The Goldstone theorem and dynamical rearrangement of symmetry in the path-integral formalism

Abstract

The spontaneous breakdown of symmetry is investigated in the path-integral formalism. The generating functional is modified by the addition of a limiting “ϵ term” which fixes the direction of the breaking. The method is illustrated by means of several models with Abelian symmetries. The Goldstone theorem is a result of Ward-Takahashi identities in which the ϵ term plays a decisive role. The problem of how the original symmetry transformation of the Heisenberg fields manifests itself at the level of observable (physical) particles, expressed in terms of the in-field operators φ in ( χ), is investigated. It is proved that, if spontaneous breakdown is present, the in-field transformation φ in ( χ) → φ in ( χ) + α ( χ), with α (χ) satisfying the same field equation as φ in ( χ) (Boson transformation), is an invariant transformation (i.e. leaves the equation of motion of the Heisenberg fields unchanged). It is also found that, when spontaneous breaking occurs, the invariant transformations of the Heisenberg and in-fields are different (dynamical rearrangement of symmetry). The form of the invariant in-field transformation is established for several models.