Bell’s Theorem, H. P. Stapp, and Process Theism by William B. Jones William B. Jones teaches philosophy at Old Dominion University, Norfolk, Virginia. He holds a Ph.D. in physics (University of Virginia, 1970) and a Ph.D. in philosophy (Vanderbilt University). The following article appeared in In a series of papers (3:1-10, 4:1303-20, 5:270-76, 6:313-23, and 7) Henry Pierce Stapp has argued that quantum mechanics is incompatible with the principle of local causes. According to this principle, the earlier of two events cannot affect the other if the distance between them is so great that a light signal cannot traverse it during the time interval separating the two events. Stapp arrives at this conclusion through a quantum mechanical analysis of the possible results of a double spin-measuring experiment, i.e., an experiment in which one measures the spins of each of two slow neutrons, say, which have just collided. (The "spin" of a particle like a neutron is roughly analogous to the spin of a top or a billiard ball.) He shows that a contradiction ensues if it is assumed that the possible results of measuring the spin of one of two such particles in various directions are independent of the direction chosen for measuring the spin of the other member of the pair of particles. Stapp’s way of handling this difficulty is to postulate (1) that the fundamental events of which the universe is composed are well-ordered as regards their coming into being (which, according to Stapp, is Charles Hartshorne (2) finds Stapp’s proposal noteworthy because making the assumption that the fundamental events of the universe are well ordered effectively disposes of a recalcitrant problem that has plagued theistic process philosophers for some time. If one accepts the relativistic view that all events are All these problems evaporate if the fundamental events of the universe are indeed well-ordered, even if relativistic considerations do prevent us human beings from determining what some of these orderings are. Of course, the crucial question is: what reasons are there for thinking that the fundamental events really are well-ordered? Stapp’s argument for this conclusion has two parts. He first presents a mathematical proof, inspired by Bell’s theorem, that a contradiction results if one assumes: (a) the principle of local causes, (b) an elementary theorem of quantum mechanics, and (c) what Stapp calls the "assumption of contrafactual definiteness" (roughly the assumption that measuring procedures which were not carried out would have yielded definite results had they been carried out and that these possible but unrealized results are restricted by the same laws that apply to the results of actual measurements). The second part of his argument is that the proper, if not the only, way to remedy this situation is to reject the principle of local causes. The intent of this paper is twofold: (i) to present Stapp’s proof in a nontechnical form so that nonphysicists can be apprised of just what it is that he has proved and (ii) to draw out and examine the assumptions of the proof so that its significance for philosophy, especially process philosophy, can be assessed. Part I below addresses the first of these concerns; part II, the second. I. Stapp’s Generalized Version of Bell’s Theorem Stapp’s proof is a modified version of what is generally referred to as "Bell’s theorem." In its original form, this theorem is addressed to the question of whether it is theoretically possible for the statistical conclusions of quantum mechanics to be explained by a hypothetical, experimentally inaccessible realm of microevents characterized by a set of "hidden variables." Bell’s conclusion is that such an explanation is not possible as long as the principle of local causes is assumed. According to this rather common-sensical principle, if two physical systems (particles, for example) have ceased interacting or have never interacted, neither is affected by changes induced in the other. Thus, two widely separated systems which have interacted in the past would each be unaffected by measurements performed upon the other. The system considered by Bell and by Stapp is made up of two particles, neutrons, for example, whose "spin" directions have become correlated through some collision process -- in somewhat the same way that the axes of rotation of two spinning billiard balls might be correlated after they have sideswiped each other (with each spinning in a direction opposite to that of the other). According to the principles of quantum mechanics, two slow" neutrons which have brushed past each other would tend to have their spins in roughly opposite directions. Stapp, unlike Bell himself, is not concerned with "hidden variables." He rather develops a version of "Bell’s Theorem" which is intended to show that, given certain very general assumptions, quantum mechanics itself, as it manifests itself in a simple theorem, is incompatible with the principle of local causes. Stapp’s argument rests directly upon a theorem in quantum mechanics concerning the behavior of a certain class of interacting pairs of particles, among which are neutrons. He focuses his attention upon the correlation that is established between the "spins" of two "slow" neutrons when they scatter off each other (collide with or brush past one another). It happens that the spins of neutrons are quantized, i.e., allowed to assume only certain discrete values. Indeed, the Stapp’s proof, however, requires that the effects of measuring the spins of the two neutrons along different axes be considered. It is easier to grasp what this involves if one employs the following model of the experiment. The model is not Stapp’s, but its use makes it easier to understand his arguments. The model is the following: think of the neutrons, after the scattering event, as traveling in opposite directions (away from each other) down a hollow axle or pipe at the ends of which spin-measuring devices are attached like thin wheels. These "wheels" thus lie in planes perpendicular to the line along which the neutrons are moving. When a neutron passes through the center of one of these devices, it measures the spin of the neutron along an axis coinciding with a radius of the "wheel." More specifically, it determines whether the neutron has spin-up or spin-down along this axis, which is, of course, perpendicular to the neutron’s line of flight. For convenience, let it be assumed that the tube through which the neutrons are traveling is lying horizontally and that initially both of the "wheels" are so turned that they will determine whether a passing neutron has spin-up or spin-down with respect to the vertical. Note that since it is with respect to a The convention followed here will be to assign a spin value of + 1 to a neutron if it is found to have spin-up with respect to a given axis and to assign a spin value of -1 if it is found to have spin-down with respect to that axis. For a given configuration of the two axes and for a given scattering event between two neutrons, the "wheel" spin-measuring devices will record a value + 1 or -1 for each neutron. From each such pair of numbers, one can then form a product, which itself can assume only the values + 1 and -1. The theorem from quantum mechanics which Stapp employs states that if one records the values of these products for a large number of neutron-scattering events (for the same configuration of the "wheels") and averages them, then the result will equal the negative of the cosine of the angle between the axes along which the spins are being measured. This is a statistical conclusion, and a large number of cases must be considered for it to be accurate. It is by applying this theorem to each of the four configurations described in the preceding paragraph that Stapp arrives at the conclusions which he calls the generalized Bell’s theorem. In his analysis, Stapp considers a set of neutron-scattering events, There are thus two possible sets of spin-values for the neutrons coming down the tubes toward the first experimenter: one set The next step in the proof is to note that each of the four sets of spin-value products must satisfy the quantum mechanical theorem referred to above. Since each set of possible spin-values is In the case of the first set of products of possible spin-values, the orientation of the axes of the spin-measuring devices is the same; the angle between them is zero; and the cosine of this angle is one (1.0). Consequently, according to the quantum-mechanical theorem in question, if one adds up the Now each statement to the effect that one of the averages just discussed equals the number given may be regarded as a mathematical equation relating the spin-values that are averaged. There would be four such equations. What Stapp shows mathematically is that these four equations lead to a contradiction. Proving this is a straight-forward exercise in algebraic manipulation. It cannot be reviewed here. The interested reader is advised to consult Stapp’s own treatment (4:1306-08). The vital question that must be addressed here is what is to be made of the fact of this contradiction. What is to be made of the fact that the four statements in the preceding paragraph about the four product averages cannot all be true? II. The Assessment of the Proof The immediate inference to be made from the fact that the four equations described just above lead to a contradiction is either (a) that one or more of the equations is false or (b) that it is somehow illegitimate to combine them algebraically in the way Stapp does in arriving at the contradiction. One can put aside, with reasonable confidence, the suggestion that all four of the equations are false. Given an experimental set-up like that described above, it is always possible actually to carry out the measurements for one of the possible configurations of the spin-measuring devices. Furthermore, experiments of this sort have been performed, and, as Stapp notes (1:938-41), their results obey the mathematical equations which are appropriate according to the quantum mechanical theorem Stapp uses in his proof. Thus, the theorem is experimentally well confirmed. But if one accepts the correctness of the theorem, is one not also obliged to accept the correctness of all four of the equations? They are all instances of the same general relationship, which is expressed by the theorem, and are thus on the same footing. Their common status is also shown by one’s being free to choose any one of the four for experimental test. As was explained in the previous section of the paper, under the right conditions, one can even make this choice after the pair of particles being considered have ceased to interact. The unavoidable conclusion would seem to be that the second of the two alternatives mentioned at the beginning of the section must be correct: some error must have been committed in combining the four equations in the way required to reach the contradiction. This is the line of argument followed by Stapp. According to him, the error lies in assuming that one is dealing with the same set of possible spin-measurement results for the particles coming out one side of the apparatus described no matter what orientation one considers for the spin-measuring device (s) on the other side of the apparatus. Put more formally, this is the assumption that in averaging the first two sets of spin-value products one is dealing with the same set of possible spin-values Stapp’s thesis about In order to understand this seemingly paradoxical situation, one must keep in mind that, for a given collision event, it is theoretically impossible to carry out more than one of the four experiments corresponding to the four possible configurations of the spin-measuring devices described above. Of course, it is manifestly impossible to give either of these devices more than one orientation at a time, but the problem goes deeper than that. Given, as noted above, that the overall magnitude of the spin of the neutron is fixed, simultaneously measuring its spin in two different directions, like the two available to each of the spin-measuring devices, would violate the Heisenberg indeterminacy principle. Thus only one of the four possible sets of spin-value products discussed in part I could ever be formed from spin-values actually obtained from experiment. The four sets of spin values, Accordingly, Stapp is careful to distinguish between (a) attributing definite spin values in more than one direction to a particle like the neutron and (b) asserting that if the spins of certain pairs of such particles are or were to be measured in this or that direction, a specific mathematical relation will or would be found to hold, on a statistical basis, between the spin values of the members of the pairs. Stapp argues that while the uncertainty principle does forbid ascribing such spin values to particles like the neutron, it does not rule out as meaningless or unphysical talk about the outcome of various possible measuring procedures, even if it is theoretically impossible for more than one of them actually to be carried out. The argument here is subtle. It is clear enough that the spin values dealt with in Stapp’s proof are not intended as hypothetical characteristics of particles; but, taken as a class, they are not all possible experimental spin values either. Each corresponding pair of members of the sets The assumption that it is meaningful and proper to employ symbols designating the results of unperformed and, in half the cases, unperformable measurements Stapp calls the "assumption of countrafactual definiteness," which he has defended in one of a series of lectures given at the University of Texas in the spring of 1977. In these lectures Stapp counters the suggestion that symbols which designate quantities that are unknowable in principle are meaningless by pointing out that philosophers of science have come to recognize the need for theories that have reference to such unknowable quantities. He cites as an example Maxwell’s electromagnetic theory with its postulated electric and magnetic fields propagating in a vacuum. The magnitude of these fields in a vacuum cannot be measured because any attempt to do so will destroy the vacuum. They are thus unknowable in principle. Stapp concludes that being unknowable in principle is not a fatal defect. He further notes that the consideration of alternative possibilities is surely meaningful and of considerable theoretical and practical usefulness. Stapp is certainly correct in arguing that talk about quantities that are unknowable in principle is not necessarily meaningless, and the example he cites surely supports this claim. However, there appears to be an important difference between the unknowable quantities found in Stapp’s proof and the kind cited in his defense of the meaningfulness of such discourse. Consequently, this argument contributes very little either toward defending the cognitive significance of Stapp’s proof or toward clarifying the status of the unknowable spin values with which the proof deals; the same can be said of his argument defending the general significance of talk about possibilities. Stapp’s use of Maxwell’s electromagnetic theory as an example to illustrate the propriety of having reference to unknowable quantities is particularly unfortunate in that it beclouds the central issue of the proper status of the spin values dealt with in the proof. The notion of an electromagnetic field propagating in a vacuum is meaningful precisely because it is part of an elaborate physical theory which has numerous well-confirmed observable implications. Clearly, the basis for the meaningfulness of Stapp’s unknowable spin values must be of a different character. He goes to considerable length to distinguish his treatment of possible spin values from such a unified, theoretical account of the behavior of particles like the neutron. He does this in order to escape the strictures of the uncertainty principle, as explained above. Furthermore, he makes it quite clear that the view that he is advancing has no real implications as regards the outcome of actual experiments. So the meaningfulness of talk about the joint class, But it does not follow from all this that one can then treat these four equations as a set of simultaneous equations expressing various relationships between the members of the joint class This same point can be put in terms of possible spin-measurements: If the three classes of spin measurements
It should be clear from the above discussion that while entertaining the possibility that the antecedents of the hypothetical statements (a), (b), (c), and (d) might all be satisfied at once would permit the four equations under discussion to be treated as simultaneous equations, merely affirming the four statements themselves does not. The meaningfulness of talk about alternative possible measurements does not in itself ensure the propriety of conjunctively combining equations which each express relationships among the results of one of the four possible measuring procedures. And since Stapp has provided no further arguments for the meaningfulness of the joint class Quite obviously, the crucial question for those who favor a Whiteheadian type of framework is precisely what status do alternative possibilities like the two sets of spin-measurement results The very possibility of this line of argument calls attention to the fundamental question of this entire matter: even if the contradiction arrived at in Stapp’s proof does constitute a genuine problem, must one accept his method of disposing of it? Is Stapp’s postulation of a well-ordered set of fundamental events with effectively instantaneous communication or causation between them the only or even the best solution to the problem posed by his modified Bell’s theorem? This is a question which will surely need to be considered by philosophers who have an interest in the very significant issues raised by Stapp’s work.
References 1. Stuart J. Freedman and John F. Clauser, "Experimental Test of Local Hidden-Variable Theories," 2. Charles Hartshorne, "Bell’s Theorem and Stapp’s Revised View of Space Time," 3. Henry Pierce Stapp, "Correlation Experiments and the Nonvalidity of Ordinary Ideas About the Physical World," LBL-5333, Berkeley, California, 1968. 4. Henry Pierce Stapp, "S-Matrix Interpretation of Quantum Theory," 5. Henry Pierce Stapp, "Bell’s Theorem and World Process," 6. Henry Pierce Stapp, "Theory of Reality," 7. Henry Pierce Stapp, "Quantum Mechanics, Local Causality, and Process Philosophy," edited by William B. Jones, Support from the Old Dominion University Research Foundation is gratefully acknowledged by the author. Viewed 8877 times. |