Abstract
We describe recent progress in understanding the continuous symmetry properties of non-Hermitian, -symmetric quantum field theories. Focussing on a simple non-Hermitian theory composed of one complex scalar and one complex pseudoscalar, we revisit the derivation of Noether’s theorem to show that the conserved currents of non-Hermitian theories correspond to transformations that do not leave the Lagrangian invariant. We illustrate the impact that this has on the consistent formulation of (Abelian) gauge theories by studying a non-Hermitian extension of scalar quantum electrodynamics. We consider the spontaneous breakdown of both global and local symmetries, and describe how the Goldstone theorem and the Englert-Brout-Higgs mechanism are borne out for non-Hermitian, -symmetric theories.
Highlights
The standard lore of quantum mechanics is that operators corresponding to real-valued observables must be Hermitian
In the context of a complex scalar model, we show that there exist conserved currents for non-Hermitian theories, but the corresponding transformations do not leave the Lagrangian invariant [6]
We must couple to a non-conserved current, and, in the case of nonHermitian scalar quantum electrodynamics, the consistency of the Maxwell equations precludes, in general, our working in Lorenz gauge [8]
Summary
The standard lore of quantum mechanics is that operators corresponding to real-valued observables must be Hermitian. We consider the spontaneous breakdown of both global and local symmetries, and describe how the Goldstone theorem and the Englert-BroutHiggs mechanism are borne out for non-Hermitian, PT -symmetric theories. In the context of a complex scalar model, we show that there exist conserved currents for non-Hermitian theories, but the corresponding transformations do not leave the Lagrangian invariant [6].
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