Bohm first achieved fame in attempting to utilize the concept of the indivisibility of all material processes with his excursions into what is dubbed the "hidden variables theory." One can approach this theory from the question of indeterminacy as depicted in the "uncertainty principle" at the subatomic level: that we cannot, for instance, know with precision the position and momentum of an electron at any one moment of time, that, in the words of Basil Hiley, "not all of actuality can be made manifest together."[1] To what is the indeterminacy due?[2] The hidden variables theory represents the category of answer which suggests, and in doing so follows the endorsement of Albert Einstein, that it is due to a temporary ignorance on the part of us humans.[3] Observed results are in fact determined by variables hidden from us, escaping our observation.
A totally new theoretical structure will arise, Bohm hoped, which will restore determinism by demonstrating that apparent randomness at one level can be described as the statistical averaging of behavior resulting from exact laws at a lower level, and that the randomness is due to inherent experimental or conceptual limitations or an indeterminism in nature itself at that particular level. It was believed that the limitations which were expressed by the leading figures in quantum theory were really inessential to the theory itself and came about simply because no one had expressed the theory in a general enough form. Furthermore, it was felt that the most obvious first step after encountering the limitations of the quantum theory would be to assume that it was incomplete and that there must exist something more general which would embrace this as a special case. Bohm's thesis concerns the possibility of using hidden variables to fill out quantum wave mechanics into a deterministic theory.[4]
Such a determinism is not a Laplacian determinism in the sense of reducing all laws to causal laws, but rather involves statistical laws arising from fluctuations at a deeper level.[5] Stanley Gudder likens it to statistical mechanics in which it is impractical if not impossible to describe the motion of each individual particle in a container.[6] One uses such macroscopic quantities as temperature, pressure and volume to make statistical predictions about each individual particle's motion. But in the case of classical statistical mechanics one uses probabilities, not because of some inherent properties of the system, but only because of human inadequacies; one would not use probabilities if it were possible to describe the state of the system of particles more precisely. The hidden variables proponents say something similar about quantum mechanics: the condition of a quantum mechanical system could be determined by certain (as yet unknown) hidden variables so precisely that exact predictions would be possible and statistical analyses would not be needed.
Bohm would have us return to the conceptual foundations of quantum mechanics in order to go beyond it to a new and deeper level; for this level we would need to accept a new range of sub-quantum entities and ideas.[7]
A variety of hidden variables theories have been put forward, Bohm's name having been closely associated with two versions. An important precursor to Bohm's is the theory of Louis de Broglie; the one-body treatment of the hidden variables theory of Bohm is very similar to that of de Broglie.[8] The first hidden variables theory of Bohm saw the light of the publishing day in 1952, and went through the usual procedure of defense and restatement, including an immediate response to W. Pauli's objections to de Broglie's theory.[9] I will in this chapter explain this proposal, its successor, criticisms of it, and its relationship to his more general holistic ideas which will be defined and developed in later chapters, and which it has been his more recent intention to expound.[10] I shall not present the mathematical details of the theories but concentrate on their philosophical and physical assumptions and consequences.
Previous to presenting his hidden variables ideas Bohm was engaged in other physics, culminating perhaps in his 1951 book Quantum Theory which is still quoted as a standard text (it "was not only orthodox in the Copenhagen sense but one of the clearest and fullest, most penetrating and critical presentations of the Copenhagen point of view ever published," writes Karl Popper).[11] Obviously I am picking up the strands of only part of his opus; however, one can look for points in Quantum Theory which indicate the beginning of his holistic ideas (despite his writing in it about the "unlikelihood of completely deterministic laws on a deeper level").[12]
He says that he wrote the book from Niels Bohr's point of view in order mainly to try to understand quantum theory. But, he adds, after he had written the book, he felt that he still did not really understand the theory, and so began looking for new approaches. One can see the marked impression Bohr's thinking has had on his holistic thinking right through to the present. After several discussions with Einstein, he felt even more certain that there was something fundamental missing in quantum theory. He shortly came upon the basic ideas of his hidden variables theory and developed them.[13]
The hidden variables theory can be said to approach the usual quantum theory as a limiting approximation, in much the same way as classical physics is approached by the quantum theory when dealing with large numbers. However, the new theory would predict "qualitatively new properties of matter" in the "deeper sub-quantum mechanical level," as Bohm calls it. Writing in 1957, and it must be remembered that much water has passed under the bridge of physics since then, he suggests that this deeper level may be of assistance in explaining phenomena which the quantum theory finds problematic, namely in the 10-13 cm or less region associated with very high energies, the domain in which elementary particles are created and destroyed. Perhaps there "is some kind of inner structure for electrons, protons and neutrons, etc., the motions of which could explain the transformation of these latter particles."[14] The usual approach had been to treat elementary particles such as electrons as points mathematically, but Bohm's approach was to see them rather as extended structures, there being hidden variables underlying them.[15] That elementary particles are extended structures has become an important contemporary research question in the guise of superstring theories.[16]
The question of the possibility of experimental proof for this challenger to the established theory became very important. Bohm writes in his initial exposition on hidden variables that while the usual interpretation of quantum theory is consistent, it is based on an experimentally untestable assumption, namely that the most complete specification possible for a system can only be in terms of a wave function, and that this can only determine probable results from actual processes of measuring. He considers that there is only one way to investigate the truth of this assumption and that is "to find some other interpretation of the quantum theory in terms of at present 'hidden' variables, which in principle determine the precise behavior of an individual system, but which are in practice averaged over in measurements of the types that can now be carried out."[17]
In a later article Bohm shows that the discrepancies between the two interpretations would show themselves at the 10-13 centimeter level and "whose cumulative effects should be felt even at the atomic level," but die out as a result of random collisions under normal circumstances making the differences negligible; the causal decays into the usual.[18] Some experimental conditions can also be anticipated, he considers, in which differences might become appreciable, although his earlier point seems to prevail, that, so long as the mathematical theory is retained in the form it was currently in, the two theories lead to precisely the same experimental results.[19]
Bohm explains in more detail why his theory cannot be tested within the confines of the current quantum theory: the detailed hidden variables fluctuation process cannot be tested experimentally if the quantum mechanical laws are valid, because it happens within a region defined by the order of Planck's constant which is also the maximum degree of accuracy possible for any measurement according to Werner Heisenberg's uncertainty principle. However, as will be elaborated below, Bohm tells us that Heisenberg's conclusion depends on the assumption that there is no sub-quantum level; thus it is possible that there is a domain in which Heisenberg's principle is not valid and that the fluctuations may be observable in detail. To show this will require us to observe properties which are different from those of the domain of quantum mechanics; in 1957 Bohm had little idea of what the entities of the sub-quantum mechanical level might be because the quantum level laws are not significantly dependent on their properties. Experiments to study the basic laws determining the behavior of matter must directly reflect these properties and therefore will be different kinds of experiments than those involving the properties of quantum-mechanical entities. It is chiefly up to the hidden variables theory to provide the guidance towards such new experiments – Bohm considers it highly unlikely that they could be discovered without the assistance of such a theory.[20]
John Bell tells us that that this first version by Bohm of nonrelativistic quantum mechanics is experimentally the same as the usual version but it does not require the division of the world into apparatus and system, or of history into measurement and nonmeasurement. It is a theory which brings out the nonlocality of quantum theory, he says; nonlocality is an important topic which we will be returning to many times.
For Bohm the deterministic hidden variables competitor to quantum theory is philosophically preferred because it allows a continuous and precise description of all processes including those at the quantum level, which in turn provides for more general mathematical formulations of quantum theory than those which the usual interpretation allows. As was mentioned above, Bohm hoped that his broader conceptual framework would resolve what he calls the insoluble difficulties which the usual mathematical formulation of quantum theory faced when extrapolated into the 10-13 cm domain. He considers that in any case just the possibility of this type of interpretation implies that it is unnecessary to forego having at the quantum level an objective description for individual systems.[21] The relationship between Bohm's hidden variables theories and the usual interpretation of the quantum mechanics formalism is an important topic to which I will return below.
Even though Bohm was able to sidestep the hammer of a theorem of John von Neumann which shows quantum theory to be inconsistent with the existence of hidden variables[22] – another topic to which I will return in detail below – his first attempt at a hidden variables or causal theory found itself sitting tidily on a shelf chiefly because it failed to be extendable to relativistic phenomena, despite his earlier optimism.[23]
The 1952 theory was proposed merely to demonstrate, by giving a
counter-example to von Neumann's proof, the possibility of a hidden variables
theory. Bohm did not seriously envisage it as an ideal
theory and consequently to his mind it suffered structurally from a lack of
elegance and simplicity. At least that is how Bohm
and Jeffrey Bub perceived it when they were
presenting Bohm's second major elaboration of a
hidden variables theory in 1966: "We now wish to propose a hidden
variables theory. . .which we consider is at least a
step toward what could seriously be considered as a tentative theory, insofar
as it suggests certain questions which we feel to be relevant."[24]
Like its predecessor, it became the focus of further examination.[25]
It is interesting to note, also, that the
In this 1966 theory Bohm and Bub proposed an equation of motion which was new and deterministic. The equation describes the wave packet collapse in a non-relativistic theory, put forth – as was the 1952 theory – in order to show that a hidden variables theory could, logically, exist.[27] In quantum theory only the probability of a collapse to a given state can be predicted – one cannot describe the collapse process itself.
Bohm and Bub proposed that a quantum system be characterized by a hidden variable vector in addition to the quantum mechanical state vector, the former being different for each individual system, so that some insight be gained into this "measurement problem," as it is called. When a measurement is made, the state vector collapses into what is termed a particular "eigenstate" which is determined by the value of the hidden variable vector. The theory describes a type of coupling between the observed system and the measuring instrument which explains and details how the wave packet is reduced in a measurement in a continuous and causally determined way. The usual statistical results of quantum mechanics can be recovered as a special case by averaging over the hidden parameters.[28]
Luc Longtin and Richard Mattuck elaborate on the significance of this theory: "It should be emphasized that BB's hidden variables are not merely a complicated way of reproducing quantum results. They were introduced to do something which quantum mechanics cannot do; i.e., they can account for the fact that after the experiment the system is generally described by a new state vector."[29] The Bohm-Bub theory also provides the possibility of being able to conceive of new experiments which could show its differences from the usual linear quantum mechanics (the collapse of the state vector in the former theory is governed by a nonlinear equation). I shall return to the question of experimental support for the Bohm-Bub theory later.
Longtin and Mattuck
raise again the characterization made by Frederik Belinfante in his survey of hidden variables theories, that
the Bohm-Bub theory is "the most promising"
of the various hidden variables theories which are designed to reproduce the
observed probabilities predicted by quantum mechanics. It does however have a
problem, which Bohm's earlier version also had,
namely that it is not relativistically covariant –
although Bohm did hope that it would be generalizable to a relativistic theory. This leads to such
difficulties as observers in different inertial frames obtaining different
results for certain experiments – a violation of the principle of relativity.
To have a meaningful description for the structure of events the authors feel
it necessary that observers should agree on the results obtained, and this is
their motivation for attempting to develop a relativistic generalization of the
Bohm-Bub equation for the collapse of the wave
function. They quote Belinfante's optimistic
statement that "there is no a priori reason why [constructing a Lorentz covariant Bohm-Bub
equation] could not be done. . . .The theory is wide open for improvements."[30]
Later I will disclose the results of this motivation. I will also raise the
measurement question again when discussing the
The Bohm-Bub version of a hidden variables theory, like Bohm's 1952 version, sidesteps the problems raised by von Neumann against a hidden variables theory in which it is supposedly shown that hidden variables are excluded from quantum mechanics.[31] Comments N. R. Hanson: "De Broglie advanced a similar argument [to Bohm's] in 1926, 1927 and 1930; he was pulverized by Pauli. Madelung made a similar suggestion, but von Neumann's critique seemed to make the future of any such position quite hopeless. But Bohm remains undaunted."[32] Von Neumann's critique leads to a class of criticisms of the hidden variables theory in addition to those to do with its being relativistically covariant, namely those attempting to produce nonexistence proofs for hidden variables of the type posed by Bohm and by Bohm and Bub, proofs which are extensions of von Neumann's theorem.[33] Despite an amount of controversy, the theory also appears to sidestep these nonexistence proofs – those raised, for instance, by A.M. Gleason (1957), Josef M. Jauch and C. Piron (1963), B. Misra (1967), S. Kochen and E. P. Specker (1967), and Gudder (1968). Jauch and Piron, for instance, claim to reduce von Neumann's assumptions "to the minimum needed to make some valid inferences concerning the hidden variables problem";[34] Gudder extended and revised this to say that the only systems which can admit hidden variables are those within classical mechanics, that there are no physically interesting quantum mechanical systems which admit hidden variables.[35]
The respondents to these attacks claim that the counter-proofs contain
circularities.[36]
Bohm and Bub, for instance,
indicate what they believe to be a circularity in Jauch
and Piron's argument: even though their assumptions
are weaker than those of von Neumann, their conclusions about the nonexistence
of hidden variables appear to follow from a false assumption which in turn
appears to be equivalent to assuming their conclusion that current quantum
mechanics is the only theory which correctly describes experimental results.[37]
The 1966 paper by
Misra addresses the same question. He summarizes
the criticisms raised against von Neumann's point that the existence of hidden
variables contradicts the empirically verified predictions of quantum
mechanics, built as it is on two assumptions both of which are open to
criticism. One criticism is that von Neumann's argument is circular because his
postulates assume the universal validity of the uncertainty principle, which in
itself will exclude the existence of hidden variables (I will elaborate on this
below when I discuss the
This criticism is all the more devastating because von Neumann's assumptions are bereft of physical justifications. The only justification [sic] of these assumptions is the a posteriori one that they lead to the usual formalism of quantum mechanics. Such a justification, which is sufficient from an empirical point of view, has little compelling force in the context of the hidden-variables problem. For one is now concerned with the possibility of generalizing the usual formalism of quantum mechanics and the mere fact that a set of postulates leads to the usual formalism cannot be a sufficient recommendation for these postulates.
Misra continues by informing us that there are only two ways of approaching the hidden variables problem once the criticism of von Neumann's analysis is accepted. The first involves attempting to reformulate quantum mechanics assuming the existence of hidden variables and the possibility of a complete determinism at the micro-level. Bohm's 1952, and Bohm and Bub's 1966 approaches are quoted as attempting this. But he considers their works to be "somewhat obscure" logically, and he provides three examples of this: he does not find that the way in which the concept of the state of a system is extended because of its having hidden variables is clearly delivered; he finds a logical confusion between the theory's kinematic and dynamical aspects; and thirdly, he expects there to be problems when the theory is confronted with the many body situation. His conclusion is that the above-mentioned works "cannot be said to have provided a definitive answer to the question of hidden variables."
The only alternative he finds is left "is to proceed axiomatically in the spirit of von Neumann," but with less stringent assumptions than those made by von Neumann. In fact his aim is to isolate the weakest postulates possible which the existence of hidden variables would violate; once this has been achieved, a decision can be made how (if indeed it is desirable) they might be changed so as to accommodate hidden variables.
The reasons Misra raises against the hidden variables theories of Bohm, and Bohm and Bub do not seem convincing; they could be just as strong in motivating a person to improve those theories. However, Misra proceeds to outline and criticize the approaches of Gleason, and of Jauch and Piron, and to build to his own version.
Gudder's later approaches are similar in the sense of showing that some hidden variables approaches are ruled out by the proofs of von Neumann and his successors, but that a type of hidden variables theory is possible for quantum mechanics (for instance, Bohm and Bub's, Einstein's, and his own) even though these theories "do not seem to evoke any simplifications or advantages and are rarely used."[42] (He may have more recently changed his mind.)[43]
To return to the question of whether Bohm's and Bohm and Bub's hidden variables theories should carry this nomenclature, perhaps J. H. Tutsch's terminological note is apt. He writes that the "name 'hidden variable' is a poor one, since it has so many meanings. I intend to use it only to refer to the new parameters introduced into quantum mechanics by Bohm and Bub in their 1966 paper."[44] Bohm and Hiley suggest that the choice of the term was an unfortunate one because "there is nothing 'hidden' about the particle variables." The term "hidden variable" is no longer used by members of the Birkbeck School to refer to their theory; nowadays they talk of it being the "quantum potential" approach or something similar, the point all along was that these variables were acted upon by a quantum potential.[45]
The terminological crisis appears to have reached a climax in 1969 after the above quoted debate over whether the Bohm and Bohm-Bub theories are susceptible to the von Neumann criticism or any one of its stronger versions. I mentioned the thoughts of Bub that the older term ought not to be used. He suggests that the 1952 hidden variables theory of Bohm and the one which Bohm and himself suggested cannot be considered hidden variables theories in the usual sense, namely, the sort of theory ruled out by von Neumann, Jauch and Piron, and Kochen and Specker. But, he claims, no hidden variables theory then in existence is so dismissed. He wants us, therefore, to abandon the use of the term as applying to the theories of Bohm, and, as an aside, mentions that Bohm has suggested the replacement term "contingent parameters."[46]
Bub reiterates the same ideas in a book a few
years later: he disowns the hidden variables approach and the Copenhagen
disturbance theory of measurement (which rejects "a God's-eye view")
in part "because they misconstrue the foundational problem of
interpretation by introducing extraneous considerations which are completely
unmotivated theoretically," and he paves the way for "a realist
interpretation of quantum mechanics" which exposes the measurement problem
as only a "pseudo-problem."[47] In the earlier article he suggests that
the use of the term hidden variables for the Bohm-Bub
type of theory is "unfortunate and misleading," and he characterizes
the latter approach as being an extension of Bohr's wholeness ideas in
opposition to von Neumann's philosophy. From here we move easily into the
holographic ideas. Chris Dewdney and Hiley prefer to
regard their
In the paragraph above I mentioned the change in the
The question is also whether the hidden variables in this theory are really hidden (as implied by many other hidden variables theories). We return also to the closely related question of experimental verification for the hidden variables theory as against that for current quantum theory. By performing certain measurements the hidden variables or contingent variables "cease to be unknown (or 'hidden')."[53] The Bohm-Bub hidden variables theory, besides reproducing the statistical predictions of quantum mechanics,[54] is intended to provide a detailed causal account of what is called the reduction of the wave packet during the process of measuring. But it makes predictions which are different from those of quantum mechanics for a short time period immediately following a measurement; Bohm and Bub entertain this possibility in their original exposition.[55] Costas Papaliolios undertook a direct and simple experimental test utilizing this feature (reported in 1967),[56] but it tended to weaken the position of the Bohm-Bub theory (it was in agreement with the results predicted by the usual quantum mechanics) while not ruling it out altogether; the time period on which the experiment is based ought to be reduced to be more decisive. Bohm thinks that this and other valid reasons allow him to conclude that it would "be premature to try to decide whether observations tend to confirm our theory or not."[57] However, according to a 1976 report, "it becomes more difficult to invent a believable physical process that might be responsible for [a suitably reduced time]. The experiment [by Papaliolios] has never been repeated; it is obviously worthwhile to corroborate and improve these results."[58]
While the discussion required to elucidate the notion of the "quantum potential" is rather technical, it is an important concept in the schemas proposed by Bohm in his attempt to understand the quantum effects. The quantum potential is an additional term to the classical Hamilton-Jacobi equation for a particle, and relates right back to the first hidden variables theory of Bohm; its present use over that of the "hidden variables" terminology is indicative of its appeal away from the controversial idea of there being something hidden, and towards the (probably equally controversial) wholeness concepts that Bohm (following Bohr) finds inherent in quantum mechanics. Write C. Philippidis, Dewdney, and Hiley: "The quantum potential suggests a radical change in our conceptual outlook and provides two new interesting possibilities that could have a direct bearing on the subsequent development of the theory." The first is that "it provides a fresh perspective on the micro-world by giving clear intuitive representations of physical processes without the need for the ambiguous relation between the individual and the wave function." The second is that it "offers a clearer insight into the quantum interconnectedness or 'quantum wholeness' that Bohr saw as the essential new feature of quantum phenomena."[59]
One of the more useful aspects of the quantum potential is that it appears alongside classical quantities in its addition to the Hamilton-Jacobi equation, which allows the retention of the localized particle with well-defined momenta and positions along with the novelties of the quantum phenomena expressed in terms of the quantum potential. This potential exhibits nonlocal features (which will become important in the discussion on the EPR experiment in the next chapter)[60] and a nondynamical nature which makes it radically different from a concept appropriate for classical physics. It expresses quantum interconnectedness, the second of the "interesting possibilities" raised in the paragraph above; for instance it expresses the essential and irreducible linking of the observed system and the observing apparatus.
Let me spell this out. The original (hidden variables) proposals are termed "relatively conservative" but did offer an explanation for a number of quantum phenomena; however they were restricted to one-body systems and did not require any really striking changes in the conceptual structure of classical physics. Bohm did recognize the interconnectedness facet of the quantum potential in his original 1952 presentation of the hidden variables theory, but he did not immediately pursue its implications. "In fact, it was only much later that these considerations led to the realization that a different kind of causality was implicit in quantum mechanics."[61] And so it was that in the mid-1970's a new conceptual structure was thought to be needed when trying to use the same approach to comprehend a many-body system. The type of analysis in which physical phenomena are seen as interacting particles has to be superseded. From this it appeared that the quantum potential is different from the classical situation for two reasons. First of all, in classical physics two particles will behave independently of each other when they are sufficiently far apart. But for such particles the quantum potential does not necessarily vanish and there may be between them a direct and strong interconnection. (This aspect will be referred to again later in this chapter.) The second reason is that the quantum potential depends on the quantum state of the system as a whole. The form of the quantum potential will change as the state of the system changes, unlike in the Newtonian tradition where the relationship between two particles is not dependent on something beyond them.[62] This can be summed up with a quotation from a paper by Philippidis, Dewdney, and Hiley:[63]
the quantum potential combines properties of all the participating elements. . .in an irreducible way and suggests that, as far as the quantum domain is concerned, space cannot be thought of simply as a neutral back cloth. It appears to be structured in a way that exerts constraints on whatever processes are embedded within it. More surprisingly still, this structure arises out of the very objects on which it acts and the minutest change in any one of the properties of the contributing objects may result in dramatic changes in the quantum potential.
In this same paper, it is shown that the quantum potential approach reproduces the form of interference patterns for particles passing through two slits as predicted by quantum mechanics, but without the usual ambiguity of whether objects at the quantum level are particles or waves, and retains the notion of a well-defined particle trajectory. "There is no longer a mystery as to how a single particle passing through one slit 'knows' the other slit is open. This information is carried by the quantum potential so that we no longer have a conceptual difficulty in understanding the results obtained in very low intensity interference experiments."[64] In another paper it is used to describe "one-dimensional time-dependent scattering of wave packets from square barriers and square wells."[65]
The insight that the quantum potential is the carrier of information is
taken more and more seriously by the members of the
the force arising from this potential is not like a mechanical force of a wave pushing on a particle with a pressure proportional to the wave intensity. Rather, it acts more like an information content. . . .We may make an analogy here to radar waves that guide a ship. These do not push the ship mechanically. If the ship is on automatic pilot it may then be regarded as a self-active system, with its activity directed by radar waves containing information about the whole surroundings.
The quantum information potential gives guidance to a particle by active information.[67]
Despite such criticisms as "we say that Bohm's theory cannot be refuted. . .however we don't believe it,"[68] this quantum potential manner of posing a hidden variables theory, as we have seen, is very much alive today. However, we ought to bear in mind what Dewdney and Hiley conclude, that, while "it is indeed possible to regard the usual quantum probabilities as arising from an ensemble average of individual processes [i.e., from hidden variables or from the quantum potential approach], . . .no new experimental results. . .can be deduced from our calculations."[69]
Recently Bohm and Hiley extended their quantum potential ideas that quantum mechanical waves refer to a sort of information which determines which path a particle will take, to the quantized Hall effect (for work on which the 1985 Nobel Prize was awarded). They suggest that there is in reality no difference between the macroscopic and microscopic worlds – particles are real in both. The two realms are different not because of their size, as the correspondence principle of Bohr and the Copenhagen interpretation would have it (the correspondence principle states that we have to use classical language and measuring devices for the microscopic world – despite their inadequacies for the job – because that is all we have, and that therefore there is a (rather elusive) boundary between the two regimes). The quantized Hall effect has shown experimentally that certain macroscopic phenomena are governed by quantum mechanical laws. This can be explained, suggest Bohm and Hiley, by the fact that on any level it is the quantum potential which governs the behavior of a particle: the behavior is according to quantum mechanics if the quantum potential is large, and if small or zero the behavior is classical-like regardless of the sizes involved.[70]
It ought to be noted also that the quantum potential approach, according to
a 1985 paper on the subject by members of the
We saw above that the Birkbeck School disagree with the correspondence principle of the
The dispute can be focused on what is called the "measurement problem." In the classical approach to field theory all packets of waves exist together within one three-dimensional space. In quantum mechanics the wave function is in a 3N-dimensional space, but only one of them can actually exist after a measurement is made – the wave packets, that is, exclude each other. This is what is meant by saying that the wave function "collapses."[77] Write Bohm and Bub: the "collapse of the wave function [is] a fundamental and irreducible phenomenon (an ultimate fact of nature which is incapable of any further analysis)."[78] A famous and vivid way of presenting this measurement problem is in terms of the paradox of Schrödinger's cat: an observation causes the wave function to collapse to produce either a situation in which the cat is dead or one in which the cat is alive. This paradox arises because of what Dewdney and Hiley call "one of the less helpful aspects of the Copenhagen school," namely its "insistence on the uniqueness of the usual interpretation which leads to the supposition that what happens to the individual between measurements is somehow inherently indescribable and is not subject to detailed analysis."[79] Bohm has also written about how this assumption of the completeness of the reigning quantum theory leads to the measurement problem:[80]
One wishes to "complete" the description given by Schrödinger's equation[s]. . .so as to describe even the "process of measurement" in terms of them. . . .[This leads to] the notion that the "quantum state" is something separate and distinct from whatever it is that may be directly observable or measurable at the macroscopic level. . .[which in turn] leads, of course, to the question of how we can "know" this "quantum state." So there arises a "problem of measurement" which contains such unfathomable and undecidable questions as to whether the [human] mind. . ., which has the quantum theory as part of the contents of its thoughts, is also a "dynamical system" that follows the very theory that is part of its contents.
There are a number of ways of attempting to get out of this problem. Some of
these ultimately resort to introducing human consciousness as fundamental (this
is a topic I will come back to); for instance, Bohm
and Hiley write, it may
appear that whether the cat is dead or alive is ultimately decided by the
observer's consciousness. Eugene Wigner suggests this
to be the case. Hugh Everett, however, avoids the collapse question and the
consciousness dependence by postulating a multiplicity of universes some of
which contain a physicist observing a live cat, and some a physicist observing
a dead cat.
In terms of the quantum potential approach it is the position of the triggering particle within the incident packet that effectively determines its future, and whatever the final outcome of the process, it will leave the cat in some definite state. In other words the cat will be dead or alive regardless of our knowledge of its final state.
The hidden variables approach answers the
Bohm and Hiley continue their discussion of the measurement problem by telling us "that all the packets of the multidimensional wave function that do not correspond to the actual result of the measurement have no effect on the particle, not only at the moment immediately after its interaction with the measuring apparatus is over, but also for all times from then on (this is seen to follow from the irreversibility of the random thermal motions of the particles constituting the apparatus). And so, such packets can be dropped from further discussion." In relation to the language of the Bohm-Bub hidden variables theory, Bohm considers the measurement problem irrelevant because measurements are no different from anything else, a process of measuring being only a particular case of a general law for determining the movement of some specific parameters.[84]
Let us return to the earlier hidden variables formulation of the theory, and
ask how it can be developed in the light of the
I also want to refer to another paper of Bohm in
which he progresses from the accepted quantum mechanical approach to that of
his hidden variables by talking about necessity and contingency.[87]
He starts by remarking that most physicists would not take seriously the
difficulties he finds with the
In my view, this idea of "order of disorder" is an inherently unclear one. Indeed, attempts to define "disorder" have failed, just because every description is some kind of order, and therefore denies what is intended to be meant by the word "disorder." Thus, the word "disorder" can have no meaning at all. What is usually called a "random" sequence has some kind of order, but one that is not relevant to the laws of the limited field that is under investigation. Nevertheless, it is relevant in a broader field where it is revealed as necessary (i.e., given by law) rather than contingent. In this way, one comes to a more clear formulation of what is to be meant by "randomness." Without some step of this general nature, there is always room for inherent incoherence in our formulations of physical law, which incoherence can. . .become relevant in the new domains that we probably have to enter, to resolve the present state of "crisis" in physics.
Bohm thought it useful to develop a new theory in which "the notion of the irreparably interwoven character of contingency and necessity" is relevant, unlike the quantum theory where it is apparently meaningless and irrelevant. However, to do this requires renouncing the idea of the completeness of a description. The Bohm-Bub hidden variables theory was conceived to illustrate the relevance of necessity and contingency in the field of enquiry at present treated by quantum theory;[88] for Bohm the theory was not intended to be definitive.
By way of summarizing the above debate, I shall quote Bohm from a review article published much later than when his initial expositions of the hidden variables theories appeared. In this he concurs with the point that his causal principle and the usual acausal principle for quantum mechanics should not be taken as having unlimited validity. "Rather, each of these positions is a methodological postulate, and considerations beyond those of scientific fact and theory are what actually make possible a decision between these alternatives, at any particular stage in the development of science." He continues by acknowledging that he did put his "own views rather dogmatically, but only at the beginning. Since then I have suggested," he writes, "that we use both interpretations to obtain a greater insight into the meaning of the quantum theory than is possible with one of them alone."[89] This could have lead to "a misunderstanding as to the intention behind the suggestion of [the hidden variables] interpretation" of quantum mechanics. It was proposed as provisional so that further insight into the meaning of the mathematics of the quantum theory might be gained, and not "as a final and received version of the ultimate nature of reality." More than one interpretation would allow, to Bohm's mind, a deeper and more thorough understanding of the theory.[90]
However, looking back on Bohm's early hidden variables work, the impression is not one of his insisting in a dogmatic fashion on the ultimate truth of his approach; for example, from 1953 (this reads very similarly to what the Birkbeck School is currently saying about the status of its quantum potential theory, as referred to above):[91]
the author [Bohm] would like to state that he does not regard his proposed interpretation of quantum mechanics as a final theory. His principal objective in proposing this interpretation has been to prove that at least one logically self-consistent causal interpretation of the quantum theory is possible, which is also consistent with all of the experimental facts that can be understood in terms of the usual interpretation. This result shows the falsity of the hitherto prevalent impression that no such interpretation is possible, on the basis of which had arisen the notion that it is necessary to give up causality at the quantum level. Naturally, as is quite normal in causal theories, more than one causal explanation of the same basic experimental facts may be possible.
In 1962 Stephen Toulmin isolated four strands in Bohm's argument. The first is the general claim "that theoretical speculation about sub-quantum physics is – for all that von Neumann may have proved – both legitimate and necessary." The second is that it is possible to give meaning to statements of the microphysical world than is allowed for by indeterminacy; Bohm believes that it is possible, moreover, to obtain different observable results from his alternative theory. The third strand concerns the sort of explanatory models Bohm utilizes, a type rejected by Heisenberg. "The 'wavicles' of the quantum world [Bohm] compares to clouds or tidal waves: they represent, for him, transient configurations with blurred edges, continually forming, dissolving and traveling across an underlying sub-stratum (or 'field') of energy." Now he can make use of statistical probabilities which involve averaging out the behavior of vast numbers of sub-quantum turbulences for the statistics and probabilities of quantum theory. And finally, Toulmin tells us, Bohm believes that there is going to be a revolution in the geometrical assumptions of physics, an abandoning of Cartesian co-ordinate geometry and replacing it with spatial concepts drawn from topology.[92]
I shall in this section look at in more depth certain aspects of two of the points made above by Toulmin.
The concept of causality has been mentioned a number of times in this chapter, especially in raising the claim that Bohm's hidden variables theories are causal.[93] This is to elaborate Toulmin's third point above. The usual interpretation of quantum mechanics says that causality has to be abandoned. Bohm suggests that quantum mechanics could just as well have been developed with a causal base (for instance, that of de Broglie) as with the acausal one. But what does Bohm mean by "causality"? "A necessary condition for causality is that a given event or state, which we call the effect, shall always be preceded by another event or state, which we call the cause (most generally by a set which we call the set of causes)." Since some event which always precedes another may not really be relevant, for instance because it is usually associated with the real cause or causes, the above statement is insufficient for defining causality. "The true causes can be distinguished as those conditions or events whose reproduction is both necessary and sufficient for the later reproduction of the effect (or effects)." This should be demonstrated experimentally if we are to sure that we have isolated the true cause. Even though in Bohm's mind all things in the universe are interconnected to some extent or other (an important idea for Bohm which will be spelled out in his later concepts), what he terms "controlling causes" can generally be isolated whose reproduction will "for all practical purposes" reproduce the effects – in this case he advises neglecting all of the other causes.[94]
There is no need, suggests Bohm in 1953, to say that future progress in quantum theory will be made with further renunciation of causality – even just the possibility of a causal presentation of quantum mechanics shows that causality at the quantum level need not be given up. He also believes that the giving up of causality has brought about no real advantages, and he finds very strong reasons for not renouncing causality especially in view of the lack of positive proof of the inconsistency between experimental fact and causality. Currently, writes Bohm, we can only study statistical averages of certain properties of the subatomic level, and we will never be able to investigate scientifically the real physical phenomena if we adhere to the usual interpretation of quantum theory because it assumes that there are no causal factors which might control such properties of individual systems at that level. It does not prove that there are no causes for such phenomena, but philosophically avoids the possibility of entities which are unobservable in terms of current theories.
Not only does Bohm find this attitude to be philosophically untenable, but also damaging to the progress of science.
when we face. . .new and as yet unexplained phenomena, the most fruitful attitude is always to assume they have a cause, which we must discover. Even if there really were no cause, no error could come from the assumption that there was one. All that would happen would be that our efforts to find the cause would not be successful. But if, as is much more likely, there really is a cause, and we assume there is not, then we may be led to overlook important new factors that are needed in the theory.
Neither does Bohm believe that one ought to wait until the causal now-hidden factors are disclosed by experimentation, because theory is needed to guide experiments, especially in the unknown depths of the 10-13 cm domain.
Bohm is apparently criticized for wanting a return to a very simple Newtonian type of causality as found in classical mechanics – even by those who would like to have some form of causality. He of course responds by disowning any form of mechanistic determinism, any unlimited extension of the causality of classical mechanics. On the other hand – and this is what Bohm may be thinking has happened to produce the attitude of current quantum-mechanical physicists to causality – one should not throw out all causality because of the inadequacies of the Newtonian brand. The difficulties he has enumerated he thinks have come from unjustifiably assuming that classical mechanics has unlimited validity, and not from the mechanics as such. Classical mechanics is in fact an approximation with a limited domain of validity; an empirical question for any domain is to ask to what extent a simple Newtonian-type of theory is valid within it, to search for the type of law which works best there by trying out every type which can be thought of. Of course one should not expect to prove experimentally that any general conceptual framework has final validity at all levels, and one ought to be aware that there could be an inexhaustible number of levels which may require quite new types of laws. That it is possible to have, Bohm suggests, a Newtonian-type causal interpretation of quantum theory implies that such a theory may be at least be partially adequate for the quantum domain.
It is thus not possible on philosophical grounds to rule out a priori a classical type of law for a limited domain; "there is nothing intrinsically wrong with classical types of laws, as long as we do not try to extrapolate them unjustifiably by imagining. . .that they furnish a final theory, or at least a final general framework, within which the details only remain to be filled in." This cautious admission applies to any simple causal laws which may be found for the sub-quantum world.
There are at least two places in the section above in which Bohm speaks from his assumption of wholeness – an elucidation of Toulmin's fourth point since, as will be brought out more clearly in a later chapter, physics has assumed the ultimacy of divisibility into parts. One of the places is the later point that both quantum theory and the quantum potential approach are both needed. The other is his parallel criticism of the assumption that quantum mechanics provides the complete understanding of reality; he objects to quantum theory being held as complete in principle for everything describable, and not as just valid in some domain. He is questioning whether "completeness" is a relevant quality for any theory at all, and would rather suggest that all theories are inherently incomplete by being relevant only within limited domains. That physics is fundamental and should cover all that does or could exist is a belief that arose because of "certain fortuitous features" within the historical development of physics itself. Bohm believes that experience is now strongly indicating otherwise for physics, that it is the same as all other forms of human knowledge in being "open ended" and incomplete, not able even in principle to describe everything.[95]
Looking further at the source of the abhorrence of completeness in physics, we could note that Bohm refers to the ideas of Bohr over the process of measurement, in particular to the possibility of completing the Schrödinger equation description so as to include even the process of measuring. Bohm writes that Bohr explicitly stated that this could not be done consistently and therefore should not be entertained. Bohr emphasized, rather, the unanalyzable wholeness of the results of observations and of descriptions of experimental conditions. There can thus be no observed system as distinct from the measuring apparatus, or a quantum state as separated from the total conditions of observation. "That is to say, the notion that we observe is one thing, while the 'wave function' or the 'quantum state' is something else that exists separately (as if it were some kind of 'object'), is in Bohr's view not compatible with what is meant by the term 'quantum'. Rather, the two interpenetrate." Such questions would be ruled out by Bohr as meaningless and irrelevant: "Bohr's language has no place in it at all for a 'measurement problem'."[96]
Bohm's attempts at a hidden variables theory imply, according to him and Hiley, "striking changes in the overall conceptual structure of physics."[97] One example is the application of the hidden variables theory to systems of more than one body. In the analysis of classical physics two particles will behave independently if they are sufficiently separated; this is necessitated by the belief that the whole of a system comprises parts which will function or exist the same whether they are independent of the system or within it. But according to the causal theory systems which are distant from each other may still have a direct and strong interconnection. Further, it means that "the relationships between any two particles depend on something going beyond what can be described in terms of these particles alone"; they may even depend "on the universe as a whole."[98] The quantum potential approach is especially useful in enabling one to see the similarities and differences between quantum mechanics and classical mechanics, as it enables both to be expressed in terms of the same language. If the wave function is multiplied by a constant there is no alteration in the quantum potential, meaning that, as we saw above, it does not drop off to nothing at great distances where the wave intensity falls off to negligible proportions. But the classical approach of analyzing a system into independent parts insists that sufficiently separated parts do not interact significantly. "This means that the quantum theory implies new kind of wholeness, in which the behavior of a particle may depend significantly on distant features of the over-all environment."[99]
This is the question of locality/nonlocality; the various hidden variables theories of Bohm are nonlocal (nonlocality means, for instance, this "incredible property" which is said to defy common sense: "the polarization of particle 1, observed with apparatus A, depends on the setting of distant apparatus B") and as such are in agreement with EPR experiments – which will be discussed in the next chapter.[100]
It must be remembered here that this wholeness property of quantum theory which was presented by Bohr in the early years of the theory is what Bohm wants to reassert and what he feels is being neglected in current quantum theory. Bohm and Hiley describe it as one of the "crucial new differences" that emerge from using the hidden variables/quantum potential approach: that there is a nonlocal interaction of particles with each other and that this is determined in general by the whole system's quantum state. It is therefore no longer valid to analyze a system classically, that is, into parts which are separate and whose relationships are not dependent on the whole. Now independent and separable parts are seen as being special and contingent, dependent on the whole in a manner indescribable purely in terms of the parts themselves. They conclude that in this way therefore their approach agrees with that of Bohr, but with a difference: "the quantum potential interpretation provide[s] a physical notion of how this wholeness may be brought about, through the actual movement of independently existent particles under the action of the quantum potential, whereas in Bohr's formulation, such questions are expressly ruled out as having no meaning."[101]
Dewdney and Hiley point out what appears to be the notion of the implicate order ("a more general underlying process"), which is an important concept of Bohm's to be discussed in later chapters, inherent in the notion of the quantum potential:[102]
it seems likely that the particle and its trajectory are but abstractions from a more general underlying process in which the quantum potential characterizes, in some general way, the average properties of the time development of this process. It is in this sense that the quantum potential is more like an organizational potential which is essential to account for the characteristic features of quantum phenomena.
Physicists were and remain dubious about the hidden variables proposals, especially because of their lack of any clear experimental support, with their empirical conclusions identical to those of conventional quantum mechanics.[103] "To the extent that Bohm's claims involve considerations of a highly technical kind," wrote Toulmin in 1962, "the theoretical physicists must be left to fight out the issues among themselves." Further:[104]
the precise form of a satisfactory sub-quantum theory. . .can be settled only by building up such a theory in detail; and this theory must provide a system of physical ideas which explains the facts of Nature more successfully than existing quantum theory. . . .Unless [Bohm] succeeds in doing this, orthodox quantum physicists. . .will continue to regard his criticisms of quantum mechanics as backward-looking – as a revival of Einstein's counter-revolution – rather than as being "the shape of things to come."
Even though the time may have come "to reconsider the legitimacy and desirability of building up some new 'sub-quantum' theory," Bohm still needs to demonstrate the merits of his ideas "in the only way [his] colleagues will find unanswerable."
This summary response by Hanson in the same volume as Toulmin's is in fact quite mild:[105]
There is only one kind of theory in the field now. It is the work of many. . . .It is a theory which is algebraically articulated, experimentally detailed and – despite renormalization – powerful in explanation. What precisely is the present alternative to this physical theory? It is a congeries of excitingly vague, bold-but-largely-formless, promising-but-as-yet-unarticulated, speculations.
He continues by stating that speculation is not working physics; there are no good reasons for suspecting that hidden variables physics will succeed in accounting for all that orthodox quantum theory can describe. (This is a different mood about Bohm's work for Hanson than the one he expressed in the conclusion to his famous 1958 publication: "At this stage it would be venturesome to try finally to settle this matter; nonetheless. . .its conceptual significance will be missed by anyone who fails to see how much was at work when physicists of the past disagreed, and missed also by anyone who thinks of the history of physics as just a march of better observations and more accurate experiments.")[106]
Support for hidden variables theories appears to be largely derived, it is claimed, from philosophical considerations.[107] In the 1950's L. Rosenfeld accused Bohm of going against the "exigencies of sound scientific method," and described those who follow the Copenhagen interpretation as possessing the "uncommitted, commonsense attitude of the true scientist."[108] Physicists referred to hidden variables as "superstition," even in 1978![109] Popper in a 1982 book refers to "David Bohm's heresy" – he cannot side with Bohm's determinist, objectivist (in some sense and subjectivist in another), realist program and theory, and he quotes Max Born: "to dream of a way back, back to the classical style of Newton-Maxwell (and it is nothing but dreams which these gentlemen [Schrödinger, Bohm, Einstein, et al.] indulge in). . .seems to me hopeless, off the way, bad taste. And we could add, 'it is not even a lovely dream'."[110] According to Toulmin, we need to exercise historical judgment: "are Bohm's ideas about a future sub-quantum microphysics based only on hunches and guesswork or can they. . .serve as the starting point for a genuine intellectual breakthrough?"[111]
Very early in the history of his hidden variables theories Bohm answered the no-experimental-proof type of objection which would say that the causal interpretation is therefore purely "metaphysical."[112] While agreeing that any hypothesis which in principle can never be verified experimentally can be classed as metaphysical (i.e., by not having any physical consequences it makes no difference whether or not it is included in theory), Bohm thinks that this does not apply to his hidden variables theories because verification of their existence cannot in principle be ruled out. As we saw above, they cannot be observed precisely within the limits of precise measurement as delimited by the usual quantum theory, but it may still be possible within the context of another theory to observe them precisely. The Bohm-Bub theory, furthermore, could be verified if shorter measurement processes were employed than have been up to the present. Of course it is not adequate in Bohm's eyes to counter with the charge that an hypothesis is metaphysical if it cannot be verified within the confines of currently accepted theories; Bohm suggests that this restriction is not satisfied even by the usual interpretation, and it also assumes something which human experience of theory making in the past would rule out, namely, that "the world is so constituted that nothing could be gained by assuming the existence of entities before one can give a prescription for how their existence is to be verified." Apparently it has frequently been fruitful – for instance in the atomic theory – to hypothesize the existence of something before it was known even in principle how its existence might be verified. In fact it would seem that often the means for verification arise from research using the hypothesis itself.
Bohm hoped to establish a case for a hidden variables type of theory, and for his general philosophical outlook which underlies his hidden variables theories, chiefly by pointing out the philosophical problems that he finds in the usual interpretation of the quantum theory.[113] As we saw, he especially singles out for criticism the indeterminacy principle when taken as absolute and final both for present theory and for all future theories of physics. He is not prepared to give up all determinism for the pure chance nature of reality which this principle implies, and seeks the determinism in a lower level than that at which quantum mechanics is applied.
Of course the vital question remains as to whether experimental proof for the hidden variables/quantum potential can in fact be found. Bohm and Bub's hidden variables theory raised Tutsch's hopes "that the hidden variables need not remain hidden, and. . .that the validity of the Bohm-Bub theory may be tested in the laboratory."[114] As we saw above, there is presently strong interest in pursuing this. However, Bernard d'Espagnat recently intimated that the "present day fruitlessness of the hidden-parameters theories is undeniable."[115] Jauch recently commented that "no useful results have come from such attempts."[116] The feelings expressed by detractors about the hopelessness of pursuing a hidden variables theory do not deter the Birkbeck School from searching for experimental support and closing over other supposed inadequacies. The abstract to a 1984 paper by Bohm and Hiley reads this way:[117]
We review briefly the quantum potential approach to quantum theory, and show that it yields a completely consistent account of the measurement process, including especially what has been called the "collapse of the wave function." This is done with the aid of a new concept of active information, which enables us to describe the evolution of a physical system as a unique actuality, in principle independent of any observer (so that we can, for example, provide a simple and coherent answer to the Schrödinger cat paradox). Finally, we extend this approach to relativistic quantum field theories, and show that it leads to results that are consistent with all the known experimental implications of the theory of relativity, despite the nonlocality which this approach entails.
Thus the hidden variable/quantum potential approach is made to conquer old uncertainties about it. It is also made to explain away a possible "irrationality" such as J.A. Wheeler's "'delayed-choice' experiment designed to show that. . .the choice to measure one or another complementary variable at a particular time can apparently affect the course of events for considerable periods of time before such a decision is made." (This is related to the manner in which the hidden variables approach is made to answer the measurement problem, as related above, and thus is a way of avoiding an approach such as Wigner's. I return to this topic below.)[118]
A few months after the above 1984 Bohm and Hiley paper appeared, Longtin and Mattuck published an article in which they claim to "present a simple first step toward a relativistically covariant generalization of the Bohm-Bub hidden variables theory," albeit a generalization to a rather simple case.[119] This could lead to satisfying a long-recognized inadequacy in Bohm's hidden variables theories. There are, of course, other hidden variables theories both before and after Bohm's proposals, and these continue to engender a good deal of research interest.[120]
Perhaps I could summarize this section by saying that I find Bohm's ideas very interesting, but they need to move beyond being almost totally speculative if they are to challenge seriously the reigning quantum theory. (1) The theoretical questions raised against his theory need to be answered. (2) There needs to be experimental support for his theory over and above that which is explicable in the current theory. It needs to solve problems that the present theory cannot do. (3) Is it true that the indeterminacy principle is given such an absolute status in current quantum theory as Bohm tells us? If not, many of his arguments against the current theory fail. (4) It would also appear that Bohm's arguments for a hidden variables theory being necessary because of problems in physics at the 10-13 cm level have been largely superseded with contemporary quark et al. theory – it may be more useful thus to drop the hidden variables terminology in favor of that of quantum potential, but it does make one wonder about the applicability of the whole theory since the two approaches are in fact different names for the same theory.
It is interesting to read Bohm's more general philosophical/metaphysical ideas of the qualitative infinity of nature. The hidden variables theory is one attempt to express it, but so are his later implicate order ideas.
Most of the remainder of this first part of this work will be the spelling out of what is evident in or underlies, at least in the eyes of Bohm and his fellow workers, the hidden variables theories (and the other later theories they have constructed). These further thrusts into controversial physics may appear as a guiding light for some philosophically-religiously or spiritually minded people, but their support within the world of physics is still very tenuous.
[1]. Hiley 1980: 80-1.
[2]. Barbour 1966: 300-1.
[3]. Barbour 1966: 299; Bohm and Hiley 1982; Toulmin 1962: 17.
[4]. Jeffreys and von Neumann, respectively, quoted in Hanson 1958: 153 and 230; see also ibid., p. 172.
[5]. Vigier 1957: 77. R.B. Lindsay 1957: 32 accuses Bohm of identifying causality with determinism. As in its title, Bohm 1957a deals in depth with the questions of causality and chance. For a discussion of causality and science, see Puterman 1977.
[6]. Gudder 1979: 59.
[7]. Toulmin 1962: 9.
[8]. Bohm 1957a: ix-xi; 1980b: 76; Bohm and Hiley 1982; 1984: 256, referring to de Broglie 1960. See also de Broglie 1963; 1970; and Flato, Maric, Milojevic, Sternheimer, and Vigier 1976. E. Madelung could be added to de Broglie's name as a precursor to Bohm's theory; see Wesley 1983: 105-8. Bohm has written also on the many-body problem and plasma physics; see, for instance, Bohm 1959; Bohm and Carmi 1964; and Carmi and Bohm 1964.
[9]. Bohm 1952a; 1952b; 1952c; 1953a; 1953b; 1953c; 1953d; 1962a (as 1980b: 76-110); Bohm, Schiller, and Tiomno 1955; Bohm and Vigier 1954; and 1958. See also Bohm and Hiley 1975: 96-101; Belinfante 1973: Part II, Chap. 2; de Broglie 1960: Part 2; d'Espagnat 1983: 90-1; and Freistadt 1957. With regard to Pauli and de Broglie, see Bohm 1952b; Bohm 1953b: 283; Bohm and Hiley 1982: 1003, 1014; Bohm and Vigier 1954; de Broglie 1960: Chap. 14; 1963: 131-3; and Dewdney and Hiley 1982: 28.
[10]. Bohm and Hiley 1975: 95.
[11]. Popper 1982: 36.
[12]. Bohm 1951: 29; see also pp. 101, 114-5, 139, and 622-3.
[13]. Bohm and Hiley 1982: 1014. See also comments by Yuval Ne'eman and Abraham Paris in Woolf 1980: 267.
[14]. Bohm 1957b: 33.
[15]. De Broglie, Bohm, Hillion, Halbwachs, Takabayasi, and Vigier 1963: 438-9; and Bohm, Hillion, Takabayasi, and Vigier 1960.
[16]. See Freedman and van Niewehuizen 1985; and Anthony and Green 1985.
[17]. Bohm 1952a: 166. For an elaboration on the notion of probability utilized here, see Vigier 1957.
[18]. Bohm 1953c: 458. See also Bohm 1980b: 65; Bohm and Vigier 1954.
[19]. Bohm 1952a: 166; see also Bohm 1980b: 78.
[20]. Bohm 1957b: 33, 35-7.
[21]. Bohm 1952a: 166.
[22]. See
Belinfante 1973: 94; and
[23]. Bohm 1952c; Bohm and Vigier 1954: 215. Keller 1953; Pauli 1953 are two publications posing important difficulties to which Bohm and Vigier 1954 reply; Halpern 1952 is replied to in Bohm 1952c (see also Belinfante 1973: 118-21). For a list of criticisms, also see Rosenfeld 1957.
[24]. Bohm and Bub 1966a: 462; see also their 1966b; 1968; Bohm 1969a: 443-6; 1971c: 102-16; 1980b: 80-110; Belinfante 1973: Part II, Chap. 4. Cerofolini considers the Bohm-Bub hidden variables theory to be formally equivalent to his sub-quantum mechanics; see Cerofolini 1982.
[25]. For instance by Tutsch 1968 and Belinfante 1973.
[26]. See Bohm and Hiley 1975: 97.
[27]. Tutsch 1968: 232. See also Tutsch 1969.
[28]. Bohm and Bub 1966a: 453.
[29]. Longtin and Mattuck 1984: 685-6.
[30]. Bohm and Bub 1966a: 467; Belinfante 1973: 162.
[31]. Von
Neumann 1955: x, 209-11, 323-8; Bohm 1971c: 97-102; Bohm 1980b: 70-1; Bohm and Bub 1966a: 460-2. A summary of von Neumann's ideas can be
found in
[32]. Hanson 1958: 173. See also his 1963: Chap. 6.
[33]. Belinfante 1973: 162. For other difficulties it faces, see Tutsch 1969: 1116ff.
[34]. Jauch and Piron 1968: 228. See also Jauch 1968: Chap. 7; and Jauch and Piron 1963, which is based on a formulation of quantum mechanics by Piron published in 1964.
[35]. Gudder 1968a: 231; see also Gudder 1968b; 1979: 60-1; and 1983: 114-6.
[36]. Tutsch 1968: 232. See also Bohm and Bub 1968; Hübner 1983: 80-1; Jauch and Piron 1968; and Turner 1968.
[37]. Bohm and Bub 1966b: 470. Bub 1974b purviews this whole question; see also Bub 1977.
[38].
[39]. Bub 1969: 101.
[40]. Bub 1969: 101; Kochen and Specker 1965; 1967; Specker 1975; see also Demopoulos 1977.
[41]. Misra 1967: 841-3. Bohm 1957a puts this criticism thoroughly.
[42]. Gudder 1979: 59-65; see also Gudder 1970: 431-2: a hidden variables "theory is always possible" in quantum mechanics in its present framework.
[43]. Gudder and Armstrong 1985.
[44]. Tutsch 1969: 1116.
[45]. Bohm and Hiley 1984: 259.
[46]. Bub 1969: 102.
[47]. Bub 1974b: vii and ix.
[48]. Dewdney and Hiley 1982: 47.
[49]. Quigg 1985: 84.
[50]. Hawking 1985: 146-7.
[51]. Hiley 1971: 181.
[52]. Ibid., pp. 188-9.
[53]. Bohm 1969a: 446; see also ibid., pp. 446-7; and Bohm and Bub 1966a: 466.
[54]. But see Christensen and Mattuck 1982.
[55]. Bohm and Bub 1966a: 466; see also Bohm 1980b: 82-5, 105-9.
[56]. Papaliolios 1967.
[57]. Bohm 1969a: 446. See also Papaliolios 1967: 624; Tutsch 1968: 232; and 1969: 1116.
[58]. Freedman, Holt, and Papaliolios 1976: 47.
[59]. Philippidis, Dewdney, and Hiley 1979: 17.
[60]. See, for instance, Philippidis, Bohm, and Kaye 1982.
[61]. Philippidis, Dewdney, and Hiley 1979: 17.
[62]. Bohm and Hiley 1976b: 176-7. See also their 1984: 259-62.
[63]. Philippidis, Dewdney, and Hiley 1979: 27.
[64]. Philippidis, Dewdney, and Hiley 1979: 25; but their result has been criticized as not fitting the double slit flux that is actually observed – see Wesley 1983: 107.
[65]. Dewdney and Hiley 1982: 27.
[66]. Bohm and Hiley 1984: 260.
[67]. Bohm and Hiley 1980: 267-9 also distinguish between active information and inactive information and thus are lead to a solution of the measurement problem which will be discussed in the following section.
[68]. Bopp, quoted in Philippidis, Dewdney, and Hiley 1979: 18. Bopp also says that Bohm's theory is not "physics but metaphysics"; Körner 1957: 51.
[69]. Dewdney and Hiley 1982: 46.
[70]. Bohm and Hiley 1985; Thomsen 1986f: 27.
[71]. See a similar discussion below in the following section.
[72]. Bohm, Dewdney, and Hiley 1985: 297.
[73]. Bohm and Hiley 1976b: 175.
[74]. Bohm, Dewdney, and Hiley 1985: 297. See also Philippidis, Bohm, and Kaye 1982: 80.
[75]. Bohm 1952a: 167.
[76]. Bohm 1957a: Chap. 3; 1969a: 439; 1980b: 65-76; but see Stapp 1972.
[77]. Bohm and Hiley 1984: 257. Concerning the measurement problem, see, for instance, Wigner 1963.
[78]. Bohm and Bub 1966a: 457. See also
[79]. Dewdney and Hiley 1982: 46-7.
[80]. Bohm 1969a: 439-43.
[81]. Bohm and Hiley 1984: 257. The
[82]. Bohm and Hiley 1984: 257; see also ibid., pp. 268-9.
[83]. Bohm and Hiley 1984: 256-7; Bohm 1980b: 88-105. With regard to the Heisenberg uncertainty principle being derivable from disturbances of the measuring apparatus on the measured, the authors refer to Bohm 1952b.
[84]. Bohm 1969a: 444.
[85]. Bohm 1957b: 33-7.
[86]. Trigg 1980: 156.
[87]. Bohm 1969a: 439-43.
[88].
Related to the above discussion, see Hübner 1983:
Chap. 2 for a critique of both Bohm's and the
[89]. Bohm 1984: 781; Bohm refers to Bohm and Hiley 1982: 1001.
[90]. Bohm and Hiley 1984: 256; see also ibid., p. 260.
[91]. Bohm 1953b: 279.
[92]. Toulmin 1962: 17-9.
[93].
[94]. Bohm 1953b: 283-5.
[95]. Bohm 1969a: 439-43.
[96]. Bohm 1969a: 439-43.
[97]. Bohm and Hiley 1975: 97.
[98]. Ibid., p. 99.
[99]. Bohm and Hiley 1982: 1005-6; see also Bohm, Dewdney, and Hiley 1985: 296.
[100]. Mattuck 1981: 331; Bohm 1952b.
[101]. Bohm and Hiley 1984: 256.
[102]. Dewdney and Hiley 1982: 48.
[103]. Belinfante 1973: 164; but see Bohm 1971c: 112-3; Bohm and Bub 1966a: 466; and Fox and Rosner 1971. See also Putnam 1965: 87; Körner 1957: 46-89; and the interesting comments in Feyerabend 1960: 326, n. 1. The same statement is made much later in Bohm and Hiley 1984: 255.
[104]. Toulmin 1962: 19-22.
[105]. Hanson 1962: 91-3. See also his criticism in his 1963: Chaps 5 and 6.
[106].
Hanson 1958: 174-5. See also Heisenberg 1958: 130-3, where he argues that Bohm's hidden variables proposal says nothing about physics
that is different from what the
[107]. Barbour 1966: 299-301; Garstens 1971: 87; Körner 1957: 51.
[108]. Quoted in Feyerabend 1960: 330, n. 1. See also de Broglie and Rosenfeld 1958.
[109]. Peres 1978: 745.
[110]. Popper 1982: 174-5.
[111]. Toulmin 1962: 11.
[112]. Bohm 1953b: 280-1.
[113]. Bohm 1957a.
[114]. Tutsch 1968: 234. He elaborates his thoughts further in Tutsch 1969.
[115]. D'Espagnat 1983: 16.
[116]. Jauch 1977: 39.
[117]. Bohm and Hiley 1984: 255. See also Dewdney, Garuccio, Gueret, Kyprianidis, and Vigier 1985. Thomsen 1986d also suggests some recent developments concerning the measurement problem.
[118]. Bohm, Dewdney, and Hiley 1985: 294; Jonas 1986; Wheeler 1978.
[119]. Longtin and Mattuck 1984: 685. Two years previous to this, an article appeared written by Mattuck and Jens Peter Christensen which details the modification of the original Bohm-Bub hidden variables theory through the work of Tutsch and Belinfante, but which still was inadequate because it was not relativistically covariant; Christensen and Mattuck 1982: 348.
[120]. See, for example, Belinfante 1973; Gudder 1980; 1984a; 1984b; Gudder and Armstrong 1985: 1009 refers to some more recent attempts; and Pitowsky 1983. Vigier 1980 finds new impetus for continuing the development of a version of hidden variables theory akin to Bohm's. Wesley 1983 develops a causal theory based on the quantum potential and Bohm's early work; see Phipps 1985.
© Copyright by KOI TRUST, 1987
© Copyright by Kevin Sharpe, 2000
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