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arXiv:2109.03068

A rigorous Hermitian proof about the G-dynamics and analogy with Berry-Keating's Hamiltonian

Jack Whongius

Published Aug 24, 2021• 0 citations
arXivHTMLPDF

Abstract

Quantum covariant Hamiltonian system theory provides a coherent framework for modelling the complex dynamics of quantum systems. In this paper, we centrally deal with the Hermiticity of quantum operators that directly links to the physical observable, thusly, we give a rigorous proof to verify one-dimensional G-dynamics ${\hat{w}^{\left( cl \right)}}={\hat{w}^{\left( cl \right)\dagger }}\in Her$ that is a Hermitian operator satisfying $\left( {\hat{w}^{\left( cl \right)}}ϕ,φ\right)=\left( ϕ,{\hat{w}^{\left( cl \right)}}φ\right)$ for any two states $ϕ$ and $φ$, and its eigenvalues are real. We also prove that curvature operator is a skew-Hermitian operator as well. The act of finishing this Hermitian proof valuably enables us to ensure the non-Hermitian Hamiltonian operator ${\hat{H}^{\left( ri \right)}} ={\hat{H}^{\left( g \right)}} -{\hat{H}^{\left( \operatorname{clm} \right)}}\in NHer$ that is divided into the Hermitian operator ${\hat{H}^{\left( g \right)}} ={\hat{H}^{\left( cl \right)}}-{{E}^{\left( s \right)}}/2\in Her$ and the skew-Hermitian operator ${\hat{H}^{\left( \operatorname{clm} \right)}}=\sqrt{-1}\hbar {\hat{w}^{\left( cl \right)}}\in SHer$ generally, and ${\hat{H}^{\left( ri \right)}}$ always has the complex eigenvalues. We use the formula of the G-dynamics to evaluate the Berry-Keating's Hamiltonian operator ${\hat{H}^{\left( \text{bk}\right)}}=-\sqrt{-1}\hbar \hat{θ}/2\in Her$ and its extensive version $\hat{H}^{\left( \text{gbk}\right)}\in NHer$ as the applications of the G-dynamics, to see how the similarity appears in the light of obvious factor $\hat{θ}/2=x\frac{d}{dx}+1/2\in SHer$, etc.

DOI
10.48550/arxiv.2109.03068

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Quantum covariant Hamiltonian system theory provides a coherent framework for modelling the complex dynamics of quantum systems. In this paper, we centrally deal with the Hermiticity of quantum operators that directly links to the physical observable, thusly, we give a rigorous proof to verify one-dimensional G-dynamics w^(cl)=w^(cl)†∈Her{\hat{w}^{\left( cl \right)}}={\hat{w}^{\left( cl \right)\dagger }}\in Herw^(cl)=w^(cl)†∈Her that is a Hermitian operator satisfying (w^(cl)ϕ,φ)=(ϕ,w^(cl)φ)\left( {\hat{w}^{\left( cl \right)}}ϕ,φ\right)=\left( ϕ,{\hat{w}^{\left( cl \right)}}φ\right)(w^(cl)ϕ,φ)=(ϕ,w^(cl)φ) for any two states ϕϕϕ and φφφ, and its eigenvalues are real. We also prove that curvature operator is a skew-Hermitian operator as well. The act of finishing this Hermitian proof valuably enables us to ensure the non-Hermitian Hamiltonian operator H^(ri)=H^(g)−H^(clm⁡)∈NHer{\hat{H}^{\left( ri \right)}} ={\hat{H}^{\left( g \right)}} -{\hat{H}^{\left( \operatorname{clm} \right)}}\in NHerH^(ri)=H^(g)−H^(clm)∈NHer that is divided into the Hermitian operator H^(g)=H^(cl)−E(s)/2∈Her{\hat{H}^{\left( g \right)}} ={\hat{H}^{\left( cl \right)}}-{{E}^{\left( s \right)}}/2\in HerH^(g)=H^(cl)−E(s)/2∈Her and the skew-Hermitian operator H^(clm⁡)=−1ℏw^(cl)∈SHer{\hat{H}^{\left( \operatorname{clm} \right)}}=\sqrt{-1}\hbar {\hat{w}^{\left( cl \right)}}\in SHerH^(clm)=−1​ℏw^(cl)∈SHer generally, and H^(ri){\hat{H}^{\left( ri \right)}}H^(ri) always has the complex eigenvalues. We use the formula of the G-dynamics to evaluate the Berry-Keating's Hamiltonian operator H^(bk)=−−1ℏθ^/2∈Her{\hat{H}^{\left( \text{bk}\right)}}=-\sqrt{-1}\hbar \hat{θ}/2\in HerH^(bk)=−−1​ℏθ^/2∈Her and its extensive version H^(gbk)∈NHer\hat{H}^{\left( \text{gbk}\right)}\in NHerH^(gbk)∈NHer as the applications of the G-dynamics, to see how the similarity appears in the light of obvious factor θ^/2=xddx+1/2∈SHer\hat{θ}/2=x\frac{d}{dx}+1/2\in SHerθ^/2=xdxd​+1/2∈SHer, etc.
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\sim |ε_μ|^{-2}∼∣εμ​∣−2, while the Berry curvature shows a weaker, subleading divergence ∼∣εμ∣−3/2\sim |ε_μ|^{-3/2}∼∣εμ​∣−3/2, with εμε_μεμ​ denoting the parameter displacement from the EP along an individual parameter axis μμμ. 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.
It is well known that Hermitian operators have real eigenvalues while non-Hermitian ones may have complex eigenvalues. Recently, numerical and analytical results indicated that the spectra of many non-Hermitians Hamiltonians HHH are indeed real if they are invariant under the combined action of self-adjoint parity PPP and time reversal T.T.T. The concept of a pseudo-Hermitian operator showed that the remarkable spectral properties of the PTPTPT-symmetric Hamiltonians follow from their pseudo-Hermiticity. It is possible to explain these observations by the concept of pseudo-Hermitian operators and to formulate completeness and orthonormality relations. Most of the effort has been devoted to study time-independent non-Hermitian systems. In this paper, we study the exactly solvable time-dependent periodic pseudo-Hermitian Hamiltonians. The method introduced, to make the reality of eigenvalues and phases, is based on a Floquet decomposition of the evolution operator UH(t)=ZH(t)exp(iMHt){U}^{H}(t)={Z}^{H}(t)exp(i{M}^{H}t)UH(t)=ZH(t)exp(iMHt) associated with the periodic pseudo-hermitian Hamiltonian H(t)=H(t+T)H(t)=H(t+T)H(t)=H(t+T). One of the results found in this paper concerns a calculation of Berry's phase for periodic, but not necessarily adiabatic, pseudo-Hermitian Hamiltonians. A two-level pseudo-Hermitian system is discussed as an illustrative example.
In this paper, we further study the G-dynamics newly emerged in the covariant dynamics defined by the QCPB theory. Then we want to seek the precise formulations of the G-dynamics to calculate the practical quantum problems. We explicitly verify how the role of G-dynamics plays in quantum mechanics, and then we propose three new operators based on the G-dynamics: D-operator, T-operator and G-operator, we find that the non-Hermitian operators are inevitable to appear in the framework of the QCPB. Then the eigenvalue equation of them are discussed accordingly by using the Schrodinger equation and geometrinetic energy operator (GEO) which leads to the Ri-operator, respectively, we obtain some useful results between two quantum situations. Meanwhile, we present generalized covariant wave equation based on the G-operator. The GEO for quantum harmonic oscillator as an application is further calculated.
The description of states and dynamics in non-Hermitian systems is fundamentally linked to the choice of an appropriate theoretical framework -- a point of ongoing debate in the field. This work addresses this issue by proposing a consistent formulation that reconciles existing controversies and establishes a unified theoretical understanding. Our approach rests on two foundational premises: (i) the dynamics of both left and right-vectors of a non-Hermitian system must satisfy the Schrödinger equation; (ii) the theoretical framework must preserve gauge invariance, ensuring that physical quantities are independent of unobservable phase choices. Building on these physically motivated assumptions, we refine the biorthogonal framework, leading to a gauge-invariant non-Hermitian quantum theory. Our framework naturally encompasses the open-system effective non-Hermitian evolution as a special case, and can naturally reduce to standard quantum mechanics in the Hermitian limit. As a concrete application, we analyze the dynamical phase transition in a one-dimensional Su-Schrieffer-Heeger (SSH) model within this gauge-invariant non-Hermitian quantum theory. Notably, our formulation naturally generalizes the known condition for such transitions in Hermitian two-band systems, namely, dki⋅dkf=0\mathbf{d}_{k}^i\cdot\mathbf{d}_{k}^f=0dki​⋅dkf​=0, to the non-Hermitian case, where it takes the form Re[dkidki⋅dkfdkf]=0\mathrm{Re}\Bigl[\frac{\mathbf{d}_{k}^i}{d_{k}^i}\cdot\frac{\mathbf{d}_{k}^f}{d_{k}^f}\Bigr]=0Re[dki​dki​​⋅dkf​dkf​​]=0. Furthermore, we identify entirely new dynamical phase transitions that cannot be characterized by the winding number. We hope that this gauge-invariant non-Hermitian quantum theory will find broad applications in the study of non-Hermitian systems.
The relevance in physics of non-Hermitian operators with real eigenvalues is being widely recognized not only in quantum mechanics but also in other areas, such as quantum optics, quantum fluid dynamics, and quantum field theory. In this note, a quantum system described by a non-Hermitian Hamiltonian, which is constituted by two types of interacting bosons, is investigated. The real eigenvalues of the Hamiltonian are explicitly determined, as well as complete biorthogonal sets of eigenfunctions of the Hamiltonian and its adjoint. The diagonal representation of H is obtained using pseudo-bosonic operators.
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.