Mutual compatibility/incompatibility of quasi-Hermitian quantum observables
Miloslav Znojil
TL;DR
The paper addresses whether two non-Hermitian observables in quasi-Hermitian quantum mechanics can share a single physical inner-product metric $\Theta$. It develops a constructive criterion: after expressing each observable via its own Dyson map, compatibility reduces to finding positive diagonal scalings $\mathfrak{c}_1,\mathfrak{c}_2$ such that $\mathcal{U}(\mathfrak{c}_1,\mathfrak{c}_2)=\mathfrak{c}_1 M \mathfrak{c}_2^{-1}$ is unitary, with $M=\Omega_1\Omega_2^{-1}$. The authors illustrate the method with an explicit $N=3$ example, where the resulting equations yield a non-real, non-positive solution for the scaling factors, proving that the two observables cannot be simultaneously quasi-Hermitian under a common $\Theta$. This establishes concrete limits on the simultaneous observability of multiple non-Hermitian operators and offers a pathway to generalize the compatibility test to more observables, with potential implications for quantum gravity and background-independent formulations.
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 $Θ\neq I$ guaranteeing the quasi-Hermiticity $A_j^\dagger \,Θ=Θ\,A_j$. The task is easy at $K=1$ because there are many eligible metrics $Θ=Θ(A_1)$. In our paper the next case with $K=2$ is analyzed. The criteria of the existence of a shared metric $Θ=Θ(A_1,A_2)$ are presented and discussed.