Quantum realizations of Hilbert-Palatini second-class constraints

TL;DR

This paper develops a novel quantization approach for Hilbert-Palatini gravity that allows a rescaling of the wavefunctional, capturing the topological role of the Barbero-Immirzi parameter in quantum gravity.

Contribution

It introduces a new method to handle second-class constraints in Hilbert-Palatini theory, enabling a topological interpretation of the Barbero-Immirzi parameter in quantum gravity.

Findings

A general rescaling procedure for the wavefunctional is established.

The approach applies to gravity with or without matter.

It works for any gauge choice, including the time gauge.

Abstract

In a classical theory of gravity, the Barbero-Immirzi parameter ($\eta$) appears as a topological coupling constant through the Lagrangian density containing the Hilbert-Palatini term and the Nieh-Yan invariant. In a quantum framework, the topological interpretation of $\eta$ can be captured through a rescaling of the wavefunctional representing the Hilbert-Palatini theory, as in the case of the QCD vacuum angle. However, such a rescaling cannot be realized for pure gravity within the standard (Dirac) quantization procedure where the second-class constraints of Hilbert-Palatini theory are eliminated beforehand. Here we present a different treatment of the Hilbert-Palatini second-class constraints in order to set up a general rescaling procedure (a) for gravity with or without matter and (b) for any choice of gauge (e.g. time gauge). The analysis is developed using the Gupta-Bleuler and the coherent state quantization methods.