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

TL;DR

This paper explores the properties of weakly pseudo-Hermitian Hamiltonians with non-self-adjoint metric operators, highlighting the role of normalization factors and pseudo-metrics in restoring standard quantum mechanics.

Contribution

It introduces the concept of an alternative involutive quasiparity operator in pseudo-Hermitian models with non-self-adjoint metrics, expanding the understanding of inner products in quantum theory.

Findings

Identification of the quasiparity operator ${ m Q}$ as an alternative to ${ m C}$

Demonstration of how standard quantum mechanics is recovered with two inner products

Analysis of the impact of non-self-adjoint ${ m P}$ on the structure of pseudo-Hermitian models

Abstract

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