The problem of coexistence of several non-Hermitian observables in PT-symmetric quantum mechanics

TL;DR

This paper explores the challenges of simultaneously representing multiple non-Hermitian observables in PT-symmetric quantum mechanics, highlighting mathematical constraints and the need for modified inner products.

Contribution

It provides a linear algebraic analysis of the conditions under which multiple non-Hermitian observables can coexist in PT-symmetric quantum systems.

Findings

Not all sets of non-Hermitian observables can be simultaneously realized with a consistent inner product.

The feasibility depends on specific linear algebraic conditions.

The paper clarifies the limitations and mathematical structure of such quantum models.

Abstract

During the recent developments of quantum theory it has been clarified that the observable quantities (like energy or position) may be represented by operators (with real spectra) which are manifestly non-Hermitian. The mathematical consistency of the resulting models of stable quantum systems requires a reconstruction of an alternative, amended, physical inner product of states. We point out the less known fact that for more than one observable the task is not always feasible. The difficulty is re-analyzed and its elementary linear-algebraic interpretation and treatment are outlined.