Effective Actions for Regge Piecewise Flat Quantum Gravity

TL;DR

This paper reviews the path integral and effective action in Regge quantum gravity, proposing a modified measure to ensure finiteness while maintaining desirable properties of the original measure.

Contribution

It introduces a new measure for the Regge path integral that guarantees finiteness without losing key features of the original measure.

Findings

Original measure does not produce a finite path integral.

Modified measure achieves finiteness of the path integral.

Preserves key features of the original measure.

Abstract

We review the construction of the path integral and the corresponding effective action for the Regge formulation of General Relativity under the assumption that the short-distance structure of the spacetime is not a smooth 4-manifold, but a picewise linear manifold based on a triangulation of a smooth 4-manifold. We point out that the exponentially damped 4-volume path-integral measure does not give a finite path integral, although it can be used for the construction of the perturbative effective action. We modify the 4-volume measure by multiplying it by an inverse power of the product of the edge-lengths such that the new measure gives a finite path integral while it retains all the nice features of the unmodified measure.