The Immirzi parameter from an external scalar field

TL;DR

This paper treats the Immirzi parameter as a scalar field in Loop Quantum Gravity, exploring its Hamiltonian constraints, defining connections, and predicting a dynamical relaxation to a non-zero vacuum value.

Contribution

It introduces a minimally coupled scalar field version of the Immirzi parameter and analyzes its implications in Loop Quantum Gravity without the time gauge.

Findings

Proper SU(2) connections can be defined with derivative terms of the scalar field.

Boost degrees of freedom are non-dynamical in this framework.

The model predicts a dynamical relaxation to a non-zero vacuum expectation value.

Abstract

We promote the Immirzi parameter to be a minimally coupled scalar field and we analyzed the Hamiltonian constraints in the framework of Loop Quantum Gravity without the time gauge. Proper SU(2) connections can be defined and a term containing derivatives of the field $\beta$ enters into their definition. Furthermore, boost degrees of freedom are non-dynamical, while the super-momentum constraints coincide with the scalar field case. Hence, the kinematical Hilbert space can be defined as for gravity in presence of a minimally coupled scalar field. Then, we analyzed the dynamical implications of this scenario and we outlined how a the dynamical relaxation to a non-vanishing vacuum expectation value is predicted.