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Annalen der Physik

Semiclassical trace formulae using coherent states

B. Mehlig, Michael Wilkinson

Published Jun 1, 2001• 5 citations
Publisher

Abstract

Abstract We derive semiclassical trace formulae including Gutzwiller's trace formula using coherent states. This formulation has several advantages over the usual coordinate‐space formulation. Using a coherent‐state basis makes it immediately obvious that classical periodic orbits make separate contributions to the trace of the quantum‐mechanical time evolution operator. In addition, our approach is manifestly canonically invariant at all stages, and leads to the simplest possible derivation of Gutzwiller's formula.

DOI
10.1002/andp.200151306-705

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Abstract We derive semiclassical trace formulae including Gutzwiller's trace formula using coherent states. This formulation has several advantages over the usual coordinate‐space formulation. Using a coherent‐state basis makes it immediately obvious that classical periodic orbits make separate contributions to the trace of the quantum‐mechanical time evolution operator. In addition, our approach is manifestly canonically invariant at all stages, and leads to the simplest possible derivation of Gutzwiller's formula.
We derive semiclassical trace formulae including Gutzwiller's trace formula using coherent states. This formulation has several advantages over the usual coordinate-space formulation. Using a coherent-state basis makes it immediately obvious that classical periodic orbits make separate contributions to the trace of the quantum-mechanical time evolution operator. In addition, our approach is manifestly canonically invariant at all stages, and leads to the simplest possible derivation of Gutzwiller's formula.
Gutzwiller's trace formula has a central place in quantum chaos because it provides semiclassical approximations for quantum energy levels in classically chaotic systems by linking them to classical periodic orbits. In this didactic article, we discuss a derivation of the trace formula starting from the Feynman path integral. We then describe how the trace formula is used to explain universal features in the distribution of the quantum energy levels that are described by random matrix theory, and we give an overview of related work.
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.
We present a detailed derivation of the trace formula for a general Hamiltonian with two degrees of freedom where one of them is canonical and the other a spin. Our derivation starts from the semiclassical formula for the propagator in a basis formed by the product of a canonical and a spin coherent states and is valid in the limit ℏ→0, j→∞ with jℏ constant. The trace formula, obtained by taking the trace and the Fourier transform of the coherent state propagator, is compared to others found in the literature.