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

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. We pass from the classical expression to a quantum operator through a regularization procedure, in which spatial derivatives of the triad are discretized as finite differences 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.

Scalar curvature operator for quantum-reduced loop gravity

“…However, if this procedure is applied to the curvature operator of [13], one finds that the action of the resulting operator is trivially vanishing on the reduced Hilbert space. In contrast, the new operator of [4] does give rise to a non-trivial curvature operator for the quantum-reduced model.…”

“…In a previous article [4], we have introduced a new operator representing the scalar curvature in loop quantum gravity. The operator is constructed within the kinematical framework of loop quantum gravity, but its definition is limited to the Hilbert space of states based on a fixed cubical graph.…”

“…In the context of quantum-reduced loop gravity, the new curvature operator of [4] represents a definite improvement over the operator introduced earlier in [13], where the basic ideas of Regge calculus were used to quantize the Ricci scalar in terms of lengths and angles. In general, operators for quantum-reduced loop gravity can be derived from the corresponding operators of full loop quantum gravity by applying the operators of the full theory on states in the Hilbert space of the quantum-reduced model and discarding certain small terms in the resulting expressions [14].…”

“…In this article we study the curvature operator introduced in [4] in the framework of quantum-reduced loop gravity. The main part of the work consists of performing the calculations required to establish the form of the curvature operator as an operator on the Hilbert space of the quantum-reduced model.…”

“…the reduced Hilbert space and the elementary operators thereon. In section 3 we review the definition of the curvature operator given in our earlier article [4]. In section 4, the form of the curvature operator as an operator of quantum-reduced loop gravity is derived by computing the action of the operator on states in the reduced Hilbert space.…”