Topological Interpretation of Barbero-Immirzi Parameter

TL;DR

This paper provides a topological interpretation of the Barbero-Immirzi parameter in loop quantum gravity by formulating a Hamiltonian theory using the Nieh-Yan density, linking the parameter to a topological invariant.

Contribution

It introduces a Hamiltonian formulation of gravity with the Nieh-Yan density, establishing a topological origin for the Barbero-Immirzi parameter.

Findings

Identifies the Barbero-Immirzi parameter as the inverse of the coefficient of the Nieh-Yan density.

Derives a real SU(2) connection formulation of gravity consistent with previous approaches.

Shows matter coupling does not alter the topological nature of the Nieh-Yan term.

Abstract

We set up a canonical Hamiltonian formulation for a theory of gravity based on a Lagrangian density made up of the Hilbert-Palatini term and, instead of the Holst term, the Nieh-Yan topological density. The resulting set of constraints in the time gauge are shown to lead to a theory in terms of a real SU(2) connection which is exactly the same as that of Barbero and Immirzi with the coefficient of the Nieh-Yan term identified as the inverse of Barbero-Immirzi parameter. This provides a topological interpretation for this parameter. Matter coupling can then be introduced in the usual manner, {\em without} changing the universal topological Nieh-Yan term.