Repeatability of measurements: Equivalence of hermitian and non-hermitian observables

TL;DR

This paper demonstrates that non-hermitian operators with real spectra can be incorporated into quantum mechanics and introduces a transformation linking hermitian and non-hermitian systems, highlighting their equivalence.

Contribution

It shows the compatibility of non-hermitian operators with real spectra within standard quantum mechanics and proposes a canonical transformation between hermitian and non-hermitian systems.

Findings

Non-hermitian operators with real spectra can be treated within quantum mechanics.

A quantum canonical transformation maps hermitian systems to non-hermitian ones.

The transformation involves an energetic cost similar to classical inertial forces.

Abstract

A non-commuting measurement transfers, via the apparatus, information encoded in a system's state to the external "observer". Classical measurements determine properties of physical objects. In the quantum realm, the very same notion restricts the recording process to orthogonal states as only those are distinguishable by measurements. Therefore, even a possibility to describe physical reality by means of non-hermitian operators should \emph{volens nolens} be excluded as their eigenstates are not orthogonal. Here, we show that non-hermitian operators with real spectrum can be treated within the standard framework of quantum mechanics. Furthermore, we propose a quantum canonical transformation that maps hermitian systems onto non-hermitian ones. Similar to classical inertial forces this transformation is accompanied by an energetic cost pinning the system on the unitary path.