Superconcise, six-paged review of the recent innovative trend, in
quantum mechanics, of using nontrivial metric $\eta \neq I$ in its
Hilbert space of states. Two prototype Hamiltonians discussed [cf.
eqs. (1) - the imaginary cubic anharmonic oscillator - and (7) - a
Bogoliubov equivalent of harmonic oscillator]. The author
emphasizes the ambiguity of the choice of $\eta$ and pays
attention to the construction of observables in the form proposed
by Mostafazadeh.

All physicists with a deeper interest in the present state of art
might extend their reading to all the neighboring pages 1005 -
1148 of ref. [8]. They form the volume of proceedings of the
Workshop mentioned in the Acknowledgements [I apologize for
recommending them being, at the same time, their Editor].

Mathematicians could, perhaps, appreciate knowing a few
complementary references concerning pre-history [one of the best
reviews of the $\eta-$related (usually called quasi-Hermitian)
representants of observables was written, in the highly relevant
context of nuclear physics applications and with the problem of
ambiguity of $\eta$ explicitly addressed, by F. G. Scholz, H. B.
Geyer and F. J. Hahne, Ann. Phys. (NY) 213 (1992) 74]. In this
light I would also modify slightly the Jones' proofreading and/or
terminology here and there [e.g., pseudo-Hermitian ->
quasi-Hermitian before eq. (3), PC -> P immediately after eq. (3),
metric -> scalar product before eq. (4)].

Concerning the Jones' first example [my favorite review is G.
Alvarez, J. Phys. A: Math. Gen. 27 (1995) 4589], let me finally
add a remarkable detail: This oscillator became famous (due to the
letter [1] by Bender and Boettcher) not only a few years before
even being attributed any correct $\eta$ (i.e., a correct
quantum-mechanical interpretation [6]) but even a few years before
the rigorous proof of the reality of its spectrum became available
[2]. Paradoxically, the contemporary knowledge of its (at that
time, the only available) ``wrong metric" (i.e., of the parity P
which plays the role of a mere auxiliary and indefinite
pseudo-metric) still gave the widespread nickname of PT-symmetric
quantum mechanics to all the subject.

MR2119183 Jones, H. F. On pseudo-Hermitian Hamiltonians and their
Hermitian counterparts. J. Phys. A 38 (2005), no. 8, 1741--1746.