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NONLOCALITY AS AN AXIOM FOR THEORY*

Daniel Rohrlich and Sandu Pop escu

School of Physics and Astronomy, Tel-Aviv University

Ramat-Aviv, Tel-Aviv 69978 Israel

ABSTRACT

Quantum and relativistic causality together imply nonlo cality: nonlo-

cal correlations (that violate the CHSH inequality) and nonlo cal equations of (the

Aharonov-Bohm e ect). Can weinvert the logical order? We consider a conjecture that

nonlo cality and relativistic causality together imply . We show that

correlations preserving relativistic causality can violate the CHSH inequality more strongly

than quantum correlations. Also, we describ e nonlo cal equations of motion, preserving rel-

ativistic causality, that do not arise in quantum mechanics. In these nonlo cal equations of

motion, an exp erimenter \jams" nonlo cal correlations b etween quantum systems.

1. INTRODUCTION

Two asp ects of quantum nonlo cality are nonlo cal correlations and nonlo cal equations

of motion. Nonlo cal correlations arise in settings such as the one discussed by Einstein,

1 2

Po dolsky and Rosen . As Bell showed (and Asp ect has reviewed in his lecture here) no

3

theory of lo cal variables can repro duce these correlations. The Aharonov-Bohm e ect is

also nonlo cal in that an electromagnetic eld in uences an electron in a region where the

eld vanishes. The eld induces a relative phase b etween two sets of paths available to

an electron, displacing the interference pattern b etween the two sets of paths. Thus, the

4

Aharonov-Bohm e ect implies nonlo cal equations of motion. Both asp ects of quantum non-

lo cality arise within nonrelativistic quantum theory.However, the very de nition of a lo cal

variable is relativistic: a lo cal variable can b e in uenced only byevents in its backward

light cone, and can in uence events only in its forward light cone. In this sense, quantum

mechanics and relativity together imply nonlo cality. They co exist b ecause quantum correla-

tions preserve relativistic causality (i.e. they do not allow us to transmit signals faster than

*Talk presented at 60 Years of E.P.R. (Workshop on the Foundations of Quantum

Mechanics, in honor of ), Technion, Israel, March 1995 1

light). But quantum mechanics do es not allow us to consider isolated systems as separate,

1

as Einstein, Po dolsky and Rosen assumed. This violation of not the letter but the spirit of

sp ecial relativity has left manyphysicists (including Bell) deeply unsettled. Today, quantum

nonlo cality seems as fundamental|and as unsettling|as ever. If nonlo cality is fundamen-

tal, why not make nonlo cality an axiom of quantum theory rather than a consequence? Can

we then invert the logical order, showing that nonlo cality and relativistic causality together

imply quantum theory?

2. NONLOCALITY I: NONLOCAL CORRELATIONS

Quantum mechanics and relativistic causality together give rise to nonlo cal corre-

lations, which manyphysicists regard as a negative asp ect of quantum theory. Here, we

regard quantum nonlo cality as a p ositive asp ect of quantum theory. What new p ossibilities

do es quantum nonlo cality o er us? In particular, if we make nonlo cality an axiom, what

57

b ecomes of the logical structure of quantum theory? The sp ecial theory of relativity can

b e deduced in its entirety from two axioms: the equivalence of inertial reference frames, and

7

the constancy of the sp eed of light. Aharonov has prop osed such a logical structure for

quantum theory. Let us take, as axioms of quantum theory, relativistic causality and nonlo-

cality. As an initial, immediate result, we deduce that quantum theory is not deterministic,

7

otherwise these two axioms would b e incompatible. Two \negative" asp ects of quantum

mechanics|indeterminacy and limits on measurements|then app ear as a consequence of

a fundamental \p ositive" asp ect: the p ossibility of nonlo cal action. Moreover, by taking

nonlo cality as an axiom, we free ourselves of the need to explain it.

Wehave not yet de ned the axiom of nonlo cality. Relativistic causalityiswell de ned,

but quantum nonlo cality arises b oth in nonlo cal correlations and in the Aharonov-Bohm

e ect. In this section we consider nonlo cal correlations. We ask which theories yield nonlo cal

correlations while preserving causality. Our result is indep endent of quantum mechanics or

8

any particular mo del. We nd that quantum mechanics is only one of a class of theories

consistent with our two axioms, and, in a certain sense, not even the most nonlo cal theory.

9

The Clauser, Horne, Shimony, and Holt (CHSH) form of Bell's inequality, holds in

any classical theory (that is, any theory of lo cal hidden variables). It states that a certain 2

combination of correlations lies b etween -2 and 2:

0 0 0 0

2  E (A; B )+E(A; B )+E(A ;B) E(A ;B )  2 : (1)

p

2 and 4, are imp ortant b ounds on the CHSH sum of Besides 2, two other numb ers, 2

correlations. If the four correlations in Eq. (1) were indep endent, the absolute value of

the sum could b e as muchas4. For quantum correlations, however, the CHSH sum of

p

10

2. Where do es this b ound come from? correlations is b ounded in absolute value by2

Rather than asking why quantum correlations violate the CHSH inequality,we might ask

why they do not violate it more.

Let us say that of the two axioms prop osed ab ove, the axiom of nonlo cality implies

that quantum correlations violate the CHSH inequality at least sometimes. Wemay then

guess that the other axiom, relativistic causality, might imply that quantum correlations

do not violate it maximally. Could it b e that relativistic causality restricts the violation

p

2 instead of 4? If so, then the two axioms determine the quantum violation of the to 2

CHSH inequality. To answer this question, we ask what restrictions relativistic causality

imp oses on joint probabilities. Relativistic causality forbids sending messages faster than

light. Thus, if one observer measures the A, the probabilities for the outcomes

A = 1 and A = 1must b e indep endent of whether the other observer cho oses to measure

0 8;11

B or B .However, it can b e shown that this constraint do es not limit the CHSH sum of

p

2. For example, imagine a \sup erquantum" correlation function quantum correlations to 2

E for measurements along given axes. Assume E dep ends only on the relative angle

 between axes. For any pair of axes, the outcomes j ""i and j ##i are equally likely, and

similarly for j "#i and j #"i. These four probabilities sum to 1, so the probabilities for j "#i

and j ##i sum to 1=2. In any direction, the probabilityof j"i or j#i is 1=2 irresp ectiveof

a measurement on the other particle. Measurements on one particle yield no information

ab out measurements on the other, so relativistic causality holds. The correlation function

then satis es E (  )=E(). Now let E ( )have the form

(i) E ( ) = 1 for 0    =4;

(ii) E ( ) decreases monotonically and smo othly from 1 to -1 as  increases from =4

to 3=4; 3

(iii) E ( )=1 for 3=4    .

0 0

^ ^

Consider four measurements along axes de ned by unit vectorsa ^ , b,^a, and b sepa-

rated by successive angles of =4 and lying in a plane. If wenow apply the CHSH inequality

Eq. (1) to these directions, we nd that the sum of correlations

0 0 0 0

^ ^ ^ ^

E (^a; b)+E(^a ; b)+E(^a; b ) E (^a ; b )=3E(=4) E (3=4) = 4 (2)

violates the CHSH inequality with the maximal value 4. Thus, a correlation function could

satisfy relativistic causality and still violate the CHSH inequality with the maximal value 4.

3. NONLOCALITY I I: NONLOCAL EQUATIONS OF MOTION

In one version of the Aharonov-Bohm e ect, an isolated magnetic ux, inserted b e-

tween two slits, shifts the interference pattern of electrons passing through the slits. It

thereby a ects the electron's momentum, since the electron arrives at a di erent p oint than

it would without the electromagnetic eld. Thus, the Aharonov-Bohm e ect implies non-

4 7

lo cal equations of motion. Aharonov has shown that a physical quantity, the modular

12

momentum of the ux, is uncertain exactly as required to keep us from seeing a violation

of causality. In general, mo dular momentum is measurable and ob eys a nonlo cal equation of

motion. But when the ux is lo cated b etween the slits, its mo dular momentum is completely

uncertain.

Is quantum mechanics the only relativistically causal theory with nonlo cal equations

of motion? As in the last section, wemay approach this question by lo oking for a theory

13

not equivalent to quantum mechanics that ob eys relativisitic causality and nonlo cality.

Wehave considered a mo del in which action by an exp erimenter a ects (\jams") nonlo cal

correlations b etween systems measured at spacelike separations from the action. For exam-

5

ple, Shimony considers the e ect of a laser b eam crossing the path of one of the photons in

a singlet pair, after the photon has already passed. We nd that while nonlo cal \jamming"

is not p ossible in quantum mechanics, it could b e consistent with relativisitic causality.If

jamming is realized in , then p erhaps, as suggested by Grunhaus, it is p ossible to jam

nonlo cal quantum correlations.

14

We brie y summarize the mo del. Two exp erimenters, call them Alice and Bob, make

measurements on systems that have lo cally interacted in the past. Alice's measurements are 4

spacelike separate from Bob's. A third exp erimenter, Jim (the jammer), presses a button

on a blackbox. This event is spacelike separate from Alice's measurements and from Bob's.

The blackbox acts at a distance on the correlations b etween the two sets of systems. We

nd no con ict with relativistic causality if jamming satis es two conditions. The unary

condition requires that neither Alice, from her results alone, nor Bob, from his, can tell

whether Jim has pressed the button. Then jamming cannot carry a signal to either Alice or

Bob. The unary condition implies indeterminism. The binary condition restricts the range

of jamming. If A and B denote Alice's and Bob's measurements, and J Jim's pressing of

the button, the overlap of the forward light cones of A and B must lie entirely within the

forward light cone of J .

4. SUMMARY

Wehave seen that quantum mechanics is not the only theory combining relativistic

causality with nonlo cality, nor even, in a sense, the most nonlo cal one. We found that

b oth \sup erquantum" correlations and a mo del for nonlo cal \jamming"|a stronger form of

nonlo cality than arises in quantum mechanics|can b e consistent with relativistic causality.

The question remains, from what minimal set of physical principles can we derive quantum

mechanics?

Acknowledgement. The researchofD.R.was supp orted by the State of Israel,

Ministry of Immigrant Absorption, Center for Absorption in Science, and by the Ticho Fund.

REFERENCES

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14. J. Grunhaus, S. Pop escu and D. Rohrlich, in preparation. 6