A version of the connection representation of Regge action

Abstract

We define for any 4-tetrahedron (4-simplex) the simplest finite closed piecewise flat manifold consisting of this 4-tetrahedron and of one other 4-tetrahedron identical up to reflection to the present one (call it a bisimplex built on the given 4-simplex, or a two-sided 4-simplex). We consider an arbitrary piecewise flat manifold. The gravity action for it can be expressed in terms of the sum of the actions for the bisimplices built on the 4-simplices constituting this manifold. We use a representation of each bisimplex action in terms of rotation matrices (connections) and area tensors. This gives some representation of any piecewise flat gravity action in terms of connections. The action is a sum of terms each depending on the connection variables referring to a single 4-tetrahedron. Application of this representation to the path integral formalism is considered. Integrations over connections in the path integral reduce to independent integrations over finite sets of connections on separate 4-simplices. One of the consequences is exponential suppression of the result at large areas or lengths (compared to the Plank scale). This is important for the consistency of the simplicial description of spacetime.