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 ten 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.
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