Discrete spacetime symmetries and particle mixing in non-Hermitian scalar quantum field theories

Abstract

We discuss second quantization, discrete symmetry transformations, and inner products in free non-Hermitian scalar quantum field theories with $\mathcal{P}\mathcal{T}$ symmetry, focusing on a prototype model of two complex scalar fields with anti-Hermitian mass mixing. Whereas the definition of the inner product is unique for theories described by Hermitian Hamiltonians, its formulation is not unique for non-Hermitian Hamiltonians. Energy eigenstates are not orthogonal with respect to the conventional Dirac inner product, so we must consider additional discrete transformations to define a positive-definite norm. We clarify the relationship between canonical-conjugate operators and introduce the additional discrete symmetry ${\mathcal{C}}^{\ensuremath{'}}$, previously introduced for quantum-mechanical systems, and show that the ${\mathcal{C}}^{\ensuremath{'}}\mathcal{P}\mathcal{T}$ inner product does yield a positive-definite norm, and hence is appropriate for defining the Fock space in non-Hermitian models with $\mathcal{P}\mathcal{T}$ symmetry in terms of energy eigenstates. We also discuss similarity transformations between $\mathcal{P}\mathcal{T}$-symmetric non-Hermitian scalar quantum field theories and Hermitian theories, showing that they would require modification in the presence of interactions. As an illustration of our discussion, we compare particle mixing in a Hermitian theory and in the corresponding non-Hermitian model with $\mathcal{P}\mathcal{T}$ symmetry, showing how the latter maintains unitarity and exhibits mixing between scalar and pseudoscalar bosons.

Topics & Concepts