Krein-space formulation of mathcal{P}mathcal{T} symmetry, mathcal{C}mathcal{P}mathcal{T}-inner products, and pseudo-Hermiticity

Emphasizing the physical constraints on the formulation of the quantum theory, based on the standard measurement axiom and the Schrödinger equation, we comment on some conceptual issues arising in the formulation of the $mathcal{P}mathcal{T}$ -symmetric quantum mechanics. In particular, we elaborate on the requirements of the boundedness of the metric operator and the diagonalizability of the Hamiltonian. We also provide an accessible account of a Krein-space derivation of the $mathcal{C}mathcal{P}mathcal{T}$ -inner product, that was widely known to mathematicians since 1950’s. We show how this derivation is linked with the pseudo-Hermitian formulation of the $mathcal{P}mathcal{T}$ -symmetric quantum mechanics.