The key message of this paper is that the search for the shortest
time needed for the passage of a classical particle from point A
to point B (= the classical brachistochrone problem) can find
several non-equivalent reformulations within quantum theory. A
compact review of these alternative possibilities is provided and
discussed via elementary matrix illustrative examples. In this
sense the paper may be read as a systematic completion of the
recent provocative letter [6] which conjectured that a ``faster
than Hermitian" evolution can exist in standard Quantum Mechanics.
The apparent paradox seems to be clarified and resolved at present
(for details one may consult, e.g., the Mostafazadeh's preprint
cited in footnote 1 on page 4). In essence, one simply has to
distinguish between the closed quantum systems (where the
generator of the time evolution must obligatorily be Hermitian so
that the passage time between two orthogonal states has its well
known non-vanishing minimum) and the open quantum systems (there,
the passage time can really be made arbitrarily short). All
interested readers are recommended to check arXiv and pay
attention also to the subsequent rather lengthy discussion of this
interesting subject by many other authors (cf., e.g., Geyer, H B;
Heiss, W D; Scholtz, F G: ``The physical interpretation of
non-Hermitian Hamiltonians and other observables." Canadian
Journal of Physics 86, 1195-1201 (2008) or the very recent letter
by Uwe Guenther and Boris F. Samsonov: ``The Naimark dilated
PT-symmetric brachistochrone." Phys. Rev. Lett. 101, 230404 (2008)
[arXiv:0807.3643]).
MR2455800 Assis, Paulo E. G.; Fring, Andreas: ``The quantum
brachistochrone problem for non-Hermitian Hamiltonians." J. Phys.
A: Math. Theor. 41 (2008), no. 24, 244002, 12 pp. 81Q05