Paul Federbush

Partially Alternate Derivation of a Result of Nelson

The result of Nelson that the total Hamiltonian is semibounded for a self‐interacting Boson field in two dimensions in a periodic box is derived by an alternate method. It is more elementary in so far as functional integration is not used.

A phase cell cluster expansion for Euclidean field theories

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 4; 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 4: 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.

A Mass Zero Cluster Expansion. Part 1. The Expansion

AbstractA cluster expansion is developed and applied to study the perturbation λ(Δφ)4 of the massless lattice field φ in dimension 3. The method is loosely inspired by the work of Gawedzki and Kupiainen on block spin techniques for the

A Lower Bound for the Mass of a Random Gaussian Lattice

\lambda (\mathop \nabla \limits^ \to \phi )^4

Ondelettes and phase cell cluster expansions, a vindication

system. The cluster expansion is given in terms of expansion coefficients for the field as a sum of certain special block spin functions. These functions are chosen with a large number of moments zero, so that the interaction couples spatially separated functions with an interaction falling off as a high inverse power of the separation distance. The present techniques, with some technical development, should work for broad classes of other models, including the lattice dipole gas and the

A Phase Cell Approach to Yang-Mills Theory III. Local Stability, Modified Renormalization Group Transformation

\lambda (\mathop \nabla \limits^ \to \phi )^4