$\mathcal{P}\mathcal{T}$-symmetric brachistochrone problem, Lorentz boosts, and nonunitary operator equivalence classes

The $\mathcal{P}\mathcal{T}$-symmetric (PTS) quantum brachistochrone problem is re-analyzed as a quantum system consisting of a non-Hermitian PTS component and a purely Hermitian component simultaneously. Interpreting this specific setup as a subsystem of a larger Hermitian system, we find nonunitary operator equivalence classes (conjugacy classes) as natural ingredients 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.

Figure 1

(Color) The transformation $\rho$ maps the initial and final states $\mid {\psi }_{i,f}⟩∊\mathcal{H}$ as well as the energy eigenstates $\mid E_{\pm }⟩∊\mathcal{H}$ into $\mid {\phi }_{i,f}⟩,\mid e_{\pm }⟩∊\tilde{\mathcal{H}}$, respectively, and leaves the EP-related fixed point states $\mid I_{\pm }⟩$ invariant. The contraction or dilation properties of the evolution paths (highlighted red or green curves) are defined by their location relative to the originally nonorthogonal energy eigenstates $\mid E_{\pm }⟩$.