The cca 80 years of development of quantum mechanics may be
characterized by the sharp contrast between the robust stability of
its abstract principles and the permanent emergence of dramatic
changes in their concrete interpretation and use. The Mostafazadeh's
extensive review of what he calls pseudo-Hermitian quantum mechanics
may be read as one of the most recent reconfirmations of this
permanent truth.

The brief history of development of this subfield of research has a
well defined starting point, viz, the Bender's and Boettcher's
thoughts-provoking 1998 letter [32]. In it, the authors conjectured
the reality of spectrum of certain Hamiltonians (explicitly, of $H =
p^2 + g(x)x^2$ with $g(x)=(ix)^\delta $ and with $\delta \geq 0$)
which {\em seem} manifestly non-Hermitian (note and emphasize: in an
irrelevant, unphysical, auxiliary, ``friendly" as well as ``false"
representation ${\cal H}^{(F)}$ of the Hilbert space of states) but
which are, in fact, Hermitian (note and emphasize: in another,
physical,  ``standard" though a bit ``sophisticated" representation
${\cal H}^{(S)}$ of the Hilbert space of states [this notation
convention is mine]).

Also the start of activity of Ali Mostafazadeh in the field has a
strictly defined date [viz., my own seminar ``Recent renewal of
interest in non-Hermitian Hamiltonians in quantum mechanics"
(Ko\c{c} University, May 3, 2001, 4.45 pm)] and motivation (viz.,
``a critical assessment" and ``a lucid and precise formulation of a
unitary quantum theory based on a non-Hermitian Hamiltonian" -- in
this sense, his present review paper is just a climax and summary of
his efforts documented by the list of his own more than 40 papers
reviewed and listed in References).

Needless to say, it is always admirable and exceptionally useful
when somebody like AM decides to write an extensive review paper.
This is a hard work, just partially rewarded by the implicit
privilege of promoting the author's own views and interpretations of
the whole story. On the side of readers, for a compensation, let me
recommend, therefore, a complementary parallel reading of the
perceivably older and, in many a respect, ``re-discovered" review
[207] by Scholtz et al [who would rather call the subject a
``quasi-Hermitian representation of Quantum Mechanics" in the
message which I also recently retold in MZ, ``Three-Hilbert-space
formulation of Quantum Mechanics", SIGMA 5 (2009), 001,
arXiv:0901.0700], or of the equally important alternative recent
review-style long summaries of the subject by Bender [30] and/or by
Dorey et al [80].

Every author of the review of a similar quickly developing branch of
physics always takes the risk of finding his text getting obsolete
after a few months (note, e.g., that AM hanged the first version of
his review on web almost two years before it appeared in print). In
the present case one may notice, for example, the current and
unexpectedly quick shift of emphasis from the concept of
``quasi-Hermiticity" (i.e., from the study of the ``hiddenly
Hermitian" observables exhibiting real spectra, as almost
exclusively covered by the AM's text) to the problems of phase
transitions (often called a ``spontaneous breakdown of PT symmetry",
strongly experimentally supported and motivated) and to the
intensified theoretical study of the ``horizons" (i.e., e.g., of the
Kato's ``exceptional points" in optical systems -- cf. the
pioneering paper [134]). One might also mention here the emergence
of new perspectives (e.g., in classical electrodynamics - cf. [176])
and of many fresh results on open systems as reported during the
tenth PHHQP international workshop
[http://www.pks.mpg.de/~phhqpx11/] (this meeting proved largely
dedicated to the truly experimental applicability of the theoretical
and mathematical concept of pseudo-Hermiticity {\em beyond} its
hiddenly Hermitian, real-spectra-exhibiting subdomain).

I do not deny that globally I admire the AMs work as presented in
refs.  [138] + [140] - [180] as well as in the present review. I
would also endorse the vast majority of his assertions. In more
detail, I would certainly see many things slightly differently,
nevertheless. {\em Pars pro toto}, I missed the comment on the role
of the ``charge" ${\cal C}$ in making the metric unique. Also my own
presentation of the whole formalism would not start from the
definition of `` a unitary quantum system \ldots [where] one
modifies the inner product of the Hilbert space" but rather, in the
spirit of Scholtz et al [207], from the fundamental role played by
the Riesz bases and by the Dyson operator (which may be easily made
time-dependent). In this sense I even completely disagree with the
unnecessarily narrowed perspective and conclusions presented in
chapter 7. My own point of view may be found described in MZ,
``Time-dependent version of cryptohermitian quantum theory", Phys.
Rev. D 78 (2008) 085003.

This being said, the AM's review must be recommended as an
obligatory reading. In a compact and mathematically disciplined form
he covers a big portion of the basic ideas and techniques in the
field. On purpose he avoided virtually all of the formal results
(notably, on exactly solvable models) and speculative ideas
(including, regrettably, also all of the underlying extensive
mathematics of Krein spaces) as not having, by the author's opinion,
any direct or concrete implications for the development of quantum
mechanics. In the same vein, just a few references are provided on
supersymmetry or weak pseudo-Hermiticity. No coverage is provided of
the recently quite popular cryptohermitian systems on lattices or
graphs (for a complementary reading in this direction see MZ,
``Fundamental length in quantum theories with PT-symmetric
Hamiltonians II: The case of quantum graphs", Phys. Rev. D. 80
(2009) 105004).

The acceptance of all of the above-mentioned limitations is quite
natural - the space provided by the Journal makes even the coverage
of some of the selected subjects very sketchy (cf. e.g., just the
four lines devoted to the MHD systems). One should rather underline
the high quality of the ``tours de force" of the report (e.g.,
chapter 4 on the calculations of metric operators, or the compact
chapter 6 which clarifies may subtleties of the quantum-classical
correspondence). At the same time, interested readers should read,
e.g., MZ, ``Scattering theory using smeared non-Hermitian
potentials", Phys. Rev. D. 80 (2009) 045009 (containing also further
references) in order to fill one of the most serious gaps in the
presented theory and to see the feasibility of the extension of the
pseudo-Hermitian quantum mechanics to the dynamical regime of the
causal and {\em unitary} scattering in which the conservation of a
global or asymptotic flow of probability is achieved in spite of the
manifest non-Hermiticity of the Hamiltonian in ${\cal H}^{(F)}$.


MR2749385 Mostafazadeh, Ali Pseudo-Hermitian representation of
quantum mechanics. Int. J. Geom. Methods Mod. Phys. 7 (2010), no. 7,
1191--1306. 81Q12 (47N50)