On the Role of the Normalization Factors κn and of the Pseudo-Metric P ≠ P† in Crypto-Hermitian Quantum Models

Abstract

Among P-pseudo-Hermitian Hamiltonians H = P 1 H † P with real spectra, the weakly pseudo-Hermitian ones (i.e., those employing non-self-adjoint P 6 P † ) form a remarkable subfamily. We list some reasons why it deserves a special attention. In par- ticular we show that whenever P 6 P † , the current involutive operator of charge C gets complemented by a nonequivalent alternative involutive quasiparity operator 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 h 1 |PQ| 2i = h 1 |CP | 2i.