Mutual compatibility/incompatibility of quasi-Hermitian quantum observables

TL;DR

This paper investigates the conditions under which two non-Hermitian quantum observables can share a common inner-product metric, extending the understanding of quasi-Hermitian quantum mechanics beyond the single-observable case.

Contribution

It provides criteria for the existence of a shared metric for two quasi-Hermitian observables, advancing the theoretical framework of non-Hermitian quantum mechanics.

Findings

Criteria for shared metric existence are established.

Analysis extends quasi-Hermitian framework from one to two observables.

Discussion of implications for quantum theory consistency.

Abstract

In the framework of quasi-Hermitian quantum mechanics the eligible operators of observables may be non-Hermitian, $A_j\neq A_j^\dagger$, $j=1,2, \ldots,K$. In principle, the standard probabilistic interpretation of the theory can be re-established via a reconstruction of physical inner-product metric $\Theta\neq I$ guaranteeing the quasi-Hermiticity $A_j^\dagger \,\Theta=\Theta\,A_j$. The task is easy at $K=1$ because there are many eligible metrics $\Theta=\Theta(A_1)$. In our paper the next case with $K=2$ is analyzed. The criteria of the existence of a shared metric $\Theta=\Theta(A_1,A_2)$ are presented and discussed.