Path integral quantization of parametrised field theory

TL;DR

This paper develops a path integral quantization for parametrised field theory, revealing a non-trivial measure and non-standard Wick rotations, and explores implications for quantum gravity and the problem of time.

Contribution

It constructs a covariant path integral measure for parametrised field theory and introduces a framework to analyze the problem of time in quantum gravity.

Findings

The Lorentzian path integral measure is non-trivial, akin to the Fradkin-Vilkovisky measure.

Euclidean integrals imply non-standard Wick rotations of the scalar field 2-point function.

Explicit computation in 1+1 dimensions illustrates the framework's application.

Abstract

Free scalar field theory on a flat spacetime can be cast into a generally covariant form known as parametrised field theory in which the action is a functional of the scalar field as well as the embedding variables which describe arbitrary, in general curved, foliations of the flat spacetime. We construct the path integral quantization of parametrised field theory in order to analyse issues at the interface of quantum field theory and general covariance in a path integral context. We show that the measure in the Lorentzian path integral is non-trivial and is the analog of the Fradkin- Vilkovisky measure for quantum gravity. We construct Euclidean functional integrals in the generally covariant setting of parametrised field theory using key ideas of Schleich and show that our constructions imply the existence of non-standard `Wick rotations' of the standard free scalar field 2 point function. We develop a framework to study the problem of time through computations of scalar field 2 point functions. We illustrate our ideas through explicit computation for a time independent 1+1 dimensional foliation. Although the problem of time seems to be absent in this simple example, the general case is still open. We discuss our results in the contexts of the path integral formulation of quantum gravity and the canonical quantization of parametrised field theory.