## Summary of Local Baker-Wightman Invariance

for Euclidean Functional Field Equations^{*}

### David N. Williams

*Randall Laboratory of Physics*

The University of Michigan

Ann Arbor, MI 48109-1120

November 27, 1985

## 1 Introduction

Baker and Wightman^{1} introduced the
idea that the class of solutions of the functional field
equation may be enlarged, and the triviality problem for four
dimensions possibly avoided, by exploiting its invariance under
the choice of contours in the lattice version of the Euclidean
path space integration. We call this *Baker-Wightman
invariance*. They discussed a ferromagnetic and
antiferromagnetic mixture, and showed that unfortunately it
violated the cluster property.

These notes summarize our local extension of their idea, called
*local Baker-Wightman invariance*, which aims to restore
the cluster property. It was the starting point for an honors
thesis by Michael K.~Weiss,^{2} and was
used by the author to explain why, in an ultralocal model, the
continuum limit from the lattice was trivial, but the Euclidean
functional field equation nevertheless had a nontrivial solution
obeying the cluster property.^{3}

^{*}
March 13, 2008: The body of this document is an almost literal
transcription of the original manuscript, with the above date,
entitled “Notes on the Euclidean φ^{4}
functional field equation.” The table of contents was
added in April, 1994, and the introduction in March, 2008.
^{1}
G. A. Baker and A. S. Wightman, “Trying to Violate
Coupling Constant Bounds in φ_{ν}^{4}
Quantum Field Theory,” *in* Progress in Quantum Field
Theory, eds. H. Ezawa and S. Kamefuchi, (Elsevier Science
Publishers, 1986), pp. 15–29.

^{2}
Michael K. Weiss, “Standard and Nonstandard
φ^{4} Theories with An Introduction to Quantum Field
Theory,” honors thesis for the Bachelor of Science degree,
University of Michigan, April 15, 1994.

^{3}
David N. Williams, “Triviality and Nontriviality of
Ultralocal, Euclidean φ^{4},” unreleased
manuscript, 1985.

Back to webprints

Back to home page.