SO(4,C)-covariant Ashtekar-Barbero gravity and the Immirzi parameter

TL;DR

This paper develops an SO(4,C)-covariant Hamiltonian formulation of generalized gravity including the Immirzi parameter, explores its Dirac quantization, and investigates the parameter's influence on quantum theory, finding no dependence in the path integral approach.

Contribution

It introduces an SO(4,C)-covariant formalism for generalized gravity with the Immirzi parameter and analyzes its quantization and parameter dependence.

Findings

Path integral quantization shows no dependence on the Immirzi parameter.

The formalism encompasses Ashtekar-Barbero gravity used in loop quantum gravity.

A potential modification of the loop approach consistent with this formalism is proposed.

Abstract

An so(4,C)-covariant hamiltonian formulation of a family of generalized Hilbert-Palatini actions depending on a parameter (the so called Immirzi parameter) is developed. It encompasses the Ashtekar-Barbero gravity which serves as a basis of quantum loop gravity. Dirac quantization of this system is constructed. Next we study dependence of the quantum system on the Immirzi parameter. The path integral quantization shows no dependence on it. A way to modify the loop approach in the accordance with the formalism developed here is briefly outlined.