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

F. Cianfrani^1(a), A. Marchini² and G. Montani^2,3,4(b)

¹ Institute for Quantum Gravity (IQG), FAU Erlangen-Nürnberg - Staudtstr. 7, 91058 Erlangen, Germany, EU
² Dipartimento di Fisica, Università di Roma “Sapienza” - Piazzale Aldo Moro 5, 00185 Roma, Italy, EU
³ ENEA, Centro Ricerche Frascati, U.T. Fus. (Fus. Mag. Lab.) - Via Enrico Fermi 45, 00044 Frascati, Roma, Italy, EU
⁴ INFN, Sezione Roma1 - Piazzale Aldo Moro 5, 00185 Roma, Italy, EU

^(a) francesco.cianfrani@gravity.fau.de
^(b) giovanni.montani@frascati.enea.it

Received: 12 January 2012
Accepted: 13 June 2012

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. In 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.