The rationale for introducing non-Hermitian Hamiltonians and other observables is reviewed and open issues identified. We present a new approach based on Moyal products to compute the metric for quasi-Hermitian systems. This approach is not only an efficient method of computation, but also suggests a new perspective on quasi-Hermitian quantum mechanics which invites further exploration. In particular, we present some first results which link the Berry connection and curvature to non-perturbative properties and the metric.
The rationale for introducing non-Hermitian Hamiltonians and other observables is reviewed and open issues identified. We present a new approach based on Moyal products to compute the metric for quasi-Hermitian systems. This approach is not only an efficient method of computation, but also suggests a new perspective on quasi-Hermitian quantum mechanics which invites further exploration. In particular, we present some first results which link the Berry connection and curvature to non-perturbative properties and the metric.
Despite acute interest in the dynamics of non-Hermitian systems, there is a lack of consensus in the mathematical formulation of non-Hermitian quantum mechanics in the community. Different methodologies are used in the literature to study non-Hermitian dynamics. This ranges from consistent frameworks like biorthogonal quantum mechanics and metric approach characterized by modified inner products, to normalization by time-dependent norms inspired by open quantum systems. In this work, we systematically explore the similarities and differences among these various methods. Utilizing illustrative models with exact solutions, we demonstrate that these methods produce not only quantitatively different results but also distinct physical interpretations. For dissipative systems where non-Hermiticity arises as an approximation, we find that simply dividing by the norm in the PT -broken regime closely aligns with the full master equation solutions. In contrast, for quantum systems where non-Hermiticity can be engineered exactly, incorporating metric dynamics is crucial for the probabilistic interpretation of quantum mechanics, necessitating the generalizations of similarity transformations and unitarity to non-Hermitian systems. This study lays the groundwork for further exploration of non-Hermitian Hamiltonians, potentially leveraging generalized transformations for novel physical phenomena.
In the conventional Schrödinger's formulation of quantum mechanics the unitary evolution of a state ψ is controlled, in Hilbert space L, by a Hamiltonian h which must be self-adjoint. In the recent, ``quasi-Hermitian'' reformulation of the theory one replaces h by its isospectral but non-Hermitian avatar H=Ω−1hΩ with Ω†Ω=Θ=I. Although acting in another, manifestly unphysical Hilbert space H, the amended Hamiltonian H=H† can be perceived as self-adjoint with respect to the amended inner-product metric Θ. In our paper motivated by a generic technical ``user-unfriendliness'' of the non-Hermiticity of H we introduce and describe a specific new family of Hamiltonians H for which the metrics Θ become available in closed form.
Abstract In the conventional Schrödinger’s formulation of quantum mechanics the unitary evolution of a state ψ is controlled, in Hilbert space <?CDATA L?> L , by a Hamiltonian ɧ which must be self-adjoint. In the recent, “quasi-Hermitian” reformulation of the theory one replaces ɧ by its isospectral but non-Hermitian avatar H = Ω−1 𝖍Ω with Ω†Ω = Θ ≠ I. Although acting in another, manifestly unphysical Hilbert space <?CDATA H?> H , the amended Hamiltonian H ≠ H † can be perceived as self-adjoint with respect to the amended inner-product metric Θ. In our paper motivated by a generic technical “user-unfriendliness” of the non-Hermiticity of H we introduce and describe a specific new family of Hamiltonians H for which the metrics Θ become available in closed form.
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.
We develop a comprehensive, gauge-covariant geometric framework for non-Hermitian quantum systems in the quasi-Hermitian regime, that is, the region of parameter space where the non-Hermitian Hamiltonian admits a real spectrum and a positive-definite metric operator. We build this framework by elevating the Dyson map to a central geometric object. This map is the transformation that converts a non-Hermitian Hamiltonian into an equivalent Hermitian one. From it we construct the Dyson connection and decompose it into Hermitian and anti-Hermitian parts, identified respectively as {\it stretching } and {\it rotation } components. This decomposition cleanly separates the genuine physical metric deformations from the unitary gauge redundancies. Working with manifestly gauge-covariant states, we then derive the complex non-Hermitian Berry phase and the quantum geometric tensor (QGT), and show that the non-Hermitian geometric curvature originates from the non-commutativity of the stretching components at the operator level. We further analyse the geometric singularities near an exceptional point (EP) and uncover a distinct hierarchy of divergences. For a general two-level non-Hermitian model, the quantum metric tensor (QMT) exhibits a leading-order divergence ∼∣εμ∣−2, while the Berry curvature shows a weaker, subleading divergence ∼∣εμ∣−3/2, with εμ denoting the parameter displacement from the EP along an individual parameter axis μ. Finally, we examine physical realizations of this model, including the non-Hermitian Su--Schrieffer--Heeger (SSH) and Hatano--Nelson (HN) models, where exact analytical results confirm the predicted critical scaling laws and illustrate the metric-deformation-driven non-Hermitian geometries.