On the symmetry of the vacuum in theories with spontaneous symmetry breaking

TL;DR

This paper clarifies misconceptions about spontaneous symmetry breaking, emphasizing that the vacuum generally retains the symmetries of the Lagrangian, and explores implications for mass generation and topological defect formation.

Contribution

It provides a detailed critique of common explanations of SSB, especially for global symmetries, and introduces a symmetric vacuum approach to mass generation and defect formation.

Findings

Vacuum generally shares the symmetries of the Lagrangian.

Symmetric vacuum can still lead to mass generation.

Implications for topological defect formation in the early universe.

Abstract

We review the usual account of the phenomena of spontaneous symmetry breaking (SSB), pointing out the common misunderstandings surrounding the issue, in particular within the context of quantum field theory. In fact, the common explanations one finds in this context, indicate that under certain conditions corresponding to the situation called SSB, the vacuum of the theory does not share the symmetries of the Lagrangian. We explain in detail why this statement is incorrect in general, and in what limited set of circumstances such situation could arise. We concentrate on the case of global symmetries, for which we found no satisfactory exposition in the existing literature, and briefly comment on the case of gauge symmetries where, although insufficiently publicized, accurate and complete descriptions exist. We briefly discuss the implications for the phenomenological manifestations usually attributed to the phenomena of spontaneous symmetry breaking, analyzing which might be affected by our analysis and which are not. In particular we describe the mass generation mechanism in a fully symmetric scheme (i.e., with a totally symmetric vacuum), and briefly discuss the implications of this analysis to the problem of formation of topological defects in the early universe.