Bogolyubov invariant via relative spectral invariants on manifolds

TL;DR

This paper introduces a new spectral invariant for pairs of elliptic operators on manifolds, linking eigenvalues and eigensections, and explores its asymptotic behavior related to particle creation in quantum field theory.

Contribution

It presents a novel spectral invariant that incorporates both eigenvalues and eigensections, expanding the understanding of spectral geometry and quantum field dynamics on manifolds.

Findings

Derived the asymptotic expansion of the invariant for small adiabatic parameters.

Explicitly computed the first two coefficients of the expansion.

Connected the invariant to the regularized particle number in quantum field theory.

Abstract

We introduce and study new spectral invariant of two elliptic partial differential operators of Laplace and Dirac type on compact smooth manifolds without boundary that depends on both the eigenvalues and the eigensections of the operators, which is a equal to the regularized number of created particles from the vacuum when the dynamical operator depends on time. We study the asymptotic expansion of this invariant for small adiabatic parameter and compute explicitly the first two coefficients of the asymptotic expansion.