Matter in Loop Quantum Gravity without time gauge: a non-minimally coupled scalar field

TL;DR

This paper extends loop quantum gravity to include a non-minimally coupled scalar field without fixing the time gauge, showing that key structures like the SU(2) gauge symmetry and area spectrum are preserved.

Contribution

It demonstrates the extension of loop quantum gravity formalism to non-minimally coupled scalar fields without time gauge fixing, maintaining core quantum geometric features.

Findings

SU(2) gauge structure is recovered without time gauge.

The super-momentum and super-Hamiltonian match those of minimally coupled scalar fields.

Area spectrum differences are consistent with the time-gauge approach.

Abstract

We analyze the phase space of gravity non-minimally coupled to a scalar field in a generic local Lorentz frame. We reduce the set of constraints to a first-class one by fixing a specific hypersurfaces in the phase space. The main issue of our analysis is to extend the features of the vacuum case to the presence of scalar matter by recovering the emergence of an SU(2) gauge structure and the non-dynamical role of boost variables. Within this scheme, the super-momentum and the super-Hamiltonian are those ones associated with a scalar field minimally coupled to the metric in the Einstein frame. Hence, the kinematical Hilbert space is defined as in canonical Loop Quantum Gravity with a scalar field, but the differences in the area spectrum are outlined to be the same as in the time-gauge approach.