Foundations of Physics Letters, VoL 4, No. 5, 1991
N O N L O C A L I T Y AS A F E A T U R E OF T H E P H Y S I C A L WORLI~ 1)
C. I. J. M. Stuart
Centre for Quantum Field Theory and Complex Systems, 10-102 Clinical Sciences Building, University of Alberta, Edmonton, Alberta, T6G 2G3, Canada Received August 7, 1989; revised April 30, 1991
A prediction puzzle leads to a form of necessary realism which forces the rejection of a central tenet of the Copenhagen interpretation. This leads to reconsidering conceptual difficulties related to Bell's locality premise. It is shown that a paradox of elementary probability theory puts new light on Bell's assumption that causality and statistical independence are mutually incompatible. Key words: prediction puzzle, necessary realism, Bell's theorem, locality, reciprocal causality. 1.THE PHYSICAL WORLD: EMPIRICAL AND PRIMITIVE REALTY
If a physical theory exhibits predictive success, then it is possible to construct a non-empty set of ordered pairs of statements, {(T,M)i}, in which the components Ti are derived from the theory in question and the Mi are obtained from experimental findings, components with the same index agreeing in their content (up to specified tolerances of experimental error). The resulting construction defines the predictive relation P = TxM,
(1:1)
meaning that the theory T has an external model or interpretation in M. According to an argument put forward by Einstein [1], any physical theory can, without logical objection, be understood in either of two ways, namely: Ea: physical theory pertains to an independent primitive reality R; 479 0894-9875/91/I000-047956.50/0 (~) 1991 Plenum Publishing Corporation
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E2: physical theory pertains only to the contents of our empirical knowledge defined with respect to empirical reality E ¢ R. If we accept El, we can then restate relation (1) as P = TxR.
(1:2)
The second alternative E2 is the one adopted by the Copenhagen interpretation of quantum mechanics; according to that interpretation, the set M in relation (1) is defined only on E. Clearly, M will depend on T in the obvious sense that different physical theories will have models in different sets of experimental findings. But, whereas each such theory is an intellectual construction having reference to an intended external model, each such model is an empirical construction defined on E; so, for each M we have E ~ M. On that understanding, the Copenhagen requirement for the predictive relation can be expressed by rewriting relation (1) as P = T × E,
with (E ~ R).
(1:3)
As regards relations (2) and (3), we note that every physical theory has a dual role; with respect to the empirical reality E, physical theory prescribes what can (and, by implication, what cannot) belong to E; on the other hand, every physical theory also defines what we are prepared to believe about the primitive physical reality R. In this way, the mathematical structure of physical theory shapes a system of empirical knowledge through the predictive relation (3) and, simultaneously, a system of theoretical belief through the conjectural relation (2). The requirement for the dual role of theory is not arbitrary; it is a necessary and sufficient condition if prediction is to test the external consistency of physical theory. This can be established as follows. Relation (1) is obviously a necessary condition, for there is simply no model external to the theory if we have no experimental findings or observations. Since E _~M, it then follows that relation (3) is a necessary condition. Hence, if (3) is not a sufficient condition, then neither is (1). The following prediction puzzle establishes that relation (3) is not a sufficient condition. The basis of the puzzle lies in this: If a theory's predictive success is not due simply to chance, then it must depend on an element of regularity or consistency in the character of our empirical experiences and, hence, in the content of our empirical knowledge; objective interpersonal testabifity of such experiences can attest to their consistency, but it cannot provide the source for the consistency itself. We are therefore led to the puzzle of accounting for consistent predictive success through theories which assume our empirical experiences form an internally consistent system on their own merit. Thus, the predictive relation not only tests the external consistency of theories, it also presupposes the existence of some independent and selfconsistent source of regularity for empirical phenomena. Successful
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prediction therefore implies that the theory has a model or interpretation in this independent source and, thereby, the predictive relation is simultaneously an indirect test of the theory's internal consistency also. Put differently, in mathematics we can attest to the internal consistency of a given system by showing that it has a model in an external system about which there is no question. This involves an act of mathematical faith since, otherwise, we are led to an infinite regression in pursuit of a system that can be attested as self-consistent without external appeal. In parallel, the situation in physical science rests on the fundamental assumption that nature, i.e., the primitive reality R, is an independent and self-consistent system. The prediction puzzle makes this assumption methodologically inescapable. If the system of beliefs expressed through relation (2) were not included in the domain of reference for theory, i.e., if T did not have a putative model in R, then we should not be able to escape the prediction puzzle. On the other hand, although relation (2) is thereby a necessary condition for the external consistency of physical theory, it cannot be a sufficient condition, since our access to R must be mediated by experimental findings defined on E. We are then directly led to the conclusion that relations (2) and (3) together constitute necessary and sufficient conditions for the external consistency of physical theory. The dual role of theory is then a necessary consequence. It follows that any physical theory which fails to escape the prediction puzzle is necessarily incomplete, since the predictive relation for such a theory cannot provide a necessary and sufficient condition for the external consistency of the theory in question. Since the Copenhagen requirement for the predictive relation, i.e., relation (3), provides a necessary but not sufficient condition for external consistency, it follows that the Copenhagen interpretation yields an incomplete theory of quantum mechanics. Another consequence of the prediction puzzle is that it precludes the idea that the only role of physical theory is to provide a mathematical structure establishing correlations between actual phenomena so as to yield successful predictions concerning possible new phenomena. That idea is itself an expression of the belief that empirical phenomena constitute the sole domain of reference for physical theory and, accordingly, it fails to provide the necessary and sufficient conditions for successful prediction. In contemporary physics, the Copenhagen interpretation is the sole barrier against the acceptance of relation (2). However, the foregoing demonstration of the incompleteness resulting from the Copenhagen interpretation forces a reconsideration of (2), but now necessarily in conjunction with relation (3). On the other hand, Einstein's argument E2 for the logical consistency of the Copenhagen interpretation is itself shown to be incomplete. For, to say that there is nothing logically inconsistent about quantum mechanics being restricted to an account of our empirical knowledge is not the same as saying that the Copenhagen
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scheme carries out the restriction in a logically consistent way. It is therefore relevant to mention that it has been shown elsewhere that the Copenhagen scheme is self-inconsistent and, in particular, that the inconsistencies arise directly in connection with the realism issue [2]. If it is correct to say that the Copenhagen interpretation acts to define the physical content of the quantum mechanical formalism so as to exclude realistic interpretation, and if it is also correct to say that it does so in an internally inconsistent way, it then follows that we can disregard the Copenhagen disavowal of realism. Moreover, since the prediction puzzle and the dual role of theory both apply to any physical theory, one sees that relations (2) and (3) jointly define a form of necessary realism. 2. NONLOCALITY In this section we make several passes over familiar ground so as to lay a foundation for Section 3. As is well known, Bell's inequalities assume realism and, further, that the physical world has the property of Einstein locality: Events that are spacelike separated are also independent events. On the joint premises of local realism, Bell's inequalities are referred to a physical system composed of two spin-dependent particles a and b, the combined system being, say, in the singlet state (with total spin equal to zero); for statistical purposes, we can assume that an ensemble of such systems is prepared. Since the individual particles thus have opposed spins (or polarizations, given a suitable choice of coordinates), measurer ments on their spin (or polarization) components along a common axis can be represented in terms of the dichotomic variable +_1. Such measurements may be carried out, as in the Orsay experiments [3], on a photon pair travelling in opposite directions from a common atomic source; any correlation created by a measurement on one member of the separating pair would thus violate Bell's locality premise. Quantum mechanics requires that the combined system be represented by a single wave function ~ in the singlet state I~to). If we know the object system to have zero total spin before any measurement collapses the wave function, then the system must have been prepared in some way. Thus, preparations yield knowledge of a certain kind; in general, this will be statistical in characer since the preparation may be done on an ensemble of object systems. On the other hand, knowledge gained from a measurement carried out on either of the particles in a given pair will be definite, taking one of the values +1. We also recall that the combined wave function has a space part, ~t(r), as well as a spin part ~t(c). In arrangements like those of the Orsay experiments, the spatial coordinates of particles a and b have become nonintersecting before any measurements are done on the particles; so we have
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{ra} n {rb} = O,
(2:1)
by which the separability of the particles can be assured. Thus, applied to photon cascade experiments, Eq.(2:l) means that the particles are spacelike. Bell's locality premise then means that a measurement done on the one should not affect the other and, hence, the corresponding observables are represented by application of the product rule for joint probabilities, i.e., for mutually independent occurrences. Thus, assuming real but "hidden" variables ~ with probability density 9(~) over domain A, Bell's expectation value, or combined probability, takes the form P(a, 13) = S P(~) A(a, k)B(13,7~) dZ..
(2:2)
Here, A represents a set of measurement results on particle a in terms of fixed experimental parameters a (e.g., polarizer settings), with A(~) as an observable having values +1; the terms B and 13have similar meanings with respect to particle b. Locality is intrinsic to Eq. (2:2), since A does not depend on 13, nor B on 5. Unlike the situation in quantum mechanics, observables in hiddenvariable theories are not represented by self-adjoint operators in Hilbert space, but instead by real functions on & On the other hand, it is usual to assume that the crux of the matter is whether the one can exactly reproduce the predictions of the other; i.e., all expectation values should agree. As is well known, Bell's theorem leads in certain cases to predictions contrary to those of quantum mechanics and, in particular, the experimental evidence stands decisively in favour of the latter. However, in connection with these conflicting predictions, we note the complication described in the next three paragraphs. Bell's inequalities set limits on the degree of correlation possible between the outcomes of the observations A and B, the outcomes depending only on the respective experimental parameters c~and [3 and the hidden variables Z. Neither the nature nor the number of the ~ is specified; nor is it made explicit whether the locality property enters specifically through L Thus, one might argue that the parameters a and 13are local, or that Z.is intrinsically local, or that both ~. and the experimental parameters are local. Since a and 13represent macroscopic entities, it then becomes unclear as to whether Bell's inequalities refer exclusively to the microscopic domain. On the other hand, whereas the wave functions and operators of quantum mechanics are given in mathematical detail, there is an element of vagueness surrounding their physical meanings. The operators are most often presented as corresponding to classical variables and, in that sense, they are said to represent quantum observables. Under the Copenhagen interpretation, our knowledge of object systems is, in general, only probabilistic in character before a measurement is carried out, the act of measurement then putting the object system into a definite state about
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which we have definite knowledge. Thus, measurements put one state (of knowledge) into another state (of knowledge). The corresponding formal statement is to say that operating with the operator A on a state IxlO yields a different state Ixlr'} = AI~0. So one may argue that the operators correspond to the act of measurement. Clearly, the two interpretations of the operators, the one equating them with observables, the other with the act of observing, are mutually incompatible. The situation is no better with the wave functions upon which the operators act. The most common view is that they represent the states of object systems. A contending view is that they represent the event: preparation of object systems [e.g., Ref.4 and citations therein]. In either case, they are said to represent our knowledge of object systems rather than the object systems themselves; in that respect, they are computational devices and, specifically, not in hypothetical correspondence with either states or preparations. But in that event, their predictive role simply gives rise to the prediction puzzle discussed in Sec. 1. The Copenhagen interpretation makes the complication described in the preceding three paragraphs directly relevant to the conflicting predictions of quantum mechanics and those derived from Bell's theorem. For, according to the Copenhagen scheme, the observed correlations do not involve nonlocal realism, which is to say that the nonlocal property cannot be imputed to the primitive reality R. On the other hand, quantum mechanics provides no explanation for the correlations which, instead, are viewed simply as consequences of the mathematical formalism. By prescinding from explanation, quantum mechanics not only encounters the prediction puzzle, it also fails to provide any significance for the conflicting predictions. Instead, we get conflicting interpretations of the quantum formalism, such as those noted above, none of which escapes the prediction puzzle since the physical meaning of the formalism remains unfixed. Not surprisingly, this situation can give rise to conceptual anomalies. On one view, a pair of particles represented by a single wave function not only constitutes a single physical system in a combined (e.g., singlet) state, it is also required that the system constitutes a single indivisible entity, the idea that two distinct particles are involved being only a mistaken idea based on injecting the outlook of classical physics into quantum mechanics [5]. This view is based on the circumstance that the wave function for combined states cannot be factorized as a direct product of two distinct wave functions, one for each of the particles. According to the Copenhagen interpretation, such factorizability is a necessary condition for having the particles in a well-defined quantum state. The argument then goes that we cannot ascribe a definite reality to entities having no definite state. Hence, the observed correlations for singlet states cannot be said to manifest nonlocal realism. The conceptual anomaly arises from confounding objects with their states: Being unable to ascribe a definite spin state to some object does not
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imply the nonexistence of that object. In physical terms, the anomaly is directly revealed by taking into account Eq. (2:1), bearing in mind that the complete wave function for the singlet state has a position part as well as a spin part. Since any measurement on a spin component of the singlet state must necessarily be carded out somewhere, on at least one of the particles, it is then self-contradictory to assert that the measurement is done on an object having no definite existence. Also, since the spin components along a common axis must have opposite signs, it is nonsense to attribute such mutually exclusive properties to one and the same entity. The foregoing illustration suffices to establish the conceptual inadequacies arising from attempts to evade the prediction puzzle in quantum mechanics. Such attempts serve only to impede progress toward an understanding of the predictive conflict between quantum mechanics and Bell's inequalities. There is, of course, one sense in which the nature of the conflict is quite clear. Bell's theorem assumes locality through an essential use of the product rule for mutually independent events. Because the quantum singlet state is nonfactorizable, and so cannot be written as a direct product of a separate wave function for each particle, the predictive conflict follows automatically. However, one cannot thereby say that quantum mechanics is a nonlocal theory, for we must take into account that Bell's locality premise is explicitly tied to the premise of realism, whereas quantum mechanics is not. Moreover, the preceding discussion of interpretational discrepancies concerning the wave functions and their operators reveals something of the difficulties standing in the way of an explicitly realistic quantum mechanics. On the other hand, we can reasonably take the view that those difficulties are really nothing but historical accumulations following in the wake of the original, and anti-realistic, Bohr-Heisenberg scheme. For, according to the arguments developed in Sec. 1, every physical theory is committed to a form of necessary realism. On that basis--and so far as I am aware, on that basis alone--one can properly conclude that if phenomena predicted by quantum mechanics have nonlocal character, then the theoretical implication is that we live in a nonlocal physical world. This does not dispose of the interpretational problem, since it does not allow us to say in what precise manner quantum mechanics tells us that the physical world is nonlocal. Failing that, we cannot with sufficient clarity assert that quantum mechanics conflicts with special relativity theory. That question is not simply a matter of clarifying the content of quantum mechanics; as pointed out elsewhere, one also has to be clear about what special relativity has to say regarding physical causality [6]. For our present needs, we shall simply put the matter as: Events that are independent by virtue of their spacelike separation are not causally related events. The foregoing italicised expression goes together with that for Einstein locality (given at the beginning of the present section) and with
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Bell's expression of locality through the product rule for probabilities in Eq. (2:2). The exclusion of the direct product rule for the singlet state then means that the corresponding spin components of the two particles are causally related by quantum mechanical theory. Since the observed spin correlations conform to the statistical predictions of that theory, the arguments of Sec. 1 allow us to consider the possibility that the physical worId may entail causalities which conflict with a fundamental requirement of special relativity. [For reasons given in Ref.6, we here disregard the idea that special relativity theory restricts causal linkages to signal transmissions.] 3.
A PHYSICO-MATHEMATICAL
PARADOX
It is often said that the violation of Bell's inequalities comes from the use of a single wave function to describe properties of n-particle systems. But this cannot be correct, because the spin correlations are empirically observed and experimental results cannot be consequences of the quantum mechanical formalism! It is also often said that the correlations arise because of some peculiarity of the spin variable. This cannot be correct either, because the Einstein-Podolsky-Rosen (EPR) correlations --which started all this business in the first place--were derived from considering the position-momemtum variables. As it turns out, though, the spin variable yields an especially simple illustration of the source of the problem. However, we need first to deal with a complication arising from our earlier specification of local causality. The complication arises because we have remained neutral on the question as to whether relativistic causality is restricted to the forward light cone of the causal event. The restriction would, of course, place relativistic causality on the same logical footing as the classical principle of causality, namely, that effects cannot preceed their causes. On the other hand, the restriction is not universally accepted. In particular, Costa de Beauregard has proposed a solution for the EPR correlations on the basis of time-reversible conditional probabilities, which he identifies as a mathematical specification of causality. [See, e,g., Ref. 7 and the citations therein; the proposal is discussed, e.g., by Selleri in his introductory chapter to Ref.8; the same volume contains a further account of time-symmetric causality by Rayski.] It has been shown elsewhere that conditional probabilities are not invariant under time reversal [9]; but that does not end the matter, because there remains another feature of Costa de Beauregard's identification of causality with conditional probabilities. Leaving aside the issue of time reversal, I stated in an earlier paper that Costa de Beauregard's formulation is unacceptable, because the occurrences it supposes to be causally related turn out to be statistically independent [10]. On reflection, however, his proposal was seen to reveal a paradox intrinsic to elementary probability theory [11]. As we now go
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on to see, the paradox in question has a direct connection with Bell's theorem. To avoid possible misunderstandings, we first recall some elementary points of classical, Laplacian, probability theory. As a matter of designation, we use joint probability with reference to the product rule for independent occurrences, and combined probability for the general case that applies equally for independence and dependence between occurrences A and B. There is no need to dedicate a symbol signifying that we are indeed talking about probabilities; so, for notational simplicity, we write the joint probability as (A)(B) and the combined probability as (A,B). In this notation, the combined probability is defined by (A,B) = (AIB)(B) = (BIA)(A), where (XIY) is the conditional probability for X given Y. In elementary quantum mechanics we have the following definitions: D(i) the quantum singlet state has total spin (or polarization) equal to zero; D(ii) a single electron (or photon) has a nonzero spin (polarization) as an intrinsic property. It then follows that, if the singlet state I~o) describes a single and irreducible entity, then that entity is neither an electron nor a photon. Therefore, one cannot strictly speaking perform a spin measurement on the singlet state; indeed, Ig~o)is represented by a nonfactorizable superposition describing the system before any measurement is made. Thus, ~Zl~o) is a meaningless expression, unless we take into account that we can perform a spin measurement on either of the particles entailed, so that ozl~o) is the same as either ~zlgta) or GzlWb). But, by definition, we also have it that Gzl a) = sign l a);
ozl b) = --sign I a),
(3:1)
so that (al ~z I b) = (b I ~z I a).
(3:2)
With a common axis of quantization understood, we can then omit the operator in Eq. (3:2), writing the amplitude for the spin state la) given Ib) simply as (a I b}, with possible values _+1. Squaring then gives I(alb)l 2 = 1 = I(b l a}l 2.
(3:3)
Self-evidently, Eq. (3:3) may be understood as an expression of symmetric causal relation between the spins of states la) and Ib); we shall designate this as reciprocal causality. With the amplitudes written as above, we can write the combined
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density as (a I b)lb), which is symmetric on interchange of the arguments a and b under all allowable combinations of their signs. There is need, however, to be careful about writing the combined density in particular cases; this is because of the meaning required for the prior probabilities. For example, with the amplitude (a+ I b_ ) one might be tempted to write the combined density as (a+ I b. }lb. ). But prior to making any measurement--which is what the prior probability must be--the singlet state combines both particles as a superposition of spin states; so the correct probabilistic description for either particle is given as a sum of possible states. Accordingly, the prior probability for Ib), and similarly for la), is given by IIb)l2 -= lib+) + Ib_)l 2.
(3:4)
The right-hand side must be equal to unity because, on measurement, each particle must have one or the other sign, exclusively. Hence, IIb)l2 = 1 = IIb__+)l2 ,
(3:5)
and similarly for Ila)l2. Taking into account Eq. (3:3) and symmetry considerations, we have, quite generally, Ila)l2 = I(alb)l 2
(3:6a)
Ila)lb)l2 = I(ai b)lb)l 2,
(3:6b)
and
so that the spin states la) and Ib) are statistically independent. The extreme simplicity by which Eqs. (3:6) are derived from (3:3) does not reduce the result to triviality, for it reveals a paradox of statistical independence between states that are related by reciprocal causality. It is immediately obvious that the paradox arises because of the superposition principle expressed through Eq.(3:4). Had we written instead the classical probability form IIb)l2 = lib+)[2 + lib_ )l 2 ,
(3:7)
then the independence of la) and Ib) would simply not have followed. It is therefore clear that the concepts of causality and independence come into paradoxical relationship in connection with singlet states directly in consequence of the role played by the principle superposition in quantum probabilities. It is equally obvious that the distinction between Eqs. (3:4) and (3:7) has bearing on Bell's theorem. In his 1964 formulation of the EPR argument, Bell drew on two crucial ideas [12]. On the one hand, and using the labels 1 and 2 for the particles we have labelled a and b, he says,"Since we can predict in advance the result of measuring any chosen component of ~2 by previously measuring the same component of gl, it
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follows that the result of any such measurement must be actually predetermined." In Eq. (3:3) we expressed this argument in terms of reciprocal causality. In the second place, Bell also requires that,"If two measurements are made at places remote from one another the orientation of one [measurement device] does not influence the result obtained by the other." This is, of course, the premise of local independence which is expressed through use of the product rule in Eq. (2:2) for the combined probability. The central feature lies in assuming the mutual incompatibility of the two crucial ideas, (I) mutual causality, (II) mutual independence; for it is by assuming their incompatibility that Bell's theorem establishes the nonlocal property of the quantum mechanical correlations for the singlet state. Nevertheless, we have seen that (I) and (II) are mathematically compatible in that (II) is derivable from (I). The seeming paradox of this mathematical relationship arises precisely because of the apparent reasonableness of Bell's assumption as to the mutual incompatibility of (I) and (II). Quite evidently then, the sense of paradox arises because the mathematical relationship scandalizes intuitions founded on our ordinary notions of causality. Interestingly, Bell's assumption as to the incompatibility of (I) and (II) is not forced even by classical Laplacian probability theory. If we stipulate unit probability for the converse conditionals (AIB) = (BIA), the prior probabilities (A) and (B) are then equal, so that the combined probability (A,B) = (AIB)(B) can just as well be written with (A) substituted for (B). This yields the combined probability as equal to (A), since (AIB) = 1 by initial stipulation; i.e., we have (A,B)
=
(A)(AIB)
=
(A).
But there are physical circumstances where we also have (A) = 1 = (B); so, from the definition of combined probability, we can then write (A,B) (B)
_
(A)(AIB),
so that (A,B) = (A)(AIB)(B) = (A)(B)
(3:8a)
and (A) -
(A,B) (B)
-
(AIB),
(3:8b)
thus establishing, within the framework of classical probability theory, the statistical independence of occurrences A and B. With unit probability everywhere, it might be objected that this case is simply a pathological one. We note, however, that it does fit with the quantum singlet state, so it is a physically meaningful case and, moreover, one that quantum mechanics treats in terms of probabilities.
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A second objection might be that there is no causal relation in the equations (AIB) = 1= (BIA) if we also have (A) = 1 = (B), since it is then clear that the conditionals merely reflect the certainties associated with the individual occurrences A and B. However, this would be to overlook the extreme simplicity of the situation we have been considering. In more complex situations more evidently probabilistic conditionals arise from quasi-independence among the elementary events; even so, the fundamental logic is the same as for the situation we have considered. Thus, if statistical correlations are not simply fortuitous, then there must be some causal linkages between the elementary events. But those linkages will not be so strong as those of the reciprocal causality identified by Eq. (3:3) or its classical equivalent. Because we usually have no knowledge about such weak causal interactions, probability theory simply leaves them out of explicit account and, instead, treats the elementary events as "quasi-independent"; clearly, if they were fully independent, we should obtain only the statistics of random behaviour. In domains where classical probability theory is usually applied, unit probabilities enter only through the normalization condition Zpi = 1; apart from that condition, one usually does not resort to probabilities when dealing with certainties. On the other hand, quantum mechanics is fundamentally a probabilistic theory and yet, as is clear in the singlet state, it also has to deal with certainties and with statistical correlations founded on causal relations. Independence comes also into the picture because we can assign a probability to an elementary event for either particle independently of knowing the state of the other, In another direction, chaos theory shows the danger of confounding certainty with exactness and, in like vein, we have here avoided confounding certainty with causality; instead, we have shown an instance in which the two notions find a common expression. We note that the left-hand equation in (3:8a) is formally the same as Costa de Beauregard's in his formulation of retrocausality. The difference in content is that, whereas Costa de Beauregard assumes invariance of the conditionals under time-reversal, we have drawn on symmetry considerations independently of the time parameter. Thus, if we assume one-way time together with spacelike separation for the events A and B, then the derivation of Eqs. (3:8) establishes the nonlocal property as being intrinsic to Costa de Beauregard's premise (AIB) = (BIA). Since this emergence of nonlocality, together with the causal paradox, is shown to be derivable entirely within the framework of the classical probabifity theory of Laplace, it follows that our previous result for certain quantum systems is also applicable, in principle, to macroscopic entities. This is, I believe, also allowed by Costa de Beauregard's proposal for retrocausality. I should wish to add that, temporal considerations aside, my initial criticism of his proposal for identifying causality with conditional probabilities quite overlooked the subtleties entailed by the symmetry features of his formulation.
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GENERALIZATION
Of course, the causality paradox in quantum mechanics is not restricted to combined states such as the singlet spin state where, as already noted, the quantum mechanical source of the paradox lies in the principle of superposition. The same source yields the same paradox in the well-known double slit experiment where the superposition principle leads to interference patterns attributable to phase interactions among the probabilities. The intrinsically probabilistic character of this is clearly revealed through the destruction of the interference effects by observations. For it has long been recognized that making an observation amounts to eliminating one of the possible events needed to specify the initial probability space. In saying that much, we are thereby saying that the observation-induced obliteration of the interference pattem is a consequence of the validity of the initial specification for the probability space. But then this persistence of the effects of the probability space is itself a direct manifestation of the causality paradox. We can perhaps express this by saying that the probability space acts as a causality field. For if we do not make the observation, we then obtain the interference pattern with unit probability and, symmetrically, if we obtain interference, then the probability for both slits must enter into the initial specification. The question of independence comes in when we consider the pattern of interference as being obtained from the accumulated effects of separate particles transmitted with macroscopic time interval separating each transmission; the individual transmissions are then mutually independent in the local sense. That independence, however, is subordinate to the causal field introduced through the preparation stage which itself determines the appropriate probability space. Thus, by defining both paths as possible routes for each independent transmission, the probability space becomes actualized in the causal paradox. One also recognizes that the causal paradox is manifested through the Pauli exclusion principle. A given fermion, otherwise having an existence independent of a second fermion that is already assigned a given quantum state, is nevertheless constrained, with probability equal to unity, from forming a state-bound pair with the second fermion. Symmetrically, the second fermion cannot accept the independent first particle into its own quantum state. Underlying this is the same paradoxical relation between reciprocal causality and mutual independence. In turn, we wish to insist that the foregoing examples are subject to the argument given in Sec.1. In that event, the causality paradox and, hence, nonlocality, are features attributable to the physical world in both its quantum and, by Eqs. (3:8), its macroscopic aspects. Clearly, the arguments put forward here make the foregoing conclusion incompatible with any continued rejection of a physical interpretation for the quantum wave function.
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REFERENCES
1.
A. Einstein, letter to E. Schr&linger, in Albert Einstein: Letters on Wave Mechanics: Correspondence with H.A. Lorenz, Max Planck, and Erwin Schr6dinger, K.Prizibram, ed. (Philosophical Library, New York, 1967; paperback reprint, 1986). 2. C.I.J.M. Stuart, Found. Phys. 21, 591 (1991). 3. A. Aspect, J. Dalibard, and G. Roger, Phys. Rev. Lett. 49,i804 (1982). 4. A. Peres, Am. J. Phys. 52, 644 (1984). 5. A. Peres, Phys. Essays 2, 288 (1989). 6. C.I.J.M. Stuart, "On quantum idealization," to appear in Phys. Essays, December 1991. 7. O. Costa de Beauregard, Found. Phys. 17, 775 (1987), 8. F. Selleri, ed., Quantum Mechanics versus Local Realism: The Einstein-Podolsky-Rosen Paradox (Plenum, New York, 1988). 9. C.I.J.M.Stuart,"Temporal irreversibility of conditional probabilities," to appear in Phys. Essays, September 1991. 10. C.I.J.M. Stuart, Found. Phys. Lett. 4, 37 (1991). 11. C.I.J.M. Stuart, Found. Phys. Lett. 4, 265 (1991). 12. J.S. Bell, Physics 1, 195 (1964). NOTE
Nemo dat quod non habet : "No one can offer more than he has ability to give;" in that spirit, this paper is dedicated to the memory of J.S. Bell.