Path integral for the Hilbert-Palatini and Ashtekar gravity

TL;DR

This paper derives a BRST path integral formulation for Hilbert-Palatini gravity and demonstrates its relation to Ashtekar variables, addressing key issues like gauge compatibility and reality conditions in complex contour integration.

Contribution

It provides a novel derivation of the path integral for Hilbert-Palatini gravity and connects it to Ashtekar variables under specific gauge conditions.

Findings

Path integral for Hilbert-Palatini gravity derived.

Reformulation in terms of Ashtekar variables shown.

Ghost terms align with naive expectations under certain gauges.

Abstract

To write down a path integral for the Ashtekar gravity one must solve three fundamental problems. First, one must understand rules of complex contour functional integration with holomorphic action. Second, one should find which gauges are compatible with reality conditions. Third, one should evaluate the Faddeev-Popov determinant produced by these conditions. In the present paper we derive the BRST path integral for the Hilbert-Palatini gravity. We show, that for certain class of gauge conditions this path integral can be re-written in terms of the Ashtekar variables. Reality conditions define contours of integration. For our class of gauges all ghost terms coincide with what one could write naively just ignoring any Jacobian factors arising from the reality conditions.