Barbero-Immirzi parameter as a solution of the simplicity constraints

TL;DR

This paper derives the Barbero-Immirzi parameter values as solutions to simplicity constraints, revealing that for certain values, the associated Lorentz group representations become finite dimensional and non-unitary.

Contribution

It presents a novel approach where the Barbero-Immirzi parameter is determined by the simplicity constraints rather than being arbitrarily set.

Findings

For b3 = \u00b1i, simplicity constraints imply finite-dimensional, non-unitary Lorentz representations.

The approach links the Barbero-Immirzi parameter to fundamental representation properties.

Provides a new perspective on the role of the Barbero-Immirzi parameter in quantum gravity.

Abstract

In this paper we naturally obtain the values of the Barbero-Immirzi parameter as the solution of the simplicity constraints rather than setting it a priori. Particularly the Main theorem shows that if $\gamma = \pm i$ then the simplicity constraints require that the corresponding Lorentz group representations be necessary finite dimensional and therefore non-unitary.