Chaos and the Quantum: How Nonlinear Dynamics Can Produce Classical Correlations Analogous to Quantum Entanglement

Departments of Chemistry and Physics/Astronomy
Michigan State University
East Lansing, MI 48824, USA

In recent years I have suggested that many of the so-called paradoxes resulting from the Copenhagen interpretation of quantum mechanics might well have more logically satisfying parallels based in nonlinear dynamics and chaos theory [WCM, Complexity 12(4), 12 (2007); “Proceedings of the 5th International Workshop DICE2010: Space-Time-Matter—Current Issues in Quantum Mechanics and Beyond,” J. Phys.: Conf. Ser. 306, 012050 (2011)]. From such arguments one can infer that quantum mechanics might not be the strictly linear science that orthodoxy postulates. Indeed, quite recently experimentalists reported definite signatures of chaotic behavior in an unquestionably quantum system [S. Chadhury et al., Nature 461, 768 (8 Oct 2009)]. As an example of what can go amiss when quantum behavior is forced into a linear interpretation, I examine Bell-type inequalities. In conventional derivations of such inequalities, a classical system imposes an upper limit on statistically correlated properties of a pair of entangled particles as measured by observers separated by an effectively infinite, incommunicado distance. (Most often these are correlations in spin orientation for particles in a Bell singlet state.) Quantum mechanics, conversely, allows greater statistical correlations — and numerous sophisticated experiments have upheld the quantum mechanical predictions! The inference is that a measurement made on particle 1 instantaneously collapses the wave function of particle 2, i.e., superluminal signals are transmitted between the particles, violating the tenets of relativity. Upon closer examination, I argue that the fault lies not with the quantum mechanical side of such derivations (the usual point of attack by those seeking to debunk Belltype arguments), but that implicit in the derivations for the so-called classical side (such as for the ubiquitous CHSH inequality [J. Clauser, M. A. Horne, A. Shimony, and R. Holt, Phys. Rev. Lett. 23, 880 (1969)]) is the assumption of independent, uncorrelated particles. Thus, one winds up comparing independent probabilities against conditional probabilities rather than comparing classical mechanics versus quantum mechanics. Now, if one examines nonlinear dynamics and chaotic systems, including emergent classical systems, one finds that many such systems exhibit correlations that can easily be as great as those found in quantum systems. Various explanations have been proposed, including the fact that nonergodic behavior — nonuniform or preferential visitation of various regions in phase space by nonlinear trajectories — can easily ape action-at-a-distance. Nonextensive thermodynamics has verified this with numerous examples [Nonextensive Entropy: Interdisciplinary Applications, M. Gell-Mann and C. Tsallis, eds., Oxford: Oxford Univ. Press (2004)]. In short, classical nonlinear dynamics involves many of the same sort of superficially counterintuitive, imponderable paradoxes as does quantum mechanics; when examined in depth, however, the classical “paradoxes” dissolve into relatively logical explanations. This encourages one to raise the question as to whether or not quantum mechanics might contain nonlinear elements. When quantum mechanics was being developed, nonlinear dynamics, and especially chaos theory, were for all practical purposes terra incognita. Thus, quantum mechanics was necessarily shaped according to a linear template. If, however, one were to allow the possibility that nonlinear dynamics and chaos theory could lead toward an emergent quantum mechanics, then they could well provide a bridge between the determinism so dear to Einstein and the statistical interpretation essential to the Copenhagen interpretation. Perhaps both Einstein and Bohr were correct in their debates over the fundamental meaning and completeness of quantum mechanics.