Non-linearity and the emergence of coherence in strongly interacting many-body systems near a critical point

Abstract

A new approach is described which provides an accurate analytical method for the study of strongly interacting many-body systems. We draw attention to the universal character of particular second quantised Hamiltonians and common physical properties which exemplify them near a critical point. From the heisenberg equations of motion for annihilators and creators we demonstrate how they may be transformed very accurately into highly non-linear equations for the classical field. It is pointed out that these equations may be solved exactly in multidimensional space using recent developments in non-linear analysis. We show that interactions may lead to strong non-linearity and hence to the formation of localised classical field solutions in a number of geometries which can be found from symmetry considerations. Unlike the standard approach, this method introduces quantum phenomena by a linearisation in ħ about the classical solutions. The resulting Schrödinger equations are exactly soluble.