  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.
