Bohm's Metaphors, Causality, and the Quantum Potential | SpringerLink

MARCELLO GUARINI

BOHM’S METAPHORS, CAUSALITY, AND THE QUANTUM POTENTIAL

ABSTRACT. David Bohm’s interpretation of quantum mechanics yields a quantum potential, Q. In his early work, the effects of Q are understood in causal terms as acting through a real (quantum) field which pushes particles around. In his later work (with Basil Hiley), the causal understanding of Q appears to have been abandoned. The purpose of this paper is to understand how the use of certain metaphors leads Bohm away from a causal treatment of Q, and to evaluate the use of those metaphors.

1. INTRODUCTION

In his 1952 paper ‘A Suggested Interpretation of the Quantum Theory in Terms of “Hidden” Variables’, David Bohm develops a deterministic interpretation of quantum mechanics which not only contains the classical potential, V , but also contains a new, quantum mechanical potential, Q. In that paper, the effects of Q are understood in causal terms. In other words, the quantum force can be understood as mechanically pushing particles around. In his last book on quantum mechanics, written with Basil Hiley, there are places where the authors appear to have renounced a causal analysis of the quantum potential. The purpose of this paper is twofold: (a) to try to understand the motivation for the switch from a causal to a noncausal understanding of Q, and (b) to evaluate some of the metaphors used to explain the sort(s) of understanding(s) of quantum mechanics that Bohm offered. I will argue that different metaphors are guiding the move away from a causal interpretation of Q, and that these metaphors offer inconsistent guidance on how to interpret Q. In short, different metaphors are leading to different ways of understanding quantum mechanics, and this was never acknowledged by Bohm, hence the parenthetical pluralization above. It will be shown that while one of his metaphors may be compatible with a thoroughly non-causal interpretation of Q, another is not. The purpose of this paper is not to examine all of Bohm’s metaphors. For example, while a fair bit of attention is given to the hologram and holomovement metaphors in Wholeness and the Implicate Order, they are not given as much attention in The Undivided Universe, where the concentric Erkenntnis 59: 77–95, 2003. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.

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cylinders metaphor appears to have pride of place in illustrating the nature of the implicate order. The hologram and holomovement metaphors will not be discussed in this paper, but the concentric cylinders metaphor will be examined in some detail. The metaphors selected for consideration herein are (a) those which Bohm appears to have thought would be most useful in explicating his views, and (b) those which clearly bear on the issue of causal versus non-causal interpretations of the effects of Q. Before getting to all this, a quick review of Bohm’s interpretation of quantum mechanics would be useful. Let us begin with an examination of Bohm’s formalism for a onebody system and some general remarks about his interpretation of quantum mechanics. Bohm begins by rewriting the wave function in polar form: (1)

ψ = Rexp(iS/ h),

where R(x, t) and S(x, t) are real functions. Rewriting for R and S, we get the following equations: (2)

∂R/∂t = −1/2m[R∇ 2 S + 2∇R · ∇S]

and (3)

∂S/∂t = −{[(∇S)2 /2m] + V (x) − (32 /2m)(∇ 2 R/R)}.

Equations (2) and (3) are equivalent to Schrödinger’s equation. Equation (3) expresses the energy balance of the system. Not only does it contain the classical potential V (x), it also contains a quantum potential: (4)

Q = −(h2 /2m)(∇ 2 R/R).

A similar procedure may be used to derive Q for a many-body system (Bohm and Hiley 1993, 56–72). The quantum potential has a number of interesting, non-classical features. First, the effect of Q does not diminish as the distance between two particles increases. Second, Q acts instantaneously. This strong nonlocality means that particles separated by great (even space-like) distances can continue to behave in a strongly correlated manner. Third, the effect of Q depends on the quantum state of the system as a whole. This means that the way the quantum potential mediates interaction between two particles depends on more than the two particles. In the limit, it may even depend on the quantum mechanical state of the universe as a whole. This last feature is what leads Bohm to emphasize the holistic nature of the quantum mechanical realm. Finally, Q has no point-like source. The field from which


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the potential is derived satisfies a homogeneous equation, meaning that it does not radiate. This is importantly different from electromagnetic fields, which do radiate. However, Bohm is careful to point out that the restriction to a homogeneous equation is an assumption of the theory in the same way that p = S(x) is an assumption (Bohm 1952, 171). Moreover, he is quite open to the possibility of allowing for inhomogeneities, especially if it leads to new predictions (Bohm 1952, 171, 178–179). So Q is strange. However, it was not so strange as to keep Bohm from providing a causal interpretation of the quantum field in his early work. Indeed, he provides an equation of motion for one particle acted on by both classical and quantum potentials which greatly resembles Newton’s second law: (5)

m(d 2 x/dt 2 ) = −∇[V (x) − Q(x)].

Equation (5), on analogy with F = ma, positively invites an interpretation of Q according to which the quantum field pushes particles around. Given that even in his early work Bohm was aware of the aforementioned nonclassical properties of Q, why did he initially accept a causal interpretation and then go on to renounce it?

2. THE BOBBING CORK AND RADIO METAPHORS

Thus far, I have suggested that Bohm renounces causality outright in his latter work. This is somewhat misleading. As we will see, one of his metaphors (concentric cylinders) suggests abandoning a causal interpretation of Q, while another metaphor (the radio metaphor, when properly understood) suggests a rather subtle reworking of the causal impact of Q. This section of the paper will explore the radio metaphor; the next section will explore the concentric cylinders metaphor. To understand Bohm and Hiley’s views in The Undivided Universe, we must notice that they are very impressed with the fact that the effect of the quantum potential varies with the form of the wave function and not its intensity (Bohm and Hiley 1993, 31). The force that classical waves or fields exert on a body varies with the amplitude or intensity of the wave. For example one may consider a water wave which causes a cork to bob. The further the cork is from the centre of the wave the less it will move. But with the quantum field, it is as if the cork could bob with full strength even far from the source of the wave. (Bohm and Hiley 1993, 31)

Bohm is right to point out that the quantum field behaves non-classically. Indeed, once we have the picture of the bobbing cork in mind, it might

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become tempting to think that the quantum field is not pushing particles around in some mechanically causal fashion. It is very difficult to picture a wave changing in amplitude (say, the height of a water wave) without changing the influence it has on a body (such as the cork), and Bohm very much likes graphic metaphors which allow us to picture what is going on. To use a Wittgensteinean turn of phrase, I think this picture of classical waves is holding Bohm’s imagination captive. After the bobbing cork example, we are asked to compare a ship being remotely guided by radio waves with the quantum field and an electron (Bohm and Hiley 1993, 31–32). The idea is that just as the behaviour of a ship varies with the information encoded in the guiding radio signal and not with the amplitude of that signal (provided the amplitude is above a certain minimum level so that it can be detected), so too the behaviour of any particle influenced by the quantum field varies with the form of the wave function and not the probability amplitude. This is why Bohm and Hiley go on to say that Although Equation [5] may look like a classical law implying pushing or pulling by the quantum potential, this would not be understandable because a very weak field can produce the full effect which depends only on the form of the wave. We therefore emphasize that the quantum filed is not pushing or pulling the particle mechanically, any more that the radio wave is pushing or pulling the ship that it guides. So the ability to do work does not originate in the quantum field, but must have some other origin. (Bohm and Hiley 1993, 37)

Notice: this passage seems to indicate that Bohm and Hiley think they have a reason not to treat the quantum field as pushing or pulling a particle which is independent of considerations supporting speculation about the implicate and explicate orders. The considerations supporting the implicate-explicate model include the reconciliation of relativity with quantum mechanics and the extension of physics beyond the Planck length. I will have more to say about the implicate-explicate model (together with the metaphor of the nested cylinders separated by glycerine) latter on. For now, we need to see that Bohm and Hiley think that the variation of the effects of the quantum field with the form and not the probability amplitude of the wave function is sufficient for renouncing a mechanically causal analysis of the quantum field. It is also important to notice that the radio metaphor is clearly guiding this move away from a mechanically causal interpretation of the quantum potential. Before going further, a few remarks on causality are in order. While I will not attempt to define causality, some remarks about a weaker notion – causal implication1 – will prove helpful. Let us say that X causing Y entails that X and Y are causally implicated, but X and Y being causally implicated entails neither that X causes Y nor that Y causes X. A rough sufficient condition for causal implication can be stated as follows: given


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events X and Y , if Y would have been different, then X would have been different.2 To show how this condition works, let us consider an example. Say that Jill has to sink the 8-ball to win a pool game. We will stipulate that the ball is situated such that the only way to sink it is to bank it off a specific part of the cushion. Under these circumstances, the following counter-factuals are true: (a) if the ball had not hit a specific part of the cushion, it would not have gone in the pocket; and (b) if the ball had not gone into the pocket, then it would not have hit the specific part of the cushion. The truth of either (a) or (b) is enough to guarantee that the ball hitting the cushion is causally implicated with the ball going into the pocket. The truth of (a) demonstrates that the sufficient condition for the causal implication of X and Y is not a sufficient condition for X causing Y , otherwise we would be allowed to say that the ball going into the pocket caused the ball to hit the cushion where it did. I do not think Bohm and Hiley would want to deny that the ship’s movements and the radio waves are causally implicated with one another; they are making the weaker and more plausible claim that the ship is not pushed or pulled by radio waves. The problem comes when they try to import their classical observations into the quantum realm. They are making the following assumption (or something like it) in their reasoning. (P :)

a necessary condition of a wave exerting a force on a particle is that variations in the amplitude of the wave or field are causally implicated (in the technical sense discussed earlier) with the movement of the particle.

Bohm and Hiley – motivated by the bobbing cork example – must think that something like P is true in order for them to reason the way they do. While P appears to be true of classical waves, it is far from obvious why it should be true of the quantum field. If we take the appropriate partial derivative of the classical potential for classical fields such as the electromagnetic field, we get the force exerted by that field. Moreover, we see that the classical potential of and the force exerted by electromagnetic waves varies with the amplitude of those waves. If we take the appropriate partial derivative of the quantum potential, we get the force exerted by the quantum field. In other words, the same mathematical procedure can be used to compute a force from either the classical or quantum potentials. Since P is true of classical waves, we know that the only way radio waves can be involved in moving a ship around (given that intensity of these waves is so weak) is if there are mechanisms on the ship which can interpret the information in the waves and generate enough energy to move the ship in accordance with the information in the waves. As a result of

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assuming that P (or something like it) is true for all fields, Bohm and Hiley suggest that the ability to do work does not originate in the quantum field (as stated in the last extended quotation). The idea is that particles in the quantum realm act like radios (Bohm and Hiley 1993, 35–38). Just as a radio (or our remote controlled ship) can receive information in the form of radio waves, decode it, and then put that information to work using its own source of energy, so too electrons (for example) have an internal structure which allows them to decode information sent by the quantum field and the ability to put that information to work. To help render the preceding conjecture plausible, Bohm and Hiley remind us that there is as much space between the shortest distance we can presently measure, 10−16 cm, and the shortest distance in which current notions of space-time have meaning, 10−33 cm, as there is between our size and the shortest distance presently measurable (Bohm and Hiley 1993, 38). The idea is that there is plenty of unexplored space in which to discover more structure for so-called “elementary” particles. The radio metaphor is worrisome for a number of reasons. First, there is the concern about where the electron (or other particles) are getting the energy to put the information they receive to work. Radios have batteries or some other power source to draw on. Metaphorically speaking, where are the electron’s batteries? Second, the radio metaphor suggests that just as radio waves are too weak to move a ship, so too the force given by taking the appropriate partial derivative of Q is too weak to move an electron (or some other particle). But this is false (and Bohm knew that). The quantum potential is such that when the appropriate partial derivative is taken, we arrive at the required force to move the particle. Bohm does not think that the quantum field can be exerting this force since the force does not vary with probability amplitude. But why should that matter? The quantum force is non-local and does not radiate; why is it so hard to believe that it could have other non-classical properties as well? There was a time when people thought that the wave-like behaviour of light could only be explained in terms of light travelling in some sort of underlying medium – an ether. After all, waves were traditionally thought of as travelling in a medium. Of course, physicists eventually abandoned the idea that the only way to understand a wave was to understand it as travelling in an underlying medium. Perhaps something like the preceding could be said about the quantum field. Traditionally, the force imparted by wave fronts has been understood to vary with their amplitude. Perhaps we simply need to get use to the idea that since the probability amplitude of the wave function is defined over configuration space, the force engendered by Q can be applied in a way that does not vary with the probability amplitude.


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Some clarifications are in order. While I have suggested that P may be true of classical waves, I do not want to suggest that a necessary condition of exerting a classical force is that variations in amplitude are casually implicated in the movement of the particle being acted on. There is a straight forward counter example to such a claim – the classical Coulomb potential is not a wave form, and yet it engenders a force. My focus on classical waves is motivated by the analogies Bohm used – the cork bobbing in water waves is one example, and the ship and radio being “informed” by radio waves are further examples. Also, I want to make it clear that P is not designed to suggest that the only way to encode guiding information is through variations in amplitude. Regardless of whether information is encoded in the amplitude of, say, an electromagnetic wave, changes in the amplitude of that wave are causally implicated in the movement of particles in the reception equipment, say, of a radio. That the reception equipment may not be designed to amplify variations in amplitude does not alter the fact that such variations will change the causal force being exerted on the radio equipment by the waves. The idea that changes in amplitude necessarily entail a change in the force exerted by a wave is what P is designed to catch. I argued that P cannot be applied to Q. There can be variations in the amplitude of the wave function without a change in the force exerted by the quantum field, and – contra Bohm – this does not undermine a causal treatment of Q. Of course, Bohm and Hiley have other reasons for thinking that the quantum field cannot be analyzed in causal terms. One of them is given in the following passage: The fact that the wave function is in configuration space clearly prevents us from regarding the quantum field as one that carries energy and momentum that was simply transferred to the particles with which it interacted (thus effectively pushing or pulling mechanically on the latter). This is a further factor in addition to the form dependence of the activity of the field which leads us to consider the interpretation of this field as active information. (Bohm and Hiley 1993, 61)

This is very intriguing passage. The first thing that we should notice is that it quite different from what Bohm says in his 1952 paper: we have effectively been led to regard the wave function of an individual electron as a mathematical representation of an objectively real field. This field exerts a force on the particle in a way that is analogous to, but not identical with, the way in which an electromagnetic field exerts a force on a charge, and a meson field exerts a force on a nucleon. (Bohm 1952, 170)

The view that there is an objectively real field that applies force to particles to push them around is repeated again.

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In our interpretation of quantum theory, [the double-slit] experiment is described causally and continuously in terms of a single precisely definable conceptual model. As we have already shown, we must use the same wave function as is used in the usual interpretation; but instead we regard it as a mathematical representation of an objectively real field that determines part of the force acting on the particle. (Bohm 1952, 174)

In his early work, the differences Bohm points out between the quantum field and other fields do not involve denying that the quantum field exerts a force (or pushes particles around). The fact that the wave function is in configuration space is an interesting observation, but I am not sure why Bohm says such a fact “clearly” prevents us from treating the quantum field as if it transfers energy and momentum. (If it is so clear, why is it that Bohm did not draw this conclusion in 1952?) I admit that the wave function being in configuration space as opposed to Cartesian space (or a closely related space) makes it difficult to visualize what is going on, but Bohm’s point needs more development. Unfortunately, he says virtually nothing else on this issue. The reason he needs to say more is that we need an explanation of how the electron gets the active information he is talking about. The remote controlled ship and radio metaphors suggest that physically real waves transmit the information these objects need. If there is no physically real quantum field to transmit the active information the electron uses, then what transmits the information to the electron? It might be objected that the preceding question is unfair. Perhaps Bohm need not be interpreted as denying that the quantum field has more than a mathematical existence; perhaps Bohm could concede the physical reality of the quantum field and claim that the existence of the wave function in configuration space simply precludes an interpretation of the quantum field as exerting a force. Such a move is logically possible, but it suffers from a lack of motivation. Once the physical reality of the quantum field has been conceded, why should it not act like other fields and exert a force (especially since the mathematics – see Equation (5) – invites this interpretation)? What is the point of conceding the physical reality of the quantum field and then denying that it is causally efficacious? Indeed, if it is not causally efficacious, in what sense is it physically real? On the other hand, if the quantum field does not exist in physical space and is a mathematical abstraction, then how do we make sense of the notion of active information? The radio metaphor explains active information transfer using a physically real entity that carries and transmits the information. Denying the physical reality of the quantum field would be to deny the existence of a physically real entity, meaning the notion of active information needs to be rethought. The second law of thermodynamics requires work to be done if information is to be transferred. If the quantum field must do work on a particle in order to guide it, then it is interacting


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causally with it (just as a radio wave must interact causally with a receiver). But if the quantum field is interacting causally with something, it must have physical reality. In response, it might be objected that Bohm should be interpreted as suggesting that the quantum field is physically real, but that its causal powers are restricted simply to the transfer of information, not to mechanically pushing particles around. This view, too, is difficult to motivate simply from the claim that the wave function is in configuration space. Once it has been conceded, in spite of the fact that the wave function is defined over configuration space, that the quantum field has causal efficacy enough to transfer information, thereby doing work or exerting force on a particle, then the existence of the function in configuration space seems insufficient to motivate the view that the quantum field is not pushing particles around. In other words, if the quantum field can exert enough force to transfer information, why should the wave function being in configuration space prevent us from saying that the quantum field exerts all the force required to move a particle (which is what the mathematics invites us to say when we take the appropriate partial derivative of Q)? It should be noted that the point of much of the preceding has been to score dialectical points against Bohm. Someone may very well come up with a reason why the existence of the wave function in configuration space precludes us from interpreting the quantum field as being physically real or causally efficacious; I have not offered any principled argument against that view. Since the purpose of this paper is to examine Bohm’s reasons for denying the causal efficacy of the quantum field, I will remain content with making a dialectical point against him: He needs the physical reality and the causal efficacy of the quantum field if his metaphors explaining active information are to be at all useful, and conceding the physical reality of the field while denying that it is causally pushing particles around is unmotivated. I would like to add one last dialectical point against Bohm. He never considers that possibility that the wave function being in configuration space could be used to defend the mechanically causal nature of the quantum field. Consider the following argument: since the probability amplitude is in configuration space, we should treat talk of “force varying with form and not amplitude” with a grain of salt; comparing the amplitude of a wave in configuration space with the amplitude of a classical (for example, water) wave is a bit like comparing apples and oranges since these waves and their amplitudes are defined over different types of spaces; when it is said that “with the quantum field, it is as if the cork could bob with full strength even far from the source of the wave” (Bohm and Hiley 1993, 31), we have to understand that apples and oranges are being compared;

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consequently, we should not be prepared to infer simply from the fact that the force engendered by Q does not vary with probability amplitude, that this force does not act in a mechanically causal manner. The preceding is an argument sketch, and I am not sure how strong an argument it would be if fully developed. Still, it does suggest that Bohm did not say enough about configuration space and causal efficacy. If one is inclined to take the existence of probability amplitudes in configuration space as explaining away an apparent oddity associated with the force of the quantum field varying with form and not probability amplitude, then the whole argument that this apparent oddity undermines the mechanically causal nature of the quantum field is undermined. We simply have not been given compelling reasons for why we should infer the causal impotency of the quantum field from the fact that it is defined over configuration space.

3. THE CONCENTRIC CYLINDERS METAPHOR

The radio metaphor suggests that there is more structure to be discovered in so-called “elementary” particles. The concentric cylinders metaphor and the theory of implicate and explicate orders also suggest that there is much more order or structure to be discovered. However, we should not be mislead by this superficial similarity. The argument in this section of the paper will be that the types of structures or orders postulated by radio and concentric cylinder metaphors are quite different. Before getting to the argument, a brief summary of the concentric cylinders metaphor is in order. Bohm’s concentric cylinders model of the implicate and explicate orders is his most thought provoking metaphor. Imagine that we had concentric cylinders separated by a viscous fluid such as glycerine (see Figure 1). Into the fluid we insert a drop of insoluble ink. By rotating the inner cylinder clockwise, we can get the droplet to thin-out to the point where it is no longer visible to the unaided eye. By rotating the inner cylinder counter-clockwise, we can get the spread-out ink particles to collect and reform the droplet. (The high viscosity of the fluid makes dispersion negligible.) When the ink is spread-out, it is enfolded or implicate or part of the implicate order; when it is visible, it is unfolded or explicate or part of the explicate order. Bohm makes the interesting observation that we can enfold a series of ink drops in such a way that when we unfold them (or reverse the motion of the inner cylinder), we get the illusion that a single particle is in motion. Indeed, it is possible to build on this example in such a way that we can see why Bohm was inclined to think that implicate-explicate explanation undermined causal explanation. For example, we can enfold


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Figure 1.

two series of droplets so that the process of unfolding them will give the illusion that there are two particles engaging in a collision. In such a case, we would not be inclined to say that there are two bodies which engaged in causal contact, exchanging momentum and energy. Similarly, Bohm seems inclined to think that the interactions of particles in the quantum realm is really non-causal because their behaviour is the explicate manifestation of the implicate realm. Roughly, the explicate realm is space-time and that which exists in it, and the implicate realm is the set of structures and processes which Bohm hypothesizes gives rise to the explicate realm.3 Bohm hopes that his implicate-explicate metaphysics will contain the resources necessary to conceptually unify Relativity with Quantum Mechanics; he also hopes that it will lead to new predictions. There is a straightforward objection to Bohm’s concentric cylinders model of the implicate and explicate orders. The behaviour of an ink drop in the glycerine can be completely explained in terms of the mechanically causal interaction between the particles making up the ink and the particles making up the glycerine. This would seem to undermine the claim that the concentric cylinders model exemplifies a non-causal form of explanation.

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However, Bohm inverts the metaphor is such a way that he can escape from the preceding charge. He claims that rather than explaining enfoldment and unfoldment in terms of extended matter and its motion, we need to explain matter and motion in terms of enfoldment and unfoldment (Bohm 1987, 41). Chapter 15 of The Undivided Universe is an attempt to start working out a mathematical model for such a view of nature, and the algebra of this view attempts to make more precise notions such as degree of implication or degree of enfoldment. For the sake of argument, I will assume that some sort of implicateexplicate metaphysics can be rendered coherent. Doing so will allow us to see what follows from such a view of nature. In a sense, what is at issue here is how we explain non-local interactions. If an implicate-explicate metaphysics can be made to work, we might have an account of nonlocal interaction which would not be mechanically causal. I say this since matter and movement are supposed to be explained in terms of enfoldment and unfoldment, not forces and causality. In short, Bohm is proposing a radically new approach to dynamics. This is not wholly unmotivated since (a) non-local interactions are non-classical, and (b) his earlier theory was explicitly non-relativistic, and it is not clear if it could be reconciled with relativity without significant modifications. It might be thought that Bohm is undermining the idea of mechanically causal explanation altogether; after all, if matter and movement are explained in terms of enfoldment and unfoldment, then, in principle, even our everyday objects – tables, chairs, coffee mugs, books, etc. – and their movements are fully explained in terms of enfoldment and unfoldment. In a strict sense, this may be true; however, since the implicate processes which lead to non-locality cancel out at higher levels of physical study, Bohm is surely in position to claim that, for practical reasons, we can and should continue to use higher level (causal) explanatory strategies. To explain fully explain quantum phenomena such as non-locality, new phenomena at sub-quantum levels, to derive quantum level processes from these sub-quantum level processes, and to reconcile these levels with Relativity, we invoke implicate-explicate explanation. There are two hopes Bohm attempts to realize with his new metaphysics – empirically, it is hoped that such an approach will be compatible with existing predictions and lead to new predictions; conceptually, it is hoped that this sort of explanation will unify ‘Relativistic and Quantum Mechanics’. What we should notice is that if the Bohmean story of an implicateexplicate metaphysics can be told, then the metaphors we examined earlier are misleading. The radio metaphor suggests that information is being transmitted or is travelling over a distance (or classical space). This differs greatly from what Bohm tries to convey with the double cylinder model.


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Imagine that we enfolded ink drops in such a way that when we unfolded them, we would get a motion picture (or an animation) of a delayed choice experiment. There would be nothing (including information) travelling from one “photon” (or ink) detector to another. Rather, the picture unfolds in the manner that it does because of the way the ink drops were enfolded. The non-locality in the explicate realm results from an encoding or enfoldment of structures in the implicate realm which cannot all be unfolded simultaneously. To beings who can only perceive the explicate realm, it might appear as if information is being transmitted and received. This appearance is the result of not taking into consideration the implicate order out of which the explicate arises. So, if an implicate–explicate metaphysics turns out to be the best way to model nature, then Bohm’s abandonment of a mechanically causal analysis of Q may be vindicated; however, this sort of vindication will demonstrate the inadequacy of the radio metaphor. This is not surprising since the radio metaphor buys into a largely classical view of nature, which leads to an interpretation of the force engendered by Q as carrying and transmitting information to a particle. The concentric cylinders metaphor rejects the idea that information is carried or transmitted by the quantum field. Indeed, if the concentric cylinders metaphor is correct, then there is no quantum field which pushes particles around. Rather, the movement of particles arises from a process of unfolding and enfolding of particles from the implicate into explicate, back into the implicate, and so on. There is no need for a quantum field of force to push particles around. This is not a point which Bohm and Hiley have appreciated. After going to some trouble (Bohm and Hiley 1993, 357–361) to explain the implicate order by using the concentric cylinders metaphor, the following statement is made: Clearly, the manifest world of common sense experience refined where necessary with the aid of the concepts and laws of classical physics is basically in the explicate order. But the motion of particles at the quantum level is evidently also in an explicate order. However . . . this latter order is not always at the manifest level because it is profoundly affected by the active information represented by the quantum potential. This latter operates in a subtle way . . . this operation is in an implicate order. (Bohm and Hiley 1993, 362)

The notion of active information is first brought up in the context of discussing the radio (and other4 ) metaphors. “The information in the radio wave is potentially active everywhere, but it is actually active, only where and when it can give form to the electrical energy which, in this case, is in the radio” (Bohm and Hiley 1993, 36). In the block quote above, we are invited to treat active information as operating at the implicate level, but something odd is going on here. One cannot make use of both the concentric cylinders and radio metaphors to make sense of the workings

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of the implicate order. Information of the type suggested by the radio metaphor travels across space and causally impacts on the receiving entity; the concentric cylinders metaphor treats particles as emerging and dissolving into a more fundamental or implicate order, and no information is transmitted. From the perspective of a being who only has perceptual access to the explicate order, it might appear as if information is being transmitted between elementary particles in circumstances such as delayed choice experiments, but if the concentric cylinders model of the implicate order is correct, this is simply an appearance since there is no transmission or reception of information. Moreover, the deeper structure postulated by the concentric cylinders model is not to be found in the particle; it is an order into which and out of which the particle emerges. If the radio metaphor is correct, there are deeper structures to reality, but they are to be found by treating “elementary” particles as having structures capable of interpreting and putting signals to work – just like a radio. It might be objected that it is logically possible to reconcile the radio and concentric cylinders metaphors. For example, consider a series of ink drops which are enfolded into the glycerine so as look like a structured object. Imagine that other ink drops are enfolded in such a way that when unfolding takes place, it appears as if a wave is travelling toward the structured object and received by it, and as if the structured object is putting the information to work. This may appear to be a way of reconciling the radio and concentric cylinders metaphors (by building the former into the latter), but it comes at the cost of robbing the radio metaphor of explanatory work. On such an approach, the concentric cylinders metaphor suggests that there is no information travelling from one point to another. Recall: the deeper structure of “elementary” particles (radio metaphor) and the ability of the implicate order to enfold and unfold structures (concentric cylinders metaphor) were both postulated to give a deeper conceptual model of quantum (and other) phenomena. They are both offered as explanations of the same phenomena. If we take the structured object in the hybrid concentric-cylinders-radio metaphor to represent a structured electron where the structure helps to explain what makes various quantumlevel phenomena (such as non-locality) possible, then the processes of dissolving into and reconstituting from the glycerine must be something even more fundamental than the structure of “elementary” particles which makes quantum-level phenomena possible. Given the way Bohm leaves open the possibility of a various levels of implicate order (see note 3), this way of reconciling the concentric cylinders and radio metaphors may appear to be in keeping with the spirit of Bohm’s ideas. However, the hybrid does not appear to be what Bohm had in mind. He appears to have offered


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both metaphors as conceptual models for the same phenomena. Moreover, even if we did combine the two metaphors, the radio metaphor would still encounters the difficulties raised in the Section 2 of this paper. Bohm cannot have both his radio metaphor and his concentric cylinders metaphor as illustrative of the same phenomena. If we take the concentric cylinders metaphor seriously, then we have to reject the radio metaphor. The radio metaphor requires us to understand particles as interpreting information; the concentric cylinders metaphor does not. The radio metaphor buys into a largely classical conception of dynamics and asks us to treat the force engendered by Q as passing information on to particles to be put to work. The concentric cylinders metaphor takes a radically new approach to dynamics, inviting us to completely rethink the way particles behave. If one of these metaphors is correct regarding how we need to think about dynamics, then the other is wrong. To be sure, Bohm’s work was incomplete. Some might make use of the incompleteness of his theories as a reason not to take his metaphors seriously. However, as I have attempted to show, the metaphors played a guiding role in the development and interpretation of his theories. If Bohmian theories are to be developed and interpreted in a coherent manner, it is important to be aware of the incompatibilities of metaphors which may be used to carry out that development.

4. CONCLUSION

The bulk of this paper has been concerned with the problems attaching to Bohm’s metaphors. However, it would be a mistake to infer from the preceding that metaphors or analogies5 have no useful role to play in scientific reasoning. The works of Norman Campbell (1957) and Marry Hesse (1966, 1974) make some strong arguments for the legitimate role played by analogy in scientific reasoning; also, their work has laid the foundation for much of the contemporary enthusiasm about analogy. Two recent exponents of the importance of analogy in physics include Andrew Pickering and James Cushing. For example, Pickering has argued at some length that the recycling of analogies plays an important role in developing and establishing a theory (Pickering 1984, 406–408), and Cushing discusses many examples of the use of analogies (Cushing 1982, 1990, 213, 219, 244–245). In discussing Bohr’s quantization of angular momentum (Bohr 1913) Cushing points out that it was not the radical break with accepted physics of the time that it is often portrayed to be. In fact, his first argument in that paper has nothing to do with quantizing the angular momentum, but rather quantizes the energy of the emitted light, in a somewhat ad hoc manner, yet still very much in the spirit of Planck’s by-then-accepted program . . . . This

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episode illustrates the importance of argument by analogy, the need to cement even new departures into a presently accepted background . . . . (Cushing 1990, 219)

Cushing is also well-known for arguing that the dominance of Copenhagen interpretations over Bohmian interpretations is largely a matter of historical contingency. He claims that the former achieved its major successes before the latter, and that is an important part of (if not the only reason) why it is the dominant interpretation of quantum mechanics; the Copenhagen approach is not preferable to Bohmian approaches on evidential or epistemic grounds. It might be thought that the contents of this paper could be used to challenge such a view. After all, I have argued that many of Bohm’s analogies are filled with problems, and if analogies help to “cement” a theory into established views (as Cushing suggests), then perhaps the reason Bohm’s approach is not dominant is that it failed to use analogy to cement itself into existing views. Discussing why such a view is incorrect will allow me make some important qualifications to this paper. There are many analogies between classical physics and Bohm’s interpretation of quantum mechanics. First and foremost, there is the analogy, mentioned in section one, between f = ma and Equation (5). The force engendered by the quantum potential that pushes particles around (as per Bohm’s 1952 paper) provides us with a more “picturable” understanding of the quantum formalism than non-Bohmian approaches. There is no argument in this paper against such an approach. Even though Bohmians like Detlef Dürr, Sheldon Goldstein, and Nino Zanghi do argue that Bohm’s original picture of a quantum force pushing particles around was ad hoc, they do not abandon the attempt to arrive at more picturable universe, one where uncertainty relations are given an epistemic interpretation rather than an ontic one (Dürr et al. 1992a, b, 1993). This more picturable universe that is characteristic of Bohmian approaches is more analogous to classical physics than anything the Copenhagen approaches have on offer. Consequently, in spite of the fact that many of Bohm’s analogies are problematic, it can still be argued that Bohmian approaches tend to be more analogous to classical physics than the Copenhagen approaches. The details of the analogy between classical physics and a Bohmian interpretation of quantum mechanics will depend on the particular interpretation in question. For example, Bohm’s work with Jean-Pierre Vigier makes use of a hydrodynamic formalism that would make certain kinds of analogies with classical physics (Brownian motion, for example) appropriate (Bohm and Vigier 1954). The same can be said of the work of Edward Nelson (1966, 1967). While the more recent work of Bohm and Hiley has a hydrodynamic flavour (think of the concentric cylinders in the viscous fluid), the analogy with Brownian motion disappears in favour of other similarities


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with other picturable, classical properties (the enfolding and unfolding of the ink dot, for example). The work of Antony Valentini, though, is not in this hydrodynamical tradition and is more in keeping with Bohm’s 1952 approach (Valentini 1991a, b). Consequently, hydrodynamical analogies cannot be drawn, but other analogies can. See, for example, Cushing’s comparison of Valentini’s work with the Maxwell-Boltzmann work on equilibrium distribution (Cushing 1994, 163–165). What has been argued for herein is not that analogy has no role to play in physics; nor has it been argued that attempts to make analogies between quantum physics and the physics that predated it is inappropriate. What has been argued is that some of David Bohm’s metaphors appear to have lead him away from a causal interpretation of the quantum potential, and his use of metaphorical or analogical reasoning to justify moving away from a causal interpretation is problematic. It might very well be quite legitimate to draw analogies (in ways other than those criticized herein) between Bohmian interpretations of quantum mechanics and pre-existing physics. However, whether or not such analogies can be justified depends on the details of the particular Bohmian interpretation in question. Assessing these different interpretations is a topic beyond the scope of this paper.

ACKNOWLEDGEMENTS I thank Kent Peacock and an anonymous Erkenntnis referee for their comments on earlier versions of this paper.

NOTES 1 The account of causal implication given here draws heavily from the work of Tim

Maudlin (1994). 2 This is a “rough” condition because it captures too much. If Jack and Jill are married,

then Jack being killed means that Jill will becomes a widow. If Jill did not become a widow, then Jack would not have been killed; this is enough (according to our sufficient condition) to causally implicate Jill becoming a widow with Jack being killed, yet there is no physical causation at work. When talking loosely, we might be inclined to say that Jack being killed caused Jill to become a widow, but what we have in mind is different from one physical event, such as the radiation of a field, causing another physical event, such as the movement of a particle. Jill becoming a widow is a change in her relational properties, not a change in her intrinsic physical properties such as position and momentum (within a single frame of reference). Our sufficient condition should be understood to apply to a change in the intrinsic physical properties of events. This is still somewhat rough, but it will serve our purposes.

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3 I say “roughly” since Bohm and Hiley are prepared to countenance super-implicate

orders (Bohm and Hiley 1993, 378–381). A super-implicate order is related to the implicate order as the implicate order is related to the explicate order. Bohm and Hiley leave open the possibility that there may be a super-implicate order, a super-super-implicate order, and so on. They are non-committal on how many levels implicate orders there may be. 4 The radio metaphor is not the only one Bohm and Hiley use to explain how a structure can put information to work. They also discuss RNA making use of information encoded in DNA (Bohm and Hiley 1993, 36). I have chosen to focus on the radio metaphor since it is given more attention by Bohm and Hiley and since it is more obviously applicable to the case of a particle and an information carrying field, but the DNA metaphor is just as difficult to reconcile with the concentric cylinders metaphor as the radio metaphor. The DNA metaphor still requires a complex, structured entity (RNA) to “decode” and put information to work. None of this is required by the concentric cylinders metaphor. 5 I am using the expressions “analogy” and “metaphor” interchangeably, a common practice in the philosophy of science literature on analogy and metaphor.

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Hesse, M. B.: 1966, Models and Analogies in Science, University of Notre Dame Press, Indiana. Hesse, M. B.: 1974, The Structure of Scientific Inference, University of California Press, Berkeley and Los Angeles. Maudlin, T.: 1994, Quantum Non-Locality and Relativity: Metaphysical Intimations of Modern Physics, Blackwell, Oxford UK and Cambridge USA. Nelson, E.: 1966, ‘Derivation of the Schrodinger Equation from Newtonian Mechanics’, Physical Review 150, 1079–1085. Nelson, E.: 1967, Dynamical Theories of Brownian Motion, Princeton University Press, Princeton, NJ. Pickering, A.: 1980, ‘Exemplars and Analogies: A Comment on Crane’s Study of Kuhnian Paradigms in High Energy Physics’ and ‘Reply to Crane’, Social Studies of Science 10, 497–502, 507–508. Pickering, A.: 1984, Constructing Quarks: A Sociological History of Particle Physics, Edinburgh University Press, Edinburgh. Valentini, A.: 1991a, ‘Signal-Locality, Uncertainty, and the Subquantum H-Theorem. I’, Physics Letters A 156, 5–11. Valentini, A.: 1991b, ‘Signal-Locality, Uncertainty, and the Subquantum H-Theorem, II’, Physics Letters A 158, 1–8. Department of Philosophy Faculty of Arts The University of Windsor 401 Sunset Ave. Windsor, Ontario N9B 3P4 E-mail: [email protected] Manuscript submitted 10 September 2001 Final version received 28 November 2002