Spectral methods in quantum field theory and quantum cosmology

TL;DR

This paper reviews how spectral zeta-functions are used to analyze one-loop quantum effects in quantum gravity and cosmology, focusing on boundary conditions, ellipticity issues, and implications for universe models.

Contribution

It discusses the application of spectral zeta-functions to quantum field theory on manifolds with boundary, addressing boundary condition invariance and ellipticity challenges in quantum gravity.

Findings

Evaluation of zeta(0) on Euclidean 4-ball background.

Universe with small 3-geometry has vanishing probability of singularity.

Analysis of boundary condition invariance and ellipticity issues.

Abstract

We review the application of the spectral zeta-function to the 1- loop properties of quantum field theories on manifolds with boundary, with emphasis on Euclidean quantum gravity and quantum cosmology. As was shown in the literature some time ago, the only boundary conditions that are completely invariant under infinitesimal diffeomorphisms on metric perturbations suffer from a drawback, i.e. lack of strong ellipticity of the resulting boundary-value problem. Nevertheless, at least on the Euclidean 4-ball background, it remains possible to evaluate the zeta(0) value, which describes in this case a universe which, in the limit of small 3-geometry, has vanishing probability of approaching the cosmological singularity. An assessment of this result is here performed, discussing its physical and mathematical implications.