Corrections to Wigner-Eckart Relations by Spontaneous Symmetry Breaking

TL;DR

This paper investigates how spontaneous symmetry breaking modifies the Wigner-Eckart relations in quantum systems, showing that corrections are linked to Ward identities and involve Goldstone bosons, with implications for systems with explicit symmetry breaking.

Contribution

It extends the understanding of Wigner-Eckart relations to systems with spontaneous and explicit symmetry breaking, connecting corrections to Ward identities and Goldstone bosons.

Findings

Corrections are given by symmetry breaking Ward identities.

In spontaneously broken systems, corrections involve Goldstone bosons.

Explicit breaking introduces pseudo Goldstone bosons into the corrections.

Abstract

The matrix elements of operators transforming as irreducible representations of an unbroken symmetry group $G$ are governed by the well-known Wigner-Eckart relations. In the case of infinitely-extended systems, with $G$ spontaneously broken, we prove that the corrections to such relations are provided by symmetry breaking Ward identities, and simply reduce to a tadpole term involving Goldstone bosons. The analysis extends to the case in which an explicit symmetry breaking term is present in the Hamiltonian, with the tadpole term now involving pseudo Goldstone bosons. An explicit example is discussed, illustrating the two cases.