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NONLOCALITY AS AN AXIOM FOR QUANTUM THEORY*
Daniel Rohrlich and Sandu Pop escu
School of Physics and Astronomy, Tel-Aviv University
Ramat-Aviv, Tel-Aviv 69978 Israel
ABSTRACT
Quantum mechanics and relativistic causality together imply nonlo cality: nonlo-
cal correlations (that violate the CHSH inequality) and nonlo cal equations of motion (the
Aharonov-Bohm e ect). Can weinvert the logical order? We consider a conjecture that
nonlo cality and relativistic causality together imply quantum mechanics. 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
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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
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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 Nathan Rosen), 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
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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
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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,
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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
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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.
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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