Pseudo-Hermitian Representation of Quantum Mechanics

TL;DR

This paper reviews the mathematical framework and physical implications of pseudo-Hermitian quantum mechanics, emphasizing the role of inner product modifications and symmetries in ensuring unitarity and real spectra for non-Hermitian Hamiltonians.

Contribution

It provides a comprehensive, self-contained survey of pseudo-Hermitian quantum mechanics, including mathematical tools, physical interpretations, and diverse applications across multiple physics domains.

Findings

Clarifies the role of antilinear symmetries like PT in real spectra

Explores the duality between local-non-Hermitian and nonlocal-Hermitian models

Demonstrates applications in nuclear physics, condensed matter, and quantum cosmology

Abstract

A diagonalizable non-Hermitian Hamiltonian having a real spectrum may be used to define a unitary quantum system, if one modifies the inner product of the Hilbert space properly. We give a comprehensive and essentially self-contained review of the basic ideas and techniques responsible for the recent developments in this subject. We provide a critical assessment of the role of the geometry of the Hilbert space in conventional quantum mechanics to reveal the basic physical principle motivating our study. We then offer a survey of the necessary mathematical tools and elaborate on a number of relevant issues of fundamental importance. In particular, we discuss the role of the antilinear symmetries such as PT, the true meaning and significance of the charge operators C and the CPT-inner products, the nature of the physical observables, the equivalent description of such models using ordinary Hermitian quantum mechanics, the pertaining duality between local-non-Hermitian versus nonlocal-Hermitian descriptions of their dynamics, the corresponding classical systems, the pseudo-Hermitian canonical quantization scheme, various methods of calculating the (pseudo-) metric operators, subtleties of dealing with time-dependent quasi-Hermitian Hamiltonians and the path-integral formulation of the theory, and the structure of the state space and its ramifications for the quantum Brachistochrone problem. We also explore some concrete physical applications of the abstract concepts and tools that have been developed in the course of this investigation. These include applications in nuclear physics, condensed matter physics, relativistic quantum mechanics and quantum field theory, quantum cosmology, electromagnetic wave propagation, open quantum systems, magnetohydrodynamics, quantum chaos, and biophysics.