Conceptual aspects of \mathcal{P}\mathcal{T} -symmetry and pseudo-Hermiticity: a status report

Abstract

We survey some of the main conceptual developments in the study of PT-symmetric and pseudo-Hermitian Hamiltonian operators that have taken place during the past 10 years or so. We offer a precise mathematical description of a quantum system and its representations that allows us to describe the idea of unitarization of a quantum system by modifying the inner product of the Hilbert space. We discuss the role and importance of the quantum-to-classical correspondence principle that provides the physical interpretation of the observables in quantum mechanics. Finally, we address the problem of constructing an underlying classical Hamiltonian for a unitary quantum system defined by an a priori non-Hermitian Hamiltonian.