Covariant singularities in Quantum Field Theory and Quantum Gravity

TL;DR

This paper extends a covariant approach to singularities from classical gravity to quantum gravity, classifying and analyzing covariant singularities in field space and their implications for quantum theories of gravity.

Contribution

It introduces a covariant classification of quantum covariant singularities and analyzes their impact on the path integral measure in quantum gravity.

Findings

Path-integral measure is regular in certain gravity theories.

Two types of covariant singularities identified: geodesic incompleteness and ill-defined path integrals.

Topological classification of functional singularities provided.

Abstract

It is rather well-known that spacetime singularities are not covariant under field redefinitions. A manifestly covariant approach to singularities in classical gravity was proposed in arXiv:2008.09387. In this paper, we start to extend this analysis to the quantum realm. We identify two types of covariant singularities in field space corresponding to geodesic incompleteness and ill-defined path integrals (hereby dubbed functional singularities). We argue that the former might not be harmful after all, whilst the latter makes all observables undefined. We show that the path-integral measure is regular in any four-dimensional theory of gravity without matter or in any theory in which gravity is either absent or treated semi-classically. This might suggest the absence of functional singularities in these cases, however it can only be confirmed with a thorough analysis, case by case, of the path integral. We provide a topological and model-independent classification of functional singularities using homotopy groups and we discuss examples of theories with and without such singularities.