Factorized Hilbert-space metrics and non-commutative quasi-Hermitian observables

The Czech Academy of Sciences, Nuclear Physics Institute - Hlavní 130, 250 68 Řež, Czech Republic, Department of Physics, Faculty of Science, University of Hradec Králové - Rokitanského 62, 50003 Hradec Králové, Czech Republic and Institute of System Science, Durban University of Technology - Durban, South Africa

^(a) znojil@ujf.cas.cz (corresponding author)

Received: 5 April 2022
Accepted: 5 July 2022

Abstract

In 1992, Scholtz et al. (Ann. Phys., 213 (1992) 74) showed that a set of non-Hermitian operators can represent observables of a closed unitary quantum system, provided only that its elements are quasi-Hermitian (i.e., roughly speaking, Hermitian with respect to an ad hoc inner-product metric). We show that such a version of quantum mechanics admits a simultaneous closed-form representation of the metric and of the observables , in terms of auxiliary operators Z_k with . At N = 2 the formalism degenerates to the well-known quantum mechanics using factorized metric , where is parity and where is charge.