Manifest symplecticity in classical scattering

TL;DR

This paper explores two formulations of classical mechanics that manifest Liouville's theorem, establishing a relation between them and providing a classical scattering theory framework.

Contribution

It introduces a classical scattering theory based on two formulations of mechanics and clarifies their relationship, including a classical derivation of the exponential generator.

Findings

Demonstrates the relation between on-shell action and exponential generator

Provides a classical derivation of scattering theory

Clarifies the connection between in-out and in-in formalisms

Abstract

The Liouville theorem states that classical time evolution is an incompressible flow in phase space. We investigate two formulations of classical mechanics in which this property is manifested. First, the traditional Hamilton-Jacobi theory provides an in-out formalism. Second, a recent idea employing an exponential representation of time evolution provides an in-in formalism. Through concrete examples, it is demonstrated that the on-shell action in the former and the exponential generator in the latter are disparate objects. Still, a concrete relation between the two is identified in terms of a matching calculation. A strictly classical derivation and formulation of classical scattering theory is provided.