Scalar curvature operator for models of loop quantum gravity on a cubical graph

TL;DR

This paper introduces a new scalar curvature operator for loop quantum gravity, defined on fixed cubical graphs, enabling applications in quantum-reduced models and effective dynamics.

Contribution

The work presents a novel discretized scalar curvature operator for loop quantum gravity on cubical graphs, bridging classical Ricci scalar expressions with quantum operators.

Findings

Operator is well-defined on fixed cubical graph Hilbert space

Enables application in quantum-reduced loop gravity

Facilitates effective dynamics studies in LQG

Abstract

In this article we introduce a new operator representing the three-dimensional scalar curvature in loop quantum gravity. Our construction does not apply to the entire kinematical Hilbert space of loop quantum gravity; instead, the operator is defined on the Hilbert space of a fixed cubical graph. The starting point of our work is to write the spatial Ricci scalar classically as a function of the densitized triad and its SU(2)-covariant derivatives. We pass from the classical expression to a quantum operator through a regularization procedure, in which covariant derivatives of the triad are discretized as finite differences of gauge covariant flux variables on the cubical lattice provided by the graph. While more work is needed in order to extend our construction to encompass states based on all possible graphs, the operator presented here can be applied in models such as quantum-reduced loop gravity and effective dynamics, which are derived from the kinematical framework of full loop quantum gravity, and are formulated in terms of states defined on cubical graphs.