PT-symmetric brachistochrone problem, Lorentz boosts and nonunitary operator equivalence classes


PT-symmetric brachistochrone problem, Lorentz boosts and nonunitary operator equivalence classes

Günther, U.; Samsonov, B.

The PT-symmetric (PTS) quantum brachistochrone problem is re-analyzed as a composite quantum system consisting of a non-Hermitian PTS component and a purely Hermitian component simultaneously. A general approach is proposed for the construction of partially PTS systems which are not reducible to purely Hermitian ones. A natural ingredient of these systems are non-unitary operator equivalence classes (conjugacy orbits) with at least one 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. Via embedding of the PTS Hamiltonian into a Dirac Hamiltonian the vanishing passage time solution is related to an ultra-relativistic regime.

Keywords: PT-symmetric Quantum Mechanics; quantum brachistochrone problem; exceptional point; singularity; Dirac equation; chiral spinors; ultra-relativistic limit; Krein space

Permalink: https://www.hzdr.de/publications/Publ-10445
Publ.-Id: 10445