We demonstrate the extension to parity-time ($\mathcal{PT}$)-symmetric field theories of the Goldstone theorem, confirming that the spontaneous appearance of a field vacuum expectation value via minimization 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 quantization of the path-integral formulation of quantum field theory can be extended consistently to a non-Hermitian model by considering $\mathcal{PT}$ conjugation instead of Hermitian conjugation. The extension of the Goldstone theorem to a $\mathcal{PT}$-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 $\mathcal{PT}$-symmetric theory, we exhibit a specific example in which we verify the existence of a massless boson at the tree and one-loop levels.

• Received 29 May 2018

DOI:https://doi.org/10.1103/PhysRevD.98.045001

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Published by the American Physical Society