Finiteness of quantum gravity with matter on a PL spacetime

Abstract

We study the convergence of the path integral for General Relativity with matter on a picewise linear (PL) spacetime that corresponds to a triangulation of a smooth manifold by using a path-integral measure that renders the pure gravity path integral finite. This measure depends on a parameter p, and in the case when the matter content is just scalar fields, we show that the path integral is absolutely convergent for p > 0,5 and not more than 2 scalar fields. In the case of Yang-Mills fields, we show that the path integral is absolutely convergent for the U(1) group and p > 0,5. In the case of Dirac fermions, we show that the path integral is absolutely convergent for any number of fermions and a sufficiently large p. When the matter content is given by scalars, Yang-Mills fields and fermions, as in the case of the Standard Model, we show that the path integral is absolutely convergent for p > 46,5. Hence one can construct a finite quantum gravity theory on a PL spacetime such that the classical limit is General Relativity coupled to the Standard Model.