DEFINING INTEGRALS OVER CONNECTIONS IN THE DISCRETIZED GRAVITATIONAL FUNCTIONAL INTEGRALS

Abstract

Integration over connection type variables in the path integral for the discrete form of the first-order formulation of general relativity theory is studied. The result (a generalized function of the rest of variables of the type of tetrad or elementary areas) can be defined through its moments, i.e. integrals of it with the area tensor monomials. In our previous paper these moments have been defined by deforming integration contours in the complex plane as if we had passed to a Euclidean-like region. In this paper we define and evaluate the moments in the genuine Minkowski region. The distribution of interest resulting from these moments in this non-positively defined region contains the divergences. We prove that the latter contribute only to the singular (δ-function like) part of this distribution with support in the non-physical region of the complex plane of area tensors while in the physical region this distribution (usual function) confirms that defined in our previous paper which decays exponentially at large areas. Besides that, we evaluate the basic integrals over which the integral over connections in the general path integral can be expanded.