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

TL;DR

This paper explores how non-Hermitian PT-symmetric scalar quantum field theories can be consistently formulated with a positive-definite inner product, and examines particle mixing phenomena within these models.

Contribution

It introduces the C'PT inner product for non-Hermitian theories, clarifies the relationship between operators, and compares particle mixing in Hermitian and non-Hermitian models.

Findings

C'PT inner product yields positive-definite norm

Non-Hermitian models maintain unitarity with PT symmetry

Particle mixing persists in non-Hermitian PT-symmetric theories

Abstract

We discuss second quantization, discrete symmetry transformations and inner products in free non-Hermitian scalar quantum field theories with PT 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 C', previously introduced for quantum-mechanical systems, and show that the C'PT inner product does yield a positive-definite norm, and hence is appropriate for defining the Fock space in non-Hermitian models with PT symmetry in terms of energy eigenstates. We also discuss similarity transformations between PT-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 PT symmetry, showing how the latter maintains unitarity and exhibits mixing between scalar and pseudoscalar bosons.