On the Role of the Normalization Factors ${\kappa }_n$ and of the Pseudo-Metric $\mathcal{P}\ne {\mathcal{P}}^{†}$ in Crypto-Hermitian Quantum Models

Abstract

Among $\mathcal{P}$-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 $\mathcal{P}\ne {\mathcal{P}}^{†}$) form a remarkable subfamily. We list some reasons why it deserves a special attention. In particular we show that whenever $\mathcal{P}\ne {\mathcal{P}}^{†}$, the current involutive operator of charge $\mathcal{C}$ gets complemented by a nonequivalent alternative involutive quasiparity operator $\mathcal{Q}$. 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 $<{\psi }_1|\mathcal{P}\mathcal{Q}|{\psi }_2>=<{\psi }_1|\mathcal{C}\mathcal{P}|{\psi }_2>$.

arXiv commentThis is a contribution to the Proc. of the 3-rd Microconference "Analytic and Algebraic Methods III"(June 19, 2007, Prague, Czech Republic), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

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Cited by in corpus (15)

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