On the classical geometry of chaotic Green functions and Wigner functions

TL;DR

This paper develops a classical geometric framework for semiclassical representations of quantum states and operators in chaotic systems, using Legendre transforms and double phase space, revealing complex structures and connections to periodic orbits.

Contribution

It introduces a Legendre transform-based resolvent surface in double phase space for semiclassical analysis, applicable to chaotic systems, and links it to periodic orbit resummation techniques.

Findings

Resolvant surface growth follows trajectories in double phase space.

Complexity of the resolvent surface resembles a multidimensional sponge.

Secondary periodic orbits approximate higher time actions.

Abstract

Semiclassical approximations for various representations of a quantum state are constructed on a single (Lagrangian) surface in phase space, but it is not available for chaotic systems. An analogous evolution surface underlies semiclassical representations of the evolution operator, albeit in a doubled phase space. It is here shown that, corresponding to the Fourier transform on a unitary operator, represented as a Green function or spectral Wigner function, a Legendre transform generates a resolvent surface as the classical basis for semiclassical representations of the resolvent operator in double phase space, independently of the integrable or chaotic nature of the system. This surface coincides with derivatives of action functions (or generating functions) depending on the choice of appropriate coordinates and its growth departs from the energy shell following trajectories in double phase space. In an initial study of the resolvent surface based on its caustics, its complex nature is revealed to be analogous to a multidimensional sponge. Resummation of the trace of the resolvent in terms of linear combinations of periodic orbits, known as pseudo orbits or composite orbits, provides a cutoff to the semiclassical sum at the Heisenberg time. It is here shown that the corresponding actions for higher times can be approximately included within true secondary periodic orbits, in which multiple windings of short periodic orbits are joined by heteroclinic orbits into larger circuits.