Path integral measure for first order and metric gravities

TL;DR

This paper derives the correct path integral measure for metric gravity from the first order formulation, establishing their equivalence in 3+1 dimensions.

Contribution

It provides a systematic derivation of the measure linking first order and metric gravity path integrals in four-dimensional spacetime.

Findings

Established the equivalence of path integrals for first order and metric gravity.

Derived the correct measure for the metric gravity path integral.

Clarified the role of torsion in the path integral formulation.

Abstract

The equivalence between the path integrals for first order gravity and the standard torsion-free, metric gravity in 3+1 dimensions is analyzed. Starting with the path integral for first order gravity, the correct measure for the path integral of the metric theory is obtained.