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.
Highlights
The conventional formulations of quantum mechanics and quantum field theory (QFT) have generally been based on Hermitian Hamiltonians and Lagrangians, respectively
We demonstrate the extension to parity-time (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
We prove an extension of the Goldstone theorem for this non-Hermitian QFT, showing that the spontaneous appearance of a field vacuum expectation value via minimization of the effective potential is accompanied by the appearance of a massless scalar mode, whose existence is linked to the presence of a conserved current in this PT -symmetric QFT
Summary
The conventional formulations of quantum mechanics and quantum field theory (QFT) have generally been based on Hermitian Hamiltonians and Lagrangians, respectively. Revisiting Noether’s derivation, one finds that there exist conserved currents for non-Hermitiam theories, but these correspond to transformations that must effect a particular nontrivial variation of the Lagrangian, which vanishes only in the Hermitian limit This observation raises the interesting question of whether there is an analogue in PT -symmetric QFT of spontaneous symmetry breaking and, if so, whether the breaking of a global symmetry is accompanied by a massless Goldstone mode, 2470-0010=2018=98(4)=045001(11). Before addressing these questions, we first discuss some basic issues in the formulation of a nonHermitian QFT, which require a consistent procedure for quantization of the path integral This is based on the existence of a complete set of real energy eigenstates, which allow the introduction of a saddle point about which the integration of quantum fluctuations is well defined.
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