A phase cell approach to Yang-Mills theory

Abstract

In this paper the basic local stability result is obtained, in a form valid in both small field and large field regions. To achieve this, some modifications are made in both the action and the renormalization group transformation. Though there is some sacrifice of elegance in these modifications, the establishment of this local stability estimate yields the most basic ingredient of the phase cell cluster expansion, good estimates for all the actions.Incidental to the estimates of this paper we establish some results on “lattice geometry,” interesting in their own right. A bound on the “minimum area” of a loop of lengthl, ind dimensions, is obtained asl 2/8(1−1/d). This, a best possible bound, was obtained for us by A. Blass. We also construct a “radial” maximal tree for the lattice ind dimensions. We hope to stimulate someone to find a better construction of “radial” trees.