Dyson's Instability in Lattice Gauge Theory

TL;DR

This paper investigates Dyson's instability argument within lattice gauge theory, analyzing the behavior of the partition function and plaquette at negative coupling, and introduces new methods to locate Fisher's zeros relevant for perturbation theory.

Contribution

It presents new techniques to locate Fisher's zeros in lattice gauge theories and discusses the implications of Dyson's instability for vacuum structure and perturbation theory.

Findings

Partition function remains well-defined at negative g^2 for compact gauge groups.

Discontinuity in the average plaquette indicates a change of vacuum, not instability.

New numerical methods successfully locate Fisher's zeros in SU(2) and U(1) gauge theories.

Abstract

We discuss Dyson's argument that the vacuum is unstable under a change g^2 -> - g^2, in the context of lattice gauge theory. For compact gauge groups, the partition function is well defined at negative g^2, but the average plaquette P has a discontinuity when g^2 changes sign. This reflects a change of vacuum rather than a loss of vacuum. In addition, P has poles in the complex g^2 plane, located at the complex zeros of the partition function (Fisher's zeros). We discuss the relevance of these singularities for lattice perturbation theory. We present new methods to locate Fisher's zeros using numerical values for the density of state in SU(2) and U(1) pure gauge theory. We briefly discuss similar issues for O(N) nonlinear sigma models where the local integrals are also over compact spaces.