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

Abstract

In 1992, Scholtz et ad. (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 Theta(N) and of the observables Lambda(k), k = 0, 1, ..., N + 1 in terms of auxiliary operators Z(k) with k = 0, 1, ..., N. At N = 2 the formalism degenerates to the well-known PT-symmetric quantum mechanics using factorized metric Theta(2) Z(2)Z(1), where Z(2) = P is parity and where Z(1) = C is charge.