On the Role of the Normalization Factors $\kappa_n$ and of the Pseudo-Metric ${\cal P} \neq {\cal P}^\dagger$ in Crypto-Hermitian Quantum Models

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 $<\psi_1|{ \cal PQ}|\psi_2>=< \psi_1|{ \cal CP}|\psi_2>$.

Categories: math-ph math.MP quant-ph