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Historically, in physics, hidden variable theories were espoused by a minority of physicists, including Albert Einstein, who argued that the statistical nature of quantum mechanics indicates that the theory is an incomplete description of the nature of the universe. It was thought that if hidden variables exist, unknown physical phenomena beyond quantum mechanics are needed to explain the universe as we know it.

{Citation needed|reason=Local determinism can be provided by non-local hidden variables|date=September 2009}

Later, Bell's theorem suggested (in the opinion of most physicists and contrary to Einstein's assertion) that local hidden variables are impossible.

The most famous such theory (because it gives the same answers as quantum mechanics, thus invalidating the famous theorem by von Neumann that no hidden variable theory reproducing the statistical predictions of QM is possible) is that of David Bohm. It is most commonly known as the Bohm interpretation or the Causal Interpretation of quantum mechanics. Bohm's (nonlocal) hidden variable is called the quantum potential. Nowadays Bohm's theory is considered to be one of many interpretations of quantum mechanics which give a realist interpretation, and not merely a positivistic one, to quantum-mechanical calculations. It is in fact just a reformulation of conventional quantum mechanics obtained by rearranging the equations and renaming the variables. Nevertheless it is a hidden variable theory.

The major reference for Bohm's theory today is his posthumous book with Basil Hiley.

Motivation

Quantum mechanics tells us only what the probabilities of the outcomes of measurements are: the same measurement done on two apparently identical systems can have different results. This raises the question of whether there might be some deeper reality hidden beneath quantum mechanics, to be described by a more fundamental theory. In other words, is there a way to determine the exact properties of every subatomic particle so that the outcome of each measurement can be predicted with certainty, as is the case in classical physics? If this can be done, then quantum mechanics is an incomplete description of reality.

According to the Copenhagen interpretation, the answer is "no": the quantum-mechanical behavior of the universe is inherently nondeterministic, so it is not possible to predict the outcome of any measurement with certainty. Physicists supporting hidden-variable interpretations believe the answer is "yes": some kind of objective foundation underlies the apparently probabilistic behavior of quantum-mechanical systems. Einstein, the most famous proponent of hidden variables, insisted that, "I am convinced God does not play dice" — meaning that he believed that physical theories must be deterministic to be complete.

Although determinism was initially a major motivation for physicists looking for hidden variable theories, nondeterministic theories trying to explain what the supposed reality underlying the quantum mechanics formalism looks like are also considered hidden variable theories; for example Edward Nelson's stochastic mechanics.

EPR Paradox & Bell's Theorem

In 1935, Einstein, Podolsky and Rosen wrote a four-page paper titled "Can quantum-mechanical description of physical reality be considered complete?" that argued that such a theory was in fact necessary, proposing the EPR Paradox as proof. In 1964, John Bell showed through his famous theorem that if local hidden variables exist, certain experiments could be performed where the result would satisfy a Bell inequality. If, on the other hand, Quantum entanglement is correct the Bell inequality would be violated. Another no-go theorem concerning hidden variable theories is the Kochen-Specker theorem.

Physicists such as Alain Aspect and Paul Kwiat have performed experiments that have found violations of these inequalities up to 242 standard deviations(excellent scientific certainty). This rules out local hidden variable theories, but does not rule out non-local ones (which would refute quantum entanglement). Theoretically, there could be experimental problems that affect the validity of the experimental findings.

Some hidden-variable theories

The Bohmian interpretation of quantum mechanics maintain that underlying the probabilistic nature of the universe is an objective foundation/property — the hidden variable. Others, however, believe that there is no deeper reality in quantum mechanics — experiments have shown a vast class of hidden variable theories to be incompatible with observations. Kirchmair and colleagues show that, in a system of trapped ions, quantum mechanics conflicts with hidden variable theories regardless of the quantum state of the system.

Assuming the validity of Bell's theorem, any hidden-variable theory which is consistent with quantum mechanics would have to be non-local, maintaining the existence of instantaneous or faster than light acausal relations (correlations) between physically separated entities. The first hidden-variable theory was the pilot wave theory of Louis de Broglie, dating from the late 1920s. The currently best-known hidden-variable theory, the Causal Interpretation, of the physicist and philosopher David Bohm, created in 1952, is a non-local hidden variable theory. Those who believe the Bohm interpretation to be actually true (rather than a mere model or interpretation), and the quantum potential to be real, refer to Bohmian mechanics.

What Bohm did, on the basis of an idea of Louis de Broglie, was to posit both the quantum particle, e.g. an electron, and a hidden 'guiding wave' that governs its motion. Thus, in this theory electrons are quite clearly particles. When you perform a double-slit experiment (see wave-particle duality), they go through one slit rather than the other. However, their choice of slit is not random but is governed by the guiding wave, resulting in the wave pattern that is observed.

Such a view does not contradict the idea of local events that is used in both classical atomism and relativity theory as Bohm's theory (and indeed quantum mechanics, with which it is exactly equivalent) are still locally causal but allow nonlocal correlations (that is information travel is still restricted to the speed of light). It points to a view of a more holistic, mutually interpenetrating and interacting world. Indeed Bohm himself stressed the holistic aspect of quantum theory in his later years, when he became interested in the ideas of Jiddu Krishnamurti. The Bohm interpretation (as well as others) has also been the basis of some books which attempt to connect physics with Eastern mysticism and consciousness. Nevertheless this nonlocality is seen as a weakness of Bohm's theory by some physicists.

Another possible weakness of Bohm's theory is that some feel that it looks contrived. It was deliberately designed to give predictions which are in all details identical to conventional quantum mechanics. Bohm's aim was not to make a serious counterproposal but simply to demonstrate that hidden-variable theories are indeed possible. His hope was that this could lead to new insights and experiments that would lead beyond the current quantum theories.

Gerard 't Hooft has disputed the validity of Bell's theorem on the basis of the superdeterminism loophole and proposed some ideas to construct local deterministic models.

See also

• Local hidden variable theory
• Bell's theorem
• Bell test experiments
• Quantum mechanics
• Bohm interpretation