Nonlinearities in Quantum Mechanics

Many of the paradoxes encountered in the Copenhagen interpretation of quantum mechanics can be shown to have plausible, more logical parallels in terms of nonlinear dynamics and chaos. These include the statistical exponential decay laws, interpretations of Bell’s inequalities, spontaneous symmetry breaking, and perhaps diffractive behavior and even quantization itself. Many of the so-called alternative explanations of quantum mechanics have toyed with ideas that approach chaotic behavior, but as they were formulated before the advent of modern chaos theory, they remained within linear systems or at most nonlinear perturbations to linear systems; however, only strongly nonlinear systems can provide the proper parallels to the Copenhagen paradoxes. Several examples of these will be covered qualitatively. Strongly nonlinear behavior related to quantum mechanics does not involve “hidden variables,” but chaos provides a bridge between the statistical behavior of quantum mechanics and deterministic behavior of classical mechanics. Perhaps both Einstein and Bohr were correct in their debates—chaos fundamentally provides the determinism so dear to Einstein, but in practice it must be interpreted statistically in the manner of Bohr.