Embedding, simulation and consistency of PT-symmetric quantum theory

A physical requirement on the Hamiltonian operator in quantum mechanics is that it must generate real energy spectrum and unitary time evolution. While the Hamiltonians are Dirac Hermitian in conventional quantum mechanics, they observe PT-symmetry in PT-symmetric quantum theory. The embedding property was first studied by Günther and Samsonov [1] to visualise the evolution of unbroken PT-symmetric Hamiltonians on C2 by Hermitian Hamiltonians on C4. This paper investigates the properties of PT-symmetric quantum systems including the embedding property. We provide a full characterization of the embedding property in the general case and show that only unbroken PT-symmetric quantum systems admit this property in a finite dimensional space. Furthermore, utilizing this property, we establish a physically realizable simulation process of the unbroken PT-symmetric Hamiltonians. An observation that the unbroken PT-symmetric quantum systems can be viewed as open systems in the conventional quantum mechanics accounts for the consistency of PT-symmetric quantum theory.