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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Quantum theory with hidden variables and dynamic chaos

Yu. I. Bogdanov

Published Mar 3, 2008• 0 citations
Publisher

Abstract

We consider dynamics of hidden variables for measurements in a generalized bell-type model for a single spin using natural assumptions. The evolution of the system, which can be expressed as dynamic chaos is studied. The equilibrium state that the system evolves to asymptotically is consistent with the predictions of quantum theory.
DOI
10.1117/12.801894

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We consider dynamics of hidden variables for measurements in a generalized bell-type model for a single spin using natural assumptions. The evolution of the system, which can be expressed as dynamic chaos is studied. The equilibrium state that the system evolves to asymptotically is consistent with the predictions of quantum theory. The thesis of incompleteness of quantum mechanics in dynamic interpretation and completeness in statistical interpretation are developed. Conceptual problems of quantum mechanics such as violations of Bell inequalities, negative probabilities, complementarity principle, Einstein's locality and others are discussed.
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A number of the paradoxes encountered in the Copenhagen interpretation of quantum mechanics have apparent resolutions through parallels based in nonlinear dynamics and chaos theory. An overview of these is presented, with Bell’s theorem treated in somewhat greater detail. In these conferences (both “Quantum Theory: Reconsiderations of Foundations” and “Foundations of Probability and Physics”) Bell’s theorem has already been attacked by a number of authors in a number of ways, but here I demonstrate that the so‐called “classical” (or hidden variables) side of the derivation of Bell‐type inequalities is flawed because it is usually based on standard, non‐correlated statistics. When correlated statistics (including nonextensive thermodynamics) is considered, the classical upper limit on correlations is raised, overlapping with quantum predictions using entangled states. Thus, many of the Bell‐type experiments are rendered moot in ruling out “local reality.” Nonlinear dynamics/chaos could well provide a bridge between the determinism of Einstein and the probability of the Bohr school. Possible nonlinear underpinnings for quantum mechanics could also have implications lowering the plausibility of quantum computing.
We clarify the meaning of Bell's theorem and its implications for the construction of hidden variable theories by considering an example system consisting of two entangled spin-1/2 particles. Using this example, we present a simplified version of Bell's theorem and describe several hidden variable theories that agree with the predictions of quantum mechanics. These example theories clarify some subtle points, which are often misunderstood, regarding what it is that Bell's theorem actually establishes. \n
Many of the so‐called paradoxes of orthodox quantum mechanics can be shown to have parallel, more logical interpretations in the realm of nonlinear dynamics and chaos theory. Among these are violations of Bell‐type inequalities, which in comparing “classical” mechanics with quantum mechanics implicitly compare uncorrelated and correlated statistics. During the past decade research in the field of nonextensive thermodynamics (including Tsallis entropy) has demonstrated the existence of many statistical correlations in classical, nonlinear systems. When such correlations exist, the conventional classical upper limit on statistical correlations in Bell‐type experiments can easily be raised to overlap with quantum mechanical predictions involving correlated states such as the Bell singlet state, a favorite for deriving Bell inequalities. Thus, arguments based on experimental violations of Bell‐type inequalities, which rule out the existence of “local reality,” become moot. Perhaps quantum mechanics does have a deterministic, ontological basis, albeit one based in nonlinear dynamics and chaos theory. If so, deterministic chaos could provide Einstein’s longed‐for fundamental determinism, but because chaotic systems must be interpreted statistically, this also fits in quite well with the ideas of Bohr — Einstein and Bohr both could have been correct! It should be emphasized that the concept of nonlinear dynamics and chaos underpinning quantum mechanics does not involve hidden variables, nor does the fact that chaos is deterministic interlope on the existence of free will.
The Statistical Interpretation of quantum theory is formulated for the purpose of providing a sound interpretation using a minimum of assumptions. Several arguments are advanced in favor of considering the quantum state description to apply only to an ensemble of similarily prepared systems, rather than supposing, as is often done, that it exhaustively represents an individual physical system. Most of the problems associated with the quantum theory of measurement are artifacts of the attempt to maintain the latter interpretation. The introduction of hidden variables to determine the outcome of individual events is fully compatible with the statistical predictions of quantum theory. However, a theorem due to Bell seems to require that any such hidden-variable theory which reproduces all of quantum mechanics exactly (i.e., not merely in some limiting case) must possess a rather pathological character with respect to correlated, but spacially separated, systems.