Quantum-reduced loop gravity: New perspectives on the kinematics and dynamics

Abstract

We present a systematic approach to the kinematics of quantum-reduced loop gravity, a model originally proposed by Alesci and Cianfrani as an attempt to probe the physical implications of loop quantum gravity. We implement the quantum gauge-fixing procedure underlying quantum-reduced loop gravity by introducing a master constraint operator on the kinematical Hilbert space of loop quantum gravity, representing a set of gauge conditions which classically constrain the densitized triad to be diagonal. The standard Hilbert space of quantum-reduced loop gravity can be recovered as a space of solutions of the master constraint operator, while on the other hand the master constraint approach provides a useful starting point for considering possible generalizations of the standard construction. We also examine the quantum dynamics of states consisting of a single six-valent node in the quantum-reduced framework. We find that the Hamiltonian which governs the dynamics of such states bears a close formal resemblance to the Hamiltonian constraint of Bianchi I models in loop quantum cosmology.