Impossibility of spontaneously breaking local symmetries and the sign problem

TL;DR

This paper reexamines Elitzur's theorem on the impossibility of spontaneous local symmetry breaking, especially in gauge theories with non-positive Euclidean measures, and provides a general criterion for symmetry breaking.

Contribution

It demonstrates that Elitzur's theorem does not hold solely based on gauge invariance and introduces a criterion applicable to theories with non-positive measures.

Findings

Elitzur's theorem relies on positivity of the Euclidean measure.

Spontaneous breaking of local symmetries can occur if measure positivity is violated.

Theoretical criterion for symmetry breaking in gauge theories is established.

Abstract

Elitzur's theorem stating the impossibility of spontaneous breaking of local symmetries in a gauge theory is reexamined. The existing proofs of this theorem rely on gauge invariance as well as positivity of the weight in the Euclidean partition function. We examine the validity of Elitzur's theorem in gauge theories for which the Euclidean measure of the partition function is not positive definite. We find that Elitzur's theorem does not follow from gauge invariance alone. We formulate a general criterion under which spontaneous breaking of local symmetries in a gauge theory is excluded. Finally we illustrate the results in an exactly solvable two dimensional abelian gauge theory.