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

TL;DR

This paper rigorously proves the Hermiticity of certain quantum operators within a covariant Hamiltonian framework, clarifies the nature of complex eigenvalues, and explores their relation to the Berry-Keating Hamiltonian, advancing the mathematical foundation of quantum dynamics.

Contribution

It provides a rigorous proof of the Hermiticity of one-dimensional G-dynamics operators and analyzes their role in the structure of quantum Hamiltonians, linking to Berry-Keating's Hamiltonian.

Findings

G-dynamics operator is Hermitian with real eigenvalues.

Curvature operator is skew-Hermitian.

The non-Hermitian Hamiltonian has complex eigenvalues.

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 $\left( {{{\hat{w}}}^{\left( cl \right)}}\phi ,\varphi \right)=\left( \phi ,{{{\hat{w}}}^{\left( cl \right)}}\varphi \right)$ for any two states $\phi$ and $\varphi$, 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 ${{\hat{H}}^{\left( \text{bk}\right)}}=-\sqrt{-1}\hbar \hat{\theta }/2\in Her$ 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 $\hat{\theta }/2=x\frac{d}{dx}+1/2\in SHer$, etc.