Impossibility of spontaneously breaking local symmetries and the sign problem

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.