Semiclassical Field Theory Approach to Quantum Chaos

TL;DR

This paper develops a semiclassical field theory framework to analyze quantum chaos, connecting classical dynamics with quantum statistical properties and unifying long-time universal behavior with short-time orbit theory.

Contribution

It introduces a supermatrix nonlinear sigma-model incorporating classical evolution operators to describe quantum chaos across energy scales.

Findings

Recovers universal long-time behavior of random matrix theory

Accurately describes short-time dynamics matching periodic orbit theory

Links classical irreversibility with quantum statistical properties

Abstract

We construct a field theory to describe energy averaged quantum statistical properties of systems which are chaotic in their classical limit. An expression for the generating function of general statistical correlators is presented in the form of a functional supermatrix nonlinear $\sigma$-model where the effective action involves the evolution operator of the classical dynamics. Low-lying degrees of freedom of the field theory are shown to reflect the irreversible classical dynamics describing relaxation of phase space distributions. The validity of this approach is investigated over a wide range of energy scales. As well as recovering the universal long-time behavior characteristic of random matrix ensembles, this approach accounts correctly for the short-time limit yielding results which agree with the diagonal approximation of periodic orbit theory.