Discrete spacetime symmetries, second quantization, and inner products in a non-Hermitian Dirac fermionic field theory

We extend to a non-Hermitian fermionic quantum field theory with <math display="inline"><mi mathvariant="script">PT</mi></math> symmetry our previous discussion of second quantization, discrete symmetry transformations, and inner products in a scalar field theory [Phys. Rev. D 102, 125030 (2020)]. For illustration, we consider a prototype model containing a single Dirac fermion with a parity-odd, anti-Hermitian mass term. In the phase of unbroken <math display="inline"><mi mathvariant="script">PT</mi></math> symmetry, this Dirac fermion model is equivalent to a Hermitian theory under a similarity transformation, with the non-Hermitian nature of the model residing only in the spinor structure, whereas the algebra of the creation and annihilation operators is just that of a Hermitian theory.