Covariant quantization of the Einstein-Hilbert theory in first-order form

TL;DR

This paper develops a covariant quantization method for the first-order Einstein-Hilbert theory, analyzing gauge structure, quantum identities, and equivalence with the second-order formulation using path integral and BV formalisms.

Contribution

It introduces a covariant quantization approach for the first-order Einstein-Hilbert action, clarifies gauge algebra, and establishes quantum equivalence with the second-order formulation.

Findings

Gauge algebra is closed and irreducible.

Dyson-Schwinger equations constrain Green's functions.

First- and second-order formulations are quantum mechanically equivalent.

Abstract

We present a covariant quantization of the first-order formulation of the Einstein-Hilbert theory using the path integral and BV formalisms. In this approach, the metric $g^{\mu\nu}$ and the connection $\Gamma^\lambda_{\mu\nu}$ are treated as independent, with the connection playing the role of an auxiliary field. We show that the gauge algebra is closed and irreducible. We further demonstrate that the Dyson-Schwinger equations in the first-order formulation lead to structural identities that constrain the Green's functions of the auxiliary field and encode the classical equations of motion at the quantum level. We revisit the quantum equivalence between the first- and second-order formulations of the Einstein-Hilbert theory. By employing a suitable trick, a manifestly covariant form of the Senjanovi\'c measure is derived. We also show that the two formulations are equivalent at the level of the effective action when the auxiliary field is on-shell.