Exact Quantum Trace Formula from Complex Periodic Orbits

TL;DR

This paper develops an exact quantum trace formula using complex periodic orbits and Lefschetz thimbles, extending classical and tunneling methods to fully account for quantum spectra.

Contribution

It introduces a fully quantum trace formula via complexification of periodic orbits and periods, unifying real-time and imaginary-time approaches.

Findings

Connects quantum spectrum to all complex time contributions

Unifies Gutzwiller's real-time and instanton imaginary-time methods

Provides a comprehensive quantum trace formula incorporating complex cycles

Abstract

The Gutzwiller trace formula establishes a profound connection between the quantum spectrum and classical periodic orbits. However, its application is limited by its reliance on the semiclassical saddle point approximation. In this work, we explore the full quantum version of the trace formula using the Lefschetz thimble method by incorporating complexified periodic orbits. Upon complexification, classical real periodic orbits are transformed into cycles on compact Riemann surfaces. Our key innovation lies in the simultaneous complexification of the periods of cycles, resulting in a fully quantum trace formula that accounts for all contributions classified by the homology classes of the associated Riemann surfaces. This formulation connects the quantum spectrum to contributions across all complex time directions, encompassing all relevant homology classes. Our approach naturally unifies and extends two established methodologies: periodic orbits in real time, as in Gutzwiller's original work, and quantum tunneling in imaginary time, as in the instanton method.