SU(2)-invariant reduction of the 3+1 dimensional Ashtekar's gravity

TL;DR

This paper performs a non-perturbative path integral quantization of SU(2)-invariant reductions of 3+1 dimensional Ashtekar's gravity, providing exact calculations of Green functions in a finite-dimensional phase space.

Contribution

It introduces a novel SU(2)-invariant reduction of Ashtekar's gravity and computes the generating functional exactly, challenging previous assumptions about the path integral measure.

Findings

Path integral measure is non-singular near configurations with additional Killing vectors.

Exact calculation of Green functions in the reduced phase space.

Demonstrates a finite-dimensional phase space for the model.

Abstract

We consider a space-time with spatial sections isomorphic to the group manifold of SU(2). Triad and connection fluctuations are assumed to be SU(2)-invariant. Thus, they form a finite dimensional phase space. We perform non-perturbative path integral quantization of the model. Contarary to previous claims the path integral measure appeared to be non-singular near configurations admitting additional Killing vectors. In this model we are able to calculate the generating functional of Green functions of the reduced phase space variables exactly.