Non-separability without Non-separability in Nonlinear Quantum Mechanics

Abstract

We show an example of benign non-separability in an apparently separable system consisting of $n$ free non-correlated quantum particles, solitonic solutions to the nonlinear phase modification of the Schrödinger equation proposed recently. The non-separability manifests itself in the wave function of a single particle being influenced by the very presence of other particles. In the simplest case of identical particles, it is the number of particles that affects the wave function of each particle and, in particular, the width of its Gaussian probability density. As a result, this width, a local property, is directly linked to the mass of the entire Universe in a very Machian manner. In the realistic limit of large $n$ if the width in question is to be microscopic, the coupling constant must be very small resulting in an ``almost linear'' theory. This provides a model explanation of why the linearity of quantum mechanics can be accepted with such a high degree of certainty even if the more fundamental underlying theory could be nonlinear. We also demonstrate that when such non-correlated solitons are coupled to harmonic oscilators they lead to a faster-than-light nonlocal telegraph since changing the frequency of one oscillator affects instantaneously the probability density of particles associated with other oscillators. This effect can be alleviated by fine-tuning the parameters of the solution. Exclusion rules of a novel kind that we term supersuperselection rules also emerge from these solutions. They are similar to the mass and the univalence superselection rules in linear quantum mechanics. The effects in question and the exclusion rules do not appear if a weakly separable extension to $n$-particles is employed.