Small Oscillation Theory of the Interaction of a Particle and Scalar Field

Abstract

We study the interaction of a nonrelativistic particle with a scalar field, with particular application to the theory of polarons. The approach is based on a general classical method for the integration of equations of motion. The Hamiltonian is transformed by successive canonical transformations, the first corresponding to describing the motion relative to special solutions of the equations of motion. This stage as applied to suitably ordered Heisenberg equations of motion is identical with intermediate coupling theory. The second transformation treats the coupled small oscillations of particle and field oscillators about the chosen special solution. This affords a natural extension of intermediate coupling theory for this problem. Differences between the classical and quantum theories arise in the ordering of operators; the differences play a crucial role in determining the effective cutoff in wave vector space.