Scalar curvature operator for loop quantum gravity on a cubical graph

Abstract

We introduce a new operator representing the three-dimensional scalar curvature in loop quantum gravity. The operator is constructed by writing the Ricci scalar classically as a function of the Ashtekar variables and regularizing the resulting expression on a cubical spin network graph. While our construction does not apply to the entire Hilbert space of loop quantum gravity, the proposed operator can be applied to concrete calculations in various approaches which are derived from the framework of full loop quantum gravity using states defined on cubical graphs.