A phase cell cluster expansion for Euclidean field theories

Abstract

We adapt the cluster expansion first used to treat infrared problems for lattice models (a mass zero cluster expansion) to the usual field theory situation. The field is expanded in terms of special block spin functions and the cluster expansion given in terms of the expansion coefficients (phase cell variables); the cluster expansion expresses correlation functions in terms of contributions from finite coupled subsets of these variables. Most of the present work is carried through in d space time dimensions (for [phi]24 the details of the cluster expansion are pursued and convergence is proven). Thus most of the results in the present work will apply to a treatment of [phi]34 to which we hope to return in a succeeding paper. Of particular interest in this paper is a substitute for the stability of the vacuum bound appropriate to this cluster expansion (for d = 2 and D = 3), and a new method for performing estimates with tree graphs. The phase cell cluster expansions have the renormalization group incorporated intimately into their structure. We hope they will be useful ultimately in treating four dimensional field theories.