Symmetries and conservation laws in non-Hermitian field theories

Anti-Hermitian mass terms are considered, in addition to Hermitian ones, for $\mathcal{P}\mathcal{T}$-symmetric complex-scalar and fermionic field theories. In both cases, the Lagrangian can be written in a manifestly symmetric form in terms of the $\mathcal{P}\mathcal{T}$-conjugate variables, allowing for an unambiguous definition of the equations of motion. After discussing the resulting constraints on the consistency of the variational procedure, we show that the invariance of a non-Hermitian Lagrangian under a continuous symmetry transformation does not imply the existence of a corresponding conserved current. Conserved currents exist, but these are associated with transformations under which the Lagrangian is not invariant and which reflect the well-known interpretation of $\mathcal{P}\mathcal{T}$-symmetric theories in terms of systems with gain and loss. A formal understanding of this unusual feature of non-Hermitian theories requires a careful treatment of Noether’s theorem, and we give specific examples for illustration.