A rigorous Hermitian proof about the Gdynamics and analogy with BerryKeating's Hamiltonian
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 onedimensional Gdynamics ${\hat{w}^{\left( cl \right)}}={\hat{w}^{\left( cl \right)\dagger }}\in Her$ 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 skewHermitian operator as well. The act of finishing this Hermitian proof valuably enables us to ensure the nonHermitian 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 skewHermitian 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 Gdynamics to evaluate the BerryKeating'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 Gdynamics, to see how the similarity appears in the light of obvious factor $\hat{\theta }/2=x\frac{d}{dx}+1/2\in SHer$, etc.
 Publication:

arXiv eprints
 Pub Date:
 August 2021
 arXiv:
 arXiv:2109.03068
 Bibcode:
 2021arXiv210903068W
 Keywords:

 Physics  General Physics
 EPrint:
 25 pages, Any feedback is welcome, 25 pages with appendix