Finiteness of quantum gravity with matter on a PL spacetime

We study the convergence of the path integral (PI) for general relativity with matter on a picewise linear (PL) spacetime that corresponds to a triangulation of a smooth manifold by using a PI measure that renders the pure gravity PI finite. This measure depends on a parameter p, and in the case when the matter content is just scalar fields, we show that the PI is absolutely convergent for p > 0,5 and not more than two scalar fields. In the case of Yang-Mills (YM) fields, we show that the PI is absolutely convergent for the U(1) group and p > 0,5. In the case of Dirac fermions, we show that the PI is absolutely convergent for any number of fermions and a sufficiently large p. When the matter content is given by scalars, YM fields and fermions, as in the case of the standard model (SM), we show that the PI is absolutely convergent for p > 52,5. Hence one can construct a finite quantum gravity theory on a PL spacetime such that the classical limit is general relativity coupled to the SM.