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

Jean Alexandre, John Ellis, Peter Millington

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Article number125030
Number of pages20
JournalPhysical Review D
Issue number12
Publication statusPublished - 28 Dec 2020


  • Formal aspects of field theory
  • field theory in curved space
  • field theory: scalar
  • parity: time reversal
  • symmetry: discrete
  • symmetry: transformation
  • space-time: discrete
  • symmetry: space-time
  • mixing
  • PT symmetry
  • energy eigenstate
  • Hamiltonian
  • quantum mechanics
  • quantization
  • pseudoscalar
  • Fock space
  • unitarity


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