave function, we define a transformation function S = Im (lnψ), which latter determines certain constants of the motion implicitly.

In order to see more clearly how the constants of the motion are determined by the S of Eq. (4) let us construct the phase integral IC = k Cpkδqk . (14) hidden variables in the quantum theory 119 The integral is taken around some circuit C, representing a set of displacement, δqk (virtual or real), in the configuration space of the system. If Eq. (13) applies, we then obtain IC = k S qk δqk = δSC (15) where δSC is the change of S in going around the circuit C. It is well known that the IC, which are the so-called ‘action variables’ of classical mechanics, generally represent the constants of the motion. (For example, in the case of a set of coupled oscillators, harmonic or otherwise, the basic constants of the motion can be obtained by evaluating the IC with suitably defined circuits.)26 The wave function, ψ, which defines a certain function, S, therefore implies a corresponding set of constants of the motion. Now, according to the current quantum theory, the wave function, ψ = ReiS/ , is a single-valued function of all its dynamical coordinates, qk. Thus, we must have δSC = 2ηπ = nh (16) where n is an integer. The actual functions, S, obtained from the wave function, ψ, therefore imply that the basic constants of the motion for the system are discrete and quantized. If the integer, n, is not zero, then as a simple calculation shows, there must be a discontinuity somewhere inside the circuit. But since S = Im (lnψ), and since ψ is a continuous function, a discontinuity of S will generally occur where ψ (and therefore R2 ) has a zero. As we shall see presently, R2 is the probability density for the system to be at a certain point in configuration space. The system will therefore have no probability of being at a 120 wholeness and the implicate order zero of ψ, with the result that the singularities of S will imply no inconsistencies in the theory.

In many ways, the quantization described above resembles the old Bohr-Sommerfeld rule; yet it is basically different in its meaning. Here, the action variable, IC, that is quantized is not obtained by using the simple expression of classical mechanics for the pk in Eq. (14). Rather, it is obtained by using the expression (12), which involves the transformation functions, S, a function that depends on the non-denumerable infinity of variables, qk. In a certain sense, we can say that the old BohrSommerfeld rule would be exactly correct if it were made to refer to the non-denumerable infinity of field variables, instead of just to the values of the variables that one obtains by solving the simple classical equations of motion for a small number of abstracted coordinates, Qn.

Before going ahead to suggest an explanation of why δSC should be restricted to the discrete values denoted by Eq. (16), we shall sum up and develop in a systematic way the main physical ideas to which we have thus far been led.

1. We abstract from the non-denumerable infinity of variables a set of ‘collective’ constants of the motion, Jn and their canonically conjugate quantities, Qn.

2. The Jn can be consistently restricted to discrete integral multiples of h. Thus, action can be quantized.

3. If this set of coordinates determined itself completely, the Qn would (as happens in typical classical theories) increase linearly with time. However, because of fluctuations due to the variables left out of the theory, the Qn will fluctuate at random over the range accessible to them.

4. This fluctuation will imply a certain probability distribution of the Qn having a dimensionality equal to 1 per degree of freedom (and not 2, as is the case for typical classical statistical distributions in phase space). When this distribution is transformed to the configuration space of the qk there will be a hidden variables in the quantum theory 121 corresponding probability function, p(q1 . . . qk . . .), which also has a dimensionality of 1 per degree of freedom (the momenta, pk being always determined in terms of qk by Eq. (12)).

5. We then interpret the wave function ψ = Reis/ , by setting p(q1 . . . qk . . .) = R2 (q1 . . . qk . . .) and by letting S be the transformation function that defines the constants of the motion of the system. It is clear that we have in this way given the wave function a meaning quite different from the one that was suggested in the preliminary interpretation of section 5, even though the two interpretations stand in a fairly definite relation to each other.

6. Because of the effects of the neglected lower-level field variables, the quantities In will, in general, remain constant only for some limited period of time. Indeed, as the wave function changes, the integral around a given circuit, Σk c pk δqk = δSc will change abruptly, whenever a singularity of S (and therefore a zero of ψ) crosses the circuit, C. Hence discrete changes, by some multiple of h will occur in the action variables for non-stationary states.

12 EXPLANATION OF QUANTIZATION OF ACTION

In the previous section, we developed a theory involving a non-denumerable infinity of field variables that has room for the quantization of action according to the usual rules of the quantum theory. We shall now suggest a more definite theory, which will give possible physical reasons explaining why action is quantized by the rules described above, and which will show possible limitations on the domain of validity of these rules.

Our basic problem evidently is to propose some direct physical interpretation of the function, S which appears in the phase of the wave function (as ψ = Reis/ ) and which is, according to our theory, also the transformation function defining the basic constants of the motion (see Eq. (15)); for if we are to explain 122 wholeness and the implicate order why the change of S around a circuit is restricted to discrete multiples of h we must evidently assume that S is somehow related to some physical system, in such a way that e is/ cannot be other than single-valued.

To give S a physical meaning that leads to the property described above, we shall begin with certain modifications of an idea originally suggested by de Broglie.27 Let us suppose that the infinity of non-linearly coupled field variables is in reality so organized that in each region of space and time associated with any given level of size there is taking place a periodic inner process. The precise nature of this process is not important for our discussion here, as long as it is periodic (e.g., it could be an oscillation or a rotation). This periodic process would determine a kind of inner time for each region of space, and it would therefore effectively constitute a kind of local ‘clock’

Now every localized periodic process has, by definition, some Lorentz frame in which it remains at rest, at least for some time (i.e., in which it does not significantly change its mean position during this time). We shall further assume that, in this frame, neighbouring clocks of the same level of size will tend to be nearly at rest. Such an assumption is equivalent to the requirement that, in every level of size, the division of a given region into small regions, each containing its effective clock, has a certain regularity and permanence, at least for some time. If these clocks are considered in another frame (e.g., that of the laboratory), every effective clock will then have a certain velocity, which can be represented by a continuous function v(x, t).

It is now quite natural to suppose: (1) that in its own rest frame each clock oscillates with a uniform angular frequency, which is the same for all clocks, and (2), that all clocks in the same neighbourhood are, on the average, in phase with each other. In homogeneous space there can be no reason to favour one clock over another, nor can there be a favoured direction of space (as hidden variables in the quantum theory 123 would be implied by a non-zero average value for ∇ in the rest frame). We can therefore write δ = ω0δτ (17) where δτ is the change of proper time of the clock, and where δ is independent of δx in this frame

The reason for the equality of clock phases in the rest frame for a neighbourhood can be understood more deeply as a natural consequence of the non-linearity, of the coupling of the neighbouring clocks (implied by the general non-linearity of the field equations). It is well known that two oscillators of the same natural frequency tend to come into phase with each other when there is such a coupling.28 Of course, the relative phase will oscillate somewhat, but in the long run, and on the average, these oscillations will cancel out.

Let us now consider the problem in some fixed Lorentz frame, e.g., that of the laboratory

We see, then, that quantization of action can, at least in this special case, arise out of certain topological conditions, implied by the need for single-valuedness of the clock phases. The above idea provides a starting point for a deeper understanding of the meaning of the quantum conditions, but it needs to be supplemented in two ways. First, we must consider the further fluctuations in the field, associated with the nondenumerable infinity of degrees of freedom. Second, we shall have to justify the assumption that the ratio m0c 2 /ω0 in Eq. (20) is universal for all the local clocks and equal to h

To begin with, we recall that each local clock of a given level exists in a certain region of space and time, which is made up of still smaller regions, and so on without limit. We shall see that we can obtain the universability of the quantum of action, h, at all levels, if we assume that each of the above sub-regions contains an effective clock of a similar kind, related to the other effective clocks of its level in a similar way, and that this effective clock structure continues indefinitely with the analysis of space and time into sub-regions. We stress that this is only a preliminary assumption, and that later we will show that the notion of the indefinite continuation of the above clock structure can be given up

To treat this problem, we introduce an ordered infinity of dynamic coordinates, x l i , and the conjugate momenta, p l i . The mean position of the ith clock at the lth level of size is represented by x l i , and p l i represents the corresponding momentum. To a first approximation the quantities of each level can be treated as collective coordinates of the next lower level set of variables; but this treatment cannot in general be completely exact because each level will to some extent be influenced directly by all the other levels, in a way that cannot fully be expressed in terms of their effects on the next lower level quantities alone. Thus, while each level is strongly correlated to the mean behaviour of the next lower level, it has some degree of independence.

The above discussion leads us to a certain ordering of the infinity of field variables that is indicated by the nature of the problem itself. In this ordering, we consider the series of quantities, x l i and p l i , defined above as, in principle, all independent coordinates and momenta which are, however, usually connected and correlated by suitable interactions

We can now treat this problem by means of a canonical transformation. We introduce an action function, S, which depends on all the variables, x l i , of the infinity of clocks within clocks. As before, we then write Pl k = S x l k (x l i . . . x l k . . .) (21) where l′ represents all possible levels. For the constants of the motion, we write IC = k,l p l kδxl k = δSC (22) where the integrals are carried over suitable contours. Each of these constants of the motion is now built up out of 126 wholeness and the implicate order circuit integrals involving pi δxi , but as we saw, each one of these clocks must satisfy the phase condition pµ δx µ = 2nπ around any circuit. Hence the sum satisfies such a condition, which in turn must be satisfied not only in real circuits actually traversed by the clocks but also in any virtual circuit that is consistent with a given set of values for the constants of the motion. Because of fluctuations coming from lower levels, there is always the possibility that any clock may move on any one of the circuits in question; and unless the constants of the motion are determined such that δSc = 2nπ, clocks that reach the same position after having followed different randomly fluctuating paths will not, in general, agree with each other in their phases. Thus, the agreement of the phases of all clocks that reach the same point in space and time is equivalent to the quantum condition.

The self-consistency of the above treatment can now be verified in a further analysis, which also eliminates the need to introduce the special assumption that m0c 2 /ω0 is universally constant and equal to for all clocks. Each clock is now regarded as a composite system made up of smaller clocks. Indeed, to an adequate degree of approximation, each clock phase can be treated as a collective variable associated with the space coordinates of the smaller clocks (which then represent the inner structure of the clock in question). Now the action variable Ic = c k,l p l kδq l k is canonically invariant, in the sense that it takes the same form with every set of canonical variables, and is not changed in its value by a canonical transformation. Hence, if we transformed to the collective coordinates of any given level, we would still obtain the same kind of restriction Ic to integral multiples of h, even if Ic were expressed in terms of the collective variables. Thus hidden variables in the quantum theory 127 the collective variables of a given level will generally be subject to the same quantum restriction as those satisfied by the original variables of that level. In order that it be consistent for variables of a given level to be essentially equal to collective variables for the next lower level, it is sufficient that the variables of all levels be quantized in terms of the same unit of action, h. In this way, an overall consistent ordering of the non-denumerable infinity of variables becomes possible.

Each clock will then have a quantized value for the action variable, Ic , associated with its inner motion (i.e., of its phase changes). This inner motion was, however, assumed to be effectively that of a harmonic oscillator.

To finish this stage of the development of the theory, we must show that the model discussed above leads to a fluctuation in the phase space of the variables of a given level, in accordance with that implied by Heisenberg’s principle. In other words, the quantum of action, h, must also be shown to yield a correct estimate of the limitation on the degree of self-determination of the quantities of any level

To prove the above conjecture, we must note that each variable fluctuates because it depends on the lower-level quantities (of which it is a collective coordinate). The lower-level quantities can change their action variables only by discrete multiples of h. It is therefore not implausible that the domain of fluctuation of a given variable would be closely related to the size of the possible discrete changes in its constituent lower-level variables.

We shall prove the theorem stated above for the special case that all the degrees of freedom can be represented as coupled harmonic oscillators. This is a simplification of the real problem (which is non-linear). The real motions will consist of small systematic perturbations on top of an infinitely turbulent background. These systematic perturbations can be treated as collective coordinates, representing the overall behaviour of the constituent local clocks of a given level. In general, such a collective motion will take the form of a wavelike oscillation, which to a certain degree of approximation undergoes simple harmonic motion.

Let us represent the action variables and angle variables of the nth harmonic oscillator by Jn and n respectively. To the extent that the linear approximation is correct, Jn will be a constant of the motion, and n will increase linearly with time according to the equation n = ωnt + 0n, where ωn is the angular frequency of the nth oscillator. Jn and n will be related to the clock variables by a canonical transformation, such as (12). Because the generalized Bohr-Sommerfeld correlation (16) is invariant to a canonical transformation, it follows that Jn = Sh, where S is an integer. Moreover, the coordinates and momenta of these oscillators can be written as29 pn = 2 J n cosn, qn = 2 J n sinn. We now consider a higher level canonical set of variables, a hidden variables in the quantum theory 129 specific pair of which we denote by Ql i and πl i . In principle, these would be determined by the totality of all the other levels. To be sure, the next lower level will be the main one that enters into this determination; nevertheless, the others will still have some effect. Hence in accordance with our earlier discussions, we must regard πl i and Ql i as being, in principle, independent of any given set of lower-level variables, including, of course, those of the next lower level. To the extent that the linear approximation is valid, we can write30 Ql i =n αinpn = 2 n αin Jn cosn πl i =n βinqn = 2 n βin J n sinn (25) where αin and βin are constant coefficients, and where, as we recall, n is assumed to cover all levels other than l. In order that it be consistent to suppose that Ql i and πl i are canonical conjugates, it is necessary that their Poisson bracket be unity or that n πl i Jn Ql i n − πl i n Ql i Jn = 1. With the aid of Eq. (25), this becomes Σαn βn = 1. (26) Eq. (25) implies a very complex motion for Ql i and πl i , for in a typical system of coupled oscillators the ωn are in general all different and are not integral multiples of each other (except for 130 wholeness and the implicate order possible sets of measure zero). Thus, the motion will be a ‘spacefilling’ (quasi-ergodic) curve in phase space, being a generalization of the two-dimensional Lissajou figures for perpendicular harmonic oscillators, with periods that are not rational multiples of each other. During a time interval, τ, which is fairly long compared with the periods, 2π/ωn, of the lower-level oscillators, the trajectory of Ql i and πl i in the phase space will, in essence, fill a certain region, even though the orbit is definite at all times. We shall now calculate the mean fluctuation of Ql i and πl i in this region by taking averages over the time, τ. Noting that Ql i = πl i = 0 for such averages, we have for these fluctuations, (∆Ql i ) 2 = 4 mn αmαn Jm Jn cosm cosn = 2 m (αm) 2 Jm (27) (∆πl i ) 2 = 4 mn βmβn JmJn sinm sinn = 2 n (βn) 2 Jn (28) where we have used the result that cos δm cos δn = sin δm sin δn = 0 for m ≠ n (except for the set of zero measure, mentioned above, in which ωm and ωn are rational multiples of each other). We now suppose that all the oscillators are in their lowest states (with J = h) except for a set of zero measure. This set represents a denumerable number of excitations relative to the ‘vacuum’ state. Because of their small number, these make a negligible contribution to (∆Ql i ) 2 and (∆πl i ) 2 . We therefore set Jn = h in Eq. (28) and obtain (∆Ql i ) 2 = 2 m (αm) 2 h; (∆πl i ) 2 = 2 n (βn) 2 h. We then use the Schwarz inequality hidden variables in the quantum theory 131 mn (αm) 2 (βn) 2 | m αmβm |2 . (29) Combining the above with Eqs. (26), (27) and (28), we obtain (∆πl i ) 2 (∆Ql i ) 2 4h2 . (30) The above relations are, in essence, those of Heisenberg. ∆πl i and ∆Ql i will effectively represent limitations on the degree of self-determination of the lth level, because all quantities of this level will evidently have to be averaged over periods of time long compared with 2π/ωn. Thus, we have deduced Heisenberg’s principle from the assumption of the quantum of action. We note that Eq. (30) has already been obtained in section 10 in a very different way – by assuming simple random field fluctuations resembling those of particles undergoing Brownian motion. Hence, an infinity of lower-level variables satisfying the conditions that Jn is discrete and equal to the same constant, h, for all the variables, will yield a long-run pattern of motions that reproduces certain essential features of a random Brownian-type fluctuation

We have thus completed our task of proposing a general physical model that explains the quantization rules along with the Heisenberg indeterminacy relations. But now, it can easily be seen that our basic physical model, involving an infinity of clocks within clocks, leaves room for fundamental changes, which would go outside the scope of the current quantum theory. To illustrate these possibilities, suppose that such a structure were to continue only for some characteristic time, τ0, after which it would cease to exist and would be replaced by another kind of structure. Then, in processes that involve times much greater then τ0, the clocks will still be restricted in essentially the same ways as before, since their motions would not significantly 132 wholeness and the implicate order be changed by the deeper substructure. Nevertheless, in processes involving times shorter than τ0, there will be no reasons for such restrictions to apply, since the structure is no longer the same. In this way, we see how Jn will be restricted to discrete values in certain levels, while they are not necessarily restricted in this way in other levels.

13 DISCUSSION OF EXPERIMENTS TO PROBE SUB-QUANTUM LEVEL We are now ready to discuss, at least in general terms, the conditions under which it might be possible to test for a sub-quantum level experimentally, and in this way to complete our answers to the criticisms of the suggestion of hidden variables made by Heisenberg and Bohr.

We first recall that the proof of Heisenberg’s relations, concerning the maximum possible accuracy of measurement of canonically conjugate variables, made use of the implicit assumption that measurements must involve only processes satisfying the general laws of the current quantum theory. Thus, in the well-known example of the gamma-ray microscope, he supposed that the position of an electron was to be measured by scattering a gamma ray from the particle in question into a lens and on to a photographic plate. This scattering is essentially a hidden variables in the quantum theory 133 case of the Compton effect; and the proof of Heisenberg’s principle depended essentially on the assumption that the Compton effect satisfies the laws of the quantum theory (i.e., conservation of energy and momentum in an ‘indivisible’ scattering process, wavelike character of the scattered quantum as it goes through the lens, and incomplete determinism of the particle-like spot on the photographic plate). More generally, any such proof must be based on the assumption that at every stage the process of measurement will satisfy the laws of the quantum theory. Thus to suppose that Heisenberg’s principle has a universal validity is, in the last analysis, the same as to suppose that the general laws of the quantum theory are universally valid. But this supposition is now expressed in terms of the external relations of the particle to a measuring apparatus, and not in terms of the inner characteristics of the particle itself

In our point of view, Heisenberg’s principle should not be regarded as primarily an external relation, expressing the impossibility of making measurements of unlimited precision in the quantum domain. Rather, it should be regarded as basically an expression of the incomplete degree of self-determination characteristic of all entities that can be defined in the quantummechanical level. It follows that if we measure such entities, we will also use processes taking place in the quantum-mechanical level, so that the process of measurement will have the same limits on its degree of self-determination as every other process in this level. It is rather as if we were measuring Brownian motion with microscopes that were subject to the same degree of random fluctuation as that of the systems that we were trying to observe.

As we saw in sections 10 and 12, however, it is possible and indeed rather plausible to suppose that sub-quantummechanical processes involving very small intervals of time and space will not be subject to the same limitations of their degree of self-determination as those of quantum-mechanical 134 wholeness and the implicate order processes. Of course, these sub-quantum processes will very probably involve basically new kinds of entities as different from electrons, protons, etc., as the latter are from macroscopic systems. Hence, entirely new methods would have to be developed to observe them (as new methods also had to be developed to observe atoms, electrons, neutrons, etc.). These methods will depend on using interactions involving sub-quantum laws. In other words, just as the ‘gamma-ray microscope’ was based on the existence of the Compton effect, a ‘sub-quantum microscope’ would be based on new effects, not limited in their degree of self-determination by the laws of the quantum theory. These effects would then make possible a correlation between an observable large-scale event and the state of some sub-quantum variable that is more accurate than is permitted in Heisenberg’s relations.

Of course, one does not expect, in the way described above, to actually determine all the sub-quantum variables and thus to predict the future in full detail. Rather, one aims only by a few crucial experiments to show that the sub-quantum level is there, to investigate its laws, and to use these laws to explain and predict the properties of higher-level systems in more detail, and with greater precision, than the current quantum theory does. To treat this question in more detail, we now recall a conclusion of the previous section, i.e. that if in lower levels the action variable should be divisible in units smaller than h, then the limits on the degree of self-determination of these lower levels can be less severe than those given by Heisenberg’s relations. Thus, there may well be relatively divisible and self-determined processes going on at lower levels. But how can we observe them on our level? To answer the above question, we refer to Eq. (25), which indicates in typical case how the variables of a given level depend to some extent on all the lower-level variables. Thus if πl i and Ql i represent the classical level, then they would, in general, be hidden variables in the quantum theory 135 determined mainly by the p l i and q l i of the quantum level; but there would be some effects due to sub-quantum levels. Usually these would be quite small. However, in special cases (e.g., with special arrangements of apparatus) the πl i and Ql i might depend significantly on the p l i and q l i of a sub-quantum level. Of course, this would mean the coupling of some new kind of subquantum process (as yet unknown, but perhaps to be discovered later) to the observable large-scale classical phenomena. Such a process would presumably involve high frequencies and therefore high energies, but in a new way

Even when the effects of the sub-quantum level on πl i and Ql i are small, they are not identically zero. Thus, room is created for testing for such effects by doing old kinds of experiments with extremely high precision. For example, the relation Jn = nh was obtained in Eq. (24) only if the quantum of action was supposed to be universally equal to h (at all levels). Sub-quantum deviations from this rule would therefore be reflected in the classical level as a minute error in the relation E = nhv for a harmonic oscillator. In this connection, recall that in classical theory there is no special relation between energy and frequency at all. This situation may to some extent be restored in the sub-quantum domain. As a result, one would discover a small fluctuation in the relation between En and nhv. For example, one would have En = nhν + ∈ where ∈ is a very small randomly fluctuating quantity (which gets larger and larger as we go to higher and higher frequencies). To test for such a fluctuation, one could perform an experiment in which the frequency of a light beam was observed to an accuracy, ν. If the observed energy fluctuated by more than ν, and if no source could be found for the fluctuation in the quantum level, this experiment could be taken as an indication of sub-quantum fluctuations.

With this discussion, we complete our answer to the criticisms of Bohr and Heisenberg, who argue that a deeper level of hidden variables in which the quantum of action was divisible could never be revealed in any experimental phenomena. This also means that there are no valid arguments justifying the conclusion of Bohr that the concept of the detailed behaviour of matter as a unique and self-determining process must be restricted to the classical level only (where one can observe fairly directly the behaviour of the large-scale phenomena). Indeed we are also able to apply such notions in a sub-quantum level, whose relations with the classical level are relatively indirect, and yet capable, in principle, of revealing the existence and the properties of the lower level through its effects on the classical level.

Finally, we consider the paradox of Einstein, Rosen and Podolsky. As we saw in section 4, we can easily explain the peculiar quantum-mechanical correlations of distant systems by supposing hidden interactions between such systems, carried in the sub-quantum level. With an infinity of fluctuating field variables in this lower level, there are ample motions going on that might explain such a correlation. The only real difficulty is to explain how the correlations are maintained if, while the two systems are still flying apart, we suddenly change the variable that is going to be measured by changing the measuring apparatus for one of the systems. How, then, does the far-away system receive instantaneously a ‘signal’ showing that a new variable is going to be measured, so that it will respond accordingly? To answer this question, we first note that the characteristic quantum-mechanical correlations have been observed experimentally with distant systems only when the various pieces of observing apparatus have been standing around so long that there has been plenty of opportunity for them to come to equilibrium with the original system through sub-quantummechanical interactions.31 For example, in the case of the molecule described in section 4, there would be time for many hidden variables in the quantum theory 137 impulses to travel back and forth between the molecule and the spin-measuring devices, even before the molecule disintegrated. Thus, the actions of the molecule could be ‘triggered’ by signals from the apparatus, so that it would emit atoms with spins already properly lined up for the apparatus that was going to measure them. In order to test the essential point here, one would have to use measuring systems that were changed rapidly compared with the time needed for a signal to go from the apparatus to the observed system and vice versa. What would really happen if this were done is not yet known. It is possible that the experiments would disclose a failure of the typical quantum-mechanical correlations. If this were to happen, it would be a proof that the basic principles of the quantum are breaking down here, for the quantum theory could not explain such a behaviour, while a sub-quantum theory could quite easily explain it as an effect of the failure of sub-quantum connections to relate the systems rapidly enough to guarantee correlations when the apparatus was changed very suddenly. On the other hand, if the predicted quantum-mechanical correlations are still found in such a measurement, this is no proof that a sub-quantum level does not exist, for even the mechanical device that suddenly changes the observing apparatus must have sub-quantum connections with all parts of the system, and through these a ‘signal’ might still be transmitted to the molecule that a certain observable was eventually going to be measured. Of course, we would expect that at some level of complexity of the apparatus, the sub-quantum connections would cease to be able to do this. Nevertheless, in the absence of a more detailed sub-quantum-mechanical theory, where this will happen cannot be known a priori. In any case, the results of such an experiment would surely be very interesting.

14 CONCLUSION

In conclusion, we have carried the theory far enough to show that we can explain the essential features of the quantum mechanics in terms of a sub-quantum-mechanical level involving hidden variables

In conclusion, we have carried the theory far enough to show that we can explain the essential features of the quantum mechanics in terms of a sub-quantum-mechanical level involving hidden variables. Such a theory is capable of having a new experimental content, especially in connection with the domain of very short distances and very high energies, where there are new phenomena not very well treated in terms of present theories (and also in connection with the experimental verification of certain features of the correlations of distant systems). Moreover, we have seen that this type of theory opens up new possibilities for elimination of divergence in present theories also associated with the domain of short distances and high energies. (For example, as shown in section 10, the breakdown of Heisenberg’s principle for short time could eliminate the infinite effects of quantum fluctuations.) Of course, the theory as developed here is far from complete. It is necessary at least to show how one obtains the many-body Dirac equation for fermions, and the usual wave equations for bosons. On these problems much progress has been made but there is no space to enter into a discussion of them here. In addition, further progress is being made on the systematic treatment of the new kinds of particles (mesons, hyperons, etc.) in terms of our scheme. All of this will be published later and elsewhere. Nevertheless, even in its present incomplete form, the theory does answer the basic criticisms of those who regarded such a theory as impossible, or who felt that it could never concern itself with any real experimental problems. At the very least, it does seem to have promise of being able to throw some light on a number of such experimental problems, as well as on those arising in connection with the lack of internal consistency of the current theory. hidden variables in the quantum theory 139 For the reasons described above, it seems that some consideration of theories involving hidden variables is at present needed to help us to avoid dogmatic preconceptions. Such preconceptions not only restrict our thinking in an unjustifiable way but also similarly restrict the kinds of experiments that we are likely to perform (since a considerable fraction of all experiments is, after all, designed to answer questions raised in some theory). Of course, it would be equally dogmatic to insist that the usual interpretation has already exhausted all of its possible usefulness for these problems. What is necessary at the present time is that many avenues of research be pursued, since it is not possible to know beforehand which is the right one. In addition, the demonstration of the possibility of theories of hidden variables may serve in a more general philosophical sense to remind us of the unreliability of conclusions based on the assumption of the complete universality of certain features of a given theory, however general their domain of validity seems to be.