Curvatures and discrete Gauss-Codazzi equation in (2+1)-dimensional loop quantum gravity

Abstract

We derive the Gauss-Codazzi equation in the holonomy and plane-angle representations and we use the result to write a Gauss-Codazzi equation for a discrete (2+1)-dimensional manifold, triangulated by isosceles tetrahedra. This allows us to write operators acting on spin network states in (2+1)-dimensional loop quantum gravity, representing the 3-dimensional intrinsic, 2-dimensional intrinsic, and 2-dimensional extrinsic curvatures.

16 pages, 10 figures

References in corpus (5)

• The Ponzano-Regge model
• Area-angle variables for general relativity
• On the geometry of loop quantum gravity on a graph
• Regge calculus from a new angle
• From covariant to canonical formulations of discrete gravity