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.
Highlights
Recent years have witnessed growing interest in nonHermitian quantum theories [1], those with PT symmetry, where P and T denote parity and time reversal, respectively [2]
There are strong arguments for the consistency of PT symmetric quantum field theory, a number of theoretical issues merit further attention. These include the analysis of discrete symmetries, which requires in turn a careful analysis of the Fock spaces of non-Hermitian quantum field theories with PT symmetry and their inner products
We have addressed in this paper some basic issues in the formulation of non-Hermitian bosonic quantum field theories, discussing in particular the treatment of discrete symmetries and the definition of the inner product in Fock space
Summary
Recent years have witnessed growing interest in nonHermitian quantum theories [1], those with PT symmetry, where P and T denote parity and time reversal, respectively [2]. There are strong arguments for the consistency of PT symmetric quantum field theory, a number of theoretical issues merit further attention These include the analysis of discrete symmetries, which requires in turn a careful analysis of the Fock spaces of non-Hermitian quantum field theories with PT symmetry and their inner products..
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