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 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.

44 pages, revtex format; to match published version, including a revised discussion of particle mixing and oscillations

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• Non-Hermitian gauge field theories and BPS limits
• Discrete spacetime symmetries, second quantization, and inner products in a non-Hermitian Dirac fermionic field theory
• IR/UV mixing from local similarity maps of scalar non-Hermitian field theories
• Complex BPS solitons with real energies from duality
• Asymptotic freedom in a strongly interacting scalar quantum field theory in four Euclidean dimensions
• The phase diagram and vortex properties of PT-symmetric non-Hermitian two-component superfluid
• Poincaré symmetries and representations in pseudo-Hermitian quantum field theory
• Schwinger-Dyson equations and mass generation for an axion theory with a PT symmetric Yukawa fermion interaction
• Renormalisation group flows connecting a $4-\epsilon$ dimensional Hermitian field theory to a $\mathcal{P}\mathcal{T}$-symmetric theory for a fermion coupled to an axion
• Non-Hermiticity: a new paradigm for model building in particle physics
• Wilsonian approach to the interaction ${\varphi }^2(i\varphi {)}^{\epsilon }$
• Decay of the Higgs boson $h\to {\tau }^{-}{\tau }^{+}\to {\pi }^{-}{\nu }_{\tau }{\pi }^{+}{\bar{\nu }}_{\tau }$ for a non-Hermitian Yukawa interaction
• Exceptional points in perturbed dielectric spheres: A resonant-state expansion study
• Complex topological soliton with real energy in particle physics
• Oscillators with imaginary coupling: spectral functions in quantum mechanics and quantum field theory
• The operational foundations of PT-symmetric and quasi-Hermitian quantum theory
• Oscillation probabilities for a PT-symmetric non-Hermitian two-state system