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Abstract
We study the interplay between spontaneously breaking global continuous and discrete antilinear symmetries in a newly proposed general class of non-Hermitian quantum field theories containing a mixture of complex and real scalar fields. We analyse the model for different types of global symmetry preserving and breaking vacua. In addition, the models are symmetric under various types of discrete antilinear symmetries composed out of nonstandard simultaneous charge conjugations, time-reversals and parity transformations; CPT. While the global symmetry governs the existence of massless Goldstone bosons, the discrete one controls the precise expression of the Goldstone bosons in terms of the original fields in the model and its physical regimes. We show that even when the CPT-symmetries are broken on the level of the action expanded around different types of vacua, the mass spectra might still be real when the symmetry is preserved at the tree approximation and the breaking only occurs at higher order. We discuss the parameter space of some of the models in the proposed class and identify physical regimes in which massless Goldstone bosons emerge when the vacuum spontaneously breaks the global symmetry or equivalently when the corresponding Noether currents are conserved. The physical regions are bounded by exceptional points in different ways. There exist special points in parameter space for which massless bosons may occur already before breaking the global symmetry. However, when the global symmetry is broken at these points they can no longer be distinguished from genuine Goldstone bosons.
Highlights
It is quite well understood how to extend the conventional framework of Hermitian classical and quantum mechanics [1, 2, 3] to allow for the inclusion of non-Hermitian systems
Based on the formal analogy between the Schrodinger equation and the propagation of light in the paraxial approximation described by the Helmholtz equation many of the findings obtained in the quantum mechanical description have been confirmed experimentally and further developed in classical optical settings with the refractive index playing the role of a complex potential [5, 6, 7, 8, 9]
We find some vacua that break the CPT -symmetries on the level of the action, but still possess physically meaningful mass spectra, as the symmetry breaking occurs at higher order couplings than at the tree approximation
Summary
It is quite well understood how to extend the conventional framework of Hermitian classical and quantum mechanics [1, 2, 3] to allow for the inclusion of non-Hermitian systems When the latter systems admit an antilinear symmetry [4], such as for instance being invariant under a simultaneous reflection in time and space, referred to as PT -symmetry, this can be achieved in a self-consistent manner. We distinguish here between a standard exceptional point, corresponding to two nonzero eigenvalues coalescing, and a zero-exceptional point defined as the point when a zero eigenvalue coalesces with a nonzero eigenvalue The problem that both groups have tried to overcome at first is the feature that the equations of motion obtained from functionally varying the action with respect to the scalar fields on one hand and on the other separately with respect to its complex conjugate field are not compatible.
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