The work presented here explores the possibility of introducing non-Hermitian scalar and fermion mass terms, in addition to the usual Hermitian ones. The consistency of the resulting description requires a reworking of all the fundamental properties in Field Theory from the beginning, which is challenging but appears to be possible.

Motivated by extending the parameter space of the Standard Model of Particle Physics, I will describe how to understand the effects of an imaginary mixing mass term in a non-Hermitian but PT-symmetric extension of scalar QED. The classical theory is already not trivial, and requires a new interpretation of the equations of motion, Noether's theorem and gauge invariance. The path integral can be defined with appropriate field variables, and a consistent picture emerges, opening the way for potential alternative descriptions of the Higgs sector.

A specific configuration of the bosonic string is discussed, in time-dependent backgrounds, which satisfies Weyl invariance to all orders in the alpha'-expansion. Conformal invariance, valid in any space time dimension, is based on the homogeneity (in time) of the Weyl-beta functions which, using field redefinitions, can be made to vanish.