Spontaneous symmetry breaking and the Goldstone theorem in non-Hermitian field theories

TL;DR

This paper extends the Goldstone theorem to PT-symmetric non-Hermitian quantum field theories, showing that spontaneous symmetry breaking still results in a massless boson, supported by a specific example at multiple loop levels.

Contribution

It demonstrates the extension of the Goldstone theorem to non-Hermitian PT-symmetric field theories and shows how to consistently quantize such models using PT conjugation.

Findings

Spontaneous symmetry breaking in PT-symmetric models produces a massless boson.

The path-integral quantization can be extended to non-Hermitian models with PT symmetry.

Verification of a massless boson at tree and one-loop levels in a specific example.

Abstract

We demonstrate the extension to PT-symmetric field theories of the Goldstone theorem, confirming that the spontaneous appearance of a field vacuum expectation value via minimisation of the effective potential in a non-Hermitian model is accompanied by a massless scalar boson. Laying a basis for our analysis, we first show how the conventional quantisation of the path-integral formulation of quantum field theory can be extended consistently to a non-Hermitian model by considering PT conjugation instead of Hermitian conjugation. The extension of the Goldstone theorem to a PT-symmetric field theory is made possible by the existence of a conserved current that does not, however, correspond to a symmetry of the non-Hermitian Lagrangian. In addition to extending the proof of the Goldstone theorem to a PT-symmetric theory, we exhibit a specific example in which we verify the existence of a massless boson at the tree and one-loop levels.