.

Feasibility and method of multi-step Hermitization of crypto-Hermitian quantum
Hamiltonians

Miloslav Znojil

The Czech Academy of Sciences, Nuclear Physics Institute,

Hlavní 130, 250 68 Řež, Czech Republic

and

Department of Physics, Faculty of Science, University of Hradec Králové,

Rokitanského 62, 50003 Hradec Králové, Czech Republic

and

Institute of System Science, Durban University of Technology, Durban, South
Africa

e-mail<EMAIL_ADDRESS>
## Abstract

Even the most economical conventional formulation of quantum mechanics called
Schrödinger picture (SP) need not be always computationally or conceptually
optimal. A Dyson-inspired remedy can sometimes be sought in a transfer of the
calculations from the conventional and correct SP Hilbert space ${\cal H}$
into a “false but friendlier” Hilbert space ${\cal R}$. The difference (i.e.,
non-equivalence) between ${\cal R}$ and ${\cal H}$ is given by a judicious
simplification of the inner product. Although the Hamiltonian $H$ itself
appears non-Hermitian in ${\cal R}$, its necessary Hermitization (mediated by
a return to ${\cal H}$) is usually feasible. If not, another eligible remedy
is described in the present paper. Our basic idea lies in the introduction of
a “chained” auxiliary manipulation multiplet of inner-product spaces ${\cal
R}_{1}$, ${\cal R}_{2}$, …, ${\cal R}_{N}$. At arbitrary $N$, the structure
and properties of the resulting generalized SP (GSP) formulation of quantum
mechanics of unitary systems (characterized by “hiddenly Hermitian”
observables) are described and discussed in detail. Nontrivial nature of the
related multi-step Dyson-map composition laws and correspondences to the
conventional Hermitian models are also revealed and clarified.

### Keywords

.

quantum dynamics in Schrödinger picture;

sequences of auxiliary inner-product spaces;

apparently non-Hermitian generators of unitary evolution;

factorization of the physical Hilbert-space metrics;

##  1 Introduction

In conventional textbooks the basics of quantum theory of unitary systems are
usually explained in Schrödinger representation alias  Schrödinger picture
(SP, [1]). The pure state is represented there by a time-dependent ket-vector
element of a Hilbert space ${\cal L}$. The evolution is assumed generated by a
self-adjoint Hamiltonian $\mathfrak{h}$. The one-to-one correspondence between
the self-adjointness of $\mathfrak{h}$ and the unitarity of the evolution was
given the rigorous mathematical form by Stone [2]. In this sense, the Bender’s
and Boettcher’s claim [3] that the unitary evolution could be also generated
by a non-Hermitian Hamiltonian $H\neq H^{\dagger}$ sounded, initially,
contradictory [4].

Fortunately, the apparent paradox found quickly its origin in an elementary
terminological misunderstanding (see, e.g., reviews [5, 6, 7] or section 2
below). It has been revealed that the Bender’s and Boettcher’s Hamiltonians
$H$ are only non-Hermitian in a “theoretically redundant”, i.e.,
mathematically preferable but manifestly unphysical auxiliary Hilbert space
${\cal H}_{math}$. In this sense, these operators have to be Hermitized in a
way explained, by Scholtz et al, in review [8]. In their words, given a
suitable non-Hermitian Hamiltonian $H$, one can still work in an innovated
version of quantum theory which “allows for the normal quantum-mechanical
interpretation“.

In such a setting the unitary evolution is found generated by a non-Hermitian
Hamiltonian $H$ because such an operator (called “quasi-Hermitian” [8] alias
“crypto-Hermitian” [9]) appears Hermitizable. In applications the
Hermitization has to be visualized as a constructive transfer of
representation of the quantum bound states of interest from the auxiliary
Hilbert space ${\cal H}_{math}$ to the traditional physical Hilbert space
${\cal L}$ of textbooks or, better [7], to one of its other suitable
representations (say, ${\cal H}_{phys}$).

The phenomenological relevance as well as the feasibility of the procedure of
Hermitization was illustrated, using an exactly solvable model, by Buslaev and
Grecchi [10]. Rigorously, these authors managed to show that the unstable
anharmonic-oscillator toy-model Hamiltonian

$H^{(BG)}=H({\rm
i}g,j)=-\frac{d^{2}}{dx^{2}}+\frac{j^{2}-1}{4\,r^{2}(x)}+\frac{r^{2}(x)}{4}-\frac{g^{2}\,r^{4}(x)}{4}\,,\
\ \ r(x)=x-{\rm i}{\eta}\,,\ \ \ \eta>0\,$ (1)

defined as acting and manifestly non-Hermitian in an unphysical Hilbert space
${\cal H}_{math}=L^{2}(\mathbb{R})$ can be assigned, at any real $j$ and
positive $g>0$, the entirely conventional self-adjoint isospectral  partner

$\mathfrak{h}^{(BG)}=Q(g,j)=-\frac{d^{2}}{dx^{2}}-(gx-1/2)\,j+(gx-1)^{2}\,x^{2}\,,\
\ \ \ \ x\in\mathbb{R}\,$ (2)

defined as acting in another, physical Hilbert space ${\cal
L}=L^{2}(\mathbb{R})$.

Remarkably enough, even such a most elementary sample of Hermitization
$H^{(BG)}\to\mathfrak{h}^{(BG)}$ appeared to require the introduction of
several intermediate auxiliary isospectral operators (their list contains cca
six items, the explicit form of which may be found in loc. cit.). On this
ground one may suspect that the feasible  process of Hermitization should also
have a multi-step, $N-$step structure in general. This is the idea which also
served as the main motivation of our present study.

In the first stage of development (cf. section 2) we will keep $N=1$ and
simplify and reformulate the very concept of the Hermitization. For this
purpose we will just review the two complementary existing forms of the
theory. In its older version (cf. review [8] or subsection 2.1 below) one
manages to circumvent the (in general, difficult) description of the direct
change $H\to\mathfrak{h}$ of the Hamiltonian. This is mediated by introduction
of another, third Hilbert space ${\cal H}_{phys}$ [6]. In subsection 2.2 we
will briefly review the newer and better known specific approach to the
technicalities known as ${\cal PT}-$symmetric quantum mechanics [5]. The
practical feasibility of the Hermitization process is enhanced there simply by
a judicious factorization of the inner-product metric (cf. formula (9) below).
Naturally, the ultimate amended physical space ${\cal H}_{phys}$ must be
constructed as equivalent to its “user-unfriendly” textbook predecessor ${\cal
L}$. By construction [5, 7], Hilbert space ${\cal H}_{phys}$ only differs from
manifestly unphysical ${\cal H}_{math}$ by an amendment and factorization of
the inner-product metric (i.e., in our present notation, of operator $\Theta$)
in a way illustrated by the diagram of Eq. (7) below.

In section 3 we will emphasize that both of the latter amendments are
essential. They will be shown to lead to a decisive ultimate simplification of
correspondence between spaces ${\cal L}$ and ${\cal H}_{phys}$. In the same
section we will also recall our recent letter [11] where we described the
first nontrivial implementation of the idea. Basically, we just considered
$N=2$ and proposed there a modified and slightly amended version of the
Bender-inspired ${\cal PT}-$symmetric quantum mechanics. Although we
encountered there certain conceptual obstacles (so that we simply did not
think about $N>2$), still, letter [11] can be perceived as a basic inspiration
of our present message.

Briefly, our present proposal can be characterized as an implementation of the
same non-Hermitian model-building strategy using arbitrary $N\geq 2$. The
explicit form of the innovation will be outlined in section 4, while its
physical background and consequences will be discussed in section 5. In a
complementary section 6 we will finally add a few comments on the
correspondence between the present $N-$step Hermitizations of the Hilbert-
space-changing form ${\cal H}_{math}\to{\cal H}_{phys}$ and the older, Dyson-
inspired [8, 12] and Hamiltonian-changing model-building flowcharts.

The summary and outline of some potential practical benefits of our present
generalized SP (GSP) reformulation of quantum mechanics will finally be given
in section 7.

##  2 Quantum mechanics using two inner products ($N=1$)

Even in the purely methodical context the Buslaev’s and Grecchi’s construction
connecting the asymptotically repulsive “wrong sign” complex potential in (1)
with the asymptotically well-behaved double-well potential in (2) is
exceptional. Indeed, both of the related partner Hamiltonians possessing the
discrete and real bound-state spectra are still just ordinary differential
operators of the second order. In contrast, the analogous would-be isospectral
partners $\mathfrak{h}^{(BB)}$ of all of the Bender’s and Boettcher’s [3]
models of the form $H^{(BB)}=-d^{2}/dx^{2}+V^{(BB)}(x)$ have only been
constructed approximatively [5, 7, 13]. Moreover, even this indicated that
their form (which, as required, was self-adjoint in ${\cal L}$) was non-local
and extremely complicated, represented by differential operators of infinite
order. After all, such an explicit construction experience only reconfirmed
the expectable contrast between a guaranteed simplicity (i.e., computational
tractability) of “input” $H$ and a prohibitive complexity of a generic
“output” $\mathfrak{h}$.

The same contrast can be also detected in an opposite extreme of practical
realistic calculations. In the above-cited review [8], for example, the
authors’ study of the Hermitization flowchart

$H\,\to\,\mathfrak{h}=\Omega\,H\,\Omega^{-1}\,$ (3)

was in fact inspired by the Dyson’s realistic many-body calculations [12]. In
them, the arrow in Eq. (4) had only been drawn in opposite direction,

$\mathfrak{h}^{(Dyson)}\,\to\,H^{(Dyson)}=\Omega^{-1}\,\mathfrak{h}^{(Dyson)}\,\Omega\,.$
(4)

Still, the motivation of the use of the isospectral mapping $\Omega$ remained
precisely the same: There was no doubt about the choice between the realistic
but user-unfriendly many-fermion Hamiltonian $\mathfrak{h}^{(Dyson)}$ and its
judiciously pre-conditioned isospectral bosonic partner $H^{(Dyson)}$ of Eq.
(4). Indeed, as long as the Dyson’s choice of the preconditioning operator
(or, in fact, matrix) $\Omega^{(Dyson)}$ was supported by intuition, the
mapping (4) could really lead to a simplification of Schrödinger equation,
especially when Dyson decided to work with the most general non-unitary
mappings such that

$\Omega^{\dagger}\Omega=\Theta\neq I\,.$ (5)

###  2.1 Dyson’s transformation and crypto-Hermitian Hamiltonians

During the application of the conventional textbook SP approach to certain
many-particle quantum systems the convergence of variational methods may
become prohibitively slow. In these cases the Hamiltonian is, typically, a
complicated partial differential operator $\mathfrak{h}$ acting in a
complicated many-body Hilbert space ${\cal L}=L^{2}(\mathbb{R}^{d})$ with a
large dimension-representing exponent $d$. The convergence becomes
particularly slow when the particles are fermions. In this case, indeed, the
system has to obey the Pauli exclusion principle [14] so that “it has become
customary to map Hermitian fermion operators onto non-Hermitian boson
operators” [8]. Purely empirically it has been revealed, originally by Dyson
[12], that the calculations (typically, of the low-lying spectra) may
perceivably be accelerated via a judicious a priori  simulation of the
correlations. In the language of mathematics this means that it makes sense to
replace the initial “fermionic” Hilbert space ${\cal L}$ by an auxiliary (and,
formally, non-equivalent) “bosonic” Hilbert space, say, ${\cal H}_{math}$. In
the latter (and, by assumption, user-friendlier) space the system becomes
described by a new Hamiltonian (say, $H$). Naturally, a successful (often,
just intuition-supported) estimate and simulation of the structure of the
correlations of fermions has often been found to imply a decisive acceleration
of the convergence of the variational treatment of the bosonic operator $H$.
In the language of numerical mathematics one may speak about an isospectral
preconditioning (4) of the Hamiltonian.

The difference between the “good” and “bad” choice of the operator $\Omega$
mapping ${\cal H}_{math}$ onto ${\cal L}$ is strongly model-dependent. One of
the conditions of success is that this mapping (often called Dyson mapping)
has to be as general and adaptable as possible, i.e., in particular, non-
unitary (cf. (5)). In a way explained in [6] the new, preconditioned, user-
friendlier Hamiltonian $H$ can be perceived as defined in another Hilbert
space ${\cal H}_{math}$ which is, due to property (5), not equivalent to its
physical predecessor ${\cal L}$: In the Dyson’s realistic calculations [12],
for example, the fermionic states in ${\cal L}$ were represented by the
bosonic states in ${\cal H}_{math}$.

In general, the Dyson’s rules (4) and (5) render $H$ non-Hermitian in ${\cal
H}_{math}$,

$H\neq H^{\dagger}=\Theta\,H\,\Theta^{-1}\,.$ (6)

For any preselected candidate $H$ for Hamiltonian the constraint (6) can be
read as specifying an amended, alternative, correct and physical inner product
in Hilbert space ${\cal H}_{phys}$ [7]. For this reason, the well known Stone
theorem [2] is not violated because the unitary evolution of the system is in
fact generated, by non-Hermitian $H$, in the auxiliary Hilbert space ${\cal
H}_{math}$. This space is not equivalent to its physical partner ${\cal
H}_{phys}$. Thus, one should rather call $H$ crypto-Hermitian [9].

In this setting it is probably useful to point out that the preservation or
failure of the “duality” or “complementarity” between ${\cal H}_{math}$ and
${\cal H}_{phys}$ may be fairly sensitive to the detailed properties of the
Hamiltonian in question. For a word of warning one does not even need to go to
the non-Hermitian quantum field theory where the Stone theorem need not apply.
Indeed, it is fully sufficient to see that multiple subtle formal problems
like, e.g., the possibility of the non-existence of the metric may emerge in a
finite-dimensional model (see, e.g., section Nr. 3 in review [8]).

The main aim of transformation (4) is that the description of dynamics
(originally provided by $\mathfrak{h}$) is now shared by the two operators
(viz., by $H$ and $\Omega$ or $\Theta$). In practical calculations, this makes
the Dyson-inspired SP more flexible, in principle at least. Still, all of the
mathematical operations have to be performed, exclusively, in the auxiliary
Hilbert space ${\cal H}_{math}$. Although the latter space seems to play just
a technical role, its key merit is that it can be re-read, after the mere ad
hoc  redefinition of the inner product (see [7, 8] or formula (24) below, with
$K=2$) as a new, unitarily equivalent representation ${\cal H}_{phys}$ of the
“old and missing” Hilbert space ${\cal L}$. The conventional probabilistic
interpretation of the unitary quantum dynamics gets restored.

The non-Hermiticity of $H$ in the working space ${\cal H}_{math}$ leads to the
necessity of using some less straightforward methods of solution of the
underlying Schrödinger equation. In many models, the merits of the
adaptability of the non-unitary mapping (often called, in the light of papers
[12], Dyson map) proved to prevail (cf., e.g., the success and productivity of
this approach in nuclear physics [14]). Still, one of the unavoidable
consequences of the non-unitarity of $\Omega$ is that the conventional
interpretation of the eigenstates of $H$ has to be modified. In the manner
described in reviews [6, 8] the situation may be clarified and visualized
using the following diagram

$\begin{array}[]{c}\ \ \begin{array}[]{|c|}\hline\cr\ {\rm inaccessible\
physical\ Hilbert\ space}\ {\cal L}\ {\rm of\ textbooks}\\\
\hline\cr\end{array}\\\ \ \ \ \ \ \ \ {\rm(Dyson^{\prime}s\ map}\ \Omega)\
\nearrow\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nwarrow\searrow\
{\rm(equivalence)}\\\ \ \ \begin{array}[]{|c|}\hline\cr\ {\rm friendlier\
representation\ space}\ {\cal H}_{math}\\\ \hline\cr\end{array}\
\stackrel{{\scriptstyle{\rm(metric}\ \Theta)}}{{\longrightarrow}}\
\begin{array}[]{|c|}\hline\cr\ {\rm alternative\ physical\ space}\ {\cal
H}_{phys}\\\ \hline\cr\end{array}\\\ \\\ \end{array}$ (7)

This diagram indicates that in certain specific non-Hermitian-Hamiltonian
models one has to separate the probabilistic physical predictions formulated
in ${\cal H}_{phys}$ from the explicit calculations performed, much more
efficiently, in ${\cal H}_{math}$. The space ${\cal L}$ and the SP Hamiltonian
$\mathfrak{h}$ have to be abandoned and replaced by the doublet ${\cal
H}_{math/phys}$ and by the preconditioned Hamiltonian $H$, respectively. A
very general crypto-Hermitian SP (CHSP) of review [8] is born.

The essence of the CHSP formalism has most concisely been explained by
Mostafazadeh [7]. He put emphasis upon the interplay of the inner-product
structures in ${\cal H}_{math/phys}$. He emphasized that the standard physical
role is played by ${\cal H}_{phys}$. Although the merits of the introduction
of another, apparently redundant Hilbert space ${\cal H}_{math}$ seem less
obvious, its introduction separated the interpretations (in ${\cal H}_{phys}$)
from calculations (in ${\cal H}_{math}$) and facilitated the qualitative as
well as quantitative predictions (cf., once more, the above-mentioned Dyson’s
study of ferromagnetism [12]). At the same time, during the implementation of
the general CHSP theory people encountered also serious technical obstacles
(some of them were listed on p. $1216\ $ of review [7]). For this reason, only
the most recent simplifications made the theory really widely known and
successful [15].

###  2.2 ${\cal PT}-$symmetric quantum mechanics

The CHSP approach can be perceived as equivalent to the standard quantum
mechanics, being distinguished just by the conversion of the conventional SP
Hamiltonian $\mathfrak{h}\,$ into its less standard isospectral representation
$H$ defined as acting in ${\cal H}_{math}$ and amenable to necessary
Hermitization. Nevertheless, the CHSP formalism has only recently been
converted, via decisive simplifications, into one of the most influential and
popular versions of SP, widely known as ${\cal PT}-$symmetric quantum
mechanics (PTQM, [5]).

In a slightly less general PTQM setting, the key role is still played by the
details of the metric-mediated transition from the auxiliary Hilbert space
${\cal H}_{math}$ to its correct physical partner ${\cal H}_{phys}$. In a way
indicated by the horizontal arrow in diagram (7) the fundamental message that
the Hamiltonian $H$ must be Hermitian in ${\cal H}_{phys}$ remains unchanged.
In the literature, unfortunately, this message is often obscured by the
diversity of notation conventions (see their sample in Table Nr. 1 of [16]).
We will use here, therefore, the properly modified Dirac’s bra-ket formalism
[1] and the maximally compact abbreviations for ${\cal H}_{phys}={\cal R}_{0}$
and ${\cal H}_{math}={\cal R}_{N}$. This will enable us to distinguish easily
between the quantum mechanics of textbooks (in which we may put $N=0$ to imply
that ${\cal H}_{math}\ \equiv\ {\cal H}_{phys}\ \equiv\ {\cal L}$) and the
menu of the two-space CHSP scenarios in which one may choose $N\geq 1$
emphasizing the non-equivalence between ${\cal H}_{math}={\cal R}_{N}$ and
${\cal H}_{phys}={\cal R}_{0}$.

One of the sources of the user-friendliness of the general CHSP formalism as
described by Scholtz et al [8] is that once we put, for simplicity, $N=1$, we
just have to consider the doublet

$\\{{\cal H}_{math},{\cal H}_{phys}\\}\ =\ \\{{\cal R}_{1},{\cal R}_{0}\\}\,$
(8)

of relevant Hilbert spaces. This clearly differs from the conventional quantum
theory in which one sets $N=0$. After innovation (8) the Hilbert space ${\cal
H}_{math}$ becomes unphysical and the Hamiltonian $H$ becomes manifestly non-
Hermitian in this space.

During the necessary Hermitization of $H$ the main technical task is the
construction of the correct physical metric operator $\Theta$ which would be
compatible with the Hamiltonian crypto-Hermiticity alias  pseudo-Hermiticity
[7] alias  quasi-Hermiticity [8] condition (6). It is worth a comment that the
key role of operator $\Theta$ (with the upper-case Greek-letter symbol
proposed in [6]) is in a sharp contrast with a lack of its sufficiently widely
accepted denotation. In Refs. [12], [8] or [7], for example, these physical
Hilbert-space metrics may be found denoted by the very different symbols like
$F$, $T$ or $\eta_{+}$, respectively.

In the most straightforward Bender-inspired PTQM approach [5] the metric
operator is constructed in the form of product

$\Theta_{(Bender)}={\cal PC}\,$ (9)

where ${\cal P}$ is parity and where the symbol ${\cal C}$ denotes the so
called charge [17]. The introduction of such an ansatz proved well motivated.
In many models it appeared to offer a perceivable simplification of the
calculations as well as of the subsequent necessary extraction of the model-
dependent testable and also, in principle, falsifiable predictions [7, 15].

##  3 Quantum mechanics using three inner products ($N=2$)

Our present paper will be devoted to a certain conceptual and methodical
extension and completion of the CHSP and PTQM approaches. We felt motivated by
the observation that in a broader area of physics both of these recipes
appeared comparatively difficult to implement. Various authors offered
remedies involving the restriction of attention to the bounded-operator
Hamiltonians [8] or the use of specific models in which the SP Hamiltonian $H$
is ${\cal PT}-$symmetric (see [5]). It is worth adding that the latter
approach was based on a parity-related (i.e., ${\cal P}-$related) and time-
reversal-related (${\cal T}-$related) assumption $H{\cal PT}={\cal PT}H$ known
as ${\cal PT}-$symmetry of the Hamiltonian. This was an intuitively appealing
feature which made the PTQM approach particularly popular, often even beyond
its original scope and restriction to the closed and unitary quantum systems
[18].

###  3.1 Intermediate space

Strictly speaking, several differences between the closely related CHSP and
PTQM unitary models are non-trivial [7]. This fact happened to be obscured by
the slightly misleading current terminology. Pars pro toto, let us mention
that the Hamiltonian operators $H$ which are all required to possess the real
spectrum and which are all required self-adjoint in the correct physical
Hilbert space ${\cal H}_{phys}$ are still called non-Hermitian in the
literature. Presumably, the reasons are psychological: All of the necessary
two-space calculations are, naturally, performed in just one of the frames,
viz., in the auxiliary and manifestly unphysical Hilbert space ${\cal
H}_{math}$ in which the Hamiltonians really are  non-Hermitian.

In our recent letter [11] we paid more attention to the terminology. We
decided to choose $N=2$ and to replace the CHSP Hilbert-space doublet (8) by
triplet

$\\{{\cal H}_{math},{\cal H}_{intermediate},{\cal H}_{phys}\\}\ =\ \\{{\cal
R}_{2},{\cal R}_{1},{\cal R}_{0}\\}\,.$ (10)

The introduction of the intermediate inner-product space helped us to throw
new light on the terminology as well as on the fundamental operator-product
(9). In the resulting intermediate-space SP (ISP) version of the formalism the
correct probabilistic physical interpretation of a given set of some
preselected candidates $\Lambda_{m}$ for the operators of observables with
$m=0,1,\ldots,{M}\,$ has been clarified.

On these grounds, the Hermitization did not proceed directly from ${\cal
H}_{math}$ to ${\cal H}_{phys}$ (as, for example, in Ref. [8]) but rather
indirectly, via the third inner-product space (i.e., Hilbert or Krein space)
${\cal H}_{intermediate}$. A key to the comparison of ISP and PTQM has been
found in the charge. Indeed, in the PTQM framework the theory only becomes
consistent after one guarantees the ${\cal PCT}-$symmetry $\,H{\cal PCT}={\cal
PCT}H\,$ of the Hamiltonian [17]. In parallel, in the ISP context the charge
appeared to play a double role, i.e., not only the original role of component
of the correct physical inner-product metric (9) which determines the geometry
in ${\cal H}_{phys}={\cal R}_{0}$, but also a new role of an auxiliary metric
operator in ${\cal H}_{intermediate}={\cal R}_{1}$.

For the sake of an enhancement of clarity we will now change the notation and
abbreviate ${\cal C}=Z_{1}$ (emphasizing the geometry-determining role of the
charge in ${\cal R}_{1}$) and ${\cal P}=Z_{2}$ (underlining the analogous
auxiliary-metric-operator role of the – possibly, generalized – parity in
${\cal H}_{math}={\cal R}_{2}$). Another abbreviation will be used to
emphasize the privileged status of the specific physical metric
$\Theta_{(Bender)}={\cal PC}=Z_{2}Z_{1}=Y_{2}\,$ of Eq. (9). Marginally, we
might add that in the literature the reference to the “parity” survived even
when the operator ${\cal P}=Z_{2}\,$ itself (re-denoted by Greek lower-case
$\eta$ in [7] and required to be self-adjoint in ${\cal R}_{2}$,
$Z_{2}=Z_{2}^{\dagger}$) ceased to be immediately connected with spatial
reflection (check also [19]).

###  3.2 Hermitian conjugation as an ambiguous, inner-product-dependent
concept

Needless to emphasize, the unmodified Dirac’s notation conventions can only be
used in the mathematical-space extreme of a maximal Hilbert-space subscript
$j=N$. Thus, at $N=2$, the conventional Hermitian conjugation of an operator
can be written in the conventional Dirac’s form $\Lambda\to\Lambda^{\dagger}$
in ${\cal H}_{math}={\cal R}_{2}$. Otherwise, in every other space ${\cal
R}_{j}$ with $j<N$ we will mark the generalized Hermitian conjugation of
operators as follows,

$\Lambda\to\Lambda^{\ddagger(j)}\,$

i.e., by a dedicated space-dependent superscript.

Concerning the Hermitian conjugations of the ket vectors, the basic
inspiration of its present, modified-Dirac denotation lies in a rather
elementary fact that any Hilbert space (with the inner product
$\langle\psi_{a}|\psi_{b}\rangle$ linear in the second, ket-vector argument)
can be perceived as an ordered pair $[{\cal V},{\cal V}^{\prime}]$ of a linear
topological vector space ${\cal V}$ (of conventional ket-vector elements
$|\psi\rangle$) and of the dual vector space of linear functionals marked by
the prime, ${\cal V}^{\prime}$ (see, e.g., p. 246 in textbook [1] where,
incidentally, also one of the best detailed explanations of the standard bra-
ket notation conventions can be found).

As long as the three Hilbert spaces (10) have to share the same linear ket-
vector-space component ${\cal V}$ (cf. Ref. [11]), it is necessary to
individualize, by the notation, the respective duals ${\cal V}^{\prime}$,
i.e., the respective invertible antilinear correspondences (we will write
${\cal T}:{\cal V}\to{\cal V}^{\prime}$). It would be insufficient to use the
same Dirac-recommended bra-vector symbol $\langle\chi|$ for the denotation of
the elements of all of the different dual spaces of our interest.

An appropriately adapted notation will be used here, therefore. In essence, we
will first define the inner product in ${\cal H}_{phys}={\cal R}_{0}$ via an
explicit specification of the correct physical mappings ${\cal T}_{phys}$.
Only after  such a preparatory step one can formulate the phenomenological
predictions deduced from the evaluation of the corresponding matrix elements
of the relevant observables. For this purpose we will use the matrix-
multiplication-resembling left-action convention to define

${\cal T}_{phys}:|\psi\rangle\ \to\ \langle\psi|\,{\Theta}\,,\ \ \ \
{\Theta}={\Theta}^{\dagger}\,$ (11)

Here the Hilbert-space-metric operator $\Theta$ is precisely the one entering
the crypto-Hermiticity constraint (6). In this context it is important to re-
emphasize that the choice of the metric $\Theta$ only becomes admissible when
one guarantees that this operator is self-adjoint in ${\cal H}_{math}$ [8].

In connection with Eq. (11) we should note that multiple nontrivial technical
problems may emerge when the dimension of our topological vector space ${\cal
V}$ of kets is infinite. This aspect of the theory must carefully be discussed
in the framework of functional analysis [20]. As a complementary further
reading in this direction let us mention the old Dieudonné’s paper [21], or
the recent Refs. [22] and [23]. This being said, it is useful to add,
explicitly, that in any case, the Hilbert-space metric operator $\Theta$ must
necessarily be chosen invertible and positive definite and bounded, with
bounded inverse [8, 20].

One of the other and most important related technical obstacles is that in the
general CHSP framework our choice of the “physical” inner product (i.e., of
the invertible antilinear Hermitian-conjugation correspondence (11)) is, given
the Hamiltonian, non unique. An exhaustive discussion of this topic can be
found in the literature [1, 7, 24]. The ambiguity of the inner product is only
irrelevant in ${\cal H}_{math}={\cal R}_{N}$ where we may use the conventional
Dirac notation

${\cal T}_{math}:|\psi\rangle\ \to\ \langle\psi|$ (12)

and where we may define the Hermitian conjugation operation as a mapping in
which $|\psi\rangle$ is a column vector while $\langle\psi|$ is perceived as
its row-vector transposition containing complex-conjugate elements.

Let us now fix $N=2$ and let us consider the triplet $\\{{\cal R}_{j}\\}$ of
the inner-product spaces (10) as in [11]. Then, the mathematical extreme of
Eq. (12) might optionally be marked by a subscripted bracketed $j=N$,

${\cal T}_{[N]}:|\psi\rangle\ \to\ \langle\psi|\ \equiv\
\langle_{[N]}\psi|\,.$ (13)

At the other two $j<N$ the subscripted index becomes obligatory. Thus, at the
correct and physical $j=0$ extreme of Eq. (11) with $\Theta=Y_{2}$ we will
write

${\cal T}_{[0]}:|\psi\rangle\ \to\ \langle_{[0]}\psi|\ \equiv\
\langle\psi|\,Y_{2}\,.$ (14)

As long as the auxiliary metric $Z_{2}=Z_{2}^{\dagger}\,$ in ${\cal
H}_{math}={\cal R}_{2}\,$ is tractable as the (possibly, generalized) parity
[19], we will finally define the intermediate-space Hermitian conjugation as
follows,

${\cal T}_{[1]}:|\psi\rangle\ \to\ \langle_{[1]}\psi|\ \equiv\
\langle\psi|\,Z_{2}\,.$ (15)

The specific PTQM (or rather, with $N=2$, ISP) realization of the general CHSP
framework can be now characterized by formula (9). It suppresses the ambiguity
of the inner-product metric $\Theta_{(Bender)}=Y_{2}=Z_{2}\,Z_{1}$. It also
implies that besides the Hamiltonian $H$ (alternatively denoted here as
$Z_{0}$), also the charge $Z_{1}$ is kept observable.

In the original PTQM proposal [17] the formula for $\Theta_{(Bender)}$ was
required to contain a self-adjoint  operator of charge. In the alternative ISP
framework of paper [11] the same factorization ansatz admited a more general
charge. In contrast to its status in PTQM, it was admitted non-Hermitian  in
${\cal H}_{math}$. In our present terminology the charge $Z_{1}$ acquired the
physical meaning of another non-Hermitian and crypto-Hermitian but
Hermitizable observable-representing operator.

##  4 Quantum mechanics using $K-$plets of inner products

The choice of $N=2$ which characterizes the ISP Hermitization of Ref. [11] can
be generalized. The number $N$ of the auxiliary, manifestly unphysical inner-
product spaces ${\cal R}_{j}$ with $j>0\,$ can be, in the resulting GSP
extension of the ISP formulation of quantum mechanics, arbitrarily large. The
ISP triplet (10) of the inner-product spaces sharing the same ket-vector
elements of ${\cal V}$ will be replaced by the $K-$plet

$\\{{\cal H}_{math},{\cal R}_{K-2},{\cal R}_{K-3},\ldots,{\cal R}_{1},{\cal
H}_{phys}\\}\,$ (16)

with $K=N+1$ and with alternative symbols for the extremely mathematical
inner-product space ${\cal H}_{math}={\cal R}_{K-1}$ and for the extremely
physical Hilbert space ${\cal H}_{phys}={\cal R}_{0}$.

###  4.1 The inner-product-dependent Hermitian conjugations

For any preselected operator $\Lambda$ defined as acting upon kets
$|\psi\rangle\in{\cal V}$, its Hermitian conjugate partners will be inner-
product-dependent. At any $K$ these partners will form the $K-$plet

$\Lambda^{\ddagger(0)}\,,\,\Lambda^{\ddagger(1)}\,,\,\ldots\,,\,\Lambda^{\ddagger({K}-2)}\,,\,\Lambda^{\ddagger({K-1})}\,$
(17)

with elements marked by the inner-product-dependent superscripts. In every
space ${\cal R}_{j}$ we will consider an operator $Z_{j}\ $ which will be
required self-adjoint in the respective space,

$Z_{j}=Z_{j}^{\ddagger(j)}\,,\ \ \ \ j=0,1,\ldots,{K-1}\,.$ (18)

This assumption will enable us to treat, in the GSP framework, every $Z_{j}$
as an analogue of the ISP metric $\Theta$ of Eq. (11). This means that every
such an operator will have to satisfy the consistency conditions as listed,
e.g., in Eq. Nr. (2.1) of the physics-oriented review [8]. These mathematical
conditions involve the domain-completeness (cf. Nr. (2.1a)), Hermiticity
(2.1b) (equivalent to our present Eq. (18)), positive definiteness (2.1c) and
the metric-boundedness constraint Nr. (2.1d) plus, let us add, the metric-
inverse boundedness constraint missing in [8]. Still, the authors of review
[8] formulated a consistent theory because they managed to circumvent certain
formal difficulties as mentioned by Dieudonné [21] by restricting their
attention, drastically, just to the not too realistic models possessing
bounded-operator Hamiltonians. In contrast, Dieudonné himself revealed that
once the class of Hamiltonians remains sufficiently general, the ISP metric
$\Theta$ need not exist at all.

In 2015 the situation has been reconsidered and summarized in the
mathematically oriented monograph [20] where the readers may find that for
unbounded operators, a sufficiently general and still mathematically
satisfactory formulation of sufficient conditions of the applicability of the
theory is still an open problem. At the same time, there are no formal
obstacles in all of the phenomenological applications working with finite
matrices. In our present paper we decided to pay attention, predominantly,
just to the algebraic aspects and amendments of the theory. This means that
our readers will either keep in mind the sufficiently elementary (and,
typically, finite-dimensional) implementations of the theory or,
alternatively, they will recall the highly sophisticated specialized
literature (pars pro toto, let us recommend Ref. [22] for introductory
reading).

Somewhere in between the two extremes, one may consider the separable (i.e.,
still sufficiently general) Hilbert spaces ${\cal R}_{j}=[{\cal V},{\cal
V}^{\prime}]$ marked by a subscript $j=0,1,\ldots,{K-1}$. We will assume that
the set ${\cal V}$ of the ket vectors remains the same (i.e.,
$j-$independent)) while the dual sets ${\cal V}^{\prime}$ of the linear
functionals [1] become $j-$dependent, ${\cal V}^{\prime}={\cal
V}^{\prime}_{[j]}$. Every space ${\cal R}_{j}$ is self-dual. The
correspondence between a ket $|\psi\rangle\in{\cal V}$ and its dual can be
perceived as a result of action of an invertible antilinear operator ${\cal
T}_{[j]}$. The $K-$plet of Hermitian conjugations yields the subscript-
dependent bra-vectors,

${\cal T}_{[j]}:|\psi\rangle\ \to\ \langle_{[j]}\psi|\,,\ \ \ \
j=0,1,\ldots,{K-1}\,.$ (19)

We will drop again the highest subscript $j={K-1}$ as redundant,
$\langle_{[{K-1}]}\psi|\,\equiv\,\langle\psi|$. This underlines that the
Hilbert space ${\cal R}_{K-1}={\cal H}_{math}$ is a privileged one, preferred
in the calculations.

Next, we may generalize the two ISP rules (15) and (14) and set
$\langle_{[{K-2}]}\psi|=\langle\psi|\,Z_{K-1}$ and
$\langle_{[{K-3}]}\psi|\,Y_{K-1}$, respectively. These bra-vectors are just
the first two transforms of the same preselected element. The list is easily
completed at any $K>3$. Using an antilexicographically ordered set of
arbitrary left-acting operators we may define

$\langle_{[{K}-2]}\psi|=\langle\psi|\,Z_{K-1}\,,\ \ \
\langle_{[{K}-3]}\psi|=\langle\psi|\,Y_{K-1}\,,\ \ \
\langle_{[{K}-4]}\psi|=\langle\psi|\,X_{K-1}\,,\ \ \ldots\,,$ (20)

with the last-item $\langle_{[0]}\psi|=\langle\psi|\,\Theta$.

In GSP it makes sense to replace the $(K-1)-$subscripted operators entering
Eq. (20) by the sequence of the $Z_{j}\,$ operators with
$j={K-2},{K-3},\ldots$. Such a replacement can be realized via the sequence of
linear equations

$Y_{j}=Z_{j}\,Z_{j-1}\,,\ \ \ \ X_{j}=Z_{j}\,Y_{j-1}\,,\ \ \ \
W_{j}=Z_{j}\,X_{j-1}\,,\ \ \ \ \ldots\,,\ \ \ \ j={K-1},{K-2},\ldots\,,$ (21)

i.e., via recurrences

$Z_{j-1}=Z_{j}^{-1}\,Y_{j}\,,\ \ \ \ Y_{j-1}=Z_{j}^{-1}\,X_{j}\,,\ \ \ \
X_{j-1}=Z_{j}^{-1}\,W_{j}\,,\ \ \ \ \ldots\,,\ \ \ \ j={K-1},{K-2},\ldots\,.$
(22)

The bra-vectors of Eq. (19) can be then given their ultimate, exclusively
$Z_{j}-$dependent form,

${\cal T}_{[j]}:|\psi\rangle\ \to\
\langle_{[j]}\psi|=\langle\psi|\,Z_{K-1}\,Z_{{K-2}}\,\ldots\,Z_{j+1}\,,\ \ \ \
\ j={K-2},{K-3},\ldots,1,0\,.$ (23)

All of these, by assumption, mutually non-parallel bra-vectors have the same
structure. The formula reproduces the ${K}=3$ and ${K}=2$ special cases
appearing in the quasi-Hermitian quantum mechanics of reviews [7, 8, 25]. The
“non-metric” operator $Z_{0}\ $ playing the role of the physical SP
Hamiltonian does not enter our final definition (23) of course.

We are now prepared to define, in all of our Hilbert spaces ${\cal R}_{j}$,
the respective inner products $(\psi_{1},\psi_{2})_{{\cal
R}_{j}}\,\equiv\,\langle_{[j]}\psi_{1}|\psi_{2}\rangle$ in recursive manner,

$\langle_{[j]}\psi_{1}|\psi_{2}\rangle=\langle_{[j+1]}\psi_{1}|Z_{j+1}|\psi_{2}\rangle\,,\
\ \ \ j={K}-2,{K}-3,\ldots,1,0\,.$ (24)

Only the “friendliest”, $j={K-1}$ item is allowed to be written in the
conventional Dirac’s bra-ket form without subscripts again,
$(\psi_{1},\psi_{2})_{{\cal R}_{{K-1}}}=\langle\psi_{1}|\psi_{2}\rangle$.

Table 1: List of auxiliary Hermiticity relations. |  | metric | operator product
---|---|---|---
$j$ | space | $Z_{j}$ | $Z_{j}Z_{j-1}=Y_{j}$ | $Z_{j}Y_{j-1}=X_{j}$ | $Z_{j}X_{j-1}=W_{j}$ | …
$1$ | ${\cal R}_{1}$ | $Z_{1}=Z_{1}^{\ddagger(1)}$ | $Y_{1}=Y_{1}^{\ddagger(1)}$ |  |  |
$2$ | ${\cal R}_{2}$ | $Z_{2}=Z_{2}^{\ddagger(2)}$ | $Y_{2}=Y_{2}^{\ddagger(2)}$ | $X_{2}=X_{2}^{\ddagger(2)}$ |  |
$3$ | ${\cal R}_{3}$ | $Z_{3}=Z_{3}^{\ddagger(3)}$ | $Y_{3}=Y_{3}^{\ddagger(3)}$ | $X_{3}=X_{3}^{\ddagger(3)}$ | $W_{3}=W_{3}^{\ddagger(3)}$ |
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | $\ddots$

###  4.2 Systematic replacements of Hermiticities by pseudo-Hermiticities

In every inner-product space ${\cal R}_{j}$ we postulated the $j-$dependent
Hermiticity property (18) of $Z_{j}$ (cf. also the third column in Table 1).
At $j=0$ this relation represents the most important dynamical property of the
Hamiltonian while the rest of the list concerns the $j-$dependent metrics.
From the point of view of the users working in ${\cal H}_{math}$, only the
$j=N$ item

$Z_{N}=Z^{\ddagger(N)}_{N}=Z^{\dagger}_{N}$

can directly be tested and verified. The rest of the list with $j<N$ will only
be accessible to the verification after its pull-down to ${\cal H}_{math}$.

In the first step of such a process, in the manner proposed in [11], every
Hermiticity relation for a metric (18) becomes re-interpreted as a crypto-
Hermiticity requirement

$Z_{j}^{\ddagger(j+1)}\,Z_{j+1}=Z_{j+1}\,Z_{j}\,,\ \ \ \ j=0,1,\ldots,K-2\,$
(25)

imposed upon the same operator $Z_{j}$ in the different inner-product context
of the neighboring, more friendly space ${\cal R}_{j+1}$. The series of
transformations of the picture to the more friendly space can be iterated. For
every initial choice of the subscript $j$ the ultimate goal of the iterations
is to reach the formally equivalent representation of property (18) in the
mathematically optimal space ${\cal H}_{math}={\cal R}_{K-1}$.

For illustration purposes let us now return to the ${K}=3$ scenario as
discussed in [11]. The first-step quasi-Hermiticity (25) has been shown there
to imply the charge-pseudo-Hermiticity of the Hamiltonian as well as the
parity-pseudo-Hermiticity of the charge (see Table Nr. 1 in loc. cit.). The
proof was based on the antilexicographically ordered relations (21) and on the
observation that $Z_{1}\,Z_{0}=Y_{1}=Y_{1}^{\ddagger(1)}$. The latter relation
is interesting also per se  because the operator-product quantity $Y_{1}$ is
of an immediate phenomenological interest in the random matrix theory [26] or
in the open-system physical context (cf. Ref. [27] where such a quantity has
been found conserved).

In the closed-system setting of Ref. [11] the construction has been completed
by the quasi-Hermiticity $Y_{1}^{\ddagger(2)}\,Z_{2}=Z_{2}\,Y_{1}$ holding in
${\cal R}_{2}={\cal H}_{math}$. This relation proves equivalent to the parity-
charge pseudo-Hermiticity of the admissible Hamiltonians. In this sense, the
explicit reference to the auxiliary space ${\cal R}_{1}$ and even to the
physical Hilbert space ${\cal R}_{0}$ itself becomes, from the practical
user’s point of view, redundant.

Table 2: Pseudo-Hermiticity properties of operators $Z_{k}$ at $k<j$. $j$ |  | $k=j-1$ | $k=j-2$ | $k=j-3$ | …
---|---|---|---|---|---
|  | pseudo-metric | pseudo-metric | pseudo-metric |
| space | $Z_{j}$ | $Y_{j}=Z_{j}Z_{j-1}$ | $X_{j}=Z_{j}Z_{j-1}Z_{j-2}$ | …
$1$ | ${\cal R}_{1}$ | $Z_{0}^{\ddagger(1)}\,Z_{1}=Z_{1}\,Z_{0}$ |  |  |
$2$ | ${\cal R}_{2}$ | $Z_{1}^{\ddagger(2)}\,Z_{2}=Z_{2}\,Z_{1}$ | $Z_{0}^{\ddagger(2)}\,(Z_{2}\,Z_{1})=(Z_{2}\,Z_{1})\,Z_{0}$ |  |
$3$ | ${\cal R}_{3}$ | $Z_{2}^{\ddagger(3)}\,Z_{3}=Z_{3}\,Z_{2}$ | $Z_{1}^{\ddagger(3)}\,(Z_{3}\,Z_{2})=(Z_{3}\,Z_{2})\,Z_{1}$ | $Z_{0}^{\ddagger(3)}\,(Z_{3}\,Z_{2}\,Z_{1})=(Z_{3}\,Z_{2}\,Z_{1})\,Z_{0}$ |
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | $\ddots$

####  4.2.1 $K>3$

Once we move to $K>3$, every round of the formal pseudo-Hermitization process
can be perceived as a relegation of Hermiticity to a more user-friendly space.
After a completion of the whole process one obtains an $N-$plet of the
relevant relations as sampled, in Table 2, at $j=N$. The step-by-step
derivation of these relations is straightforward. Indeed, the second round of
the relegations involves the $({K-1})-$plet of operator products

$Z_{j}\,Z_{j-1}=Y_{j}=Y_{j}^{\ddagger(j)}\,,\ \ \ \ j=1,2,\ldots,{K-1}\,.$
(26)

The manifest Hermiticity of these products in ${\cal R}_{j}$ is equivalent to
their quasi-Hermiticity property valid in the next space ${\cal R}_{j+1}$,

$Y_{j}^{\ddagger(j+1)}\,Z_{j+1}=Z_{j+1}\,Y_{j}\,,\ \ \ \
j=1,2,\ldots,{K-2}\,.$ (27)

In full analogy with the special case of ISP, each operator $Z_{j+1}\,$ plays
here the role of an intermediate, subscript-dependent Hilbert- or Krein-space
inner-product metric.

In the subsequent round of algebraic manipulations we start from the the
$Y_{j-1}-$containing operator products

$Z_{j}\,Y_{j-1}=X_{j}=X_{j}^{\ddagger(j)}\,,\ \ \ \ j=2,3,\ldots,{K-1}\,.$
(28)

We replace their Hermiticity valid in ${\cal R}_{j}$ by the quasi-Hermiticity

$X_{j}^{\ddagger(j+1)}\,Z_{j+1}=Z_{j+1}\,X_{j}\,,\ \ \ \ j=2,3,\ldots,{K}-2\,$
(29)

postulated in the next Hilbert space ${\cal R}_{j+1}$. This is in fact the key
consequence of our assumption of existence of the chain of Hermitian
conjugations, i.e., of several inner-product spaces.

The core of our message is the recommendation of a systematic and exhaustive
iterative transfer of the description of all of the operators of interest to
${\cal R}_{K-1}$. Such a process is, therefore, naturally continued by the
fourth round, with Hermiticities

$Z_{j}\,X_{j-1}=W_{j}=W_{j}^{\ddagger(j)}\,,\ \ \ \ j=3,4,\ldots,{K-1}\,$ (30)

replaced by quasi-Hermiticites

$W_{j}^{\ddagger(j+1)}\,Z_{j+1}=Z_{j+1}\,W_{j}\,,\ \ \ \
j=3,4,\ldots,{K-2}\,,$ (31)

etc. Tables 1 (implicitly) and 2 (explicit when setting $j=N$) offer a sample
of the ultimate results of these systematic replacements.

##  5 Physics behind the multi-space mathematics

The practical implementation of the GSP formalism beyond its ISP special case
where we had ${N}=K-1=2$ is a truly challenging task. In an indirect support
of its potential relevance let us mention our older paper [28]. At the time of
its publication there was, naturally, no consistent SP theory with $K=4$ at
our disposal. As long as the model in question exhibited a nonlinear
supersymmetry, we needed the charge ${\cal C}$ expressed as a product of two
operators. In the light of ansatz (9), the physical Hilbert-space metric would
have to be triply factorized, therefore, $\Theta_{3}=Z_{3}Z_{2}Z_{1}$. In
principle, the $j={N}=3$ line of our present Table 2 would apply. In
retrospective, the non-availability of a consistent GSP quantum theory with
$K=4$ was precisely the reason why the results of our study [28] remained
incomplete.

###  5.1 Observability status of the charge at arbitrary $K$

In the general$-K$ multiplet (16) one has to treat the first item ${\cal
H}_{math}={\cal R}_{K-1}$ of the sequence as the mathematically optimal,
preferred and manipulation-friendly Hilbert space. The last item ${\cal
H}_{phys}={\cal R}_{0}$ of the same sequence is, in contrast, the only Hilbert
space in which the mean values of all of the observables carry the standard
probabilistic interpretation, i.e., in which the inner product is correct and
physical. All of the other inner-product spaces play just an interpolative and
auxiliary role. In Table 2, in particular, one has to select and, in practice,
work just with the single row where $j={N}=K-1$.

At any $K$, in the manner fully consistent with Stone theorem [2],
Hamiltonians $H=Z_{0}\,$ have to be self-adjoint in ${\cal H}_{phys}={\cal
R}_{0}$. Simultaneously, all  of the relevant mathematical constructions have
to be performed in working space ${\cal R}_{N}={\cal H}_{math}$. Contextually,
the same Hamiltonian can be called Hermitian or non-Hermitian. In the
Hermitian case one simply returns to the conventional probabilistic
interpretation of the unitary system in question. In this sense the physical
interpretation of the predictions of the GSP theory using $K>1$ remains
unchanged. Only our mathematical working space ceases to be equivalent to the
correct physical Hilbert space so that the operators $\Lambda$ representing
the observables are defined, in ${\cal R}_{N}={\cal H}_{math}$, as non-
Hermitian.

The physics behind the “non-Hermitian” theory acquires the universal,
$K-$independent CHSP form, with its various phenomenological aspects
extensively discussed in Ref. [8]. In our present notation one merely
represents ${\cal H}_{phys}={\cal R}_{0}$ in ${\cal R}_{N}$ using a formal
replacement of the trivial metric in ${\cal R}_{N}={\cal H}_{math}$ by its
amended physical alternative. This enables us to work, simultaneously, with
the two inner products such that

$\langle\psi_{a}|\psi_{b}\rangle=\langle_{[N]}\psi_{a}|\psi_{b}\rangle\neq\langle_{[0]}\psi_{a}|\psi_{b}\rangle=\langle_{[N]}\psi_{a}|\Theta|\psi_{b}\rangle=\langle\psi_{a}|\Theta|\psi_{b}\rangle\,,$

keeping in mind that only the latter inner product has the standard
probabilistic interpretation.

In the most elementary nontrivial scenario we set $N=2$ and get the ISP
formalism with $\Theta=Y_{2}$. Equally well, we may choose any larger $N>2$
and get the genuine GSP formalism. Under both of these arrangements our
Hamiltonian only carries its conventional physical meaning of an observable in
${\cal R}_{0}={\cal H}_{phys}$. Hence, even the predictions of the generalized
theory remain probabilistic. The model-building process does not proceed in
the conventional physical Hilbert space ${\cal H}_{phys}$ but rather, step-by-
step, along the sequence of manifestly unphysical but perceivably user-
friendlier mathematical representation spaces ${\cal R}_{j}$ with the
decreasing subscript $j$.

There exist two main distinguishing features of the GSP models with $K>2$. The
first one is formal: the correct physical metric $\Theta$ which makes our
Hamiltonian $H$ self-adjoint is equal to an $N-$term operator product which
generalizes Eq. (9). The first few illustrative examples are displayed as the
$k=0$ items in Table 2.

The second distinguishing feature of the GSP models is rather serendipitious:
One can notice that after a tentative premultiplication by the charge ${\cal
C}=Z_{1}$ from the right, also the $k=1$ pseudo-Hermiticities as sampled in
Table 2 acquire, unexpectedly, the form equivalent to the precise crypto-
Hermiticity criterion

${\cal C}\neq{\cal C}^{\dagger}=\Theta\,{\cal C}\,\Theta^{-1}\,.$ (32)

Once we compare this relation with Eq. (6) above we see that the most
unexpected but strictly physical second characteristics of the GSP theory is
that it guarantees the observability of the crypto-Hermitian Hamiltonian
(i.e., energy) together with  the observability of the crypto-Hermitian
charge.

###  5.2 Recurrences for conjugations

The $j-$th-Hilbert-space Hermiticity [e.g., (18)] and the related $(j+1)-$th-
Hilbert-space quasi-Hermiticity of an arbitrary linear operator $\Lambda$ can
be made explicit in recurrent manner,

$\Lambda^{\ddagger(j)}=(Z_{j+1})^{-1}\,\Lambda^{\ddagger(j+1)}\,Z_{j+1}\,,\ \
\ \ \ j=0,1,\ldots,{K-2}\,.$ (33)

This pull-down of the conjugation connecting the neighboring Hilbert spaces is
an elementary consequence of definition (24) which played just a marginal role
in Ref. [11] at $K=3$ (cf. equation Nr. 10 in loc. cit.). The importance of
relation (33) grows with the growth of $K$. At any $K\geq 3$ it leads to the
closed-form definition

$\Lambda^{\ddagger(j)}=\Theta_{(K-1,j)}^{-1}\,\Lambda^{\dagger}\,\Theta_{(K-1,j)}\,,\
\ \ \ \ \Theta_{(K-1,j)}=Z_{{K-1}}\,Z_{{K-2}}\,\ldots\,Z_{j+2}\,Z_{j+1}\,$
(34)

of the $j-$th conjugation in terms of the conventional one as defined in
${\cal H}_{math}$.

For illustration one could return to Tables 1 and 2. A detailed inspection of
the Tables indicates that at any preselected ${K}$ the relegation of the
Hermiticity will terminate only after we manage to exhaust the whole set of
products (21). The systematically iterated step-by-step replacements

$Z_{0}\,(=H)\to Y_{1}\to X_{2}\to W_{3}\to\ldots\to\Theta$

will then guarantee the self-adjointness of the Hamiltonian in the correct
space ${\cal H}_{[phys}={\cal R}_{{0}}$ re-expressed as its quasi-Hermiticity
in ${\cal H}_{math}={\cal R}_{{K-1}}$.

An analogous chain of relegations of Hermiticity remains applicable to the
charge and/or to any other operator of phenomenological relevance. In the
spaces which are infinite-dimensional it would be necessary to discuss
multiple technical questions concerning the domains of operators, etc. Here,
we skipped these questions and restricted our attention just to the detailed
discussion of what could be called the underlying algebraic relations and
symmetries.

####  5.2.1 $K=3$

At ${K}=3$ the closed-form solution of the GSP recurrences was given in [11].
For methodical purposes it still makes sense to start the presentation of this
solution by the tutorial re-derivation of the first nontrivial ${K}=3$ ISP
pattern. Indeed, in our present notation the correspondence between ${\cal
H}_{phys}$ and ${\cal H}_{math}$ can be seen as mediated either by a single-
step simplification of the inner product
$\langle_{[0]}\psi_{a}|\psi_{b}\rangle\ \to\ \langle\psi_{a}|\psi_{b}\rangle$
(at $K=2$), or by a two-step realization in which the preparatory step
$\langle_{[0]}\psi_{a}|\psi_{b}\rangle\ \to\
\langle_{[1]}\psi_{a}|\psi_{b}\rangle$ is followed by the final step
$\langle_{[1]}\psi_{a}|\psi_{b}\rangle\ \to\ \langle\psi_{a}|\psi_{b}\rangle$
(at $K=3$).

Naturally, after an elementary algebraic exercise one can show that the
phenomenological consequences of the two recipes are equivalent. Indeed,
although the latter, ISP pattern with $K=3$ is prescribed by the three inner-
product-dependent Hermiticity relations (18) which read

$Z_{0}=Z_{0}^{\ddagger(0)}\,,\ \ \ \ Z_{1}=Z_{1}^{\ddagger(1)}\,,\ \ \ \
\underline{Z_{2}=Z_{2}^{\ddagger(2)}}\,,$ (35)

only the last, underlined item is in its final form living in ${\cal
H}_{math}={\cal R}_{2}$. For the other two relations we still have to find
their representation using the ${\ddagger(2)}-$marked Hermitian conjugation
defined in the user-friendliest and preferred Hilbert space ${\cal
R}_{2}={\cal H}_{math}$. For this purpose let us recall Eq. (25) and re-
express the two former relations in the respective quasi-Hermitian forms,

$Z_{0}^{\ddagger(1)}\,Z_{1}=Z_{1}\,Z_{0}\,,\ \ \ \
\underline{Z_{1}^{\ddagger(2)}\,Z_{2}=Z_{2}\,Z_{1}}\,.$ (36)

The second, underlined formula is final as it already lives in ${\cal
H}_{math}$. The former item requires a further translation using the first
formula in (26),

$Z_{1}\,Z_{0}=Y_{1}=Y_{1}^{\ddagger(1)}\,.$ (37)

This is easily transferred to ${\cal H}_{math}$ via Eq. (27),

$Y_{1}^{\ddagger(2)}\,Z_{2}=Z_{2}\,Y_{1}\,.$ (38)

What remains to be done is the insertion of definition (37),

$Z_{0}^{\ddagger(2)}\,Z_{1}^{\ddagger(2)}\,Z_{2}=Z_{2}\,Z_{1}\,Z_{0}\,$ (39)

and the incorporation of Eq. (36) yielding our final underlined equation

$\underline{Z_{0}^{\ddagger(2)}\,\Theta_{2}=\Theta_{2}\,Z_{0}}\,,\ \ \ \ \ \
\Theta_{2}=Z_{2}\,Z_{1}\,.$ (40)

This is the ultimate form of the quasi-Hermiticity of the Hamiltonian written
in terms of the Hilbert space metric $\Theta_{K-1}$ at ${K}=3$.

We see that all of the auxiliary, intermediate Hilbert spaces ${\cal R}_{j}$
with $0<j<{K-1}$ have been successfully eliminated. In terms of the operator-
product metric $\Theta_{2}$ the observability of $H$ is fully guaranteed,
strictly in the spirit of review [8], by its quasi-Hermiticity property (40)
in ${\cal H}_{math}$. A consistent quantum model can be constructed in which
all observables $\Lambda$ would have to satisfy the same quasi-Hermiticity
relation as $H$ itself does. Besides $Z_{0}$, as we already mentioned, also
operator $Z_{1}$ represents an observable quantity [for proof it is sufficient
to pre-multiply the underlined equation in Eq. (36) by $Z_{1}$ from the
right]. Analogously, the product $Z_{2}\,Z_{1}$ can represent another
observable (the proof is similar) while $Z_{2}$, when standing alone, cannot.

####  5.2.2 $K=4$

At $N=3$, the quadruplet of the Hermiticity relations (18) can be split in the
underlined final rule for $\underline{Z_{3}=Z_{3}^{\ddagger(3)}}$ in ${\cal
H}_{math}$ and the remaining auxiliary triplet

$Z_{0}=Z_{0}^{\ddagger(0)}\,,\ \ \ \ Z_{1}=Z_{1}^{\ddagger(1)}\,,\ \ \ \
Z_{2}=Z_{2}^{\ddagger(2)}\,,$ (41)

to be replaced by its quasi-Hermitian analogue (25). This yields, first of
all, the last, underlined relation
$\underline{Z_{2}^{\ddagger(3)}\,Z_{3}=Z_{3}\,Z_{2}}\ $ for $Z_{2}$ which is
already in its desired final form. In ${\cal H}_{math}={\cal R}_{3}$ this
makes the quasi- or pseudo-Hermiticity of $Z_{2}$ perceived as mediated, under
appropriate mathematical conditions, by metric $Z_{3}$. The remaining doublet
of relations

$Z_{0}^{\ddagger(1)}\,Z_{1}=Z_{1}\,Z_{0}\ (=Y_{1})\,,\ \ \ \
Z_{1}^{\ddagger(2)}\,Z_{2}=Z_{2}\,Z_{1}\ (=Y_{2})$ (42)

is restricted by the respective Hermiticity requirements
$Y_{1}=Y_{1}^{\ddagger(1)}$ and $Y_{2}=Y_{2}^{\ddagger(2)}$ [cf. Eq. (26)].
Using relation (27) they may be both quasi-Hermitized,

$Y_{1}^{\ddagger(2)}\,Z_{2}=Z_{2}\,Y_{1}\ (=X_{2})\,,\ \ \ \
\underline{Y_{2}^{\ddagger(3)}\,Z_{3}=Z_{3}\,Y_{2}\ (=X_{3})}\,.$ (43)

The second, underlined item is already written in the correct space ${\cal
H}_{math}={\cal R}_{3}$. The insertion of $Y_{2}$ from the second line of Eq.
(42) gives
$\underline{Z_{1}^{\ddagger(3)}\,(Z_{3}\,Z_{2})=(Z_{3}\,Z_{2})\,Z_{1}}$, i.e.,
the correct final rule for $Z_{1}$ formulated in ${\cal H}_{math}$.

We are left with the first item in (43). It can readily be quasi-Hermitized in
${\cal H}_{math}$,

$X_{2}^{\ddagger(3)}\,Z_{3}=Z_{3}\,X_{2}\,.$ (44)

After a series of elementary insertions we finally get

$\underline{Z_{0}^{\ddagger(3)}\,\Theta_{3}=\Theta_{3}\,Z_{0}}\,,\ \ \ \ \ \
\Theta_{3}=Z_{3}\,Z_{2}\,Z_{1}\,.$ (45)

These conclusions are again summarized in Table 2. The role of an observable
which would be manifestly self-adjoint in ${\cal H}_{phys}$ can now be played
by $Z_{1}$, by the product $Z_{2}\,Z_{1}$ and by the product
$Z_{3}\,Z_{2}\,Z_{1}$, but not by the product $Z_{3}Z_{2}$ or by the operators
$Z_{2}$ or $Z_{3}$ standing alone (the proofs are analogous to the ones given
above). Thus, in the more conventional notation one can conclude that the
“hidden Hermiticity” (i.e., the observability) status holds for $H$ and for
the charge ${\cal C}=Z_{1}$. The observability status of the “other charge”
${\cal D}=Z_{2}$ (i.e., the validity of condition ${\cal
D}^{\dagger}\,\Theta_{3}=\Theta_{3}\,{\cal D}$) could only be achieved, in a
sufficiently elementary manner, under an additional commutativity requirement
${\cal D}\,{\cal C}={\cal C}\,{\cal D}$. Otherwise, the observability status
of ${\cal D}$ would require a complicated analysis as sampled in Ref. [29].

####  5.2.3 General $K$

Among the ${K}$ Hermiticity relations (18) valid (or postulated) for the
operators $Z_{j}$, the last one (with $j=K-1$) is a ${\cal H}_{math}-$space
property of $Z_{K-1}$. Among the ${K-1}$ Hermiticity relations (25) for the
two-term products of $Z_{j}$s, the last one (with $j=K-2$) is always a valid
Hermiticity property of $Y_{K-1}$ (i.e., a valid quasi- or pseudo-Hermiticity
property of $Z_{{K-2}}$) in the same ${\cal H}_{math}-$space. Etc. Along this
line we finally arrive at $j=0$ and Hamiltonian $H=Z_{0}$. An elementary proof
by mathematical induction really yields, in the space ${\cal R}_{K-1}={\cal
H}_{math}$, the quasi-Hermiticity relation

$\underline{Z_{0}^{\ddagger({K-1})}\,\Theta_{K-1}\,\equiv\,Z_{0}^{\dagger}\,\Theta_{K-1}=\Theta_{K-1}\,Z_{0}}\,,\
\ \ \ \ \ \Theta_{K-1}=Z_{{K-1}}\,Z_{{K-2}}\,\ldots\,Z_{2}\,Z_{1}\,.$ (46)

This is our ultimate, experimentally relevant property of the Hamiltonian of
the system in question.

In the latter relation the ultimate metric operator $\Theta_{K-1}$ is
factorized in a way which is compatible with the last $j=0$ item in formula
(23) defining the ultimate and correct physical bra-vector. An internal
consistency of the GSP formalism is confirmed. The observation also opens the
possibility of a removal, in the nearest future, of our present methodical
limitation of attention to the specific SP formulation of quantum mechanics.

In this direction, two next steps of possible methodical development seem most
promising. In the first one the present stationary GSP theory could be
replaced by its non-stationary generalization. This would lead to the
implementation of the present metric-factorization idea in the non-SP context
of the so called non-Hermitian interaction picture (interested readers might
find its $K=2$ description in [16]).

In the second, alternative direction of research our present factorization of
the metric could also inspire a further progress in the context of quantum
statistical mechanics. In it, one would merely replace a pure state
characterized, in our present notation, by the elementary projector

$\pi(t)=|\psi(t)\rangle\,\frac{1}{\langle\psi_{[0]}(t)|\psi(t)\rangle}\,\langle\psi_{[0]}(t)|\,$
(47)

by the non-Hermitian density matrix

${\varrho}(t)=\sum_{k}|\psi^{(k)}(t)\rangle\,\frac{p_{k}}{\langle\psi^{(k)}_{[0]}(t)|\psi^{(k)}(t)\rangle}\,\langle\psi^{(k)}_{[0]}(t)|\,,\
\ \ \ \ \ \sum_{k}p_{k}=1\,$ (48)

which would describe the probability distribution of a statistical mixture of
states. This density matrix would have the well known physical meaning
combining the probability-based experimental preparation of a statistical
quantum system with its generalized $K>1$ theoretical description.

##  6 Generalized Dyson maps

In the majority of applications of conventional quantum theory the information
about dynamics (i.e., about the realistic self-adjoint Hamiltonian
$\mathfrak{h}$) is extracted, directly or indirectly, from the principle of
correspondence. All interest in some unconventional, crypto-Hermitian
Hamiltonians emerges only after one encounters some unsurmountable technical
difficulties. One then imagines that the conventional formulations of quantum
dynamics need not be optimal. In such a situation, Dyson [12] found, purely
empirically, that his calculations prove perceivably simplified after a
reformulation of the SP theory in which one would work with two inner-product
spaces (i.e., in our present notation, with ${K}=2$).

The motivation of the Dyson’s construction was pragmatic, based on a good
intuitive guess of the operator $\Omega$ representing correlations. In fact,
the strong dependence of the success on this guess was one of the main weak
points of the strategy. Thus, one has to ask how should one amend this aspect,
and how could one make the choice of the correlators $\Omega$ more robust and
flexible.

###  6.1 Multi-step preconditionings

In the most elementary $K=2$ CHSP scenario it is not easy to keep the physics
given by the respective candidates $\mathfrak{h}$ and $\Omega\,$ for the
Hamiltonian and mapping fully under control. One must be even more careful
when using the inverted flowchart $H\to\Theta\to\Omega$ as discussed by
Scholtz et al [8]. In an ideal situation, the determination of an optimal
$\Omega$ should not be based on the mere educated guess. One may expect an
improvement of the results, in particular, after the above-described multi-
step relegation of the Hermiticity of operators from ${\cal R}_{j}$ to ${\cal
R}_{j+1}$.

What seems lost is the reference to the idea of preconditioning, i.e., an
intuition-based insight in the dynamics. This is a shortcoming of the theory
which damages the appeal and popularity of the CHSP formalism in applications.
A step towards recovery came with PTQM, i.e., with the proposal of
factorization (9). Still, a certain theoretical weakness of the recipe
survived, lying in a somewhat mysterious status of the charge [7] as well as
in a manifestly non-Dysonian nature of the Bender’s factorization
$\Theta_{2}={\cal PC}$ of the metric.

The gap has partially been filled in [11]. We found there that the charge
${\cal C}$ becomes in fact a very natural component of the formalism. Still,
our satisfaction remained limited because the operator-product metric
$\Theta={\cal PC}$ did not seem to be easily re-factorized (i.e., in effect,
interpreted) in terms of a single Dyson map $\Omega$. Also the most common re-
factorization $\Omega=\sqrt{\Theta}$ of the metric (as recommended, e.g., in
[7]) may often prove too artificial.

In [11], precisely the latter artificiality impression caused our loss of
interest in the Hermiticity-to-quasi-Hermiticity relegations at ${K}\geq 4$.
We did not imagine that a consistent concept of the Dyson maps and of their
compositions might very naturally be connected with the notion of the space-
dependent Hermiticity.

###  6.2 Composition laws

In a way inspired by Eq. (5) let us postulate a factorization

$Z_{j}=\Omega_{j}^{\ddagger(j)}\,\Omega_{j}\,,\ \ \ \ j=1,\ldots,{K-1}\,.$
(49)

Such an ansatz is, first of all, compatible with our present $j-$dependent
self-adjointness (18) of $Z_{j}$ in ${\cal R}_{j}$. It allows us to reclassify
the new family of the invertible operators $\Omega_{j}$ as an upgraded model-
building-information input.

###### Lemma 1

At ${K}=3$ let us assume that both $Z_{2}$ and $Z_{1}$ are positive definite
and factorized via Eq. (49). In ${\cal H}_{math}$, the physical Hilbert-space
metric $\Theta_{2}=Z_{2}Z_{1}$ is then factorized as follows,

$\Theta_{2}=\Omega_{21}^{\ddagger(2)}\,\Omega_{21}=\Omega_{21}^{\dagger}\,\Omega_{21}\,,\
\ \ \ \ \Omega_{21}=\Omega_{2}\,\Omega_{1}\,.$ (50)

Proof. Relation (33) implies that in the factorization postulate
$Z_{1}=\Omega_{1}^{\ddagger(1)}\,\Omega_{1}$ we can relegate the Hermiticity
of $\Omega_{1}^{\ddagger(1)}=Z_{2}^{-1}\,\Omega_{1}^{\ddagger(2)}\,Z_{2}$. The
insertion in $\Theta_{2}=Z_{2}\,Z_{1}$ yields the result. $\square$

###### Lemma 2

At ${K}=4$ let us assume that operators $Z_{j}$ with $j=1,2$ and $3$ are
positive definite and factorized via Eq. (49). In ${\cal H}_{math}$, the
physical Hilbert-space metric is then factorized as follows,

$\Theta_{3}=Z_{3}Z_{2}Z_{1}=\Omega_{321}^{\ddagger(3)}\,\Omega_{321}=\Omega_{321}^{\dagger}\,\Omega_{321}\,,\
\ \ \ \ \Omega_{321}=\Omega_{3}\,\Omega_{2}\,\Omega_{1}\,.$ (51)

Proof. As long as the superscript ‡(j) denotes the usual Hermitian conjugation
in our unique representation space ${\cal H}_{math}$ at $j={K-1}=3$, we have
to eliminate, from the definition of $\Theta_{3}=Z_{3}\,Z_{2}\,Z_{1}$, just
the two unusual Hermitian conjugation superscripts ‡(j) with $j=1$ and $j=2$.
We need, for this purpose, just the two items in relations (33), viz.,

$Z_{2}\,\Lambda^{\ddagger(1)}=\Lambda^{\ddagger(2)}\,Z_{2}$ (52)

and

$Z_{3}\,\Lambda^{\ddagger(2)}=\Lambda^{\ddagger(3)}\,Z_{3}=\Lambda^{\dagger}\,Z_{3}\,,$
(53)

respectively. With $\Lambda=\Omega_{1}$ we get

$\Theta_{3}=Z_{3}\,\Omega_{1}^{\ddagger(2)}\,Z_{2}\,\Omega_{1}$

from the former identity, while we further get

$\Theta_{3}=\Omega_{1}^{\dagger}\,Z_{3}\,Z_{2}\,\Omega_{1}$

from the latter identity. In the third step, the factorization (49) at $j=2$
and the use of Eq. (53) with $\Lambda=\Omega_{2}$ lead immediately to the
final result. $\square$

The structure of the latter proof reveals its recursive nature. Along the same
lines it is now easy to prove the general result (the task is left to the
readers).

###### Theorem 3

At any integer ${K}\geq 2$ the existence of decompositions (49) of the
positive definite operators $Z_{j}$ at $j=1,2,\ldots,{K-1}$ is formally
equivalent to the existence of a refactorization

$\Theta_{K-1}=\Omega_{{K-1}\ldots 321}^{\dagger}\,\Omega_{{K-1}\ldots 321}\,,\
\ \ \ \ \Omega_{{K-1}\ldots
321}=\Omega_{{K-1}}\,\ldots\,\Omega_{3}\,\Omega_{2}\,\Omega_{1}\,,$ (54)

of the Hilbert-space metric

$\Theta_{K-1}=Z_{{K-1}}\,Z_{{K-2}}\,\ldots\,Z_{2}\,Z_{1}\,$ (55)

which is responsible for the quasi-Hermiticity (46) of the Hamiltonian
$H=Z_{0}$ in ${\cal H}_{math}$.

It makes good sense to emphasize that the latter result is purely algebraic,
i.e., guaranteed as valid in the finite-dimensional Hilbert spaces. For the
infinite-dimensional Hilbert spaces the statement had to be complemented by
several necessary technical assumptions like, typically, those concerning the
domains and ranges of operators, etc. In this direction of the further
development of the theory, the recommended preparatory reading may be found,
e.g., in the recent dedicated monograph [20].

##  7 Conclusions

In the manner proposed and verified by Dyson [12], a decisive simplification
of difficult calculations of certain properties of complicated quantum systems
can sometimes be achieved via introduction of an additional, auxiliary inner
product [11]. Our present paper offers, in essence, just a nontrivial
$K-$space technical extension and amendment of the Dyson-inspired CHSP quantum
theory.

We felt motivated by the observation that the fairly complicated CHSP-related
mathematics weakened the appeal and visibility of the innovative methodical
aspects of the approach [30]. For this reason, the current return of attention
to the physics with non-Hermiticites [31] was obviously motivated by the
simplification of some technicalities provided by the most popular PTQM
philosophy [5, 7].

Applications of the updated PTQM theory profited, first of all, from the
factorization of metric. Simultaneously, these applications suffered from the
loss of a direct contact with the principle of correspondence as offered,
initially, by Eq. (4). In our recent brief note [11] we addressed the problem.
We pointed out that the somewhat mysterious PTQM-mediated charge-based
relegation of the Hermiticity from the physical Hilbert space ${\cal
H}_{phys}$ to auxiliary ${\cal H}_{math}$ may be given a very natural
conceptual background when one decides to treat, on equal footing, the
observability of the Hamiltonian $H$ and the observability of the charge
${\cal C}$.

More recently we revealed that the strict restriction of the resulting ISP
formalism to the $K=3$ kinematical scenario given by Eq. (10) was in fact
based on an elementary misunderstanding. We felt discouraged by the apparently
counterintuitive observation that at $K=3$ the parity ${\cal P}=Z_{2}$ seems
to have lost its physical observability status. Only later we imagined that
such a loss is in fact mathematically very natural, especially when one builds
the models without some suitable ad hoc  additional constraints (see, e.g., an
extensive discussion of this point in [29]).

With the latter mental barrier removed, we still felt discouraged by another,
more technical observation that whenever one tries to move to any nontrivial
$K>2$ (i.e., to the kinematics using the larger inner-product-space multiplets
(16)), the properly generalized Dyson maps seem to “cease to behave nicely”.
After a more detailed analysis of the latter problem (leading, finally, to the
results sampled here by Theorem 3), fortunately, also the latter doubts have
been eliminated.

We can conclude that the present systematic $K>3$ GSP extension of the scope
of the most economical and popular SP formulation of quantum mechanics will
find efficient implementations in all of the contexts in which one finds the
reasons for the replacement of the Dyson-mapping-based metric (5) by its
generalizations, be it the metrics (50) or (51) or the entirely general
metric-factorization formulae (54) or (55) in which the number of the
auxiliary generalized-charge factors can be an arbitrary positive integer
$N=K-1$.

### Acknowledgements

The author is grateful to Excellence project PřF UHK 2211/2022-2023 for the
financial support.

### Data Availability Statement

Data sharing is not applicable to this article as no new data were created or
analyzed in this study.

### Conflicts of Interest

The author declares no conflict of interest.

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