Asymptotic Quantization of Palatini Action

TL;DR

This paper explores the quantization of the Palatini action in general relativity, focusing on the Gauss law, gauge transformations, and the structure of theta vacua within a quantum framework.

Contribution

It introduces a novel approach to quantizing the Palatini action using a spacetime split and analyzes the algebraic structure of gauge operators and vacua.

Findings

Gauss law treated on a Hilbert space with complex SL(2,C) connections

Explicit representation of theta vacua and spin-isospin mixing

Gauss law algebra replaces diffeomorphism algebra in this framework

Abstract

The Palatini action is based on vector-valued one forms or frames and SL(2,C) connections on R^4. Using the spacetime split of R^4 as a direct sum of R^3 and R^1, the Gauss law in this paper is treated on a Hilbert space. This is achieved by noting that quantum operators act on a complex Hilbert space and SL(2,C) is just the complexification of the compact SU(2) in the self-dual (1/2,0) representation used for the Ashtekar variables. This observation enables a treatment of small and large gauge transformations and superselection sectors. An explicit representation of theta vacua and their attendant 'spin-isospin mixing' are also shown. It is argued that the Gauss law algebra replaces that of diffeomorphisms in the Palatini approach : operators implementing the latter with the correct algebraic relations do not seem available. (Those obtained by multiplying Gauss law operators with fields do not have the correct commutators.)