Anomaly free quantum dynamics for Euclidean LQG

TL;DR

This paper demonstrates that a proposed quantum dynamics for Euclidean Loop Quantum Gravity, centered on the Electric Shift operator, is free of anomalies and correctly reproduces classical symmetries at the quantum level.

Contribution

It proves the anomaly freedom of a specific quantum Hamiltonian constraint in Euclidean LQG, ensuring correct classical limit and symmetry representation.

Findings

Commutator of Hamiltonian constraints mirrors classical Poisson bracket.

Finite spatial diffeomorphisms are faithfully represented.

Hamiltonian constraint is diffeomorphism covariant.

Abstract

Classical gravitational evolution admits an elegant and compact re-expression in terms of gauge covariant generalizations of Lie derivatives with respect to a spatial phase space dependent $su(2)$ valued vector field called the Electric Shift [1]. A quantum dynamics for Euclidean Loop Quantum Gravity which ascribes a central role to the Electric Shift operator is derived in [2]. $Here\; we\; show\; that\; this \;quantum\; dynamics\; is\; nontrivially\; anomaly$ $free$. Specifically, we show that on a suitable space of off shell states (a) the (non-vanishing) commutator between a pair of Hamiltonian constraint operators mirrors the Poisson bracket between their classical correspondents, (b) the group of finite spatial diffeomorphisms is faithfully represented and (c) the action of the Hamiltonian constraint operator is diffeomorphism covariant with respect to the action of spatial diffeomorphisms.