The picture of the Bianchi I model via gauge fixing in Loop Quantum Gravity

TL;DR

This paper investigates the gauge fixing and quantization of the Bianchi I model in Loop Quantum Cosmology, revealing how the holonomy-flux algebra aligns with Loop Quantum Gravity and proposing a new super-Hamiltonian regularization approach.

Contribution

It introduces a gauge fixing method that simplifies the quantization of the Bianchi I model by restricting paths and maintains local diffeomorphism invariance, offering a novel perspective on super-Hamiltonian regularization.

Findings

Holonomy-flux algebra matches Loop Quantum Gravity when paths are aligned with fiducial vectors.

Homogeneity emerges from the imposition of a relic diffeomorphism constraint.

A new perspective on super-Hamiltonian regularization is proposed.

Abstract

The implications of the SU(2) gauge fixing associated with the choice of invariant triads in Loop Quantum Cosmology are discussed for a Bianchi I model. In particular, via the analysis of Dirac brackets, it is outlined how the holonomy-flux algebra coincides with the one of Loop Quantum Gravity if paths are parallel to fiducial vectors only. This way the quantization procedure for the Bianchi I model is performed by applying the techniques developed in Loop Quantum Gravity but restricting the admissible paths. Furthermore, the local character retained by the reduced variables provides a relic diffeomorphisms constraint, whose imposition implies homogeneity on a quantum level. The resulting picture for the fundamental spatial manifold is that of a cubical knot with attached SU(2) irreducible representations. The discretization of geometric operators is outlined and a new perspective for the super-Hamiltonian regularization in Loop Quantum Cosmology is proposed.