A Gutzwiller Trace formula for Dirac Operators on a Stationary Spacetime

TL;DR

This paper extends the Gutzwiller trace formula to Dirac operators on stationary spacetimes, analyzing spectral properties and singularities related to Killing flows, with implications for quantum field theory in curved spacetime.

Contribution

It generalizes the Gutzwiller trace formula to Dirac operators on stationary spacetimes and develops microlocal analysis tools for vector bundle settings.

Findings

Spectrum of Lie derivative is discrete and real

Trace singularities occur at Killing flow periods

Weyl law asymptotics at zero period

Abstract

A Duistermaat-Guillemin-Gutzwiller trace formula for Dirac-type operators on a globally hyperbolic spatially compact stationary spacetime is achieved by generalising the recent construction by Strohmaier and Zelditch [Adv. Math. \textbf{376}, 107434 (2021)] to a vector bundle setting. We have analysed the spectrum of the Lie derivative with respect to a global timelike Killing vector field on the solution space of the Dirac equation and found that it consists of discrete real eigenvalues. The distributional trace of the time evolution operator has singularities at the periods of induced Killing flow on the manifold of lightlike geodesics. This gives rise to the Weyl law asymptotic at the vanishing period. A pivotal technical ingredient to prove these results is the Feynman propagator. In order to obtain a Fourier integral description of this propagator, we have generalised the classic work of Duistermaat and H\"{o}rmander [Acta Math. \textbf{128}, 183 (1972)] on distinguished parametrices for normally hyperbolic operators on a globally hyperbolic spacetime by propounding their microlocalisation theorem to a bundle setting. As a by-product of these analyses, another proof of the existence of Hadamard bisolutions for a normally hyperbolic operator (resp. Dirac-type operator) is reported.