Square Root Actions, Metric Signature, and the Path-Integral of Quantum Gravity

TL;DR

This paper explores a novel approach to quantum gravity quantization using a square root form of the gravitational action, emphasizing the importance of mixed signature manifolds for unitarity and connecting to the problem of time.

Contribution

It introduces a path-integral formulation based on a square root action for gravity and argues for integrating over manifolds of both Euclidean and Lorentzian signatures.

Findings

Path integral should use exp[√i S] instead of exp[i S]

Unitarity requires integration over Euclidean and Lorentzian manifolds

Extended approach to include fermions in quantum gravity

Abstract

We consider quantization of the Baierlein-Sharp-Wheeler form of the gravitational action, in which the lapse function is determined from the Hamiltonian constraint. This action has a square root form, analogous to the actions of the relativistic particle and Nambu string. We argue that path-integral quantization of the gravitational action should be based on a path integrand $\exp[ \sqrt{i} S ]$ rather than the familiar Feynman expression $\exp[ i S ]$, and that unitarity requires integration over manifolds of both Euclidean and Lorentzian signature. We discuss the relation of this path integral to our previous considerations regarding the problem of time, and extend our approach to include fermions.