Foundations o f Physics, Vol. 18, No. 4, 1988

Quantum Nonlocality 1 Henry P. Stapp 2 Received April 13, 1987 It is argued that the validity of the predictions o f quantum theory in certain spincorrelation experiments entails a violation of Einstein's locality idea that no causal influence can act outside the forward light cone. First, two preliminary arguments suggesting such a violation are reviewed. They both depend, in intermediate stages, on the idea that the results of certain unperformed experiments are physically determinate. The second argument is entangled also with the problem o f the meaning of''physical reality." A new argument having neither of these characteristics is constructed. It is based strictly on the orthodox ideas o f Bohr and Heisenberg, and has no realistic elements, or other ingredients, that are alien to orthodox quantum thinking.

1. I N T R O D U C T I O N The foundations of quantum theory is a subject of growing interest among physicists. (1) The prior neglect was due largely to a perceived lack of relevance of the issues involved to a scientific account of laboratory phenomena. The revival of interest is due partly to the remoteness from laboratory phenomena of contemporary theoretical efforts to create a consistent and adequate quantum theory. And it is logically connected also to recent developments in the field of microprocessing and cryogenics: these developments are tending to destroy the sharp distinction between microscopic and macroscopic phenomena that is essential to the orthodox formulation of quantum theory. Two central foundational issues are (1) the question of the scope, or completeness, of quantum theory, and (2) the question of its compatibility 1This work was supported by the Director, Officeof Energy Research, Officeof High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098. 2 Lawrence BerkeleyLaboratory, University of California, Berkeley,California, 94720. 427 0015-9018/88/0400-0427506.00/0 © 1988 Plenum Publishing Corporation

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with the Einstein locality idea that no causal influence can act outside the forward light cone. The question of completeness was the subject of a famous 1935 paper of Einstein, Podolsky, and Rosen. (2~ These authors assumed the locality property mentioned above, and claimed to prove the incompleteness of the quantum-mechanical description. Bohr's response (3) rejected this locality assumption, in the form used by EPR. But Einstein adhered steadfastly to his conviction regarding locality. (4) The question lay relatively dormant until 1964, when Bell (5) showed, essentially, that if certain predictions of quantum theory are valid then some of the premises of the EPR argument must be false. But these premises were carefully constructed to be in line with orthodox quantum thinking. Indeed, it was this careful formulation that forced Bohr to reject the plausible-sounding assumption that no physical reality in one spacetime region could be disturbed by an act of measurement performed in a spacelike separated region. (6) Bohr justified this rejection by revising the meaning of "physical reality." The question considered by EPR was: Can quantum-mechanical description of physical reality be considered complete? Thus their argument entangles the issue of completeness with the issue of the meaning of physical reality. Moreover, it entangled these two issues with the further question of locality. The aim of the present work, briefly stated, is to disengage, in the combined work of EPR and Bell, the question of locality from the two other issues of "completeness" and "reality," and demonstrate that the Einstein locality assumption itself is incompatible with the validity of the predictions of quantum theory. The question immediately arises whether the issue under consideration pertains to science proper, or is merely some philosophical or terminological dispute that has no bearing on the scientific description of laboratory phenomena. This question leads back to the issue of the completeness, or scope, of quantum theory. Bohr (7) and Heisenberg ~8) both emphasized that quantum theory was a theory of atomic phenomena. And Einstein admitted that "the contemporary quantum theory constitutes an optimal formulation of [certain] connections, [but] offers no useful point of departure for future development. ''°) The origin of these recognitions of limitations in the scope of contemporary quantum theory lies in the fact that contemporary orthodox quantum theory is based on approximations, or idealizations: it is based on an idealized separation of the world into a microscopic [atomic] system, which is described in terms of the quantum symbols, and the remaining parts of the world, including our measuring devices, which are described in the language of classical physics. However, the microscopic quantum

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system must be small enough to act only insignificantly upon its environment, during the interval between its preparation and measurement, in order for the Schr6dinger equation to hold, and the macroscopic devices must be strongly enough connected to their environment to function as classically describable measuring devices. These two conditions limit the scope of quantum theory, which is not applicable in its present orthodox form to intermediate situations in which quantum effects are important, but the influence of the quantum system upon its environment cannot be ignored. Such situations are beginning to appear now in laboratories, as a consequence of recent technological advances.~l°) To extend quantum theory into these new domains it appears necessary to go beyond the "rules" that work well in the idealized special cases. But passing from rules that work in special idealized cases to a comprehensive general theory requires guidance. An understanding of the causal structure of the theory, in regard to its space-time underpinnings, can be expected to provide important guidance for this impending task. The plan of the paper is to begin with a brief account of the essential background material and, in particular, of the seminal contributions of EPR, Bohr, and Bell outlined above. First, after describing the experimental arrangements, two arguments for the incompleteness of quantum theory are described. The first is called the naive argument, and the second is the EPR argument. Bohr's response to EPR is described. Then it is shown how, if the validity of certain key predictions of quantum theory is assumed, a modified form of Bell's work converts each of these two arguments for the incompleteness of quantum theory to a proof that not all of the premises of that argument can be valid. The significance of this result is clouded, however, in the approach of EPR by the central occurrence of the term "physical reality," and in both approaches by the occurrence, in intermediate steps, of the concept of the results of unperformed experiments. This concept is alien to orthodox quantum thinking. The question is thus posed whether the argument can be cast into a form that has no trace of any concept that is alien to orthodox quantum thinking. The new argument begins in Section 8 with the introduction of a criterion that identifies the presence of a causal influence. Then the assumption "Unique Results" is described: it rules out the many-worlds ontology. (11) As a temporary device, to provide a foundation for physical intuition, the idea is introduced of a stochastic mechanism that selects the result of the measurement that is performed, and that provides absolutely no trace or indication of what the results of any unperformed experiments "would have been" if they thad been performed.

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The essential properties of this stochastic mechanism are exactly represented in terms of the mathematical concept of random variables. (12) The locality assumption is introduced, first as an expression of the condition that the mechanism be subject to the constraint that causal influences can act only into the forward light cone. However, the idea of the mechanism is later shown to be completely inessential: the locality requirement can be introduced in a completely satisfactory manner directly within the mathematical formalism of random variables. It is proved that no local stochastic (i.e., random variable) theory can reproduce the spincorrelation predictions of quantum theory. Sharp constrasts are drawn between this result and the earlier result that no stochastic hidden-variable theory (13'14) can reproduce the statistical predictions of quantum theory. The hidden-variable formulation rests on numerous realistic features that are alien to orthodox quantum thinking. None of these feature are involved in the formulation of local stochastic theories. In the final section the random-variable approach to this locality question is contrasted to an earlier approach of the present author.

2. THE EXPERIMENTAL A R R A N G E M E N T

The experimental arrangement is essentially Bohm's variation ~6~of the experiment of Einstein, Podolsky, and Rosen. ~2) Two beams of low energy identical spin-½ particles are caused to intersect near the center point P of several concentric arrays of particle detectors. Some pairs of particles that scatter at approximately 90 ° escape from these concentric arrays of detectors through appropriately lined-up escape holes. The escaping pairs are electronically tagged and labelled by an integer i that runs from 1 to some very large number n. The two particles from each escaping pair i are directed so as to enter two (different) Stern-Gerlach devices, D~ and D2. Shortly before the arrival of the batch of n particles at device Dj (j ~ { 1, 2 }) an experimenter Ej chooses one of two possible angles, 0g = 0~ or 0j = 0j', for the direction of the line of deflection associated with Stern-Gerlach device Dj, and sets this device in the chosen position. Consequently, each of the n particles entering the device Dj will be deflected either in the chosen direction 0i or in the opposite direction. Particle detectors are placed in the two alternative possible paths of deflection, and a value of either + 1 or - 1 is assigned to a variable rji , according to whether the deflection in device D i of the particle from pair i is in the direction 0j or the opposite direction. These values of the variables rsi are stored in a memory device.

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The experimental quantity of primary interest is the correlation function

c(rl, r2) = - 1

n . ~=l

rlir2i

(1)

Its value must lie in the closed interval + ! >/c ~> - 1. By virtue of the Pauli principle each pair of particles scattered at 90 ° must be in a singlet spin state. Hence quantum theory predicts that for large n the observed value of c ( r l , r2) will be close to the value

~(01, 0~)= -cos(0~- 02)

(2)

provided the angles 01 and 02 are defined in an appropriately lined-up way. More precisely, for any e > 0 and 6 > 0, however small, quantum theory predicts that the total probability that c(rl, r2) will lie in the range

IC(rl, r2)-c(01,0=)1 >6

(3)

can be made less than e by taking n sufficiently large. An important special case is 01 = 02. Then c(01, 02)= -1. This means that, for each value of i, rlir2i= -1. Hence the deflections of the two particles of each pair i must be perfectly anticorrelated: one particle in each pair will be deflected "up" (i.e., in the direction 01 = 02), and the other will be deflected "down" (i.e., in the direction opposite to 01 = 02). In the arguments that follow, the index i, which labels the pairs of particles that escape from the spherical arrays of counters, will play a mathematical role similar to that played in Bell's work (5~ by the hidden variable 2. But i, unlike 2, is not a hidden variable. It is defined and manipulated by the fast electronics, and is "observable." Two space-time regions, R1 and R2, are defined, and the experiment is set up so that all of the procedures pertaining to device/)1, are performed in Rj. In particular, the choice of the angle 0j by the experimenter Ej, and the subsequent positioning of the device Dj, is made in Rj, along with the deflections and detections in Dj, and the subsequent recordings of the values of the variables rji. These two regions R i and R 2 are to be spacelike separated. This means that no point in either region can be reached from any point in the other region without moving either faster than light or backward in time. Thus if causal influences propagate only forward in time, and no faster than light (i.e., into the forward light cone), then for each j e {1, 2} the

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choice made by the experimenter in Ri cannot causally influence the values of the results rk~ appearing in R k, for k :¢ j. The word "particle" has been used in this description of the experimental arrangement. However, this usage is merely a convenient verbal shorthard. The only important ingredients of the subsequent arguments will be the space-time arrangements of the macroscopic devices, and the associated numerical predictions of quantum theory. We may now consider a first argument for the incompleteness of quantum theory. 3. THE NAIVE ARGUMENT Consider a case in which 01 = 02. Then the deflections in the two regions R~ and R2 will be perfectly anticorrelated. Thus if causally effective information can propagate only forward in time and no faster than light, then information sufficient to establish whether the deflection will be "up" in R 1 and "down" in R2 or, alternatively, "down" in R1 and "up" in R 2 must be contained in the intersection of the backward light cones of R~ and R 2. For otherwise the information needed to establish the exact anticorrelation of the results in the two regions cannot get to both regions. On the other hand, information about the experimenters' choices of the angles 0~ and 02 can propagate only into the forward light cones of Rt and R2; it canot get back into the intersection of the backward light cones of Rt and R2. Thus the information sufficient to establish whether the deflections will be "up" or "down" that needs to be contained in the intersection of the backward light cones under the condition that 0t = 02 must be present in this region independently of the experimenters' choices of these two angles, Consequently, the information sufficient to decide whether the deflection in say R~ will be "up" or "down" in the two alternative cases 0~ = 0'1 and 01 = 0'[ must be present in the intersection of the backward light cones of R~ and R2 regardless of which of the two alternative possible values of 0~ is chosen. But then the quantum mechanical description must be incomplete. For this description cannot represent simultaneously a well-defined result for the two incompatible measurements associated with angles 0~ and 0~': the structure of quantum states does not allow this. 4. ORTHODOX RESPONSE TO THE NAIVE ARGUMENT According to Niels Bohr: "In our description of nature the purpose is not to disclose the real essence of phenomena but only to track down as far

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as possible relations between the multifold aspect of our experience. ''(~5) "Strictly speaking, the mathematical formalism of quantum theory ... merely offers rules of calculation for the deduction of expectations about observations obtained under well-defined conditions specified by classical physical concepts. ''~16~ "The formalism does not allow pictorial representation along accustomed lines, but merely aims directly at establishing relations between observations obtained under well-defined conditions. ''(17) "In fact, wave mechanics, just as the matrix theory, represents on this view a symbolic transcription of the problem of motion of classical mechanics adapted to the requirements of quantum theory and only to be interpreted by an explicit use of the quantum postulate. ''(ls~ Bohr thus instructs us to regard the quantum formalism as providing merely predictions pertaining to observations, rather than pictures of what is really happening. This attitude justifies to some extent simply ignoring the naive argument: pictorial analysis does not fit the quantum way of thinking. On the other hand, the pictorial representations in the naive argument do not involve any attempt to understand in a pictorial way the workings of the quantum formalism. The formalism is used only to provide predictions pertaining to observations. No attempt is made to give any meaning to a "wave function" as anything but an integral part of the quantum formalism for making predictions. The space-time pictorializations occurring in the naive argument arise, rather, simply from the need to formulate in some way the idea that a causal influence can act only into the forward light cone.

5. THE EPR A R G U M E N T

The EPR argument (z6~ can be regarded as a sharpened version of the naive argument: superfluous pictorial elements are stripped away, and, in accordance with quantum thinking, the entire analysis is based on predictions about results of measurements. The title of the EPR paper is: "Can quantum-mechanical description of physical reality be considered complete?" Since the aim of EPR is to speak to orthodox quantum theorists about "physical reality," EPR must give to this term a precise meaning that quantum theorists cannot easily ignore or reject. This they do by their criterion of physical reality: "If, without in any way disturbing a system, we can predict with certainty (i.e., with probability unity) the value of a physical quantity then there exists an element of physical reality corresponding to that physical quantity." To adapt their argument to our experimental arrangement, (6) let the angles 0j satisfy 0] = 02 and 01 - 02, and let the variables rji corresponding t

825/18/4-3

t!

_

tt

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to a measurement with 0j = 0~- (or Oj') be denoted by r}, (or by r}~). Then the EPR argument, with appropriate replacement of symbols, is as follows: By measuring either r'j or r'( we are in a position to predict with certainty, and without in any way disturbing the system in R2, the value of either r2 or r2' (by virtue of the exact anticorrelations predicted by quantum theory). In accordance with our criterion of reality, in the first case we must consider r; as being an element of reality, and in the second case r','. Thus either r; or r;' is an element of reality according to whether we measure r 'I or rl." But to maintain that either r~ alone or r2" alone is an element of reality, depending upon what we measure in R1, would make "the reality of r~ and r;' depend upon the process of measurement carried out in R2. No reasonable definition of reality could be expected to permit this." Thus r~ and r~ must be simultaneous elements of physical reality. But if r~ and r2' are simultaneous elements of physical reality, then the quantum mechanical description of physical reality cannot be complete, because this description cannot represent simultaneously well-defined values of these two incompatible measurements.

6. BOHR'S REPLY

Bohr's reply ~3~ to the EPR paper dealt mainly with the consistency of the quantum theoretical treatment of the experimental situation considered by EPR, rather that the question of completeness raised by EPR. The core of Bohr's reply to the EPR argument itself was this: From our point of view we now see that the wording of the abovementioned criterion of physical reality proposed by Einstein, Podolsky, and Rosen contains an ambiguity as regards the meaning of the expression "without in any way disturbing a system." Of course there is, in a case like that just considered, no question of a mechanical disturbance of the system under investigation during the last critical stage of the measuring procedure. But even at this stage there is essentially the question of an

influence on the very conditions which define the possible types of predictions regarding the future behavior of the system. Since these conditions constitute an inherent element of the description of any phenomena to which the term "physical reality" can be properly attached, we see that the argumentation of the mentioned authors does not justify their conclusion that quantummechanical description is essentially incomplete. C3) The point of this rebuttal is to tie "physical reality" to what can be predicted about a system, and then to maintain that, since our future

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predictions pertaining to region R2 depend upon what we do in R~, the physical reality in R 2 is disturbed by what we do in R~. Bohr's argument entails, of course, "a radical revision of our attitude toward the problem of physical reality. ''(3) The subtlety of Bohr's argument testifies to the strength of the EPR argument: Bohr was essentially forced to admit that the "physical reality" in one region can be disturbed by what we do in the other region. But he revised the idea of "physical reality."

7. BELL'S C O N T R I B U T I O N IN M O D I F I E D F O R M

The first parts of both the naive and EPR arguments are supposed to establish that what "would appear" if the unperformed measurements were performed is determined (within nature), even though it is not revealed by the measurement in question. This property is called "counterfactual definiteness," or CFD. Bell (5~ used CFD to justify the introduction of hidden variables that determine the results of all four measurements, and then showed that no local deterministic hidden-variable theory could reproduce the spin-correlation predictions of quantum theory. However, the incompatibility of CFD, Einstein locality, and these predictions of quantum theory can be proved without introducing hidden variables. CFD allows one to define two functions r1(01, 02) and r2(01,02) over the domain 0~s{0'~,0~'} and 02s{0~,0;'} by requiring that the pair (rj(O~, 02), r2(01, 02)) be what "would appear" if the measurement characterized by the pair of angles (0~, 02) were performed. Given CFD, the Einstein locality idea demands that what "would appear" in one region cannot depend upon what the experimenter does in the other region:

rl(O~, Oz) = r1(O~)

(4a)

r2(01,

(4b)

and 02) = r2(02)

If the angles are chosen as oi = 0 °

(Sa)

0~' = 90 °

(5b)

01=0 °

(5c)

0;'

(5d)

= 45 °

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436

then the predictions approximately

of quantum

theory are, according to (1)-(3),

1 ~ 'ii(0,, 0'2) r2i(01, Or2) = --[

(6a)

g/i=l

1 ~ r!,(O,,' 02 ) r2,(0t; 0~')= --1/,~/~

17.l = l

--[ ~

r l i tka,' ul,Oi)

r:i(O,,,, Oi) = 0

(6b) (6c)

K/i= 1

and -

n i=l

'. . . . . . .

-1/

(6d)

From (6a), (4), and Irj~[= 1 one concludes that rli(O'l) = --r21(O~)

(7)

Then (4), I@ = 1, and (6b), (6c), (6d) give ,~l= (r,,(0'l) + r,~(O~) +

,/5 r2i(0;')) 2 = O

(8)

But Irj,[ = 1 entails a lower bound on the RHS, hence ( 2 - ~/2)2 ~<0

(9)

This is a contradiction. Inclusion of the arbitrarily small 6 from (2) in (6) cannot undo this contradiction. Thus the assumption that both CFD and Einstein locality hold contradicts the assumption that the predictions of quantum theory are valid. The implication of this result is that if one accepts the validity of the predictions of quantum theory, for these spin-correlation experiments, then the conclusion that may properly be drawn from the naive and EPR arguments is not that CFD holds, and hence that quantum theory is incomplete, but that each of these arguments has a false explicit or implicit premise. The premises of the EPR argument involve, besides the assumptions that certain predictions of quantum theory are valid and that no causal influence can act over a spacelike interval, a definition of "physical reality." And both the naive and EPR arguments involve the rather elusive C F D concept of the "results of unperformed experiments."

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The occurrence of these problematic concepts obscures the physical significance of the conclusion to be drawn from the arguments. One is led naturally to ask, therefore, whether the mathematical results can be set into a logical framework where no such obscurities arise, and the physical significance of all of the assumptions is clear.

8. CRITERION OF CAUSAL I N F L U E N C E David Hume has pointed out a difficulty with the idea of causal influence: invariable succession does not entail causal influence. In order to establish causal influence one needs, rather, to compare, in a single instance, the consequences of doing something with the consequences of not doing it. But in a single instance one cannot both do something and not do it. However, given a physical theory one may, within the constraints imposed by that theory, be able to compare the consequences of doing something to the consequences of not doing it. In the present case this physical theory is quantum theory, and our first premise is the validity of the predictions of quantum theory pertaining to the four alternative possible spin-correlation measurements under consideration. To analyze causal influences in the clearest way, one must have clearly identified variable causes. Here we shall identify the acts of the experimenters as the variable causes: the choice between these acts will be treated as free variables. This conceptualization is in complete accord with orthodox quantum thinking. Its acceptance does not entail that in every experimental context the choices between various possible acts of experimenters must be considered to be free variables. The commitment is only to the idea that within the specific experimental situation under consideration here these choices may be treated as free variables. A criterion must then be given for recognizing when, within a physical theory, a variable y is causally influenced by the choice of an independent variable x. Our criterion is this: a variable y is, within a physical theory, causally influenced by the choice of an independent variable x if, for some choice of values of the other independent variables, the theory does not allow the possibility that y could be left undisturbed (unchanged) as x is varied over its domain. That is, if the constraints imposed by the theory force y to change as x is varied over its domain we say that, within the theory, y is influenced by the choice of x.

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9. C O U N T E R F A C T U A L D E F I N I T E N E S S VERSUS U N I Q U E RESULTS One aim in the argument being developed here is to avoid completely the notion of counterfactual definiteness, either as an assumption, or in an intermediate step. On the other hand, I shall make essential use of the mathematical relationships described in Section 7. Those relationships were obtained there in a context where CFD holds. Thus there is a danger that CFD might slip in, as a tacit or hidden assumption. It is essential to the argument that this not happen. Let me specify in detail what C F D is, within the context of the specific situation under consideration here: CFD: "Regardless of which of the four alternative possible measurements is performed, there is determined or fixed, within nature, according to some underlying theoretical conception of nature, a quartet of values (r~, r2, r3, r4), in which rm can be identified as the value that would be obtained as the result of the measurement if measurement m were performed." I distinguish here between the physical theory under consideration, namely quantum theory, and some underlying, and perhaps not completely defined, idea of the nature to which this theory is supposed to refer. The idea of C F D described above is the one that is supposed to be entailed by the naive and EPR arguments, and that holds in Bell's local deterministic hidden-variable theories. The CFD property may be contrasted to the property "Unique Results": UR: " F o r each of the four alternative possible measurements m, if m is performed, then nature will select some unique value for the result of this measurement m, and will never fix any values for the results which the remaining three measurements would have had if they had been performed." This property UR is connected to the assumption "quantum theory": QT: "For each of the four alternative possible measurements rn - (02, 02), if m is performed, then the unique value r - (r~, r2) selected by nature in accordance with UR will with probability greater than 1 - ~ lie in a set Qm(e), where for any e > 0 we may take

Qm(e)=

{r;

IC(F1, /'2) -- C(01' 02)1 < 0,01 }

by taking n, the number of pairs, sufficiently large."

(10)

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The property CFD contradicts orthodox quantum thinking. The property UR accords with the orthodox idea that the results of unperformed measurements are indeterminate; i.e., that nature never assigns any value at all to the property in question unless it is measured. The property UR is in line also with Heisenberg's idea that a transition from "possible" to "actual" takes place when the interaction between the quantum object and the measuring device comes into play: some particular result is selected from among the previously existing possibilities. The property UR may go slightly beyond what is strictly entailed by Bohr's idea of the orthodox Copenhagen interpretation. For example, Bohr's idea might not absolutely rule out the many-worlds ontology. (~1) In that ontology nature makes no unique selection: the uniqueness of the results that appears to any particular community of communicating observers arises as a consequence of a dynamical splitting of the world into noninteracting branches, rather than from any singling-out of this unique value by nature herself. The many-worlds ontology allows one to maintain the idea that no causal influence acts outside the forward light cone. Thus, to demonstrate the failure of this idea, one must exclude the many-worlds ontology. In the present work this is done by assuming UR. This assumption provides the ontological bases of the whole argument: it introduces the notion of a selection. This selection is the thing that is influenced by the far-away act of an experimenter. 10. FAILURE OF EINSTEIN LOCALITY To provide a physical foundation for the discussion I shall proceed in two steps. First a space-time macrolocal non-CFD model will be considered. Then its logical essence will be extracted. Suppose, for any one of the four possible measurements m, if in is performed, then nature wilt construct, in some local but nonpredetermined manner, a local mechanism that will pick the unique value for the result of measurement m. Suppose that the same constituents are used to form this mechanism in each of the four alternative possible cases, so that, in view of the nonpredetermined, or stochastic, character of the process of construction of this mechanism, there will be no indication or trace of what the mechanism would have been if some other measurement had been performed. Thus the mechanism is thoroughly non-CFD. The required localizations need be maintained only to the order of, say, millimeters, with possible weak tails. For the nonlocal effects in question are large, and can extend over very large distances, with in principle no fall-off with increasing distance.

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Let us now consider the process of construction of this mechanism and, in view of the stochastic character of this process, the various conceivable possible mechanisms that might be constructed. For each of these conceivable possible mechanisms one may consider the part located in V-(Rj), the backward light cone of the region Rj. The locality requirement that causal influences can act only into the forward light cone means that the parts of the mechanism that can influence the results in either R~ and R2 must lie in V-(R1)w V (R2), the union of these two backward light cones. Thus the mechanism can be considered confined to this union. For the same reason the only parts of this mechanism that can be influenced by the acts of the experimenters in RI and R2 must lie in V+(RI) w V+(R2), the union of the forward light cones from R1 and R 2. Thus the parts of the mechanism that can be influenced by these acts are localized in the union of the two disjoint regions V (R~)n V+(R~) and V-(R2)c~ V+(R2), the intersections of the forward and backward light cones from R~ and R2, respectively. The first of these regions contains all parts of the mechanism that can be influenced by the acts of the experimenter in R~, whereas the second of these regions contains all parts of the mechanism that can be influenced by the acts of the experimenter in R 2. Since these two regions are disjoint, the influences of the acts of the two experimenters upon the mechanism must act independently: the influences of the acts of the experimenter in R/cannot be influenced by the acts of the experimenter in Rk (k # j ) . The essential features of the structure described above can be expressed symbolically by writing r, = r~(xl, 21)

(lla)

r2 = r2(x2, 2~)

(1 lb)

and

where 21 and 22 are two nondisjoint sets of stochastic variables whose domains can depend in general (e.g., for nonlocal mechanisms) upon both xl and x2. Here, for either value of j in the set {1, 2}, xj represents the choice made by the experimenter Ej that acts in Rj, and r/is the n-vector with components rji. These variables 2i are different from Bell's hidden variable 2. One difference is that they refer to the entire set of n pairs, and there is no suggestion of any decomposition of 2s into variables 2y corresponding to the individual pairs i: the selection by nature of a unique pair (r 1, r2) corresponds to a choice of a pair (x~, x2) and a selection of a unique value for the pair (21, 22). A second difference is that there is no representation of

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the values of the results of unperformed measurements. In the theory of Bell (5) the domain of the single hidden variable 2 that replaces 21 and 22 does not depend upon the values of x~ and x2, and one may therefore define a quartet of values for the pair (rl, r2) by allowing the variables xl and x2 in the equations analogous to (11) [namely rj--rj(xj, 2)] to vary over their two-valued domains, with 2 held fixed. The four values of r = (rl, r2) in this quartet are identified as the values of r that would be obtained under the four alternative possible conditions of measurement, if 2 had, in actual fact, the specific fixed value. The existence within the theory of these fixed quartets entails the C F D character of Bell's theory. The formulas ( t l ) permit no such a construction, in general, because the domains of 2~ and )~2 depend on xl and x2. Hence there is no way to hold 2i and 22 fixed as x I and x 2 are varied. This characteristic accords with the non-CFD structure of our model. The dependence in our equations of the domains of 21 and 22 upon xl and x2 arises from the fact that the choices made by experimenters can influence the construction of the mechanisms of selection. The common, or shared, parts of 21 and 22 correspond to the shared aspects of the mechanisms that select r I and r:. These latter parts are confined to V - ( R 1 ) c~ V (R2), in the model. For the case of a local (i.e., macrolocal) mechanism the disjointness of V+(Rj) and V (Rk) for j C k means that the domains of the stochastic variables in the set 2k should be independent of xj. This is so because the part of the mechanism that can influence the selection of r k is confined to V-(RD, which is a part of space-time into which no influence of the choice of x s can extend. (The locality requirement was already partly included in (11 ) when explicit dependence of r k upon xj was excluded.) Equations (11) may be written in the more detailed form rl = r l ( x l , 210. 212)

(12a)

r2 = r2(x2, 220, 2.12)

(12b)

and

where 212 represents the shared part of 21 and 22. Since the domain of 2s does not depend upon xk, k C j, the domain of 2~2 can depend on neither xl nor x2, whereas the domain of )vo can depend upon x s, but not on xk. The symbols )vo and 212 represent stochastic variables: each of their possible values carries a statistical weight. The restrictions placed by locality on the domains of these variables apply equally well to these statistical weights: the statistical weights carried by the various possible values of 2j2 cannot depend upon either x~ or x2, whereas the statistical

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weights carried by the various possible values of ;% can depend upon XJ, but not upon x~ (k C j). Indeed, if the statistical weights of the values of the variables upon which rj depends cannot be made independent of xk, then rj would manifestly be influenced by x~: in such a situation rj could not, in the stochastic sense, be left undisturbed as xk is varied over its domain. As a purely mathematical step, devoid of any physical meaning, one may, with arbitrarily small loss of accuracy, replace 21o by a stochastic variable with domain consisting of some sufficiently large number N1o of discrete points, each with statistical weight Nlo 1. For sufficiently large Nlo this same domain can be used for both values of Xl. Similarly, the domain of 22o can be replaced, with arbitrarily small loss of accuracy, by a domain with N2o discrete points, each carrying statistical weight N ~ ~. And one may take these two numbers to be equal: Nlo = N2o = No. Similarly, one may take the domain of z~2 to consist of N~2 discrete points, each of weight N~2~. Then (12) may be written in the form rl = r l ( x I , )~)

(13a)

r2 -- r2(x2, 2)

(13b)

and

where the domain of 2 consists of No x N~2 = N discrete points, each with statistical weight N 1. Equations (13) define, for each fixed 2, a quartet of values

(rl(xt, £), r2(x2, A)) This quartet is defined by letting x t and x 2 vary over their two-valued domains. Each such quartet is a local quartet: within the quartet the value of rj is independent of xk, for k ¢ j. There are N such local quartets, and each of the four members of each quartet has statistical weight N -1. Thus the statistical weight ascribed to any particular pair (r~, r2), under measurement condition (Xl, x2), is N -1 times the number of quartets whose (xl, x2)-element is (r~, r2). By virtue of the result obtained in Section 7 the statistical weights given by such a local stochastic theory cannot agree with the statistical weights predicted by quantum theory. This is because each local quartet has, according to the result proved in Section 7, at least one member that lies in a preestablished domain of (r 1, r2) space such that the total statistical weight ascribed by quantum theory to that domain is less than the arbitrarily small ~ > 0. Since each of the N local quartets has at least one such member, the total statistical weight of these members, distributed over the four alternative possible measurements, is at least unity, whereas

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the weight assigned by quantum theory is less than &. Thus no stochastic theory that satisfies the locality conditions demanded (at least within our non-CFD stochastic mechanism) by Einstein's locality idea that no causal influence can act outside the forward light cone can reproduce the statistical predictions of quantum theory for the spin-correlation experiments under consideration.

11. THREE COMMENTS 1. Although local quartets have appeared automatically, as a consequence of the locality assumption, and our mathematical manipulations, there is no assumption or implication of determinism or CFD. The theoretical structure considered by Bell (s) was motivated by the CFD result of EPR, and represented the physical idea that at some time prior to the choices of the values of xl and x2 some definite value of 2 was physically fixed, and that the four members of the corresponding quartet represented what "would happen" under each of the four alternative possible conditions if those condition were actualized. Our structure is non-CFD from its inception. Our groupings into quartets are strongly dependent upon our arbitrary mathematical procedures, and hence can have no physical significance. Moreover, the 2~ and /~2 are stochastic variables. One may imagine that under some particular condition (x~,x2) these stochastic variables would acquire particular values, and these values, together with the values of x~ and x2, would determine the result (r~,r2) that would appear under those conditions. But there is no suggestion or implication that under a different condition (x~, x2) the 2~ and 22 would necessarily be the same. What is assumed to be independent of xj is only the domain of 2k, and the statistical weights of the various possible elements in this domain, for k ¢ j. There is no implication that the value that 2 would acquire under any given physical condition is the same as the value it would acquire under a different physical condition. For this property is not entailed by the notion of a stochastic variable, and is neither needed, suggested, nor implied by our theoretical structure. Thus the appearance of local quartets in the present framework is not associated with any notion of what "would happen" under conditions not actualized; the theoretical structure represented by the equations is completely non-CFD and stochastic in character. It is, by the same token, nondeterministic. 2. In accordance with the dicta of Bohr the quantum formalism is used merely to provide predictions pertaining to observations: there is no

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attempt to give any interpretation or meaning to the quantum symbols, which have, in fact, never appeared. 3. There is no suggestion of any physical correlate to the word "particle," or any idea that there are quantum entities that are separate, or separable, from the devices we observe, or that these devices are separate or separable from their environment.

12. ABSTRACT F O R M U L A T I O N The macrolocal non-CFD mechanism provided an initial basis for thinking about the physical meaning of the symbols. But it plays no essential role. The first function of the mechanism idea was to cast in visualizable form the crucial assumption that, under any one of the four alternative possible measurement conditions nature will, if these conditions are actualized, select some unique value, identifiable as a result of the chosen measurement. It is the influence upon this selection of the act of the far-away experimenter that is the subject of the analysis. The second function of the mechanism idea is to put in concrete terms our exclusion of CFD. This exclusion was important for our narrower purpose of emphasizing that no hidden assumption, or implication, of CFD was involved. But the argument can be generalized by dropping this explicit requirement. The first two functions of the mechanism idea were, therefore, merely to put into picturesque form the assumption UR. The third function of the mechanism idea was to provide some visual imagery connected to the notion of locality. Locality involves the idea of things that can be localized. However, the localizable aspects of the stochastic mechanism, beyond the settings of the devices and the observable results, are very intangible: the only other aspects represented in the stochastic equations are domains of stochastic variables and statistical weights. In any case, the locality aspects arising from the mechanism idea can be avoided completely by formulating the locality assumption directly as the requirement that, for each k ~ { 1, 2}, if xk = 1 then the selection of a value for result rk not be influenced by the choice of value for xJ(J ¢ k) and if Xk = --1 then the selection of a value for result r k not be influenced by the choice of value for x j ( j ¢ k ) . Our concern here is with the structure of adequate theoretical ideas, and adequate theoretical ideas are expected to cover the alternative possible cases. In the context of stochastic variables, the demand that the result rk not be influenced by (the choice of) XJ means that rk has no explicit dependence

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445

upon xj and that the stochastic variables upon which rk depends can be left undisturbed, as regards both domains and statistical weights, as xj varies over its domain. The fourth function of the mechanism is to exhibit another important implication of locality, namely that the influences of the acts of the two experimenters act independently: the "order" in which the acts of the experimenters are performed has no influence upon the process of selecting values for the results rji; the act in RJ does not influence the influence of the act in Rk, for k ¢ j . This independence property arises from the disjointness of the domains influenced by the acts in R1 and R2, provided one is concerned solely with influences upon selections localizable in R1 and R2, and not with influences upon things localized at later times, at which these domains of influence can intersect. The idea of the localizable mechanism has therefore played no essential role: the conditions imposed on the stochastic variables can be justified in terms of the space-time locations of the observable quantities, and the idea that causal influences cannot act outside the forward light cone. The mathematical argument given in Section 10 is essentially equivalent to that given in connection with the so-called stochastic hiddenvariable theories. (~3'141 But the physical basis is altogether different. The physical basis of the stochastic hidden-variable theories is an idea of "physical realism," akin to CFD, in which one conceives of a physical system as having a definite set of properties independently of their being observed/~3) The physical system in question is taken to be the composite system of the two particles. This system is assumed to have classical-type "objective" states, labelled by a hidden-variable 2, which in any given physical situation is supposed to have a definite value. The value of this variable 2, combined with the values of variables that characterize the physical dispositions of the devices, is then supposed to determine the probabilities of the various possible outcomes of the measurement. The word "stochastic" arises from this injection, at the final stage, of this idea that the "objective" or "real" hidden variable determines only the probabilities of the various possible results, rather than the individual results themselves. These physical idea behind stochastic hidden-variable theories are completely contrary and opposed to orthodox quantum thinking, even before the idea of locality is introduced. Thus the incompatibility of local stochastic hidden-variable theories with the predictions of quantum theory is, according to orthodox thinking, most naturally ascribed to a failure of the general realistic hidden-variable framework: the locality postulate is, according to orthodox quantum thinking, never placed in jeopardy. The general stochastic framework introduces nothing alien to

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orthodox quantum thinking. Accepting the idea that our purpose is to explore the logical structure of theories that cover the alternative possible cases, with the choices between these cases representable by a pair (xl, x2) of independent variables, we need only the UR assumption that nature makes selections of rl and r2, and the idea that these selections should be controlled in part by stochastic elements. If one accepts these ideas, then the incompatibility with quantum predictions must apparently be attributed to a failure of the locality idea that no causal influence of any kind can act outside the forward light cone. This latter idea is certainly not a natural or necessary part of orthodox relativistic quantum theory, which allows no predictions pertaining to the results of a measurement performed in one space-time region to depend upon what is done by experimenters in regions spacelike separated from the first. The nonlocal effects established here do not violate that causality condition, which is quite sufficient to account for the normally observed light-cone limitations of causal influences. 13. F O R M U L A T I O N IN TERMS O F VALUES The analysis given above was formulated in the mathematical framework of random variables. I12) In a discrete topology a random variable is a function, with discrete range, of a variable whose domain consists of discrete points, each of which is assigned a finite measure called its statistical weight. Thus for each of the four values of m = (x~, x2) in its domain { + 1, _+ 1 }, and each of the two values of j in its domain { 1, 2 }, the function r:m(;%,) is a 2"-valued random variable. The locality condition obtained in Section 10 was derived from conditions upon the domains of the variables 2Sin, and upon the measures assigned to the points of these domains. No conditions were imposed in terms of the values of these functions. The prior works of the author on the subject/191 were formulated in terms of the values of the rim, with no mention of random variables. However, the physical assumptions were the same. In particular, the locality assumption was the demand that for each j e {1, 2}, and each xs e { + i, - 1 }, nature's selection of a value for rim = rj(xi, xk) cannot be influenced by the choice of value for xk(k ~ j). By virtue of our criterion of influence, interpreted now in terms of values, rather than domains and statistical weights, this means that for each j 6 {1, 2}, and each xj s { + 1, - 1 }, a theory is local only if it allows there to be a value rj(xj) such that nature could select this value rj(xj) if the condition (xi. + t ) were chosen, and nature could select this same value rj(xs) if the condition (XJ, - 1 ) where chosen. This formulation of the locality requirement, like

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the f o r m u l a t i o n given above, does n o t d e p e n d upon, o r entail, the validity of the C F D i d e a that the results of u n p e r f o r m e d m e a s u r e m e n t s are determined, o r defined within nature. W h a t q u a n t u m t h e o r y "allows" was specified by excluding as disallowed only the pairs (r~, r2) in the d o m a i n of a r b i t r a r i l y small total statistical weight e defined earlier. T o satisfy the necessary c o n d i t i o n for locality, it is then necessary only to find, for each x~ in { + 1, - 1 }, a n d for each x2 in { + 1, - 1 } , s o m e p a i r of values, rl(x~) a n d r:(x2), such t h a t the t h e o r y allows (rl(x~), r2(x2)) to be the value o f (r~, r2) p i c k e d b y n a t u r e u n d e r c o n d i t i o n s ( x l , x2). If no such q u a r t e t exists, then the t h e o r y f o r c e s either the selection of rt to change as x 2 is varied or the selection of r 2 to change as x~ is varied, c o n t r a r y to the locality idea that, conjunctively, x~ does n o t influence r2 a n d x2 does n o t influence r~. But the result of Section 7 shows t h a t no such q u a r t e t exists. T h u s this v a l u e - b a s e d necessary c o n d i t i o n for a t h e o r y to be local c a n n o t be m e t by q u a n t u m theory.

REFERENCES l. I cite the proceedings of three recent major international conferences in this field: Symposium on the Foundations of Modern Physics, P. Lahti and P. Mittelstaedt, eds. (World Scientific, Singapore, 1985); Microphysical Realism and Quantum Formalism, A. van der Merwe, F. Selleri, and G. Tarozzi, eds. (Kluwer, Dordrecht, 1988); New Techniques and Ideas in Quantum Measurement Theory, D. Greenberger, (New York Academy of Sciences, New York, 1987). 2. A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935), cited as EPR. 3. N. Bohr, Phys. Rev. 48, 696 (1935). 4. A. Einstein, in Albert Einsteb~: Philosopher-Scientist, P. A. Schilpp, ed. (Tudor, New York, 1951), pp, 85 and 682. 5. J. S. Bell, Physics (N.Y.) 1, 195 (1964). 6. I have adapted the EPR arguments to the variant proposed by Bohm and Aharonov: D. Bohm and Y. Aharonov, Phys. Rev. 103, 1070 (1957). 7. a. N. Bohr, Atomic Theory and the Description of Nature (Cambridge University Press, 1934); b. Atomic Theory and the Description of Nature (Cambridge University Press, Cambridge, t934); b. Atomic Physics and Human Knowledge (Wiley, New York, 1959); c. Essays 1958/1962 on Atomic. Physics and Human Knowledge (Wiley, New York, 1963). See quotations given in H. P. Stapp, Am. J. Phys. 40, 1098 (1972). 8. W. Heisenberg, Physics and Philosophy (Harper and Row, New York, 1958); see comments in Appendix B of H. P. Stapp, Am. J. Phys. 40, 1098 (1972). 9. Ref. 4, p. 87. 10. D. B. Schwartz, B. Sen, C. N. Archie, and J. E. Lukens, Phys. Rev. Lett. 55, 1547 (1985); S. Washburn, R. A. Webb, R. F. Voss, and S. M. Faris, Phys. Rev. Lett. 54, 2712 (1985). 11. H. Everett, III, Rev. Mod Phys. 29, 454 (t957). 12. J. L. Doob, Stochastic Processes (Wiley, New York, I953). t3. J. Clauser and A. Shimony, Rep. Prog. Phys. 41, 182 (1978). !4. J. Clauser and M. A. Horne, Phys. Rev. D 10, 526 (1974).

448 15. 16. 17. 18. 19.

Stapp N. Bohr, Ref. 7a, p. 18. N. Bohr, Ref. 7c, p. 60. N. Bohr, Ref. 7b, p. 71. N. Bohr, Ref. 7a, p. 75. H. P. Stapp, Phys. Rev. D 3, 1303 (1971); Am. J. Phys. 53, 306 (1985); "Are faster-thanlight influences necessary," in Quantum Mechanics versus Local Realism: The Einstein, Podolsky, and Rosen Paradox, F. Selleri, ed. (Plenum, New York, 1988); "Quantum Nonlocality and the Description of Nature," in Philosophical Consequences of Quantum Theory, J. Cushing and E. McMullin, eds. (Univ. of Notre Dame Press, 1989).