关于规范化因子$κ_n$和赝度规${\cal P} \neq {\cal P}^\dagger$在隐厄米量子模型中的作用
On the Role of the Normalization Factors $κ_n$ and of the Pseudo-Metric ${\cal P} \neq {\cal P}^\dagger$ in Crypto-Hermitian Quantum Models
摘要：
在具有实谱的${\cal P}$ -伪埃尔米特哈密顿量$H ={\cal P}^{-1} H^\dagger \cal P}$中，“弱伪埃尔米特”哈密顿量（即，采用非自伴${\cal P} \neq {\cal P}^\dagger$的那些）形成显著的亚家族。我们列出了一些值得特别关注的原因。特别是，我们表明，无论何时${\cal P} \neq {\cal P}^\dagger$，目前的对合算子的收费${\cal C}$得到补充的非等价的替代对合拟奇偶算子${\cal Q}$。我们展示了如何，在这种语言中，标准的量子力学恢复通过两个替代的内积在物理希尔伯特空间的状态，与$<ψ_1|{\cal PQ}|ψ_2>=< ψ_1|{\cal CP}|ψ_2>$。
英文摘要：
Among ${\cal P}$-pseudo-Hermitian Hamiltonians $H ={\cal P}^{-1} H^\dagger \cal P}$ with real spectra, the ''weakly pseudo-Hermitian" ones (i.e., those employing non-self-adjoint ${\cal P} \neq {\cal P}^\dagger$) form a remarkable subfamily. We list some reasons why it deserves a special attention. In particular we show that whenever ${\cal P} \neq {\cal P}^\dagger$, the current involutive operator of charge ${\cal C}$ gets complemented by a nonequivalent alternative involutive quasiparity operator ${\cal Q}$. We show how, in this language, the standard quantum mechanics is restored via the two alternative inner products in the physical Hilbert space of states, with $<ψ_1|{\cal PQ}|ψ_2>=< ψ_1|{\cal CP}|ψ_2>$.
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