Abstract

Trace formulas provide the only general relations known connecting quantum mechanics with classical mechanics in the case that the classical motion is chaotic. In particular, they connect quantal objects such as the density of states with classical periodic orbits. In this chapter, several trace formulas, including those of Gutzwiller and Balian and Bloch, Tabor, and Berry, are examined from a geometrical standpoint. New forms of the amplitude determinant in asymptotic theory are developed as tools for this examination. The meaning of caustics in these formulas is revealed in terms of intersections of Lagrangian manifolds in phase space. The periodic orbits themselves appear as caustics of an unstable kind, lying on the intersection of two Lagrangian manifolds in the appropriate phase space. New insight is obtained into the Weyl correspondence and the Wigner function, especially their caustic structures.

Introduction

This chapter concerns the trace formulas of Gutzwiller[1] and Balian and Bloch[2], which express the density of states of a bound quantal system as a sum over the periodic orbits of the corresponding classical system, and closely related trace formulas, such as that of Tabor[3] for the density of quasistates in a time-periodic system, and that of Berry[4] for the scars of Wigner functions in phase space. The purpose of this chapter is to explore the semiclassical structures of such formulas, i.e., the geometrical objects in the classical phase space associated with them and the interplay between these objects and the corresponding wave fields.