Manifest symplecticity in classical scattering

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.

Figures (1)

• Figure 1: An illustration of symplecticity: time evolution preserves area element in phase space.