hidden variables

hidden variables [′hid·ən ′ver·ē·ə·bəlz]

(quantum mechanics)

Hypothetical additional variables or parameters which would supplement quantum mechanics, making it possible to unambiguously predict the result of a single measurement on a single microscopic system.

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Hidden variables

Additional variables or parameters that would supplement quantum mechanics so as to make it like classical mechanics. Hidden variables would make it possible to unambiguously predict (as in classical mechanics) the result of a specific measurement on a single microscopic system. In contrast, quantum mechanics can give only probabilities for the various possible results of that measurement. Hidden variables would thus provide deeper insights into the quantum-mechanical probabilities. In this sense the relationship between quantum mechanics and hidden variables could be analogous to the relationship between thermodynamics (for example, temperature) and statistical mechanics (the motions of the individual molecules). See Statistical mechanics

F. J. Belinfante formulated a three-section classification scheme for hidden variable theories—zeroth kind, first kind, and second kind. Most interest, both theoretically and experimentally, has been focused on hidden variable theories of the second kind, also known as local hidden variable theories.

In 1932 J. von Neumann provided an axiomatic basis for the mathematical methods of quantum mechanics. As a sidelight to this work, he rigorously proved from the axioms that any hidden variable theory was inconsistent with quantum mechanics. This was the most famous of a number of proofs, appearing as recently as 1980 and purporting to show the impossibility of any hidden variable theory. In 1966 J. S. Bell pinpointed the difficulty with von Neumann's proof—one of his axioms was fine for a pure quantum theory which makes statistical predictions, but the axiom was inherently incompatible with any hidden variable theory. The other impossibility proofs have also been found to be based on self-contradictory theories. Such theories are called hidden variable theories of the zeroth kind.

Hidden variable theories of the first kind are constructed so as to be self-consistent and to reproduce all the statistical predictions of quantum mechanics when the hidden variables are in an “equilibrium” distribution. Hidden variables of the second kind predict deviations from the statistical predictions of quantum mechanics, even for the “equilibrium’’ situations for which theories of the first kind agree with quantum mechanics. They are generally called local hidden variable theories because they are required to satisfy a locality condition. Intuitively, this seems to be a very natural condition. Locality requires that an apparatus at one location should operate independently of any settings or actions of a second apparatus at a spatially separated location. In the strict Einstein sense of locality, the two apparatus must be independent during any time interval less than the time required for a light signal to travel from one apparatus to the other.

The focus for much of the discussion of local hidden variable theories is provided by a famous thought experiment (a hypothetical, idealized experiment in which the experimental results are deduced) that was introduced in 1935 by A. Einstein, B. Podolsky, and N. Rosen. Their thought experiment involves an examination of the correlation between measurements on two parts of a single system after the parts have become spatially separated. They used this thought experiment to argue that quantum mechanics was not a complete theory. Although they did not refer to hidden variables as such, these would presumably provide the desired completeness. The Einstein-Podolsky-Rosen (EPR) experiment led to a long-standing philosophical controversy; it has also provided the framework for a great deal of research on hidden variable theories.

New efforts were stimulated in 1952 when Bohm did the “impossible” by designing a hidden variable theory of the first kind. Bohm's theory was explicitly nonlocal, and this fact led Bell to reexamine the Einstein-Podolsky-Rosen experiment. He came to the remarkable conclusion that any hidden variable theory that satisfies the condition of locality cannot possibly reproduce all the statistical predictions of quantum mechanics. Specifically, in Einstein-Podolsky-Rosen type experiments, quantum mechanics predicts a very strong correlation between measurements on the spatially separated parts. Bell showed that there is an upper limit on the strength of these correlations in the statistical prediction of any local hidden variable theory. Bell's result can be put in the form of inequalities which must be satisfied by any local hidden variable theory but which may be violated by the statistical predictions of quantum mechanics under appropriate experimental conditions.

Experiments performed under conditions in which the statistical predictions of quantum mechanics violate Bell's inequalities can test the entire class of local hidden variable theories. However, all existing experiments have required supplementary assumptions regarding detector efficiencies. Due to the supplementary assumptions, small loopholes still remain, and experiments have been proposed to eliminate them. The overwhelming experimental evidence is against any theory that would supplement quantum mechanics with hidden variables and still retain the locality condition; that is, any hidden variable theory that reproduces all the statistical predictions of quantum mechanics must be nonlocal. The remarkable Einstein-Podolsky-Rosen correlations have defied any reasonable classical kind of explanation. See Quantum mechanics