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Physical Review Letters

Geometry underlying no-hidden-variable theorems

Daniel I. Fivel

Published Jul 15, 1991• 18 citations
Publisher

Abstract

The set of orientations of a measuring device (e.g., a Stern-Gerlach magnet) produced by the action of a Lie group constitutes a honmogeneous space S (e.g., a sphere). A hidden-variable measure determines a metric D on S, the triangle inequality being Bell's inequality. But identification of S with Hilbert-space projectors induces a locally convex metric d on S. The Einstein-Podolsky-Rosen (EPR) hypotheses imply that D=${\mathit{d}}^{2}$, which is impossible because the square of a locally convex metric cannot be a metric. This proves the Bell-EPR theorem. Classical systems avoid the contradiction by allowing only values d=0,1. The ``watchdog'' effect is shown to result from the form of the quantum-mechanical metric.

DOI
10.1103/physrevlett.67.285

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The set of orientations of a measuring device (e.g., a Stern-Gerlach magnet) produced by the action of a Lie group constitutes a honmogeneous space S (e.g., a sphere). A hidden-variable measure determines a metric D on S, the triangle inequality being Bell's inequality. But identification of S with Hilbert-space projectors induces a locally convex metric d on S. The Einstein-Podolsky-Rosen (EPR) hypotheses imply that D=d2{\mathit{d}}^{2}d2, which is impossible because the square of a locally convex metric cannot be a metric. This proves the Bell-EPR theorem. Classical systems avoid the contradiction by allowing only values d=0,1. The ``watchdog'' effect is shown to result from the form of the quantum-mechanical metric.
Generalized coherent states arise from reference states by the action of locally compact transformation groups and thereby form manifolds on which there is an invariant measure. It is shown that this implies the existence of canonically associated Bell states that serve as measuring rods by relating the metric geometry of the manifold to the observed Einstein-Podolsky-Rosen (EPR) correlations. It is further shown that these correlations can be accounted for by a hidden variable theory which is nonlocal but invariant under the stability group of the reference state.
Bell's theorem can be proved through simple geometrical reasoning, without the need for the Psi function, probability distributions, or calculus. The proof is based on N. David Mermin's explication of the Einstein-Podolsky-Rosen-Bohm experiment, which involves Stern-Gerlach detectors which flash red or green lights when detecting spin-up or spin-down. The statistics of local hidden variable theories for this experiment can be arranged in colored strips from which simple inequalities can be deduced. These inequalities lead to a demonstration of Bell's theorem. Moreover, all local hidden variable theories can be graphed in such a way as to enclose their statistics in a pyramid, with the quantum-mechanical result lying a finite distance beneath the base of the pyramid.
Einstein, Podolsky, and Rosen (EPR) have designed a gedanken experiment that suggested a theory that was more complete than quantum mechanics. The EPR design was later realized in various forms, with experimental results close to the quantum mechanical prediction. The experimental results by themselves have no bearing on the EPR claim that quantum mechanics must be incomplete nor on the existence of hidden parameters. However, the well known inequalities of Bell are based on the assumption that local hidden parameters exist and, when combined with conflicting experimental results, do appear to prove that local hidden parameters cannot exist. This fact leaves only instantaneous actions at a distance (called "spooky" by Einstein) to explain the experiments. The Bell inequalities are based on a mathematical model of the EPR experiments. They have no experimental confirmation, because they contradict the results of all EPR experiments. In addition to the assumption that hidden parameters exist, Bell tacitly makes a variety of other assumptions; for instance, he assumes that the hidden parameters are governed by a single probability measure independent of the analyzer settings. We argue that the mathematical model of Bell excludes a large set of local hidden variables and a large variety of probability densities. Our set of local hidden variables includes time-like correlated parameters and a generalized probability density. We prove that our extended space of local hidden variables does permit derivation of the quantum result and is consistent with all known experiments.
Einstein, Podolsky and Rosen (EPR) showed that it is possible to predict with certainty the value of a property without disturbing the object in question. In contrast, Quantum Mechanics (QM) holds that if different measurement setups cannot coexist, then predictions about those can neither. Using an EPR-inspired experiment with distantly separated measurements on pairs of entangled spinning particles, Bell proved that no local hidden variable (HV) theory can describe reality in more detail than QM. However, it is possible to conceive a viable HV theory based on the assumption that the perceived structure of spacetime is emergent from a hidden curved spacetime. According to this theory, locality can be maintained for each of the measurements while what is perceived as non-locality can be ascribed to the emergence of spacetime correlations between the instruments of the two parties. The theory predicts correlations that agree with QM, provided that the hidden spacetime has three spatial dimensions. If it had fewer than three dimensions, the CHSH inequality would not be violated and if more, Tsirelson's bound would be violated. According to this HV theory, the laboratory frame of reference is a corollary of correlations of the type that are the subject of Bell's thought experiment.
The phenomenon of nonlocality in quantum mechanics is one of its most fundamental features and is most strikingly exemplified in the discussion of the Einstein, Podolsky and Rosen (EPR) type of experiment. The generality of Bell's inequality and the results of experiments done up to now show that local hidden-variable theories are ruled out as a means to resolve the famous EPR paradox. What remains for further consideration are the nonlocal and the Einstein-separable hidden-variable models. Finally, an alternative and possibly successful approach in trying to 'explain' nonlocality might involve ideas of backward causation.
The celebrated inequalities of Bell are based on the assumption that local hidden parameters exist. When combined with conflicting experimental results, these inequalities appear to prove that local hidden parameters cannot exist. This contradiction suggests to many that only instantaneous action at a distance can explain the Einstein, Podolsky, and Rosen type of experiments. We show that, in addition to the assumption that hidden parameters exist, Bell tacitly makes a variety of other assumptions that contribute to his being able to obtain the desired contradiction. For instance, Bell assumes that the hidden parameters do not depend on time and are governed by a single probability measure independent of the analyzer settings. We argue that the exclusion of time has neither a physical nor a mathematical basis but is based on Bell's translation of the concept of Einstein locality into the language of probability theory. Our additional set of local hidden variables includes time-like correlated parameters and a generalized probability density. We prove that our extended space of local hidden variables does not permit Bell-type proofs to go forward.