Field Theory Methods in Classical Dynamics

TL;DR

This paper develops a perturbation theory for classical dynamics using field theory methods, introduces complexification formalisms for phase space analysis, and connects Hamilton's equations to path integrals, with applications to geometric phases.

Contribution

It introduces a Dirac picture perturbation theory for classical dynamics and extends complexification formalisms to analyze phase space and geometric phases.

Findings

Derived detailed rules for classical time evolution operator computations

Extended formalisms to two-dimensional phase space for Berry phase analysis

Connected Hamilton's equations to path integral formulations in classical mechanics

Abstract

A Dirac picture perturbation theory is developed for the time evolution operator in classical dynamics in the spirit of the Schwinger-Feynman-Dyson perturbation expansion and detailed rules are derived for computations. Complexification formalisms are given for the time evolution operator suitable for phase space analyses, and then extended to a two-dimensional setting for a study of the geometrical Berry phase as an example. Finally a direct integration of Hamilton's equations is shown to lead naturally to a path integral expression, as a resolution of the identity, as applied to arbitrary functions of generalized coordinates and momenta.