Conceptual Aspects of PT-Symmetry and Pseudo-Hermiticity: A status report

TL;DR

This paper reviews the conceptual developments in PT-symmetric and pseudo-Hermitian quantum mechanics over the past decade, focusing on mathematical descriptions, unitarization, and classical correspondence.

Contribution

It provides a precise mathematical framework for PT-symmetry and pseudo-Hermiticity, and discusses the unitarization process and classical Hamiltonian construction for non-Hermitian systems.

Findings

Mathematical description of PT-symmetric and pseudo-Hermitian systems

Method for unitarizing quantum systems via inner product modification

Insights into classical Hamiltonian construction from quantum models

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.