Path Integral Quantization of the First Order Einstein-Hilbert Action from its Canonical Structure

TL;DR

This paper explores the path integral quantization of the first order Einstein-Hilbert action, revealing differences from the Faddeev-Popov method due to tertiary constraints and non-trivial ghosts, affecting covariance.

Contribution

It demonstrates that canonical quantization yields a path integral measure distinct from the Faddeev-Popov approach for the first order Einstein-Hilbert action, highlighting the role of constraints.

Findings

Path integral measure differs from Faddeev-Popov approach.

Presence of tertiary first class constraints affects the measure.

Non-trivial ghosts arise from second class constraints.

Abstract

We consider the form of the path integral that follows from canonical quantization and apply it to the first order form of the Einstein-Hilbert action in $d > 2$ dimensions. We show that this is inequivalent to what is obtained from applying the Faddeev-Popov (FP) procedure directly. Due to the presence of tertiary first class constraints, the measure of the path integral is found to have a substantially different structure from what arises in the FP approach. In addition, the presence of second class constraints leads to non-trivial ghosts, which cannot be absorbed into the normalization of the path integral. The measure of the path integral lacks manifest covariance.