关于 G-动力学且类比 Berry-Keating 哈密顿量的严格厄米证明
综合物理
2021-09-08 v1
摘要
量子协变哈密顿系统理论为建模量子系统的复杂动力学提供了连贯框架。本文集中处理与物理可观测量直接关联的量子算符的厄米性，因此，我们给出严格证明以验证一维 G-动力学 w ^ ( c l ) = w ^ ( c l ) † ∈ H e r {{\hat{w}}^{\left( cl \right)}}={{\hat{w}}^{\left( cl \right)\dagger }}\in Her w ^ ( c l ) = w ^ ( c l ) † ∈ H er ϕ \phi ϕ φ \varphi φ ( w ^ ( c l ) ϕ , φ ) = ( ϕ , w ^ ( c l ) φ ) \left( {{{\hat{w}}}^{\left( cl \right)}}\phi ,\varphi \right)=\left( \phi ,{{{\hat{w}}}^{\left( cl \right)}}\varphi \right) ( w ^ ( c l ) ϕ , φ ) = ( ϕ , w ^ ( c l ) φ ) H ^ ( r i ) = H ^ ( g ) − H ^ ( clm ) ∈ N H e r {{\hat{H}}^{\left( ri \right)}} ={{\hat{H}}^{\left( g \right)}} -{{\hat{H}}^{\left( \operatorname{clm} \right)}}\in NHer H ^ ( r i ) = H ^ ( g ) − H ^ ( clm ) ∈ N H er H ^ ( g ) = H ^ ( c l ) − E ( s ) / 2 ∈ H e r {{\hat{H}}^{\left( g \right)}} ={{\hat{H}}^{\left( cl \right)}}-{{E}^{\left( s \right)}}/2\in Her H ^ ( g ) = H ^ ( c l ) − E ( s ) /2 ∈ H er H ^ ( clm ) = − 1 ℏ w ^ ( c l ) ∈ S H e r {{\hat{H}}^{\left( \operatorname{clm} \right)}}=\sqrt{-1}\hbar {{\hat{w}}^{\left( cl \right)}}\in SHer H ^ ( clm ) = − 1 ℏ w ^ ( c l ) ∈ S H er H ^ ( r i ) {{\hat{H}}^{\left( ri \right)}} H ^ ( r i ) H ^ ( bk ) = − − 1 ℏ θ ^ / 2 ∈ H e r {{\hat{H}}^{\left( \text{bk}\right)}}=-\sqrt{-1}\hbar \hat{\theta }/2\in Her H ^ ( bk ) = − − 1 ℏ θ ^ /2 ∈ H er H ^ ( gbk ) ∈ N H e r \hat{H}^{\left( \text{gbk}\right)}\in NHer H ^ ( gbk ) ∈ N H er θ ^ / 2 = x d d x + 1 / 2 ∈ S H e r \hat{\theta }/2=x\frac{d}{dx}+1/2\in SHer θ ^ /2 = x d x d + 1/2 ∈ S H er
引用
@article{arxiv.2109.03068,
title = {A rigorous Hermitian proof about the G-dynamics and analogy with Berry-Keating's Hamiltonian},
author = {Jack Whongius},
journal= {arXiv preprint arXiv:2109.03068},
year = {2021}
}
备注
25 pages, Any feedback is welcome, 25 pages with appendix