Complexity via Correlated Statistics in Quantum Mechanics
Wm. C. McHarris
Michigan State University
Last modified: March 6, 2006
Although chaos theory has been highly successful in most disciplines, it has experienced notable lack of complete success when applied to quantum mechanics. Nevertheless, many of the so-called paradoxes generated within the Copenhagen interpretation of quantum mechanics can be explained through logical parallels in the realm of nonlinear dynamics and chaos [WCM, in "Quantum Theory: Reconsideration of Foundations-3, Ed. by G. Adenier, A. Yu. Khrennikov, and Th. M. Nieuwenhuizen, AIP Conference Proceedings, Vol. 810, p. 367 (2006), and references therein]. There have been a fair number of attempts to tack on nonlinear behavior to quantum systems, but they have been just that — adding nonlinear perturbations to basically linear systems, resulting in "weak" nonlinear systems, in which chaos and the less intuitive phenomena of nonlinear dynamics cannot develop. Additionally, many analyses implicitly assume that classical systems exhibit uncorrelated, as opposed to the correlated statistics (enthanglement) of quantum systems, plus there has been a tendency to (mis)apply statistical arguments to individual quantum states. I shall focus on Bell-type inequalities, where it can be shown that complex classical systems indeed exhibit correlated statistics, as exemplified by the "nonextensive" thermodynamics introduced by Tsallis et al. And I shall demonstrate that deriving a Bell-type inequality using correlated statistics on the classical side can easily produce classical behavior that overlaps with quantum mechanical predictions. This puts into question the inferences from the many "action-at-a-distance" experiments that local reality must be nonexistant — indeed, such arguments are rendered moot.