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 Wightman1 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 φ4 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 φν4 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 φ4 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 φ4,” unreleased manuscript, 1985.

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