A discrete history of the Lorentzian path integral

TL;DR

This paper introduces a non-perturbative Lorentzian gravitational path integral that addresses key issues in quantum gravity, demonstrating its advantages over Euclidean approaches through discrete models and results in lower dimensions.

Contribution

It presents a new formulation of the Lorentzian path integral for quantum gravity, highlighting its well-defined properties and advantages over Euclidean methods.

Findings

Explicit example of Euclidean-Lorentzian inequivalence

Non-perturbative conformal factor cancellation

Causality as an effective regulator

Abstract

In these lecture notes, I describe the motivation behind a recent formulation of a non-perturbative gravitational path integral for Lorentzian (instead of the usual Euclidean) space-times, and give a pedagogical introduction to its main features. At the regularized, discrete level this approach solves the problems of (i) having a well-defined Wick rotation, (ii) possessing a coordinate-invariant cutoff, and (iii) leading to_convergent_ sums over geometries. Although little is known as yet about the existence and nature of an underlying continuum theory of quantum gravity in four dimensions, there are already a number of beautiful results in d=2 and d=3 where continuum limits have been found. They include an explicit example of the inequivalence of the Euclidean and Lorentzian path integrals, a non-perturbative mechanism for the cancellation of the conformal factor, and the discovery that causality can act as an effective regulator of quantum geometry.