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

TL;DR

This paper explores a non-Hermitian Dirac fermionic quantum field theory with PT symmetry, demonstrating its equivalence to a Hermitian theory in the unbroken PT phase through similarity transformations.

Contribution

It extends previous scalar field theory results to fermionic fields, analyzing discrete symmetries, second quantization, and inner products in a non-Hermitian context.

Findings

In the unbroken PT phase, the non-Hermitian fermionic model is equivalent to a Hermitian theory.

The non-Hermitian nature is confined to the spinor structure, not the algebra of creation and annihilation operators.

The model's algebra remains identical to that of a Hermitian theory.

Abstract

We extend to a non-Hermitian fermionic quantum field theory with PT symmetry our previous discussion of second quantization, discrete symmetry transformations, and inner products in a scalar field theory [arXiv:2006.06656]. 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 PT 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.