Quasi-Hermitian formulation of quantum mechanics using two conjugate
Schrödinger equations

Miloslav Znojila,b

a Department of Physics, Faculty of Science, University of Hradec Králové,
Rokitanského 62, 50003 Hradec Králové, Czech Republic

b The Czech Academy of Sciences, Nuclear Physics Institute, Hlavní 130, 250 68
Řež, Czech Republic, e-mail<EMAIL_ADDRESS>
### Abstract

To the existing list of alternative formulations of quantum mechanics a new
version of non-Hermitian interaction picture is added. What is new is that in
contrast to the more conventional non-Hermitian model-building recipes, the
primary information about the observable phenomena is provided not only by the
Hamiltonian but also by an additional operator with real spectrum (say,
$R(t)$) representing another observable. In the language of physics the
information carried by $R(t)\neq R^{\dagger}(t)$ opens the possibility of
reaching the exceptional-point degeneracy of the real eigenvalues, i.e., a
specific quantum phase transition. In parallel, the unitarity of the system
remains guaranteed, as usual, via a time-dependent inner-product metric
$\Theta(t)$. From the point of view of mathematics, the control of evolution
is provided by a pair of conjugate Schrödiner equations. This opens the
possibility od an innovative dyadic representation of pure states by which the
direct use of $\Theta(t)$ is made redundant. The implementation of the
formalism is illustrated via a schematic cosmological toy model in which the
canonical quantization leads to the necessity of working with two conjugate
Wheeler-DeWitt equations. From the point of view of physics, the “kinematical
input” operator $R(t)$ may represent either the radius of a homogeneous and
isotropic expanding empty Universe or, if you wish, its Hubble radius, or the
scale factor $a(t)$ emerging in the popular Lemaitre-Friedmann-Robertson-
Walker classical solutions, with the exceptional-point singularity of the
spectrum of $R(t)$ mimicking the birth of the Universe (“Big Bang”) at $t=0$.

### Keywords

quantum theory of unitary systems; non-Hermitian interaction representation;
non-stationary physical inner products; dyadic representation of pure states;
schematic quantum model of Big Bang;

##  1 Introduction

Around the turn of millennium it had widely been accepted that the various
existing formulations of quantum mechanics (QM) “differ dramatically in
mathematical and conceptual overview, yet each one makes identical predictions
for all experimental results” [1]. In the cited review the authors emphasized
the historical as well as methodical importance of the Heisenberg’s alias
“matrix” formulation of QM (in which the states do not change in time) as well
as the economy of the most common Schrödinger’s alias  “wavefunction”
formulation (which “shifts the focus from measurable quantity to state”).

In loc. cit. the catalogue of formulations was not exhaustive. The authors did
not mention the “universal” interaction picture (IP) in which the observables
(say, $\mathfrak{a}$) and the states (say, $\psi$) are both  allowed to vary
with time, $\mathfrak{a}^{(IP)}=\mathfrak{a}^{(IP)}(t)$ and
$\psi^{(IP)}=\psi^{(IP)}(t)$. In the general Hermitian IP framework of
conventional textbooks [2] one can easily re-derive both the Heisenberg-
picture (HP) or Schrödinger-picture (SP) methodical extremes when setting
$\psi^{(HP)}(t)=\psi^{(HP)}(0)=\psi^{(HP)}$ or
$\mathfrak{a}^{(SP)}(t)=\mathfrak{a}^{(SP)}(0)=\mathfrak{a}^{(SP)}$,
respectively.

The review also did not reflect the new quick developments in the field in the
direction initiated by Bender with Boettcher [3]. The latter innovation turned
attention to the overrestrictive role played by the Stone theorem [4]. By this
theorem, indeed, the evolution described by Schrödinger equation

${\rm
i}\frac{d}{dt}\,|\psi^{(SP)}(t)\\!\\!\succ\,\,=\mathfrak{h}\,|\psi^{(SP)}(t)\\!\\!\succ\,\,\,,\
\ \ \ \ \ |\psi^{(SP)}(t)\\!\\!\succ\,\,\in{\cal L}^{(SP)}\,$ (1)

is unitary in ${\cal L}^{(SP)}$ if and only if the Hamiltonian is self-adjoint
in ${\cal L}^{(SP)}$, $\mathfrak{h}=\mathfrak{h}^{\dagger}$. In our present
paper we intend to offer a further extension of the latter methodical
developments in which the Hermiticity restrictions imposed by the Stone
theorem were circumvented.

In introduction we have to remind the readers that the origin of the idea can
in fact be traced back to the paper by Dyson [5]. Long before the turn of
millennium this author revealed that the goal of having the theory non-
Hermitian but still unitary can be achieved via a non-unitary time-independent
preconditioning of the SP wavefunctions,

$|\psi^{(SP)}({t})\\!\\!\succ\,\,\ \to\
|\psi_{(Dyson)}({t})\rangle=\Omega_{(Dyson)}^{-1}\,|\psi^{(SP)}({t})\\!\\!\succ\,\,\
\in\ {\cal H}_{(Dyson)}\,,\ \ \ \ \
\Omega_{(Dyson)}^{-1}\neq\Omega_{(Dyson)}^{\dagger}\,.$ (2)

In applications, the non-unitarity of the stationary Dyson’s map
$\Omega_{(Dyson)}$ led to an efficient description of correlations in various
complicated many-body systems [5, 6, 7, 8].

Along this line the potentially user-unfriendly Hamiltonian $\mathfrak{h}$ has
been replaced by its user-friendlier isospectral avatar defined as acting in
the new, potentially user-friendlier Hilbert space ${\cal H}_{(Dyson)}$,

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

The conventional Hermiticity gets lost ($H\neq H^{\dagger}$) so that Hilbert
space ${\cal H}_{(Dyson)}$ has to be declared unphysical. Importantly, due to
the time-independence of $\Omega_{(Dyson)}$, the loss is just formal, with the
Hermiticity of $\mathfrak{h}$ in ${\cal L}^{(SP)}$ merely replaced by the
Dieudonné’s [9] metric-mediated quasi-Hermiticity of $H$ in ${\cal
H}_{(Dyson)}$,

$H^{\dagger}\,\Theta_{(Dyson)}=\Theta_{(Dyson)}\,H\,,\ \ \ \ \
\Theta_{(Dyson)}=\Omega_{(Dyson)}^{\dagger}\,\Omega_{(Dyson)}\neq I\,.$ (4)

On a more abstract quantum-theoretical level the isospectrality between a pre-
selected, sufficiently user-friendly non-Hermitian Hamiltonian $H\neq
H^{\dagger}$ and its self-adjoint reconstructed partner
$\mathfrak{h}=\mathfrak{h}^{\dagger}$ opened multiple new model-building
strategies, first of all, in quantum field theory [10, 11]. The possibility of
reconstruction of the “missing” physical inner-product metrics $\Theta$ from a
given non-Hermitian Hamiltonian $H$ led also, in the framework of relativistic
QM, to a completion of the years-long efforts of a consistent probabilistic
interpretation of the Klein-Gordon fields [12, 13, 14, 15] and/or of the Proca
fields [16, 17].

A few other successful applications can be found mentioned in the recent
review of the field by Mostafazadeh [18]. Still, the author had to admit there
that after a tentative transition from the Klein-Gordon equation to a formally
not too different Wheeler-DeWitt (WDW) equation of quantum gravity [19, 20],
the applicability of the reconstruction of $\Theta=\Theta(H)$ appears to be
limited. In the Mostafazadeh’s own words, “the lack of a satisfactory solution
of this problem has been one of the major obstacles in transforming canonical
quantum gravity and quantum cosmology into genuine physical theories” (cf. p.
1291 in loc. cit.).

For this reason the review [18] of quasi-Hermitian QM did only marginally
mention the WDW models. An analogous scepticism can be also found expressed in
the quantum-gravity-dedicated monographs [21, 22]). The main mathematical
obstacle can be seen in the fact that the operators “that arise in quantum
cosmological models” have to be manifestly time-dependent and that such a
choice “requires a more careful examination” [18].

More recently, the problem has been reopened and the latter challenge was re-
addressed in [23]. Still, the main methodical and conceptual challenges were,
from our present point of view, circumvented. For this reason we felt urged to
complement the theory, in our present paper, by a new analysis in which the
manifest $t-$dependence of the quasi-Hermitian operators would prove tractable
in a more satisfactory manner.

As we already indicated in Abstract, an important source of the inspiration of
our present project was that the vast majority of the conventional
applications of the non-Hermitian model-building recipes starts from the
assumption of our knowledge of the Hamiltonian $H$. In most cases, this
operator is assumed observable, i.e., constrained by relation (4). At the same
time, the more abstract theory of review [7] admits the existence of an “input
information” knowledge of at least one other, independent operator (i.e., in
our present notation, of $R(t)$) with real spectrum.

In some sense (cf. [24]), an attempt of a feasible and, at the same time,
purposeful incorporation of $R(t)$ in the formalism was one of the main
driving forces behind the present work. On the side of physics we decided to
motivate it by the needs of quantum cosmology in which the notions like Hubble
radius or scale factor play a key role in the classical-physics toy-model
descriptions of the empty, homogeneous and isotropically expanding Universe.
Still, for our present purposes we found it sufficient to speak just about an
entirely schematic observable “radius of an expanding toy-model Universe”
$r(t)$ which is allowed to vary with the so called cosmological time $t$.

The presentation of our results starts in section 2 where we briefly review
the existing stationary and non-stationary versions of the quasi-Hermitian
quantum mechanics. In section 3 we then turn attention to the WDW equation and
review and emphasize the recent progress in its study. A deeper insight in its
role is then provided in section 4 in which we introduce our present highly
schematic but instructive toy model of the quantum Universe.

For the sake of simplicity, just the radius $r(t)$ will be considered
quantized, i.e., represented, just shortly after Big Bang, by a (quasi-
Hermitian) operator $R(t)$. Subsequently, the related basic technical
questions of the construction of the physical Hilbert-space metric $\Theta(t)$
and of the evolution equations in the non-Hermitian interaction picture (NIP)
are addressed and reviewed in section 5. Several conceptual aspects of the
theory are finally discussed in section 6 and summarized in section 7.

##  2 Two quasi-Hermitian formulations of quantum theory

The introduction in quantum mechanics usually starts, in textbooks, from its
formulation in Schrödinger “representation” alias  “picture” (SP, [2]). In
this language the states are represented by the ket-vector elements
$|\psi^{(SP)}({t})\\!\\!\succ\,\,$ of a suitable Hilbert space ${\cal
L}^{(SP)}$. The unitary evolution of the system is prescribed by Eq. (1),
i.e., by Schrödinger equation in which the Hamiltonian is required self-
adjoint, $\mathfrak{h}=\mathfrak{h}^{\dagger}$. In such a setting a decisive
simplification of the solution of Eq. (1) can be achieved, in principle at
least, via the (usually, just numerical) diagonalization of the Hamiltonian.

###  2.1 Non-Hermitian Schrödinger picture (NSP)

In many realistic models the diagonalization of $\mathfrak{h}^{(SP)}$ may
happen to be prohibitively difficult. More than half a century ago,
fortunately, Freeman Dyson [5] revealed that whenever the “maximal”
simplification $\mathfrak{h}\to\mathfrak{h}_{(diagonal)}$ remains, due to its
complexity, unavailable, the underlying realistic Schrödinger Eq. (1) might
still be made tractable via a “partial” simplification of the Hamiltonian. He
just recommended that the “inaccessible” diagonalization is to be replaced by
any other (i.e., just invertible) auxiliary time-independent isospectrality
mapping $\Omega:\mathfrak{h}\to H$ which need not even be required unitary.

One just has to redefine the states as well as, whenever necessary or useful,
also the Hilbert space itself (cf. Eq. (2) above). The original conventional
Schrödinger equation becomes replaced by its equivalent representation in
${\cal H}_{(Dyson)}$,

${\rm i}\,\frac{d}{dt}\,|\psi(t)\rangle=H\,|\psi(t)\rangle\,.$ (5)

As long as $H\neq H^{\dagger}$, it makes sense to introduce the “alternative
ket vectors”

$|\psi(t)\rangle\\!\rangle=\Theta\,|\psi(t)\rangle$

evolving, in ${\cal H}_{(Dyson)}$, according to a complementary Schrödinger
equation,

${\rm
i}\,\frac{d}{dt}\,|\psi(t)\rangle\\!\rangle=H^{\dagger}\,|\psi(t)\rangle\\!\rangle\,.$
(6)

The main benefit of the resulting formalism using two Schrödinger equations
and the two different Hamiltonians (viz., $H$ and $H^{\dagger}\neq H$) it that
for pure states, the predictions of the results of measurements have the fully
analogous form in ${\cal L}$ and in ${\cal H}_{(Dyson)}$. Indeed, once one
considers any stationary and self-adjoint operator $\mathfrak{a}^{(SP)}$
representing an observable in ${\cal L}^{(SP)}$, and once one defines its NSP
avatar $A=\Omega^{-1}\,\mathfrak{a}^{(SP)}\,\Omega$ in ${\cal H}_{(Dyson)}$,
the validity of the elementary mathematical identity

$\prec\\!\\!\psi^{(SP)}(t)|\mathfrak{a}^{(SP)}|\psi^{(SP)}(t)\\!\\!\succ\,\,=\langle\\!\langle\psi(t)|A|\psi(t)\rangle$
(7)

implies the coincidence of the predictions (i.e., of the probability
densities) when computed via the single textbook Schrödinger Eq. (1) or via
the conjugate pair (5) + (6).

###  2.2 Non-Hermitian interaction picture (NIP)

Once we admit the dependence of the Dyson’s mapping on time,
$\Omega=\Omega(t)$, it is sufficient to follow the description of the
necessary (i.e., NIP) amendments of the theory as described in [25, 26]. After
such a generalization of the formalism the main changes result from the
emergence, in both of the non-Hermitian Schrödinger equations, of the non-
vanishing Coriolis-force term

$\Sigma(t)={\rm i}\,\Omega^{-1}(t)\,\dot{\Omega}(t)\,,\ \ \ \
\dot{\Omega}(t)=\frac{d}{dt}\Omega(t)\,.$ (8)

The non-stationary generalization $H(t)$ of the observable non-Hermitian
Hamiltonian (or, in the standard language of mathematics, of the quasi-
Hermitian Hamiltonian) remains, by definition, isospectral with the self-
adjoint (though now, admissibly, non-stationary) Hamiltonian $\mathfrak{h}(t)$
of textbooks. Still, the price to pay for the non-stationarity is that the
non-stationary upgrade of the two NIP Schrödinger equations reads

${\rm i}\,\frac{d}{dt}\,|\psi(t)\rangle=G(t)\,|\psi(t)\rangle\,,\ \ \ \ \
G(t)=H(t)-\Sigma(t)$ (9) ${\rm
i}\,\frac{d}{dt}\,|\psi(t)\rangle\\!\rangle=G^{\dagger}(t)\,|\psi(t)\rangle\\!\rangle\,,\
\ \ \ \ G^{\dagger}(t)=H^{\dagger}(t)-\Sigma^{\dagger}(t)$ (10)

where the generator of evolution contains the Coriolis term $\Sigma(t)$ and
ceases to be observable, therefore (cf., e.g., p. 1272 in [18]).

The admissibility of the non-stationarity of the prototype textbook
Hamiltonian $\mathfrak{h}(t)$ is, in some sense, exceptional [27]. For all of
the other non-Hermitian and non-stationary operators $A(t)$ of observables
defined, in ${\cal H}_{(Dyson)}$, by their pull-down from ${\cal L}^{(SP)}$,
the formalism would become prohibitively complicated unless we assume that
$\mathfrak{a}(t)=\mathfrak{a}(0)=\mathfrak{a}^{(SP)}$, i.e., unless we require
that

$A(t)=\Omega^{-1}(t)\,\mathfrak{a}^{(SP)}\,\Omega(t)$ (11)

where the self-adjoint avatars of the non-Hamiltonian observables remain time-
independent (see also [23] for a detailed discussion of this subtlety).

Under the latter assumption one can re-establish a complete parallelism
between the conventional Hermitian quantum mechanics in interaction picture
and its non-Hermitian alternative in which the two NIP Schrödinger Eqs. (9)
and (10) control the evolution of states. Naturally, a complete picture is
only obtained when one also takes into consideration the manifest and
necessary time-dependence of the observables. The most straightforward
guarantee of the internal consistency of the theory is then provided by the
following result.

###### Lemma 1

The time-dependence of observables (11) can be reconstructed from their
initial values by the solution of Heisenberg equation

${\rm i\,}\frac{\partial}{\partial
t}\,{A}(t)=A(t)\,\Sigma(t)-\Sigma(t)\,A(t)\,.$ (12)

Proof. Definition (11) is equivalent to relation
$\Omega(t)\,A(t)=\mathfrak{a}^{(SP)}\,\Omega(t)$ which is easily
differentiated,

$\Omega^{-1}(t)\,\frac{\partial}{\partial
t}\,\Omega(t)+\frac{\partial}{\partial
t}\,\,{A}(t)=\Omega^{-1}(t)\,\mathfrak{a}^{(SP)}\,\frac{\partial}{\partial
t}\,\Omega(t)\,.$

In the light of definition (8) of the Coriolis force this immediately yields
formula (12). $\square$

One has to notice that the role of the NIP generator of evolution is played
here by the Coriolis force. The solution of the equation specifies the (by
definition, non-stationary) operator $A(t)$ in consistent manner. At the same
time, our knowledge of this solution immediately opens the possibility of an
ultimate evaluation of the matrix elements entering the following
nonstationary upgrade of Eq. (7),

$\langle\\!\langle\psi(t_{f})|A(t_{f})|\psi(t_{f})\rangle=\prec\\!\\!\psi^{(SP)}(t_{f})|\mathfrak{a}^{(SP)}|\psi^{(SP)}(t_{f})\\!\\!\succ\,\,\,.$
(13)

This formula expresses the probabilistic contents of the non-stationary theory
and quantifies the prediction of the results of the measurement at time
$t=t_{f}$.

##  3 Samples of application

In the preface to the ambitious theoretical monograph [22] we read that
“despite an enormous effort of work by a vast amount of physicists over the
past 70 years, we still do not have a credible quantum general relativity
theory” (QGR). “What we do have today are candidate theories; …for each of
them one still has to show …that it reduces to the presently known …classical
general relativity at low energies” [22].

###  3.1 Wheeler-DeWitt equation

The incompleteness of the candidate QGR theories is best illustrated by the
canonical quantum gravity based on WDW equation. In this field, incidentally,
the progress is significant. On p. 1291 of [18], for example, it is emphasized
that “in the 1960’s the discovery of the Hamiltonian formulation of the
General Theory of Relativity …provided the necessary means to apply Dirac’s
method of constrained quantization”. We believe that in this context also the
NIP-based study of not too realistic WDW equation did not still tell us its
last word yet.

The scepticism of the theoreticians is in a sharp contrast with the
experimental  side of the QGR problem where the efforts of physicists were
amazingly successful. For example, the age of our Universe is currently widely
agreed to be finite and equal to cca 13.8 billion years [28]. It is worth
adding that the determination of the latter value belongs to one of the most
impressive recent experimental results in physics. Under a self-explanatory
name “Cosmic Background Radiation Anisotropy Satellite/Satellite for
Measurement of Background Anisotropies” (COBRAS/SAMBA, [28]) the measurement
was initiated around 1996 and operated by the European Space Agency between
the years 2009 to 2013. The necessary sensitivity and resolution were further
improved by the NASA Wilkinson Microwave Anisotropy Probe (WMAP).

This resulted in the data summarized in the so called Lambda cold dark matter
($\Lambda$CDM) model alias  “standard” cosmological model” [29]. In the
acronym the first, Greek letter refers to the cosmological constant while the
use of the word “standard” emphasizes that its parameters fit not only the
expansion of the universe or the distribution laws of the atomic nuclei and/or
galaxies but also the fairly contradictory hypothesis of existence of the
initial point-like Big-Bang singularity.

Once we reopen the question of compatibility of the $\Lambda$CDM hypotheses
with the basic principles of quantum theory we only have to return to a
moderate scepticism. The applicability of the underlying classical-physics-
based concepts finds its first natural limitation in a restriction to its far-
from-Big-Bang verifications. The experiments remain persuasively compatible
with the classical GR theory as the correct theory of gravity at macroscopic
distances.

In this context we found our basic theoretical inspiration and encouragement
in the comprehensive review paper [18]. We read there that in quantum
cosmology “the relevant …second order differential equations” (i.e., WDW
equations) resemble the Klein-Gordon equations and “have the following general
form”

$\frac{d^{2}}{dt^{2}}\,\psi(t)+D(t)\,\psi(t)=0\,$ (14)

(cf. Eq. Nr. 377 and the related comments in [18]). The symbol $\psi(t)$
denotes here a “wave function of the Universe” which would be “void of a
physical meaning” without “an appropriate inner product” $\Theta$ [18].

In the nearest future the quantum effects emerging at the singularities (as
sampled by black holes or hypothetical Big Bang) have to be re-analyzed. In
other words, there is still a broad gap between our understanding of the
correspondence between the well-confirmed classical singularities and their
internally consistent quantum analogues. A model-based description of their
mechanism and dynamics is still, in the light of Eq. (14), one of the most
important subjects of research and one of the sources of open questions which
motivated also our forthcoming considerations.

In a way explained in review [18] (cf., in particular, section Nr. 9.2) a full
formal analogy between the Klein-Gordon and WDW equations only exists when the
operator part of Eq.(14) remains time-independent, $D(t)=D(0)=D$. In this
case, indeed, the Klein-Gordon-type equation can be transformed into its NSP
equivalent (5) of section 2.1 (cf. also equation Nr. 378 in loc. cit.). The
second NSP Schrödinger Eq. (6) of section 2.1 is then easily written in terms
of the conjugate Hamiltonian operator $H^{\dagger}$. The correct physical
(i.e., probabilistic) interpretation of the evolution then follows from the
one-to-one correspondences (2) and (3) between the states and operators in the
respective Hilbert spaces ${\cal L}^{(SP)}$ and ${\cal H}_{(Dyson)}$.

After a transition (of our present interest) to the genuine WDW version of Eq.
(14) the operator $D(t)$ must necessarily be kept manifestly time-dependent.
This forces us to make use of the non-stationary NIP formalism of section 2.2.
The main innovation is that any  quasi-Hermitian observable of interest (say,
$A$ in Eq. (7)) becomes, by definition, time-dependent (cf. Eq. (11)). As a
consequence, the prediction of any  measurement (i.e., the evaluation of the
overlap (13)) requires, suddenly, not only the solution of the two
comparatively friendly Schrödinger-like evolution equations for the state
(i.e., the construction of the two vectors  in ${\cal H}_{(Dyson)}$) but,
first of all, also the solution $A(t)$ of another, maximally unser-unfriendly
Heisenberg-like evolution equation (12) for the operator.

###  3.2 Closed versus open quantum systems

Many authors proved discouraged by the latter technical obstacles and so they
have redirected their attention, typically, to the exactly solvable models
(later, we will pick up the letter [30] for illustration). Frequently, people
also simplify the model-building process by giving up the unitarity. They
declare their quantum system, in the spirit of Refs. [31, 32], “open”. In some
sense, the newly acquired freedom becomes abused because by definition of the
open quantum systems (formulated, basically, in the Feshbach’s spirit [33,
34]) the resulting “effective” non-Hermitian descriptions are, in the sense of
fundamental theory, incomplete [35, 36].

This being said one should add that the use of the effective operators of
observables really enables one to pay more attention to the ever-present noise
and fluctuations in the quantum systems living in the real world (cf., e.g.,
[37, 38]). In the future, in this sense, it will certainly be necessary to try
to move beyond the restrictive closed system models, indeed.

In our present paper the consequent fundamental-theory approach is not
abandoned. In its framework, nevertheless, even the operator-evolution nature
of the Heisenberg-like Eq. (12) need not be the main technical problem.
Indeed, difficulties also emerge in connection with the corresponding non-
stationary upgraded doublet of Schrödinger-like equations (9) and (10). As
long as the generator $G(t)$ of the evolution of the states becomes defined as
the difference between the Hamiltonian $H(t)$ (which is, by definition,
observable) and the Coriolis force $\Sigma(t)$ (which is, in general, not
observable), also the generator $G(t)$ itself is not observable (once more, we
may recall Theorem Nr. 2 in [18] for details). Moreover, many examples (cf.,
e.g., [39, 40]) show that the elements of its spectrum need not even form the
complex conjugate pairs. For this reason, it makes also hardly any sense to
try to simplify the model by imposing, upon this operator, the popular ${\cal
PT}-$symmetry constraint. Still, in the search for innovations this is what is
often being done – see, e.g., [41, 42].

Recently, new light has been thrown upon these problems by the two studies of
several solvable and manifestly time-dependent “wrong-sign” anharmonic
oscillators [30, 43]. Surprisingly enough, it has been shown there that it may
still make good sense to impose the ${\cal PT}-$symmetry constraint directly
upon the observable Hamiltonian $H(t)$. At the first sight, the motivation
seems missing because this operator only enters the pair of the NIP
Schrödinger equations in combination with Coriolis force. Nevertheless, the
toy-model studies reconfirmed that the main advantage of using the concept of
${\cal PT}-$symmetry lies in its capability of a clear separation of the
unitary dynamical regime (in which the symmetry remains unbroken) from its
unphysical non-unitary complement (in which the ${\cal PT}-$symmetry becomes
spontaneously broken).

###  3.3 Pure states in dyadic representation

A clarification of the slightly complicated NIP situation becomes provided
when one imagines that the Schrödinger’s, the Heisenberg’s and the Dirac’s
intermediate alias  interaction pictures describe the same physics. All of
them characterize the same evolution of a quantum system which is initiated by
the preparation of a pure state $\psi(t)$ at $t=t_{i}=0$ and which finally
leads to the prediction of the results of the measurement at $t=t_{f}>0$. In
the language of mathematics this means that once we complement the Dyson map
(2) (or, more precisely, its non-stationary, time-dependent amendment) by a
complementary, alternative replacement

$|\psi^{(SP)}({t})\\!\\!\succ\,\,\ \to\
|\psi_{(Dyson)}({t})\rangle\\!\rangle=\Omega_{(Dyson)}^{\dagger}(t)\,|\psi^{(SP)}({t})\\!\\!\succ\,\,\
\in\ {\cal H}_{(Dyson)}\,,$ (15)

we may immediately deduce that
$|\psi_{(Dyson)}({t})\rangle\\!\rangle=\Theta_{(Dyson)}(t)\,|\psi_{(Dyson)}({t})\rangle$.
Thus, whenever we decide to work with the two different  state-vector elements
$|\psi_{(Dyson)}({t})\rangle\\!\rangle$ and $|\psi_{(Dyson)}({t})\rangle$ of
the Hilbert space ${\cal H}_{(Dyson)}$, it appears sufficient to work just
with the information about the metric encoded in the double-ket vector
$|\psi_{(Dyson)}({t})\rangle\\!\rangle$.

The latter trick leads, in a way emphasized in review [23], to an enormous
simplification of the NIP formalism. The statement may be also given a more
compact mathematical form in which the pure state $\psi(t)$ is represented, in
${\cal H}_{(Dyson)}$, by the rank-one (i.e., dyadic) projector

$\pi_{\psi}(t)=|\psi^{(Dyson)}(t)\rangle\,\frac{1}{\langle\\!\langle\psi^{(Dyson)}(t)|\psi^{(Dyson)}(t)\rangle}\,\langle\\!\langle\psi^{(Dyson)}(t)|\,.$
(16)

With (or even without) the conventional assumptions of the biorthonormality
and bicompleteness

$\langle\\!\langle\psi^{(Dyson)}(t)|\phi^{(Dyson)}(t)\rangle=\delta_{\psi\phi}\,,\
\ \ \
\sum_{\psi}\,|\psi^{(Dyson)}(t)\rangle\langle\\!\langle\psi^{(Dyson)}(t)|=I\,$
(17)

this enables us to re-derive the fundamental measurement-prediction formula
(13).

##  4 Quantum gravity in a toy model

In the preceding context it is worth adding that in spite of a certain
resemblance of our state-representing formulae with the Aharonov’s and
Vaidman’s time-symmetric two-state-vector formalism [44, 45], the parallels
are purely formal because the present approach remains safely traditional and
time-asymmetric (see also a few other comments in section 6.1 below). This
means that we just stay in the framework of a traditional non-relativistic
quantum cosmology in which the $t-$dependent state (and, say, the pure state)
of the Universe would have to be prepared at a suitable time $t=t_{initial}$.
One can expect that the state of the Universe evolves and gets measured at
another instant $t=t_{final}>t_{initial}$. Now, our task is to explain how one
might realize such an evolution scenario, in principle at least, when making
the pure states represented by the rank-one projectors (16).

###  4.1 Classical singularities

There are not too many results in which one would really succeed in making the
space-time background of QGR quantized (i.e., represented by an operator) and,
simultaneously, time-dependent. In our present considerations we decided to
emphasize, therefore, just a few preselected methodical aspects of quantum
gravity, with our attention paid, predominantly, to the requirement of the
background independence of the theory in which even the measurements of the
distances in an empty Universe would be of a strictly quantum, probabilistic
nature [46].

We will test our ideas in the non-covariant kinematical regime in which the
time $t$ will still be treated as a parameter [47]. Moreover, even the
strictly quantum Universe will be assumed simplified and existing just in a
very small vicinity of its classical Big-Bang singularity. After such a
specification of the simplified dynamical regime we will add several further,
methodically motivated reductions of the picture.

  * •

The classical space-time geometry of the Universe has to remain “next to
trivial”. We will employ the not exceedingly revolutionary kinematics working
with the non-covariant concept of absolute time. The quantum-theory-controlled
evolution of the Universe will be then assumed unitary, i.e., unitary in the
language of the more or less conventional quantum mechanics of the so called
closed systems.

  * •

In both the classical and quantum settings, the naively physical non-
relativistic parameter of time $t$ will be assumed positive and set equal to
zero at Big Bang. On the classical non-relativistic level also the 3D spatial
coordinates will be assumed time-dependent, therefore, $x=x(t)$, $y=y(t)$ and
$z=z(t)$.

  * •

All this would lead to a still nontrivial version of the background
independence because the observable values of the spatial nodes $x=x(t)$,
$y=y(t)$ and $z=z(t)$ (i.e., say, point-particle positions [47]) have to be
defined (i.e., prepared and/or measured) as eigenvalues of operators, in
principle at least.

  * •

The last three operators have to be self-adjoint in a physical Hilbert space
${\cal H}_{phys}$ in which the inner product has the property of being time-
dependent and degenerating at $t=0$. In other words, a “non-Hermitian” NIP
version of QM will have to be used.

For the sake of simplicity of our toy-model-based considerations we will
assume that the kinematics of the expanding Universe will be just one-
parametric. The purpose of such a choice is twofold. In the context of
mathematics a maximal simplicity of our methodical considerations has to be
achieved. In this sense one can simply speak about a homogeneous and
isotropic, centrally symmetric expanding empty Universe characterized, say, by
its volume or radius $r(t)$. Thus, in our present minimal project, just such a
real function would have to be reinterpreted as an eigenvalue of an ad hoc
operator $R(t)$.

In parallel, in the context of physics the interpretation of the parameter
$r(t)$ might be made more sophisticated, with the details to be found, e.g.,
in the dedicated monograph [48]. Thus, for example, one could identify its
$t-$dependent value with the scale factor $a(t)$ emerging in the popular
Lemaitre-Friedmann-Robertson-Walker classical solvable model, or with the
closely related function of $t$ called Hubble radius, etc.

In any case, the reduction of the description of the classical dynamics to a
single real parameter implies that in the centrally symmetric picture with
$r(t)=\sqrt{x^{2}(t)+y^{2}(t)+z^{2}(t)}$ we will have to replace, firstly, the
three spatial Cartesian coordinates $x=x(t)$, $y=y(t)$ and $z=z(t)$ by the
equivalent spherical coordinates $r=r(t)$, $\theta=\theta(t)$ and
$\phi=\phi(t)$. Secondly, for the sake of simplicity this will enable us to
assume, in another reasonable approximation, the stationarity
$\theta(t)=\theta(0)$ and $\phi(t)=\phi(0)$ of the angles. Moreover, we will
treat the latter two values fixed and not quantized. Thus, both of the
spherical angular coordinates will be kept “frozen” and “irrelevant”, i.e.,
classical and time-independent.

All of the latter simplifications have just a methodical motivation. In
contrast, the radius of the Universe $r(t)$ itself will be defined, after
quantization, as one of the available real eigenvalues of a time-dependent
“dynamical-geometry” operator $R(t)$. At any suitable Hilbert-space dimension
$N\leq\infty$ we will have to write $r(t)=r_{n}(t)$ where the “multiverse-
counting” quantum number $n=1,2,\ldots,N$ specifies the hypothetical
“prepared” pure quantum state of the Universe.

The quantum radius of the Universe $r_{n}(t)$ must be time-dependent. At Big
Bang we have to guarantee the existence of an “unavoidable” degeneracy (also
called “exceptional point”, EP, [49]), $\lim_{t\to 0}r_{n}(t)=0$ for all $n$.
Our time-dependent “dynamical-geometry” operator $R(t)$ must be, in our
working Hilbert space ${\cal H}_{math}$, non-Hermitian but Hermitizable alias
quasi-Hermitian, i.e., such that

$R^{\dagger}(t)\,\Theta(t)=\Theta(t)\,R(t)\,.$ (18)

In the literature, interested readers may find a number of the generic
methodical comments on the latter equation (cf., e.g., [7, 18, 50]). In what
follows, we intend to work just with an illustrative family of certain non-
numerically tractable $N$ by $N$ matrices $R(t)=R^{(N)}(t)$. This will enable
us to keep also the related discussion sufficiently short and specific.

###  4.2 The radius of the Universe in a solvable toy model

A mathematical inspiration of our present project of the realization of a
schematic quantum model of the Universe dates back to the unpublished preprint
[51]. We considered there, in an entirely different context, a one-parametric
family of non-Hermitian (but Hermitizable, quasi-Hermitian) matrices with the
real spectra which represented the discrete bound-state energies. The purpose
of the preprint (to be cited as PI in what follows) was a study of the slow,
adiabatic unitary-evolution process resulting in a fall of the $N-$level
systems into an exceptional-point singularity (EPN, [49]).

Figure 1: The “multiverse” eigenvalues of the toy-model operator $R^{(4)}(t)$
of Eq. (19) representing the eligible instantaneous size of the quantized
Universe expanding after Big Bang.

Not the same but analogous matrices will be used here in another role, viz.,
in the role of a non-stationary operator $R(t)$ with the eigenvalues
$r_{n}(t)$ representing the observable instantaneous radii of the Universe.
For the sake of simplicity we will assume that the dimension $N$ of our
schematic Hilbert space ${\cal H}_{(Dyson)}$ is finite. We will consider
$N=2,3,\ldots$ and postulate that our “kinematical”, geometric-background-
representing input matrices $R^{(N)}(t)$ have the following respective forms,

$R^{(2)}({t})=\left[\begin{array}[]{cc}-1+{\sigma^{(2)}(t)}&{\tau^{(2)}(t)}\\\
{}-{\tau^{(2)}(t)}&1+{\sigma^{(2)}(t)}\end{array}\right]\,,\ \ \ \
R^{(3)}({t})=\left[\begin{array}[]{ccc}-2+{\sigma^{(3)}(t)}&\sqrt{2}\,{\tau^{(3)}(t)}&0\\\
-\sqrt{2}\,{\tau^{(3)}(t)}&{\sigma^{(3)}(t)}&\sqrt{2}\,{\tau^{(3)}(t)}\\\
0&-\sqrt{2}\,{\tau^{(3)}(t)}&2+{\sigma^{(3)}(t)}\end{array}\right]\,,\ \ \ \ $
$\ \ \ \ \
R^{(4)}({t})=\left[\begin{array}[]{cccc}-3+{\sigma^{(4)}(t)}&\sqrt{3}\,{\tau^{(4)}(t)}&0&0\\\
-\sqrt{3}\,{\tau^{(4)}(t)}&-1+{\sigma^{(4)}(t)}&2\,{\tau^{(4)}(t)}&0\\\
0&-2\,{\tau^{(4)}(t)}&1+{\sigma^{(4)}(t)}&\sqrt{3}\,{\tau^{(4)}(t)}\\\
0&0&-\sqrt{3}\,{\tau^{(4)}(t)}&3+{\sigma^{(4)}(t)}\end{array}\right]\,,\
\ldots\,.$ (19)

Here, $\tau=\tau^{(N)}(t)$ and $\sigma=\sigma^{(N)}(t)$ are suitable real and
smooth functions of time $t$. Thus, for illustration we may choose the shift
$\sigma^{(N)}(t)=2N\,\sqrt{1-[\tau^{(N)}(t)]^{2}}$, the dimension $N=4$ and
the parameter $\tau^{(4)}(t)=1-t\,$. This would yield the spectrum
$\\{r_{n}(t)\\}$ as displayed in Figure 1. We may see that the model is
“realistic” in the sense that at any quantum number $n$ our toy-model empty
Universe exhibits a point-like singularity (Big Bang) at $t=0$ and a quick
expansion at $t>0$.

A number of comments is to be made in advance. First, we have to keep in mind
that once we start from the hypothetical knowledge of the kinematics, it need
not be easy to combine the underlying space-evolution ansatzs (i.e., in our
toy-model case, the specification of parameters in Eq. (19)) with the
requirements of the dynamics (sampled, in our case, by the WDW Eq. (14)).

##  5 The consistent model-building process

Our model-building philosophy is based on the Big-Bang-admitting ansatz (19).
Thus, our very first task is to make the corresponding choice of the
kinematics (i.e., of the time-dependent matrix $R^{(N)}(t)$) compatible with
the unitarity of the evolution (in this sense we assume that the quantum
system under consideration is a closed system). This means that we have to
take into account, first of all, the Dieudonné’s Hermitizability constraint
(18).

###  5.1 The first step: The construction of the metric

From the point of view of physics all of the sufficiently ambitious quantum-
Big-Bang-related models have to mimic a quasi-static phase transition.
Naturally, the concept of the phase transition itself is very broad (see,
e.g., the comprehensive review paper [52] where the authors list more than 400
further relevant references). In comparison, the range of physics behind our
present Big-Bang-related project is perceivably narrower. In a way discussed,
more thoroughly, in papers [53, 54] we will only deal here with the more
specific philosophy of the quantum phase transitions the realization of which
is based on the presence, in the space of parameters, of a suitable Kato’s
[49] exceptional point. Moreover, just a marginal attention will be paid to
the energy levels and to the Hamiltonians. Our study will be redirected to the
background-representing operator (or rather non-Hermitian $N$-by-$N$-matrix)
$R^{(N)}(t)$. Depending on a real time-simulating parameter $t\in(0,1)$ (such
that $t=t^{(Big\,Bang)}=0$) and preceding the case of a more realistic time-
dependent (i.e., non-relativistic) triplet  of $N$ by $N$ matrices
$X^{(N)}_{1}(t)$, $Y^{(N)}_{2}(t)$, $Z^{(N)}_{3}(t)$ representing a dynamical,
fully quantized  (and, at finite $N<\infty$, just discretized) three-
dimensional space-time-grid background.

We will require that the spectra of all of these matrices are complex at the
negative times $t<0$ (this has to reflect the unobservable status of the space
before Big Bang), real but EPN-degenerate at $t=0$ (i.e., at the hypothetical
non-relativistic Big-Bang instant) and real and non-degenerate at $t>0$ (for
pragmatic reasons we will just keep in mind the not too large times, i.e.,
say, $t\leq 1$). Moreover, in the three-dimensional space of the hypothetical
expanding Universe we will also reparametrize the coordinates and replace
their time-dependent and system-dependent Cartesian grid
$\\{x(t),y(t),z(t)\\}$ by the spherical triplet
$\\{r(t),\theta(t),\phi(t)\\}$. For the sake of simplicity, the angular
coordinates will be assumed fixed, and just the radial one will be treated as
the expanding-Universe spatial background and quantized, i.e., treated as one
of the eligible eigenvalues of an ad hoc, kinematical-input operator
$R^{(N)}(t)$.

For the methodical purposes (as well as for the sake of definiteness) we will
assume that the latter, geometry-representing operator is given in advance,
having the form resembling closely the Hamiltonians in PI. Thus, once we
abbreviate $t=t(\tau)=1-\tau$ or introduce a new variable $\tau=\tau(t)=1-t$,
the parallels become complete.

All of our toy-model matrices will be chosen real, non-Hermitian and, whenever
$t=1-\tau(t)>0$, Hermitizable. This means that at any preselected matrix
dimension $N$ there exists an inner-product-metric operator
$\Theta=\Theta^{(N)}(t)$ (which need not be unique, see [7]) such that our
spatial-grid-simulating operator $R=R^{(N)}(t)$ satisfies the quasi-
Hermiticity condition (18). In PI we worked with the close analogues of our
present matrices (19) so that we can just recall and modify Theorem Nr 1 of PI
and formulate the following result.

###### Theorem 2

At every finite Hilbert-space dimension $N<\infty$ the metric
$\Theta^{(N)}({t})$ compatible with the respective radii (19) may be sought in
the following generic form

$\Theta^{(N)}({t})=\sum_{j=1}^{N}\,{\cal M}^{(N)}(j)\,[-{\tau}(t)]^{j-1}$ (20)

containing the sparse-matrix coefficients

${\cal M}^{(N)}(1)=\left[\begin{array}[]{cccc}\alpha_{11}(1)&0&\ldots&0\\\
{}0&\alpha_{12}(1)&\ddots&\vdots\\\ {}\vdots&\ddots&\ddots&0\\\
{}0&\ldots&0&\alpha_{1N}(1)\end{array}\right]\,,$ (21) ${\cal
M}^{(N)}(2)=\left[\begin{array}[]{cccccc}0&\alpha_{11}(2)&0&\ldots&\ldots&0\\\
{}\alpha_{21}(2)&0&\alpha_{12}(2)&0&\ldots&0\\\
{}0&\alpha_{22}(2)&0&\alpha_{13}(2)&\ddots&\vdots\\\
{}\vdots&\ddots&\ddots&\ddots&\ddots&0\\\
{}0&\ldots&0&\alpha_{2,N-2}(2)&0&\alpha_{1,N-1}(2)\\\
0&\ldots&{}\ldots&0&\alpha_{2,N-1}(2)&0\end{array}\right]\,,\ \ \ \ \ $ (22)
$\ \ \ \ \ {\cal
M}^{(N)}(3)=\left[\begin{array}[]{ccccccc}0&0&\alpha_{11}(3)&0&\ldots&\ldots&0\\\
{}0&\alpha_{21}(3)&0&\alpha_{12}(3)&0&\ldots&0\\\
{}\alpha_{31}(3)&0&\alpha_{22}(3)&0&\alpha_{13}(3)&\ddots&\vdots\\\
{}0&\alpha_{32}(3)&\ddots&\ddots&\ddots&\ddots&0\\\
{}0&\ddots&\ddots&0&\alpha_{2,N-3}(3)&0&\alpha_{1,N-2}(3)\\\
{}\vdots&\ldots&0&\alpha_{3,N-3}(3)&0&\alpha_{2,N-2}(3)&0\\\
0&\ldots&{}\ldots&0&\alpha_{3,N-2}(3)&0&0\end{array}\right]\,,$ (23)

etc.

In PI we also recommended to arrange the non-vanishing matrix elements into
the $k$ by $(N-k+1)$ arrays,

$\alpha{(k)}=\left[\begin{array}[]{ccccc}\alpha_{11}(k)&\alpha_{12}(k)&\alpha_{13}(k)&\ldots&\alpha_{1,N-k+1}(k)\\\
\alpha_{21}(k)&\alpha_{22}(k)&\alpha_{23}(k)&\ldots&\alpha_{2,N-k+1}(k)\\\
\vdots&\vdots&\vdots&&\vdots\\\
\alpha_{k1}(k)&\alpha_{k2}(k)&\alpha_{k3}(k)&\ldots&\alpha_{k,N-k+1}(k)\end{array}\right]\,.$
(24)

The values of these matrix elements had to be computed as solutions of Eq.
(18) – for the first few Hilbert-space dimensions $N$, the results may be
found in PI.

###  5.2 Coriolis force and the evolution equations

At a fixed Hilbert-space dimension $N$ the general inner-product-metric
solution of Eq. (18) is non-unique. It varies with an $N-$plet of free
parameters, the variability of which being only restricted by the condition of
the necessary positivity of the metric. In [54] we studied a special “zero-
spectral-shift” sub-family of our present time-dependent models (19) with
$\sigma^{(N)}(t)=0$. At arbitrary $N$, we described there certain “optimal”
solutions (24) which exhibited a number of desirable features. From our
present point of view the most important one was that the positivity of the
metric (i.e., of all of its time-dependent eigenvalues
$\theta_{k}^{(N)}({t})$) proved guaranteed at all of the dimensions $N>0$ and
at all of the times $t>0$ of interest.

Easily, the latter result can be re-adapted to our present needs.
Irrespectively of the radius-positivity-guaranteeing ad hoc  shift parameters
$\sigma^{(N)}(t)>0$, the following Theorem can be easily proved by
mathematical induction.

###### Theorem 3

All of the time-dependent eigenvalues $\theta_{k}^{(N)}({t})$ of the optimal-
radius-dependent inner-product metric $\Theta^{(N)}(t)$ are given, at any
matrix-dimension $N$, by the following closed formula,

$\theta_{k}^{(N)}({t})=\sum_{m=1}^{N}\,C^{(N)}_{km}\,{[\tau(t)]}^{m-1}\,,\ \ \
\ \ \ \ k=1,2,\ldots,N$

where $C_{1n}^{(N)}=\left(\begin{array}[]{c}N-1\\\ n-1\end{array}\right)\,$,
$C_{2n}^{(N)}=\left(\begin{array}[]{c}N-2\\\
n-1\end{array}\right)-\left(\begin{array}[]{c}N-2\\\ n-2\end{array}\right)\,$
and, in general,

$C_{kn}^{(N)}=\sum_{p=1}^{k}\,(-1)^{p-1}\,\left(\begin{array}[]{c}k-1\\\
p-1\end{array}\right)\,\left(\begin{array}[]{c}N-k\\\
n-p\end{array}\right)\,,\ \ \ \ k,n=1,2,\ldots,N\,.$

The main consequence of this result is that all of the eigenvalues of the
metric are positive. This means that we may recall Eq. (4) (in which we
reconstructed the “unknown” metric $\Theta$ as a product of the two “known”
Dyson maps) and that we may try to invert the recipe (assuming that the metric
is “known” and that the Dyson map $\Omega(t)$ is to be reconstructed, say, in
the form of a real square root of $\Theta$ [54]). This means that we can
factorize the metric into the product

$\Theta(t)=\Omega^{\dagger}(t)\,\Omega(t)\,$ (25)

representing a time-dependent generalization of the stationary factorization
formula of Eq. (4).

This enables us to treat also the Coriolis-force matrix (as defined by Eq.
(8)) as known. One can conclude that the construction of the toy model is
almost completed. Indeed, any $J-$plet of its other observable features can be
represented by the respective operators (say, $\Lambda_{j}(t)$, with, if
needed, $j=0$ assigned to the energy-representing Hamiltonian). All of these
operators must be, in terms of the same “correct and physical” Hilbert-space
metric, quasi-Hermitian,

$\Lambda_{j}^{\dagger}(t)\,\Theta(t)=\Theta(t)\,\Lambda_{j}(t)\,,\ \ \ \
j=0,1,\ldots,J\,.$ (26)

Secondly, any one of them (and, in particular, the Hamiltonian
$H(t)=\Lambda_{0}(t)$) can be used to define the basis which can be
biorthonormalized (cf. [55] or Eq. (17)). The purpose may be served, in the
real-spectrum dynamical regime, by the doublet of eigenvalue problems

$H(t)\,|m(t)\rangle=E_{m}(t)\,|m(t)\rangle\,,\ \ \ \ \
H^{\dagger}(t)\,|m(t)\rangle\\!\rangle=E_{m}(t)\,|m(t)\rangle\\!\rangle\,,\ \
\ \ \ m=1,2,\ldots,N\,.$ (27)

Thirdly, the knowledge of the metric also facilitates the search for the other
candidates for the observables (denoted, say, as $\Lambda(t)$ without a
subscript). Indeed, once we consider the product
$\widetilde{\Lambda}(t)=\Theta(t)\,\Lambda(t)$, we immediately see that
$\widetilde{\Lambda}(t)=\widetilde{\Lambda}^{\dagger}(t)$ is Hermitian. Thus,
any  Hermitian “input-information” matrix $\widetilde{\Lambda}(t)$ can be
treated as a set of free parameters defining a quasi-Hermitian operator
${\Lambda}(t)=\Theta^{-1}(t)\,\widetilde{\Lambda}(t)$ eligible as an
observable.

The observables of the latter type may be required to correspond to their
conventional SP avatars $\lambda^{(SP)}$ which are stationary, conserved and
time-independent. In such a case the process of the definition of the operator
(at all times) can be facilitated and replaced by the definition of the
operator just at a single instant $t=t_{initial}$, with the completion of the
construction of $\Lambda(t)$ (at all times) provided by the solution of the
corresponding Heisenberg Eq. (12).

In the last step of our considerations we may preselect a Hermitian matrix
$A(t)$ and use it as the parameters defining the energy-representing
observable Hamiltonian $H(t)=\Theta^{-1}(t)\,A(t)$. Then we may immediately
reconstruct the generator $G(t)=H(t)-\Sigma(t)$ of the evolution of the states
which enters, finally, the two conjugate Schrödinger Eqs. (9) and (10). The
construction of the model is completed.

##  6 Discussion

In a way summarized in reviews [11, 18, 56] the recent theoretical
developments in quantum mechanics threw new light on many traditional model-
building strategies. The main idea of the innovation lies in an extension of
the concept of the so called observable from its traditional form (i.e., from
its self-adjoint representation) to an unconventional alternative which is
non-Hermitian but which happens to be Hermitizable. The Hermitization is still
needed, mediated by an amended Hilbert-space inner-product metric $\Theta$
which, “it it exists” [7], varies with our choice of the observable.

In applications one often works with an observable Hamiltonian. Whenever its
most standard self-adjoint SP version $\mathfrak{h}\neq\mathfrak{h}(t)$
happens to be user-unfriendly, the desirable user-friendliness can be
recovered after transition to its suitable non-Hermitian avatar. A full
compatibility of the resulting hiddenly Hermitian NSP reformulation of quantum
mechanics is achieved when, with a suitable $\Theta=\Theta(H)$, the new, non-
Hermitian Hamiltonian $H\neq H^{\dagger}$ remains $\Theta-$quasi-Hermitian,

$H^{\dagger}=\Theta\,H\,\Theta^{-1}\,.$ (28)

We already reminded the readers that in section 9.2 of review [18] it has been
pointed out that the NSP-based construction of the stationary inner-product
matric $\Theta=\Theta(H)$ plays a particularly important role in relativistic
quantum mechanics (with $H\neq H^{\dagger}$ being the Klein-Gordon operator)
and in the various versions of application of the Dirac’s method of
constrained quantization to gravity (with $H\neq H^{\dagger}$ being the
Wheeler-DeWitt operator). It is only desirable to add now that after a
transition to the more advanced NIP version of the theory in which one decides
to work with the time-dependent Hilbert-space metrics $\Theta(t)$, most of the
above-cited statements must be thoroughly reformulated. In particular, the
most general non-stationary version of the Klein-Gordon operator of the
relativistic QM cannot remain consistently identified with the observable
operator $H(t)$ anymore.

In our present paper the same change of paradigm has been described and shown
necessary also in the NIP approach to the genuine, non-stationary Wheeler-
DeWitt equation of quantum gravity. For the purpose, naturally, multiple
technical simplifying assumptions had to be accepted.

###  6.1 Conventional time-asymmetric QM concept of the evolution

The description of a quantum system in which the observables are represented
by operators is, certainly, richer than the description of its classical limit
[57]. One of the related paradoxes is that a substantial part of the success
of quantum theory is, in some sense, serendipitious, based on a lucky choice
of one of many eligible “quantizations”. In this sense we are currently not
too lucky when trying to quantize the Einstein’s general relativity (see,
e.g., the Isham’s foreword to the Thiemann’s monograph [22]).

One of the problems is, in the Thiemann words, that the “quantum theory of the
non-gravitational interactions …completely ignores General Relativity” while
the latter classical theory alias  geometry “completely ignores quantum
mechanics” (see p. 9 in loc. cit.). In our present paper, in this sense, we
tried to stay, firmly, in the framework of non-relativistic quantum mechanics.

After such a simplification the survival of the concept of time $t$ enables
one to order the evolution in a strictly causal manner. Incidentally, the
“fixed-frame” restriction of such a type can be softened by a change of
perspective working with another, “non-time” evolution parameter [46]. In an
extreme case as presented and discussed in methodical study [47], one can even
quantize the time itself, i.e., one can treat $t$ as a “pure-state” eigenvalue
of a “quantum clock” operator.

The idea of such a type is presented also in the Rovelli’s monograph [21]. We
can read there that only in the conventional approaches one believes that “the
Schrödinger picture is only viable for theories where there is a global
observable time variable $t$”. Naturally, “this conflicts with GR [general
relativity], where no such variable exists” (cf. pp. 10 and 11 in loc. cit.).
One has to conclude that a properly covariant formulation of the unitary
quantum evolution near Big Bang is still not too well understood and
formulated at present, especially because after the replacement of quantum
mechanics by quantum field theory (QFT), one reveals that also “most of the
conventional machinery of perturbative QFT is profoundly incompatible with the
general-relativistic framework” [21]. Thus, only the traditional,
perturbation-approximation-based pragmatic approaches to the predictive
cosmology seem to be available at present [48].

In this sense we proposed here that one of the possible schematic keys to the
puzzle might be sought in the quantization of the classical GR singularities
(like Big Bang) using, on quantum level, the Kato’s [49] concept of the
exceptional-point degeneracy of the schematic, non-covariant Universe at
$t=0$.

###  6.2 More realistic frameworks like loop quantum gravity

The current progress in experimental astronomy is amazing: we already
mentioned the measurements of the cosmic microwave background [28]. This
confirmed the Big Bang hypothesis experimentally. In parallel, its
mathematically singular nature also motivated an intensification of the
efforts of making the Einstein’s classical general relativity (GR) compatible
with the first principles of quantum mechanics (QM) [21, 22].

The recent progress in this direction is remarkable. We already mentioned the
studies of the conventional canonical recipes aimed, according to Wheeler [19]
and DeWitt [20], at the constructions of a “wave-function of the Universe”.
Among the more recent related theoretical results one must mention also the
formalism of the so called loop quantum gravity (LQG, [58]). In this setting,
one is really able to work with the modified QM called “relational”, with some
basic details mentioned in section Nr. 5.6 of monograph [21]. Still, we read
there that the relational reformulation of QM “has aspects that need to be
investigated further” (cf. p. 367 in [21]).

On these grounds our interest in the problem has been born. During one of the
seminars on the subject (dedicated to the description of quantum Big Bang) we
imagined that people very often come to a quick conclusion that the classical
GR singularities (like, typically, the Big-Bang-mediated “abrupt” birth of the
Universe) must necessarily  get, according to the conventional wisdom,
“smeared” (i.e., in the mathematical sense, “regularized”) after quantization.

For a long time, the latter intuitive expectation had been widely accepted. A
replacement of the Big-Bang singularity by the so called Big Bounce was
advocated by the widest LQG community [59, 60]. Only very recently the
assertion has been reconsidered and opposed [61]. This means that the
competition between the Big Bang and Big Bounce hypotheses may currently be
considered reopened.

In our present toy model the quantum Big Bang instant remains singular.
Counterintuitive as such a possibility seems to be, one could see its multiple
analogues, say, in the physics of phase transitions. Naturally, many forms of
the description of the conventional phase transitions are more or less
standard, not requiring the use of the sophisticated mathematics of the LQG
approach. At the same time, the newly emerging undecided status of the quantum
Big Bang hypothesis represents a challenge. We believe that the new forms of
insight were also provided by our present paper.

###  6.3 A broader physical context

One of the main formal supports of optimism may be seen in the fact that one
of the key formal features of our present NIP theory is in its richer
representation of quantum dynamics. Indeed, in the conventional version of QM
the flexibility of the model-building processes is strongly restricted by the
fact that the (pure) state of a unitary quantum system of interest is merely
represented by a ket-vector element $|\psi^{(SP)}\\!\\!\succ\,\,$ of a
preselected and time-independent Hilbert space ${\cal L}^{(SP)}$. In contrast,
the mathematical and phenomenological roles of the ket-vectors in the NIP
Hilbert space ${\cal H}_{(Dyson)}$ become separated. The amended theory works
with the two non-equivalent versions of the latter space, viz., with ${\cal
H}_{math}$ (where the inner product in elementary but unphysical) and with
${\cal H}_{phys}$. In the latter case one can say that either the definition
of the correct, physical inner product contains the operator of metric, or
that the operation of the physical Hermitian conjugation is realized as the
less conventional antilinear map $|\psi\rangle\ \to\ \langle\\!\langle\psi|$.
This, indeed, simplifies the formalism because the mathematically user-
friendly space ${\cal H}_{math}$ (which must be declared “unphysical”) can
also serve as a representation space for ${\cal H}_{phys}$.

From such a perspective the NIP approach comes with the new possibility of
making the family of the gravity-related quantum field theories “background-
independent”(cf. p. 22 in [21], or the more detailed comments in [22]). From a
purely pragmatic point of view this simply means that in the conventional
models (i.e., say, in the point-particle wave functions $\psi(\vec{x})$) even
the parameters (i.e., in this case, the coordinates $\vec{x}$) have to be
perceived as eigenvalues of a suitable operator (let us note that many of the
associated technical problems are discussed in the framework of the so called
non-commutative-geometry [62]).

In our present paper, an innovative realization of the background-independence
requirement has been achieved by making the time-dependent radius of the
expanding Universe quantized, i.e., identified, in the pure-state multiverse-
philosophy spirit, with one of the eigenvalues of an ad hoc  quasi-Hermitian
operator $R(t)$.

##  7 Conclusions

At present, the use of non-Hermitian operators in quantum theory is remarkably
diversified, ranging from the traditional and pragmatic effective-operator
descriptions of the open and resonant quantum systems [31] up to the new
horizons opened by the studies of the abstract mathematical aspects of the
formalism [56].

In a narrower domain of the description of the closed (i.e., unitary) quantum
systems using non-Hermitian operators the main division line is the one which
separates the stationary and non-stationary theories. In the former subdomain
the Coriolis forces vanish so that $H=G$. There emerge no problems with
calling the Schrödinger-equation generator a Hamiltonian [18].

In the latter, non-stationary-theory subdomain the situation is different. We
have to work there with the less elementary relation

$H(t)=G(t)+\Sigma(t)$ (29)

(called, by some authors, the time-dependent Dyson equation [41, 63] – [67]).
The term “Hamiltonian” must be then allocated, interpreted and used with much
more care [68].

In the stationary NSP setting the idea of acceptability of the various non-
Hermitian forms of quantum Hamiltonians has its origin in the Dyson’s paper
[5]. The knowledge of a standard stationary self-adjoint Hamiltonian
$\mathfrak{h}$ of textbooks (which is, by definition, safely self-adjoint in
${\cal L}^{(SP)}$) was simply complemented there by a tentative, “trial and
error” choice of $\Omega$. Via the isospectrality constraint (3) one was
immediately able to define a preconditioned, friendlier stationary
representation $H$ of the conventional Hamiltonian. This made the innovative
“Dyson’s picture” of QM complete.

The encouraging experience with the $\Omega-$mediated simplifications of
multiple conventional Schrödinger equations (say, in nuclear physics [6])
inspired Scholtz et al [7] to invert the paradigm. They assumed that what we
are given are just the “tractable” time-independent operators of the
observables (including, first of all, the Hamiltonian $H$) which are non-
Hermitian but which possess the real spectra. The core of the idea (i.e., of
the “quasi-Hermitian” reformulation of quantum mechanics called non-Hermitian
Schrödinger picture (NSP)) was that once we recall the respective quasi-
Hermiticity constraints (cf., e.g., Eqs. (4) or (18) above), we may
reconstruct (not always uniquely) and factorize (also not always uniquely) the
correct physical Hilbert-space metric $\Theta=\Omega^{\dagger}\,\Omega$ “if it
exists” (cf. p. 74 in [7]). The resulting “quasi-Hermitian-input” version of
the NSP formalism is then again a consistent theory.

The authors of paper [7] were well aware of the main weaknesses of their NSP
recipe. They identified them as lying, in the sufficiently realistic models,
not only in the ambiguity of the assignment of $\Theta$ to a given Hamiltonian
$H$ but also in the technically rather complicated nature of an explicit
construction of any such a metric (cf. also a few related comments in [18]).
Fortunately, a way out of the dead end has been found by Bender with coauthors
[3, 11] who proposed to narrow the class of the eligible non-Hermitian
stationary Hamiltonians $H$. The more user-friendly subfamily of the
Hamiltonians was required ${\cal PT}-$symmetric, i.e., such that
$H^{\dagger}{\cal PT}={\cal PT}\,H$. Originally, the symbol ${\cal P}$ denoted
here just the operator of parity while the antilinear operator ${\cal T}$
mediated the time reversal. Later, it became clear that after a suitable
generalization of these concepts, the physics-motivated property of the ${\cal
PT}-$symmetry of $H$ can be also perceived as mathematically equivalent to the
self-adjointness of $H$ with respect to a suitable pseudo-metric, i.e., as the
self-adjointness of $H$ in Krein space [69, 70].

The success of the ${\cal PT}-$symmetric models was enormous [11].
Paradoxically, it also appeared to have the two not entirely pleasant
consequences. The first one was that around the year 2007 the mainstream
research left the rather narrow area of quantum physics. Beyond this area
(i.e., typically, in classical optics) the idea of ${\cal PT}-$symmetry found
a large number of new and exciting applications (for reviews see [71] or the
recent monographs [72, 73]). The second paradox connected with the deep appeal
of the idea of the ${\cal PT}-$symmetry of $H$ can be seen in the above-
mentioned narrowing of the scope and perspective. In the words written on p.
1198 of review [18], “the adopted terminology is rather unfortunate” because
the “${\cal PT}-$symmetric QM is an example of a more general class of
theories …in which ${\cal PT}-$symmetry does not play a basic role”.

As another unwanted consequence of the reduction of the scope of the ${\cal
PT}-$symmetric version of the theory there emerged (and, for a long time,
survived) several “no-go” theorems (sampled, e.g., by Theorem Nr. 2 in [18])
which claimed the impossibility of a sufficiently satisfactory non-stationary
extension of the quasi-Hermitian quantum mechanics. It took several years
before the correct and consistent non-stationary extension of the quasi-
Hermitian quantum mechanics as described in [25, 26] has finally been accepted
as correct (cf., e.g., [74]). The process of acceptance was also slowed down
by certain purely terminological misunderstandings (cf., e.g., their brief
account in [23]). At present, fortunately, the situation seems clarified.
Different groups of authors (using still very different notation conventions,
cf., e.g., papers [75] or [42]) accepted, ultimately, the same (or at least
practically the same) interpretation of the non-stationary NIP theory.

The related developments enriched the field by a number of the new and highly
relevant applications. Virtually all of them can be characterized by the role
played by the time-dependent Dyson equation (29) (cf., e.g., section Nr. 5 in
[43]). The build-up of the theory may then start either from the knowledge of
$H(t)$ (so that one can speak about a “dynamical-information” (DI) input), or
from the knowledge of $\Sigma(t)$ (one then relies upon a purely kinematical
or “Coriolis-force” (CF) input information), or, finally, from $G(t)$ (let us
call this option a “Schrödinger-generator” (SG) input knowledge).

In all of these alternative approaches their users decided to call their
preferred preselected component of Eq. (29) “the Hamiltonian”. In fact, the
above-cited words that “the adopted terminology is rather unfortunate” applied
again. The main reason is that even in the unitary evolution dynamical regime
the spectra of $\Sigma(t)$ and/or of $G(t)$ need not be real or even complex
conjugate in general [39, 40, 68]. In this sense, calling the generator $G(t)$
a Hamiltonian (which was, originally, the proposal by one of my PhD students
[76, 77]) is far from optimal because only the spectrum of the observable-
energy component $H(t)$ of $G(t)=H(t)-\Sigma(t)$ can consistently be assumed
real.

On these grounds the most natural implementations of the NIP approach seems to
be provided by its DI model-building realization. In our recent paper [43]
such a conjecture has been tested using the exactly solvable wrong-sign-
oscillator model of Fring and Tenney [30]. We came to a not quite expected
conclusion that for the model in question, by far the most convenient and
efficient construction strategy appeared to be the innocent-looking
“kinematical” CF approach.

This observation can be perceived as one of the sources of inspiration of our
present paper. It forced us to reconsider the theory and to re-read one of the
oldest studies in the field, viz., paper [7] in which the authors always kept
in mind the need of working with a complete set of observables rather than
just with a Hamiltonian. We imagined that precisely this idea offers also the
“missing source” of a deeper understanding of the non-stationary NIP theory.

The return to the roots helped us to resolve at least some of the paradoxes.
For example, once one starts thinking about the unitary systems characterized
by more than one observable [7, 24], the build-up of the theory starting from
the mere single operator $H(t)$ appears to be conceptually less satisfactory.
During the build-up of a more satisfactory theory one must keep in mind both
the dynamics (i.e., the influence of $H(t)$ upon the states $\psi(t)$ as
mediated by Schrödinger equation(s)) and the kinematics (due to the fact that
$H(t)$ only appears in Schrödinger equation(s) in combination with Coriolis
force).

In our present paper we managed to show that the initial choice of a “non-
dynamical” observable (i.e., in our present notation, of $R(t)$) simplifies
the constructions significantly. This is, after all, our present main
methodical message. We saw that our innovative strategy does not only
simplify, decisively, the “introductory-step” reconstruction of the kinematics
(i.e., of the metric as well as of the Dyson map and of $\Sigma(t)$ from
$R(t)$), but that it also leaves an entirely unrestricted space for the
subsequent choice of the “dynamics”, i.e., for an independent specification of
the instantaneous energy $H(t)$, etc.

We may only add that our other, serendipitious, physicists-addressing message
is that the independence of the initial choice of the non-dynamical observable
$R(t)$ might very well serve the purpose of the extension of the applicability
of the unitary NIP quantum theory to the “exotic”, exceptional-point-related
dynamical regimes. This is sampled, in our schematic cosmological toy model,
by the demonstration of the possibility of an internal consistence of the
hypothetical point-like Big Bang singularity even after the quantization.

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