We describe recent progress in understanding the symmetry properties of non-Hermitian, PT-symmetric quantum field theories. We start by revisiting the derivation of Noether’s theorem, showing that the conserved currents of non-Hermitian theories correspond to transformations that do not leave the Lagrangian invariant. The associated symmetry transformations instead yield families of equivalent PT-symmetric theories. These results are illustrated by means of two concrete examples: a theory of one complex scalar and one complex pseudo-scalar, and a non-Hermitian extension of quantum electrodynamics. After describing the construction of the path integral for non-Hermitian but PT-symmetric Lagrangians, we finish by considering the spontaneous breakdown of both global and local symmetries, and describe how the Goldstone theorem and Englert-Brout-Higgs mechanism are borne out.
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