Scalar curvature operator for quantum-reduced loop gravity

Abstract

In a previous article we have introduced an operator representing the three-dimensional scalar curvature in loop quantum gravity. In this article we examine the new curvature operator in the setting of quantum-reduced loop gravity. We derive the explicit form of the curvature operator as an operator on the Hilbert space of the quantum-reduced model. As a simple practical example, we study the expectation values of the operator with respect to basis states of the reduced Hilbert space.

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

“…a flat spatial geometry). If such calculations confirm that the expectation values of our operator are distributed too strongly towards the negative side, we expect that the problem could be resolved through a simple modification of the regularization of second covariant derivatives of the triad, as outlined in [18].…”

“…In the continuation article [18] calculations were performed to probe the properties of the new curvature operator on the Hilbert space of quantumreduced loop gravity. Our results indicate that the expectation values of curvature are consistently negative in certain states where one would intuitively think that neither sign of curvature should be clearly preferred over the other.…”

“…The operator R v was studied in [18] in the simplified kinematical setting of quantum-reduced loop gravity. In particular, we looked at expectation values of curvature in the standard basis states on the Hilbert space of the quantum-reduced model.…”