PROBABILITY-DISTRIBUTION FUNCTIONAL FOR EQUAL-TIME CORRELATION-FUNCTIONS IN CURVED SPACE

We present a systematic method to calculate the probability distributi on functional (PDF) for spatial configuration of an interacting field in curved space-time. As an example, we consider PDF for the minimally coupled massive lambda phi(4) theory up to the first order of the cou pling constant and evaluate it both in Minkowski and de Sitter spaceti mes. We observe that PDF has an ultraviolet divergence even after the ultraviolet renormalization. This divergence is unavoidable to reprodu ce finite expectation values; thus some kind of regularization is nece ssary to write down PDF. As an application of it, a scaling law among multipoint correlation functions in the de Sitter space is found.