Pseudo-Hermitian Representation of Quantum Mechanics

TL;DR

This work develops and systematizes pseudo-Hermitian quantum mechanics, showing how a non-Hermitian Hamiltonian with a real spectrum can yield unitary evolution by redefining the inner product with a metric operator η_+. It unpacks the mathematical foundations (Hilbert spaces, biorthonormality, and metric operators), contrasts quasi-Hermitian and pseudo-Hermitian approaches, and surveys methods to construct η_+ (spectral, perturbative, differential, and Lie-algebraic). It then extends the formalism to complex-contour systems, links to complex classical mechanics, analyzes time dependence and path integrals, and explores the geometry of state space and brachistochrone problems, concluding with a broad panorama of physical applications from nuclear physics to quantum optics and cosmology. The results demonstrate that pseudo-Hermitian QM supplies a family of dynamically equivalent, kinematically distinct representations that can simplify calculations and illuminate non-Hermitian phenomena while preserving physical consistency through a positive-definite inner product. Overall, the framework clarifies when non-Hermitian models describe unitary dynamics and how metric-operator choices impact observables, spectra, and classical-quantum correspondences. The work thereby provides both a foundational and a practical toolkit for applying pseudo-Hermitian techniques across quantum theory and related fields.

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.

Figures (2)

• Figure 1: A contour $\Gamma_\nu$ (the solid curve) lying asymptotically in the union of the Stokes wedges (the grey region). The dashed lines are the bisectors of the Stokes wedges $S^\pm_\nu$. The angle $\theta_\nu$ between the positive real axis and the bisector of $S^+_\nu$ is also depicted.
• Figure 2: Stereographic projection of the sphere $S^2$ defined by $x^2+y^2+(z-\frac{1}{2})^2=\frac{1}{4}$: $\mathbf{N}$ and $\mathbf{S}$ are respectively the north and south poles of $S^2$. $\Pi_{\mathbf{N}}$ is the tangent plane at $\mathbf{N}$. $p$ is a point on $S^2-\{\mathbf{S}\}$, and ${\hbox{\large p}}$ is its stereographic projection on $\Pi_{\mathbf{N}}$.