The PT-symmetric brachistochrone problem, Lorentz boosts and non-unitary operator equivalence classes

TL;DR

This paper investigates the PT-symmetric quantum brachistochrone problem by analyzing non-Hermitian and Hermitian components, revealing non-unitary equivalence classes and their relation to passage time bounds in quantum systems.

Contribution

It introduces a framework for understanding PT-symmetric brachistochrone problems using non-unitary operator classes within a larger Hermitian system.

Findings

Non-unitary operator equivalence classes contain Hermitian representatives.

Compatibility of zero passage time solutions with Hermitian bounds is demonstrated.

Geometric analysis supports the relation between PT-symmetric and Hermitian brachistochrone problems.

Abstract

The PT-symmetric (PTS) quantum brachistochrone problem is reanalyzed as quantum system consisting of a non-Hermitian PTS component and a purely Hermitian component simultaneously. Interpreting this specific setup as subsystem of a larger Hermitian system, we find non-unitary operator equivalence classes (conjugacy classes) as natural ingredient which contain at least one Dirac-Hermitian representative. With the help of a geometric analysis the compatibility of the vanishing passage time solution of a PTS brachistochrone with the Anandan-Aharonov lower bound for passage times of Hermitian brachistochrones is demonstrated.