Wheeler-DeWitt equation and the applicability of crypto-Hermitian interaction
representation in quantum cosmology

Miloslav Znojil

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

Rokitanského 62, 50003 Hradec Králové, Czech Republic

and

The Czech Academy of Sciences, Nuclear Physics Institute,

Hlavní 130, 250 68 Řež, Czech Republic

e-mail<EMAIL_ADDRESS>
## Abstract

In the broader methodical framework of quantization of gravity the crypto-
Hermitian (a.k.a. non-Hermitian) version of the Dirac’s interaction picture is
considered. The formalism is briefly outlined and shown well suited for an
innovative treatment of certain cosmological models. In particular, it is
demonstrated that the Wheeler-DeWitt equation could be a promising candidate
for the description of evolution of quantized Universe near its initial Big
Bang singularity.

## Keywords

.

quantum gravity and the problem of Big Bang;

hiddenly Hermitian formulations of quantum mechanics;

stationary Wheeler-DeWitt system;

physical Hilbert space metric;

non-stationary Wheeler-DeWitt system;

##  1 Introduction

The concept of the wave function $\psi$ of the Universe (introduced, 55 years
ago, as a solution of the Einstein-Schrödinger alias  Wheeler-DeWitt (WDW)
equation [1, 2]) is contradictory. On positive side this concept played a key
role during the development of the canonical quantization of gravity [3].
These efforts climaxed in the recent comparatively satisfactory and
constructive formulation of the so called loop quantum gravity (LQG, [4, 5,
6]). At the same time, Mostafazadeh pointed out, in his review of the recent
progress in quantum theory [7], that the solutions $\psi$ themselves remain
“void of a physical meaning” without “finding an appropriate inner product on
the space of solutions of the WDW equation” (see p. 1291 in review [7]). In
loc. cit. Mostafazadeh also emphasized that “the lack of a satisfactory
solution to this problem has been one of the major obstacles in transforming
canonical quantum gravity and quantum cosmology into genuine physical
theories”. Precisely this obstacle is to be addressed and discussed in what
follows.

In the cited review we can further read that “in …quantum cosmology …the
relevant field equations …are second order differential equations in a “time”
variable …[which] have the …general form

$\frac{d^{2}}{dt^{2}}\psi(t)+D(t)\,\psi(t)=0$ (1)

where $t$ denotes a dimensionless time variable, $\psi:\mathbb{R}\to{\cal L}$
is a function taking values in some separable Hilbert space ${\cal L}$ , and
$D:{\cal L}\to{\cal L}$ is a positive-definite operator that may depend on
$t$”. Treating the latter variable as “a fictitious evolution parameter in
quantum cosmology” (see p. 1292 in [7]), the same author later adds that “the
cases in which $D$ is $t-$dependent (that arises in quantum cosmological
models) require a more careful examination”. In this sense we are prepared to
discuss some of the open questions and subtleties of the theory.

In loc. cit., Mostafazadeh redirected interested readers to his earlier study
[8]. In a series of our subsequent unpublished comments on this topic [9]
(which were later finalized and summarized in papers [10, 11]) we showed that
an appropriate “dealing with these cases” is, simultaneously, less complicated
and more complicated than it seems. Less complicated in the sense that some of
the technical obstacles have later been found surmountable, and more
complicated because it appeared necessary to amend the overall quantum-
theoretical framework and to replace the non-Hermitian Schrödinger-picture
(NSP) interpretation of the evolution of $\psi$ (as presented, basically, in
[7] or [8]) by the more involved formalism called interaction picture (IP)
(a.k.a. Dirac’s representation; see a comprehensive review of its non-
Hermitian form (NIP) in [12]).

In what follows we intend to outline the implementation of the NIP approach in
the WDW case. The key purpose of our paper is to provide an explicit
explanation of the connection between several challenging and open physical
questions (a typical one concerns the quantum Big Bang problem as formulated
in section 2) and the most recent progress in the hiddenly unitary version of
quantum mechanics (the basic features of this theoretical innovation are
reviewed). Our main message (viz, the detailed description of the theory and
of its application to the WDW equation) will finally be outlined in section 3
(devoted to a specific schematic toy-model of the quantum geometry of the
Universe), in section 4 (on the full-fledged NIP formalism), and in section 5
(in which the mechanism of transition to the Big Bang singularity will be
given its ultimate model-independent construction-recipe form). Our results
will be discussed in section 6 and summarized in section 7.

##  2 Challenge: Quantum Big Bang problem

At present it is widely believed that up to the “youngest age” of the Universe
(i.e., for times $t>t_{1}$ with $t_{1}\approx 10^{32}$ seconds) the evolution
(i.e., a slow expansion) of the Universe is more or less safely controlled by
the classical theoretical cosmology. In contrast, in the interval of times
$(t_{0},t_{1})$ (where $t_{0}=0$ denotes a hypothetical time of Big Bang) we
still miss a fully consistent and rigorous quantum theory behind the early
history of the Universe [3].

###  2.1 Could the degeneracy survive quantization? Yes, it could.

In our present study we felt strongly motivated by the deep relevance of the
understanding of the evolution of the Universe near its Big Bang origin, i.e.,
in a genuine quantum dynamical regime. In this regime the theoretically most
ambitious LQG formalism still seems to lead to at least some contradictory
results. In one of the LQG predictions [13], for example, the Big Bang
singularity (compatible with the classical Einstein’s theory of gravity) has
been found to be smeared out by the quantization. In the series of papers [14,
15, 16, 17, 18, 19] or in section Nr. 8 in [5]), for example, it is claimed
that the Big Bang singularity of classical theory must necessarily be replaced
by a regularized “Big Bounce” mechanism. In contrast, more recently, Wang with
Stankiewicz [20] came with an opposite conclusion claiming that within the
scale-invariant LQG framework “the quantized Big Bang is not  replaced by a
Big Bounce”.

Figure 1: Eigenvalues of matrix (2) (avoided crossing phenomenon).

At the first sight, the latter claim looks suspicious. In Rovelli’s words, the
quantization-related “absence of singularities” is in fact “what one would
expect from a quantum theory of gravity” (see p. 297 in [5]). An elementary
support of such an intuitive expectation can be provided by the following
schematic observable

$\Lambda(t)=\left[\begin{array}[]{cccc}0&-1+i\,t&0&0\\\ \vskip 6.0pt plus
2.0pt minus 2.0pt\cr-1-i\,t&0&-1+i\,t&0\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr 0&-1-i\,t&0&-1+i\,t\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr
0&0&-1-i\,t&0\end{array}\right]$ (2)

and by the inspection of its spectrum (see Figure 1). As long as the matrix is
Hermitian, its spectrum must be real. Moreover, in the generic case (i.e.,
unless we impose a symmetry upon the matrix), the spectrum must  remain non-
degenerate. This is the reason why the levels avoid the crossing (which would
simulate the regularized Big Bounce). In our example the proof of the
phenomenon is elementary: Up to a small vicinity of the “Big Bang time”
$t^{(BB)}=0$ the matrix as well as its spectrum are dominated by their
asymptotic components which are strictly linear in $t$. One might even suspect
that the eigenvalues could cross due to an accidental symmetry emerging at
$t=0$ but such a symmetry is manifestly broken by the $t-$independent
component of the model.

We are going to show that against all expectations, the latter argument is not
foolproof. Admitting that it need not necessarily  lead to the wrong
conclusions, we will only show that the Wang’s and Stankiewicz’s alternative
scenario [20] may equally well be supported by an equally elementary toy
model. The essence of such a claim is that the Hermiticity property (cf.
relation $\Lambda=\Lambda^{\dagger}$ satisfied by our toy-model matrix (2),
with the superscript † marking the matrix transposition plus complex
conjugation) depends on a mathematically motivated a priori  specification of
the inner product in our physical Hilbert space of states [21].

A deeper abstract foundation of our “constructive scepticism” concerning the
genericity of the Big Bounce may be found in the literature on quantum
mechanics using non-Hermitian operators [7, 22, 23, 24]. In this sense, the
common requirement of the self-adjointness of the operators of observables
$\Lambda(t)$ can be weakened and replaced by the condition of their
Hermitizability alias  quasi-Hermiticity [22]. In many non-Hermitian models,
indeed, the Hermiticity may be restored by the mere ad hoc  amendment of the
inner product [25].

In our present paper, we will narrow the scope of the discussion to the Big
Bang and to the WDW equations. Simultaneously, we will broaden the theoretical
framework, emphasizing that in the genuine Big Bang spatial-degeneracy context
it is necessary to replace the most common NSP mathematics by its perceivably
more complicated NIP amendment. In a preparatory step let us now return to the
toy model (2) and let us Wick-rotate the time $\,t\to-{\rm i\,}t$ and shift
the origin, $\,t\to t-1$. The resulting new matrix

$Q(t)=\left[\begin{array}[]{cccc}0&-2+{t}&0&0\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr-{t}&0&-2+{t}&0\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr
0&-{t}&0&-2+{t}\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr
0&0&-{t}&0\end{array}\right]$ (3)

is just a hiddenly Hermitian (i.e., via an amendment of the inner product,
Hermitizable) candidate for a toy-model observable representing, in the
context of quantum cosmology, say, a potentially measurable discrete spatial
grid [26, 27, 28, 29, 30].

Figure 2: The reality of spectrum of non-Hermitian matrix (3) at not too large
$t\geq 0$.

In essence, the latter example indicates that the Big-Bang-type singularities
need not  necessarily be smeared out by the quantization. Indeed, at the not
too large values of the positive time parameter $t>0$ the spectrum of our
manifestly non-Hermitian model (3) may be shown real and non-degenerate. This
is illustrated in Figure 2. At $t=0$ the spectrum becomes degenerate and the
matrix itself ceases to be diagonalizable.

The latter simulation of the Big Bang singularity is called exceptional point
(EP) in mathematics [31, 32]. In the complementary context of physics, the
spatial-grid interpretation of the time-dependent eigenvalues $\lambda_{n}(t)$
as sampled in Figure 2 enables us to speak about the “inflation period” of the
history of the related hypothetical and highly schematic (i.e., four-point)
quantized Universe immediately after its birth. Naturally, the corresponding
internally consistent quantum theory must be reformulated accordingly [22].

###  2.2 Stationary theory (non-Hermitian Schrödinger picture, NSP)

In the pedagogically oriented and compact review of the history of quantum
mechanics [33] the authors emphasized that there exists no universal version
of quantum theory and that “no formulation produces a royal road to quantum
mechanics”. This explains an incessant emergence of the new versions of the
theory including its recent “non-selfadjoint-operator” formulations [23].

Incidentally, the “non-selfadjoint-operator” characteristics of these theories
could be misleading. As we already indicated, the mathematical concept of non-
selfadjointness (or, in the shorthand terminology used by physicists, of non-
Hermiticity) is ambiguous. Covering, in various branches of physics, both  the
generators of the unitary evolution [22] and  of the non-unitary evolution
[34]. It is necessary to emphasize that only the former (i.e., unitarity-
compatible) meaning of the word ‘non-Hermiticity’ will be considered and taken
into account in what follows.

The disambiguation really deserves the early notice because the difference is
often less clear in applications. Also the formulation of the physical
background of the problems happens to suffer of ambiguities. The details will
be discussed later (see, first of all, the introduction in the problem as
given in Appendix). Now, let us only repeat that the questions which we are
going to discuss will have their origin in the field of quantum gravity [3].
In this broad context our attention will be paid, first of all, to the
possible role played by the WDW equation and to the questions of physics near
the Big Bang (BB) singularity.

###  2.3 Stationary Wheeler-DeWitt equation

In the stationary case the WDW problem becomes formally equivalent to the
Klein-Gordon (KG) problem known in the relativistic quantum mechanics [8]. In
their simplest versions, both of these problems may be characterized, in
suitable units, by the linear differential equation. Thus, in the KG case
(where the suitable units are $\hbar=c=1$ and where one omits, for the sake of
simplicity, the electromagnetic field) we have, for example,

$\left(\frac{\partial^{2}}{\partial
t^{2}}+D^{(KG)}\right)\,\psi^{(KG)}(\vec{x},t)=0\,,\ \ \ \ \ \
D^{(KG)}=-\triangle+m^{2}\,.$ (4)

The kinetic energy is represented here by the elementary Laplacean
$\triangle$, and the dynamics can be maximally reduced to the mere scalar mass
term which may be made position dependent, $m^{2}=m^{2}(\vec{x})$.

In the simplest non-stationary WDW model the analogue of the mass term would
be a time-dependent function (cf., e.g., section Nr. 3.5 of review [7] for
further references). The KG-WDW analogy enables us to use the same
mathematical tools. The relevant literature is fairly extensive but for our
present purposes it is sufficient to cite the paper by Feshbach and Villars
([35], cf. also Ref. [36]) in which the change of variables

$\psi^{(WDW)}(\vec{x},t)\ \ \to\ \
\langle\vec{x}|\psi^{(FV)}(t)\rangle=\left(\begin{array}[]{c}{\rm
i}\partial_{t}\psi^{(WDW)}(\vec{x},t)\\\
\psi^{(WDW)}(\vec{x},t)\end{array}\right)\,$ (5)

was shown to lead to a replacement of the hyperbolic partial differential Eq.
(4) by the Schrödinger-like parabolic equation for the two-component wave
function (5),

${\rm i}\frac{\partial}{\partial
t}\,\,|\psi^{(FV)}(t)\rangle=G_{(FV)}(t)\,|\psi^{(FV)}(t)\rangle\,.$ (6)

This equation can be interpreted as controlling the unitary evolution of the
system via the generator alias  FV Hamiltonian

$\ \ \ G_{(FV)}(t)=\left(\begin{array}[]{cc}0&D(t)\\\
I&0\end{array}\right)\neq G_{(FV)}(t)\,.$ (7)

Such an operator is, in the FV Hilbert space

${\cal H}^{(FV)}={\cal L}^{2}(\mathbb{R}^{3})\,\bigoplus\,{\cal
L}^{2}(\mathbb{R}^{3})$

manifestly non-Hermitian, $G_{(FV)}\neq G_{(FV)}^{\dagger}$. Pauli with
Weisskopf [37] noticed that the same operator can in fact be treated as self-
adjoint with respect to another, indefinite inner product,

$\langle\psi_{1}|\psi_{2}\rangle\ \ \to\ \
\left(\psi_{1},\psi_{2}\right)_{(Krein)}=\langle\psi_{1}|{\cal
P}_{(FV)}|\psi_{2}\rangle\,.$ (8)

i.e., that it is self-adjoint in another, ad hoc  Krein space. In the modern
terminology one would say that this operator is non-Hermitian but ${\cal
PT}-$symmetric [38].

A decisive progress achieved under the stationarity assumption $G_{(FV)}\neq
G_{(FV)}(t)$ (or, more precisely, after its generalized form called quasi-
stationarity assumption) is due to Mostafazadeh. In his papers [8, 39] he
imagined that the FV pseudometric ${\cal P}$ could be replaced by the positive
definite metric $\Theta_{(stationary)}$ converting the Krein-space physics (in
which, during evolution, the usual norm is not conserved) into the fully
standard and norm-conserving Hilbert-space physics. In essence, just a
straightforward change of the inner product was needed,

$\left(\psi_{1},\psi_{2}\right)_{(Krein)}\ \ \to\ \
\left(\psi_{1},\psi_{2}\right)_{(Mostafazadeh)}=\langle\psi_{1}|\Theta_{(stationary)}\,|\psi_{2}\rangle\,.$
(9)

This opened the way towards a consistent picture of unitary physics in which
the stationary Hamiltonian $G_{(FV)}=H_{(FV)}$ controls the NSP quantum
evolution which is, with respect to the amended inner product (9), unitary.

After either the KG or the WDW interpretation of Eq. (4) in stationary case,
the Hilbert-space metrics in (9) can be given a formal block-diagonal-operator
structure

$\Theta_{(stationary)}=\left(\begin{array}[]{cc}1/\sqrt{D}&0\\\
0&\sqrt{D}\end{array}\right)\,.$ (10)

This leads to the first quantization of both of these systems.

##  3 Fine-tuned nature of the quantum Big Bang

The conventional mental operation called “quantization of the classical
theory” does really very naturally lead to the conclusion that the singularity
gets “smeared our” near $t\approx 0$ due to “quantum effects” [40, 41] (see
also the four-by-four Hermitian matrix (2)). In our text we pointed out that
the support of such a regularization hypothesis is only unavoidable in the
conventional “textbook” quantum mechanics. In a more general, hiddenly
Hermitian theory such an assumption is artificial and unfounded (cf. Appendix
or toy model (3)). Once one overcomes the mental barrier one reveals that the
inner product may start playing the central descriptive role.

###  3.1 The $N-$grid-point toy model of kinematics

In the literature the manifestly non-Hermitian but Hermitizable Wheeler-DeWitt
equation has only been considered in the stationary (or, better, quasi-
stationary) mathematical NSP regime (cf. [7] or section 2.2 above). In section
4 we will turn attention to the conceptual necessity of keeping the WDW-
related Hilbert space time-dependent. In an overall context of the canonical
quantization of gravity we have to be prepared to address, therefore, a number
of purely technical questions and tasks.

In the first one the point-like Big Bang must be made compatible with a
consequent theoretical unitary-evolution scenario. Thus, we have to complement
the abstract argumentation of section 2 by a detailed description of a
suitable concrete toy model. In the model the measurable values of the spatial
grid points (say, the necessarily real and time-dependent values $q_{j}(t)$
with $j=1,2,\ldots N$) will have to be assumed obtainable, in principle at
least, as eigenvalues of a suitable hiddenly Hermitian geometry-representing
“effective kinematical input” operator (say, $Q^{(N)}(t)$).

Secondly, we have to keep in mind that in a way indicated by our four-by-four
matrix (3) we will assume that the general $N$ by $N$ matrix $Q^{(N)}(t)$ will
still be real and tridiagonal. Indeed, in a way explained in [42] the reality
and tridiagonality is an important merit of any  candidate for an observable
because it enables one to construct the metric algebraically, in recurrent
manner. In this sense we may recall the existing results in linear algebra
[43] and choose the one-parametric family of our $N$ by $N$ toy-model
“effective kinematics” as follows,

$Q^{(N)}(z)=\left[\begin{array}[]{cccccc}-i\,(N-1)z&-\sqrt{N-1}&0&0&\ldots&0\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr-\sqrt{N-1}&-i\,(N-3)z&-\sqrt{2(N-2)}&0&\ddots&\vdots\\\ \vskip 6.0pt
plus 2.0pt minus 2.0pt\cr 0&-\sqrt{2(N-2)}&-i\,(N-5)z&\ddots&\ddots&0\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr
0&0&-\sqrt{3(N-3)}&\ddots&-\sqrt{2(N-2)}&0\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\vdots&\ddots&\ddots&\ddots&i\,(N-3)z&-\sqrt{N-1}\\\ \vskip 6.0pt plus
2.0pt minus 2.0pt\cr 0&\ldots&0&0&-\sqrt{N-1}&i\,(N-1)z\end{array}\right]\,.$
(11)

The non-triviality of this matrix and the arbitrariness of its dimension $N$
in combination with its non-numerical tractability [44] will enable us to show
how the requirement of the existence of the quantum Big Bang singularity
becomes supported by a consistent reconstruction of the related physical time-
dependent Hilbert-space metric. As long as $z=z(t)$ can be any suitable
function of time we may restrict our considerations to the interval of
$z\in(-1,1)$ in the interior of which the grid-point-coordinate spectrum of
$Q^{(N)}(z)$ remains non-degenerate, real and discrete, and at the boundaries
of which one can visualize the realization of the Big Bang. Thus, after the
simplest choice of $z(t)=-1+t$ we obtain an immediate $N-$level analogue of
the graphical evolution pattern of Eq. (2) where we have $N=4$.

One of the main constraints imposed upon our toy-model “geometry operator”
(11) is its compatibility with the unitarity of the quantum evolution, i.e.,
with the existence of the Hilbert-space metric. Naturally, the process of the
evolution of the corresponding schematic Universe will have to start at the
Big Bang single-point-degeneracy singularity which is such that

$\lim_{t\to 0^{+}}q_{n}(z(t))=q_{1}(z(0))\,,\ \ \ \ \ n=1,2,\ldots,N\,$ (12)

On the technical level, one can really speak about a challenge because even
the purely formal construction of a highly schematic “Big-Banging” model of
the quantum Universe must remain compatible with the basic theoretical
requirement of compatibility between the kinematical input information (12)
and the dynamical input information as represented by the WDW Hamiltonian
operator. The details will be discussed below. For the time being let us only
assume that with the kinematical spatial-grid input (12) adapted to any
phenomenological requirements, the dynamics of the WDW-related Universe will
remain reflected by a suitable non-stationary form of the operator $D$ in its
form entering the non-stationary analogue of the stationary (or, if you wish,
adiabatic) form (10) of the WDW Hilbert-space metric.

###  3.2 The fine-tuned nature of the Hilbert-space metric $\Theta(z)$

One of our most important WDW-related model-building tasks can be seen in the
generalization of the qualitative and consistent picture of the quantum Big
Bang singularity as mediated by its $N=4$ grid-point realization via Eq. (3)
above (cf. also Figure 2). In such project we encounter the two main technical
obstacles. The first one lies in the necessity of the guarantee of the
existence of the metric $\Theta$ at all times $t>0$ (i.e., in our model, at
all of the sufficiently small positive times) up to the very Big-Bang birth-
of-the-Universe EP limit  $t\to 0^{+}$. In our toy model, due to its exact
solvability [43], such a guarantee will have an exact, non-numerical form.

The way of circumventing the second technical obstacle (viz., the necessity of
a guarantee that the Hilbert-space metric remains, at all of the relevant
times, non-singular and positive definite) is equally difficult to find. In
our model we shall see that for the model in question this goal can be
achieved by the non-numerical means as well.

The respective solutions of both of the above-mentioned problems are closely
interrelated. Their essence can be identified with the necessity of the
coexistence of the singularity in the grid with the singularity-free nature of
the metric $\Theta(z)$. The most universal approach to this problem has been
promoted by Scholtz et al [22] who proposed to use a complete  information
about the set of the observables $\Lambda_{1}(z)$, $\Lambda_{2}(z)$, …. Such
an “extreme” model-building strategy yielded a unique physical metric
$\Theta^{(N)}(z)$. In principle, its applicability is strongly $N-$dependent
of course. Thus, our methodical considerations will only concern the systems
with the smallest dimensions.

####  3.2.1 The eligible Hilbert-space metrics at $N=2$

At $N=2$, the grid-point operator (11) reads

$Q^{(2)}(z)=\left[\begin{array}[]{cc}-iz&-1\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr-1&iz\end{array}\right]\,,\ \ \ \ \ \ z\in(0,1)\,.$ (13)

With the four real parameters $a,c,d$ and $\chi\in(0,2\pi)$, with, for the
sake of definiteness, positive $z\in(0,1)$, and with the general ansatz

$\Theta^{(ansatz)}(a,c,d,\chi)=\left[\begin{array}[]{cc}a&c\,e^{-{\rm
i}\chi}\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr c\,e^{{\rm
i}\chi}&d\end{array}\right]$ (14)

for the Hilbert-space metric the condition of quasi-Hermiticitiy degenerates
to the two elementary relations,

$d=a=z^{-1}\,c\,\sin\chi\,.$

Without any loss of generality we may set $c=z$ and evaluate the eigenvalues
of matrix of Eq. (14),

$\lambda_{\pm}=\sin\chi\pm z\,.$

Thus, this matrix may be declared acceptable as a metric if and only if it is
positive definite, i.e., if and only if

$\sin\chi>z\,.$ (15)

This relation clearly indicates that near the EP limit $z\to 1$ the range of
variability of the admissible parameter $\chi$ (numbering the admissible
Hilbert-space metrics) becomes extremely narrow. Moreover, whenever the
dynamics-controlling parameter $z$ moves closer to the EP singularity the
interval quickly shrinks so that our choice of the metric must be, in the Big
Bang vicinity, very precisely “fine-tuned”.

Formula (15) gets further simplified when we reparametrize the strength of the
non-Hermiticity $z=\sin\beta\ $ in terms of the new variable
$\beta\in(0,\pi/2)\ $. Now, the Hermitian limit corresponds to $\beta=0$ while
the singular EP (or, if you wish, Big Bang or Big Crunch) extreme is reached
at $\beta=\pi/2$. Ultimately, formula

$\Theta(\beta,\chi)=\left[\begin{array}[]{cc}\sin\chi&e^{-{\rm
i}\chi}\,\sin\beta\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr e^{{\rm
i}\chi}\,\sin\beta&\sin\chi\end{array}\right]\,,\ \ \ \ \ \ \
\chi\in(\beta,\pi-\beta)\,$ (16)

defines, up to an inessential overall factor, all of the eligible correct
metric operators at $N=2$.

####  3.2.2 $N=3\,$ and the requirement of positivity

Once we move to the next geometry operator (11) with $N=3$, the general ansatz
for the metric may be reduced to a six-parametric Hermitian matrix

$\Theta=\left[\begin{array}[]{ccc}a&b{e^{i\phi}}&c{e^{i\chi}}\\\ \vskip 6.0pt
plus 2.0pt minus 2.0pt\cr b{e^{-i\phi}}&f&b{e^{i\phi}}\\\ \vskip 6.0pt plus
2.0pt minus 2.0pt\cr c{e^{-i\chi}}&b{e^{-i\phi}}&a\end{array}\right]\,.$ (17)

This reveals that the construction of the metric remains a purely routine
linear-algebraic problem. At the same time, the weakness of the construction
is found to lie in the less easy determination of the domain of parameters for
which the metric operator $\Theta$ remains positive definite. Although the
domain of positivity of the metric is still implicitly defined by the $N=3$
secular determinant and by the relation

${{\it\lambda}}^{3}+\left(-f-2\,a\right){{\it\lambda}}^{2}+\left(-2\,{b}^{2}-{c}^{2}+{a}^{2}+2\,fa\right){\it\lambda}+{c}^{2}f-f{a}^{2}+2\,a{b}^{2}-2\,{b}^{2}c\cos\left(2\,\phi-\chi\right)=0\,$

the $N=3$ analogue of the $N=2$ formula (16) would be complicated for an
explicit display. The task still remains non-numerical because the secular
polynomial remains linear in the parameters $b^{2}$, $f$ and/or
$\cos\left(2\,\phi-\chi\right)$. This still leaves the determination of the
range of the admissible parameters straightforward. A typical sample of such a
determination is provided by Figure 3.

Figure 3: Eigenvalues of our singularity-free metric (17) as functions of one
of the parameters at $N=3$.

In this illustrative picture we see that the eigenvalues of the metric remain
real and non-degenerate in a large interval of one of the dynamical $N=3$
parameters $\beta$. In a small vicinity of the singular EP/BB limit
$\beta\to\pi/2$ we may deduce that the rank of the metric becomes
approximately equal to one. The picture even shows the confluence of the
eigenvalues of the metric in the trivial-metric Hermitian-system limit
$\Theta\to I$, i.e., very far from the EP/BB dynamical regime.

Naturally, the technical difficulties will grow with the dimension. At the
larger $N$ the construction has to be given an alternative, purely graphical
form. This strategy has been used in paper [45] where it has been shown that
the use of the graphical method remains feasible even for the higher-order
secular polynomials (cf. Figures Nr. 16 and 17 in loc. cit.). Stil, one has to
expect that at the truly large matrix dimensions the construction becomes
purely numerical.

###  3.3 Candidates for the other observables

For any given non-Hermitian grid-point operator $Q^{(N)}(z)$ with the real and
nondegenerate spectrum $\\{q_{n}(z)\\}$ one can construct the arbitrarily
normalized eigenvectors,

$Q^{(N)}(z)\,|n(z)\rangle=q_{n}(z)\,|n(z)\rangle\,,\ \ \ \ n=1,2,\ldots,N\,.$
(18)

For the same spectrum, the arbitrarily normalized double-bra-marked left
eigenvectors may be also defined as the standard right eigenvectors of a
Hermitian conjugate operator,

$\left[Q^{(N)}(z)\right]^{\dagger}\,|n(z)\rangle\\!\rangle=q_{n}(z)\,|n(z)\rangle\\!\rangle\,,\
\ \ \ n=1,2,\ldots,N\,.$ (19)

It is easy to deduce that $\langle\\!\langle m|n\rangle=0$ for $m\neq n$. In
the generic case the overlaps $\langle\\!\langle m|m\rangle$ will be real and
non-vanishing. Whenever $N$ is finite, the resulting biorthogonal basis can be
used in a generalized spectral representation of the operator

$Q^{(N)}(z)=\sum_{n=0}^{N-1}\,|n(z)\rangle\,\frac{q_{n}(z)}{\langle\\!\langle
n(z)|n(z)\rangle}\,\langle\\!\langle n(z)|\,.$ (20)

We may conclude that the general (though not necessarily invertible or
positive definite) $N-$parametric Hilbert-space metric can be then defined by
formula

$\Theta^{(N)}=\sum_{n=0}^{N-1}\,|n\rangle\\!\rangle\,\kappa_{n}\,\langle\\!\langle
n|\,.$ (21)

The parameters $\kappa_{n}$ must be all real. The acceptability of the matrix
in the role of the physical Hilbert-space metric (i.e., the necessary
invertibility and positivity properties) are then guaranteed if and only if
$0<\kappa_{n}<\infty$ at all $n$ [46]. In such a setting, one can easily use
an analogous generalized spectral representation to define also any other
operator of an acceptable quantum observable.

##  4  Mathematics: Non-Hermitian interaction picture (NIP)

Naturally, the (quasi-) stationarity restriction becomes, in the WDW case,
hardly acceptable, especially if one tries to deal with the quantum dynamics
near a singularity like Big Bang. In such a case, a much deeper modification
of the formalism of the non-Hermitian quantum theory is needed.

After one decides to relax the assumptions of stationarity, an increase of the
complexity of the system of equations is partially compensated by the
clarification of several conceptual problems. In this sense our main
methodical recommendation is that in analogy with the Hermitian interaction
picture of textbooks, one still keeps in mind the necessity of description of
the dynamics in terms of both the operators and wave functions. In other
words, it is necessary to avoid several existing and wide-spread
misunderstandings which can be found in the current literature. Paradoxically,
the root of these misunderstandings may be seen in an insufficiently careful
use of the terminology (see, e.g., the explanatory “Rosetta-stone-like” Table
Nr. 1 in [12]). Indeed, once we replace a stationary NSP model by its non-
stationary IP and/or NIP alternative and extension, the concept of quantum
Hamiltonian ceases to be unique and adequate.

###  4.1 Non-stationary quantum systems

In the non-stationary quantum theory the use of the time-dependent metric is
known to lead to the loss of the unitarity of the evolution or to the loss of
the observability of the NSP Hamiltonian [7]. In fact [9], the puzzle is
artificial and purely terminological. The problem disappears when one employs
the non-Hermitian version of the Dirac’s interaction picture (NIP, [12]).

####  4.1.1 Evolution law for the NIP ket vectors

In the non-stationary non-Hermitian cases there is no need of the
observability of the generator of the evolution of the ket-vectors [47, 48,
49, 50]. Easily, the stationary version of the Dyson map (58) can be replaced
by its time-dependent generalization

$|\psi(t)\\!\\!\succ\,\,=\Omega(t)\,|\psi(t)\rangle\in{\cal H}^{(T)}\,,\ \ \ \
\ \ |\psi(t)\rangle\in{\cal H}^{(F)}\,.$ (22)

In ${\cal H}^{(F)}$, similarly, Schrödinger Eq. (59) acquires the form

${\rm i}\frac{\partial}{\partial t}\,|\psi(t)\rangle=G(t)\,|\psi(t)\rangle\,$
(23)

in which the generator is only one of the two unobservable components of the
observable instantaneous-energy operator

$H(t)=G(t)+\Sigma(t)\,.$ (24)

Only the sum will be called Hamiltonian in what follows. The other component
of the Hamiltonian can be defined directly in terms of the Dyson map,

$\Sigma(t)={\rm i}\Omega^{-1}(t)\,\dot{\Omega}(t)\,,\ \ \ \ \
\dot{\Omega}(t)=\frac{d}{dt}\,\Omega(t)\,$ (25)

(see [10, 11, 12] for details).

In the unitary-evolution case the observable version of the non-Hermitian but
Hermitizable Hamiltonian (24) is connected with its self-adjoint partner by
formula,

$H(t)=\Omega^{(-1)}(t)\,\mathfrak{h}_{(NSP)}(t)\,\Omega(t)\,.$ (26)

In ${\cal H}^{(F)}$ operator (26) has the property of quasi-Hermiticity,

$H^{\dagger}(t)\,\Theta(t)=\Theta(t)\,H(t)\,,\ \ \ \
\Theta(t)=\Omega^{\dagger}(t)\,\Omega(t)\,.$ (27)

In an internally consistent theory of a unitary (or hiddenly unitary) quantum
system the Hamiltonian still has to have the real and discrete spectrum
representing the instantaneous (but still observable) bound-state energies.

It is unfortunate that in the literature, only too may people assign the name
of a Hamiltonian also to both  of the other operators $G(t)$ and $\Sigma(t)$,
neither of which represents an observable quantity [9, 11, 49]. We prefer
calling operator $G(t)$ a “generator” (which does not represent an observable
while still controlling and generating the evolution of the IP/NIP wave
functions). In parallel, we would also propose calling operator $\Sigma(t)$,
say, a “Coriolis force”.

####  4.1.2 Evolution law for the NIP bra vectors

Equation (22) has a dual-space alternative

$|\psi(t)\\!\\!\succ\,\,=\left[\Omega^{\dagger}(t)\right]^{-1}|\psi_{\Theta}(t)\rangle\in{\cal
H}^{(T)}\,,\ \ \ \ \ \ |\psi_{\Theta}(t)\rangle\ \equiv\
\Theta(t)\,|\psi(t)\rangle\in{\cal H}^{(F)}\,.$ (28)

This enables us to treat the new states $|\psi_{\Theta}(t)\rangle\ \equiv\
\Theta(t)\,|\psi(t)\rangle$ as solutions of another Schrödinger equation in
${\cal H}^{(F)}$ [10, 11],

${\rm i}\frac{\partial}{\partial
t}\,|\psi_{\Theta}(t)\rangle=G^{\dagger}(t)\,|\psi_{\Theta}(t)\rangle\,.$ (29)

The process of solution of the two Schrödinger equations is maximally
economical. The key merit of this recipe (see also more commentaries in [12])
is that it circumvents the necessity of the technically much more complicated
direct construction of the metric as used, e.g., in papers [49, 51, 52, 53,
54].

The present version of the process must be initiated by the specification of
the respective states $|\psi(t)\rangle$ and $|\psi_{\Theta}(t)\rangle$ at
$t=t_{i}=0$. Thus, equations (23) and (29) have to be complemented by the
specification of the initial values represented by the kets
$|\psi(t_{i})\rangle$ and $|\psi_{\Theta}(t_{i})\rangle$. Naturally, such
values must obey constraints (22) and (28) at $t=t_{i}=0$. This, in turn, is
closely connected with the experiment and with the preparation of the system
in question.

###  4.2  Non-Hermitian operators in interaction picture

It is well known that even in the conventional Hermitian version of IP, the
Coriolis-force operators obey the Heisenberg-type equations. These equations
control the evolution of every  relevant operator of an observable.

In the non-Hermitian NIP formalism the role of $\Sigma(t)$ is analogous. In
both of the IP and NIP cases, the ultimate goal of the theory lies in the
derivation of the predictions of the results of measurements. In our present
version of the recipe, this merely requires the evaluation of the overlaps

$\langle\psi_{\Theta}(t_{f})\,|Q(t_{f})\,|\psi(t_{f})\rangle\,.$ (30)

Due to the identity

${\rm i}\,\frac{\partial}{\partial
t}\,\Theta(t)=\Theta(t)\,\Sigma(t)-\Sigma^{\dagger}(T)\,\Theta(t)\,$ (31)

or due to its alternative version (cf. Eq. (27)),

${\rm i}\,\frac{\partial}{\partial
t}\,\Theta(t)=G^{\dagger}(t)\,\Theta(t)-\Theta(t)\,G(t)\,$ (32)

the NIP formalism is internally consistent, indeed. At the same time, one has
to keep in mind that the operators of the IP or NIP observables are manifestly
time-dependent and that their time-dependence is not arbitrary.

####  4.2.1 Evolution law for the density matrices

In the non-Hermitian but unitary pure-state quantum systems of our present
interest the state is defined by a pair  of the ket vectors, i.e., by the
projectors

$\pi_{\psi,\Theta}(t)=|\psi(t)\rangle\,\frac{1}{\langle\psi_{\Theta}(t)\,|\psi(t)\rangle}\,\langle\psi_{\Theta}(t)|\,.$
(33)

Alternatively, one can speak about the non-Hermitian density matrix

$\widehat{\varrho}(t)=\sum_{k}|\psi^{(k)}(t)\rangle\,\frac{p_{k}}{\langle\psi^{(k)}_{\Theta}(t)\,|\psi^{(k)}(t)\rangle}\,\langle\psi^{(k)}_{\Theta}(t)|\,,\
\ \ \ \ \ \sum_{k}p_{k}=1\,.$ (34)

Due to Eqs. (23) and (29), this operator has to obey the specific evolution
equation

${\rm
i\,}\partial_{t}\,\widehat{\varrho}(t)=G(t)\,\widehat{\varrho}(t)-\widehat{\varrho}(t)\,G(t)\,$
(35)

which opens the way towards the formulation of quantum statistics in non-
Hermitian Liouvillean picture [12].

####  4.2.2 The evolution of observables

The requirement

$Q^{\dagger}(t)\,\Theta(t)=\Theta(t)\,Q(t)$ (36)

guarantees the observability status of any operator $Q(t)$. This relation is
equivalent, due to Eq. (62), to the NSP Hermiticity of $\mathfrak{q}(t)$ in
${\cal H}^{(T)}$ since

$Q(t)=\Omega^{(-1)}(t)\,\mathfrak{q}_{(NSP)}(t)\,\Omega(t)\,$ (37)

The Heisenberg-type evolution equation follows,

${\rm i\,}\frac{\partial}{\partial
t}\,{Q}(t)=Q(t)\,\Sigma(t)-\Sigma(t)\,Q(t)+K(t)\,,\ \ \ \ \
K(t)=\Omega^{(-1)}(t)\,{\rm i\,}\dot{\mathfrak{q}}_{(NSP)}(t)\,\Omega(t)\,.$
(38)

It is recommendable to assume that the partial derivatives
$\dot{\mathfrak{q}}_{(NSP)}(t)$ vanish so that the related operator $K(t)$
would be vanishing as well, making the process of the solution of Eq. (38)
user-friendlier.

Given the generator $G(t)$, the choice of the Coriolis force $\Sigma(t)$ is
far from arbitrary. First of all, it is constrained by the experiment-related
initial state vectors. Secondly, it must be compatible with its relation (24)
to the initial instant energy $H^{(NIP)}(t_{i})$ and to its evolution law

${\rm i\,}\frac{\partial}{\partial
t}\,{H}^{(NIP)}(t)=H^{(NIP)}(t)\,\Sigma^{(NIP)}(t)-\Sigma^{(NIP)}(t)\,H^{(NIP)}(t)+K^{(NIP)}(t)$
(39)

or, equivalently,

${\rm i\,}\frac{\partial}{\partial
t}\,{H}^{(NIP)}(t)=G^{(NIP)}(t)\,H^{(NIP)}(t)-H^{(NIP)}(t)\,G^{(NIP)}(t)+K^{(NIP)}(t)\,.$
(40)

Next, one will also frequently decide to accept the important simplification
obtained for the vanishing NSP-time-derivative operators

$K^{(NIP)}(t)=\Omega^{(-1)}(t)\,{\rm
i\,}\dot{\mathfrak{h}}_{(NSP)}(t)\,\Omega(t)\,.$

As long as $\Sigma^{(NIP)}(t)=H^{(NIP)}(t)-G^{(NIP)}(t)$ there remains no
freedom left. In particular, as long as we have the definition,

${\rm i}\frac{\partial}{\partial
t}\,\Omega^{(NIP)}(t)\rangle=\Omega^{(NIP)}(t)\,\Sigma^{(NIP)}(t)\,,$ (41)

the only ambiguity of $\Omega^{(NIP)}(t)$ is contained in its initial-value
specification.

##  5 The construction of non-stationary WDW Universe admitting Big Bang

In our present study of applicability of the NIP approach to the various
models in cosmology we felt particularly interested in a guarantee of the Big
Bang degeneracy property

$\lim_{t\to 0^{-}}q_{j}(t)=0\,,\ \ \ \ j=0,1,\ldots\,$ (42)

which, in the formal context of quantum mechanics, prescribes and restricts
the behavior of certain time-dependent eigenvalues $q_{j}(t)$ of a suitable
operator characterizing the spatial geometry (or at least the size) of the
Universe. Sampled, say, by $Q(t)$ of Eq. (37), or by $\widetilde{\Lambda}(t)$
of Eq. (3), with the spectrum as sampled in Figure 2. For this purpose let us
now return to some less general, simplified WDW models.

###  5.1 The evolution of the WDW ket-vectors

Even in the non-stationary cases, many KG and WDW models remain formally
equivalent. For this reason let us now return to Eq. (4) replaced by its non-
stationary generalization

$\left(\frac{\partial^{2}}{\partial
t^{2}}+D(t)\right)\,\psi^{(WDW)}(\vec{x},t)=0\,,\ \ \ \ \ \
D(t)=-\triangle+m^{2}(\vec{x},t)\,.$ (43)

Using the same amendment of the wave functions as before,

$\langle\vec{x}|\psi^{(NIP)}(t)\rangle=\left(\begin{array}[]{c}{\rm
i}\partial_{t}\psi^{(WDW)}(\vec{x},t)\\\
\psi^{(WDW)}(\vec{x},t)\end{array}\right)\,$ (44)

we are able to replace Eq. (43) by an analogue of Eq. (23), i.e., by the
correct NIP Schrödinger equation

${\rm i}\frac{\partial}{\partial
t}\,|\psi^{(NIP)}(t)\rangle=\left(\begin{array}[]{cc}0&D(t)\\\
I&0\end{array}\right)\,|\psi^{(NIP)}(t)\rangle\,.$ (45)

Here, the spectrum of the WDW generator $G_{(NIP)}(t)$ need not be real of
course (see, for example, an elementary illustrative example as given in
[11]).

###  5.2 The evolution of the WDW bra-vectors

It is obvious that the time-dependence of the metric $\Theta(t)$ may be highly
sensitive to its initial value at $t=t_{i}$ [49, 51, 52, 55]. Unfortunately,
the direct analysis of this dependence via the solution of Eq. (32) is
complicated. For this reason we recommended, in [12], to follow the guidance
by papers [47, 55] and to circumvent the solution of the auxiliary operator
evolution Eq. (32) (which was characterized, in [49], as the “time-dependent
quasi-Hermiticity relation”) and to solve the second Schrödinger equation (for
the mere bra-vectors) instead.

This leads to the implementation of the NIP recipe with the evolution of

$|\psi_{\Theta}^{(NIP)}\rangle=\Theta(t)\,|\psi^{(NIP)}\rangle$ (46)

controlled by Schrödinger Eq. (29),

${\rm i}\frac{\partial}{\partial
t}\,|\psi_{\Theta}^{(NIP)}(t)\rangle=\left(\begin{array}[]{cc}0&I\\\
D^{*}(t)&0\end{array}\right)\,|\psi_{\Theta}^{(NIP)}(t)\rangle\,.$ (47)

Here, it is necessary to emphasize that once we identified the NIP generator
$G(t)$ with the WDW generator in Eqs. (45) and (47), we made, in effect, a
certain highly nontrivial decision. It has two aspects. In the
phenomenological context such a decision implies that the WDW generator does
not  represent an observable. We believe that there are all reasons for such a
preference, especially in the context of the possible quantization of gravity
because in such a context the WDW eigenstates are usually treated as a means
of specification of the Hilbert space rather than as the observable states
which would be directly connected with the energy [5].

###  5.3 Reconstruction of the metric $\Theta(t)$ from the generator $G(t)$

In the NIP framework it is sufficient to admit that only the sum (24) of the
generator $G(t)$ and of the Coriolis-force $\Sigma(t)$ (of a purely
kinematical origin) can be interpreted as the observable Hamiltonian. In such
a non-stationary NIP scenario, several open questions emerge and have to be
resolved of course.

####  5.3.1 Big Bang rendered possible by the time-dependence of the metric

Let us now accept the model-building strategy in which one is given the
kinematical input operator $G(t)$. Then, the general non-Hermitian interaction
picture can be declared exceptional because only this picture is in fact a
candidate for a realization of the quantum Big-Bang-like phase transitions via
a unitary evolution process [43, 56, 57]. Naturally, the details of such a
realization remain nontrivial even when we restrict our attention just to the
Wheeler-DeWitt form of the most elementary differential-operator generators
$G(t)$ and to the Big-Bang-like quantum phase transitions. Nevertheless, what
we achieve is that we avoid and eliminate the danger of the Big-Bounce
smearing after quantization. In Hermitian theory, this smearing is
unavoidable, caused by an effective level repulsion as sampled in Figure 1
above. In the quasi-Hermitian NIP context, the Big-Bang-related exceptional-
point degeneracy is rendered possible via the “fine-tuning” of the metric: A
few non-numerical, exactly solvable simulations of such a fine-tuning may be
found described, e.g., in [56].

A complementary word of warning has been formulated in our brief methodical
note [47]. We revealed there that in Heisenberg picture (HP), the Big-Bang
degeneracy cannot be realized at all. Indeed, the underlying constat choice of
vanishing $G^{(HP)}(t)=0$ has been shown to imply the stationarity of the HP
metric, $\Theta^{(HP)}\neq\Theta^{(HP)}(t)$ (recall Eq. (32) for the quick
proof). The HP form of Eq. (24) implies that we have
$\Sigma_{(HP)}(t)=H_{(HP)}(t)$ so that only the solution of the Heisenberg
Eqs. (38) is needed. The only advantage of using the HP simplification is that
both of the underlying Schrödinger equations remain trivial. Nevertheless, as
long as the realization of the Big Bang degeneracy necessarily requires that
the Hilbert-space metric $\Theta(t)$ has to vary with time, the use of the
full-fledged NIP formalism with nontrivial $G^{(NIP)}(t)$ is unavoidable.

Unfortunately, no help has been reached in an extended Heisenberg picture
(EHP, [55]). In a slightly amended formalism we proposed the use of a
constant-operator choice of a non-vanishing  generator
$G_{(EHP)}(t)=G_{(EHP)}(0)\neq 0$. We found that the EHP formalism can already
describe the evolution equivalent to the one generated by the manifestly time-
dependent self-adjoint quantum Hamiltonians $\mathfrak{h}(t)$ (cf. Abstract of
Ref. [55] or a rediscovery of this possibility in [54]). Still, the
description of the phase transitions (like Big Bang) remained beyond the
capacity of the amended EHP approach. The full-fledged NIP is needed.

####  5.3.2 The detailed WDW NIP recipe

In [12] we outlined some of the details of the constructive treatment of the
quantum phase transitions. We pointed out that our “dynamical input” knowledge
of the non-observable Hamiltonian $G(t)$ enables us to solve the pair of our
Schrödinger Eqs. (23) and (29) at any initial conditions. In this sense, every
initial $N-$plet

$|\psi_{1}(0)\rangle\,,|\psi_{2}(0)\rangle\,,\ldots\,,|\psi_{N}(0)\rangle\,$
(48)

and

$|\psi_{1,\Theta}(0)\rangle\,,|\psi_{2,\Theta}(0)\rangle\,,\ldots\,,|\psi_{N,\Theta}(0)\rangle\,$
(49)

chosen at $t=0$ can be used to construct the time-dependent $N-$plets of the
kets

$|\psi_{1}(t)\rangle\,,|\psi_{2}(t)\rangle\,,\ldots\,,|\psi_{N}(t)\rangle\,$

and

$|\psi_{1,\Theta}(t)\rangle\,,|\psi_{2,\Theta}(t)\rangle\,,\ldots\,,|\psi_{N,\Theta}(t)\rangle\,.$

Under an elementary working hypothesis of a finite-, $N-$dimensional Hilbert
space, the additional initial bi-orthonormality assumption

$\langle\psi_{m,\Theta}(0)\,|\psi_{n}(0)\rangle=\delta_{m,n}\,,\ \
m,n=1,2,\ldots,N\,$ (50)

and the completeness

$\sum_{n=1}^{N}|\psi_{n}(0)\rangle\langle\psi_{n,\Theta}(0)|=I\,$ (51)

become immediately extended to all times $t>0$,

$\sum_{n=1}^{N}|\psi_{n}(t)\rangle\langle\psi_{n,\Theta}(t)|=I\,,\ \ \ \ \ \
\langle\psi_{1,\Theta}(t)\,|\psi_{n}(t)\rangle=\delta_{m,n}\,,\ \
m,n=1,2,\ldots,N\,.$ (52)

Also the time-dependent metric operator $\Theta(t)$ acquires the standard
representation in ${\cal H}^{(F)}$,

$\Theta(t)=\sum_{n=1}^{N}|\psi_{n,\Theta}(t)\rangle\,\langle\psi_{n,\Theta}(t)|\,.$
(53)

This means that the choice of a suitable generator $G(t)$ and of the two
suitable initial vector sets (48) and (49) with properties (50) and (51) does
not leave too much space for the further requirements concerning the dynamics.

Fortunately, we come to the conclusion that the space left by the NIP
formalism is still sufficient for our present purposes. Indeed, in our
construction we started from the assumption of the knowledge of a preselected
WDW form of the generator $G(t)$. Such a specific “kinematical-like” input
information is still not in conflict with the Big-Bang dynamics. Indeed, such
a version of the general NIP formalism still admits the use of the formal
spectral representation of the observables. In this sense, there exist the two
most important operators of our present interest. The first one is the
“dynamical”, observable-energy-representing operator $H(t)$ called
Hamiltonian. In its spectral representation of the form

$H(t)=\sum_{n=1}^{N}\,|\psi_{n}(t)\rangle
E_{n}(t)\langle\psi_{n,\Theta}(t)|\,$ (54)

the choice of the energy eigenvalues $E_{n}(t)$ remains unrestricted.

In a climax of the story, an entirely analogous expansion should be finally
introduced in order to define the complementary, “kinematical”, background-
representing operator of a suitable “geometry” or “spatial grid” operator
(37). In its analogous spectral representation

$Q(t)=\sum_{n=1}^{N}\,|\psi_{n}(t)\rangle
q_{n}(t)\langle\psi_{n,\Theta}(t)|\,$ (55)

we are free to require the validity of the Big-Bang constraint (42) imposed
upon all of its spatial background representing eigenvalues $q_{n}(t)$.

##  6 Discussion

The non-Hermitian innovation of the NSP framework opened, in [8], the way
towards a deeper understanding of the KG- and WDW-like quantum systems in
stationary approximation. Later, the birth of the more sophisticated non-
Hermitian version of the Dirac’s interaction picture [10] seemed to be,
initially, just an artificial mathematical exercise. Nobody seemed to be
willing to admit that the NIP formalism might find an application in quantum
gravity. The main reason was that in the most advanced version of quantum
gravity (i.e., in the canonical LQG approach), virtually all the results
seemed to indicate that the classical Big Bang singularity has to be replaced
by its quantized Big Bounce alternative.

Even among mathematicians it has been firmly believed that the quantization
must necessarily smear out the singularities of the classical Einstein’s
general relativity [3]. In this sense, before any return to the quantum Big
Bang hypothesis it was necessary to wait for a renewal of its support in the
realistic LQG context [20]. Naturally, the problem is technically complicated.
In this sense also the present methodical support of the latter hypothesis is
just schematic and incomplete. In its framework we only had to leave many
important phenomenological requirements aside. Let us now mention some of them
in the form of brief comments.

###  6.1 The background-independence requirement

In the Isham’s foreword preceding the Thiemann’s comprehensive 2007 monograph
on canonical quantum gravity [3] the Hamiltonian constraint $\hat{H}\psi=0$
alias “the famous Wheeler-DeWitt equation” is characterized as “arguably one
of the most elegant equations in theoretical physics, and certainly one of the
most mathematically ill-defined.” In the introductory part of the book itself
one reads, indeed, that the sufficiently rigorous specification of a suitable
Hilbert space in which the Wheeler-DeWitt operator $\hat{H}$ would be defined
represents one of the most important unresolved theoretical challenges.

The latter Hilbert-space problem may be found thoroughly discussed in section
9.2 of the Mostafazadeh’s 2010 study [7]. Even the authors of the LQG results
admit that such an approach does not yet provide a fully consistent
description of the physical reality. Still, their approach addresses,
successfully, the necessary background independence of the theory [5]. In some
sense, such a requirement should be incorporated in any theory which pretends
to be “fundamental” rather than just “effective”.

From the point of our present approach based on the drastically simplified WDW
equation the constructions which would be background-independent were found
feasible. In some sense, such a requirement can be perceived as lying in the
very center of the NIP approach in which, admittedly, one starts from the
knowledge of the explicit WDW form of the operator $G(t)$, but in which the
theory admitsthe introduction of an “observable background”. Although our
present spectral-representation definition (55) of such an independent
kinematical background may look rather abstract, a more specific example may
be sought, say, in [58] where a nontrivial coordinate/background has
constructively been obtained in an elementary dynamical model.

In our considerations the role of a geometric background has been played by
the “dynamical input” operator $Q(t)$ sampled by a matrix in Eq. (3) with the
spectrum $q_{n}(t)$ simulating the “observable” spatial grid points and
guaranteeing the existence of the Big Bang singularity at $t=0$ (cf. Eq. (42)
or Figure 2). In discussion, one only has to emphasize the mathematical
subtlety of the correspondence between the hidden Hermiticity of $Q(t)$ and
the fine-tuned nature of the corresponding Hilbert-space metric $\Theta(t)$
which guarantees the unitarity of the system (i.e., of the evolution of the
Universe from the very beginning of its observability and existence).

In the latter considerations the truly drastic simplifications of the picture
seem still absolutely necessary at present. Skipping, typically, the Lorentz-
covariance requirements and working with the models in which the time is a
parameter and in which, for methodical reasons, the “Universe” is one-
dimensional and discretized via a finite mesh of the time-dependent grid
points $q_{j}(t)$, $j=1,2,\ldots,N$. In such a “Universe” just the degeneracy
$q_{j}(t)\to 0$ in the classical-physics Big Bang limit $t\to 0^{-}$ is asked
for.

###  6.2 Problems with terminology

The conventional belief that the avoided crossings of the eigenvalues are
generic is equivalent to the (usually, just tacit) assumption of the time-
independence of the physical inner-product metric $\Theta$. In opposite
direction, once we replace the conventional self-adjoint grid-matrix
$\mathfrak{q}(t)$ by its isospectral but merely hiddenly Hermitian partner
${Q(t)}$, we discover the existence of a new freedom in the formalism as
carried by the Dyson map $\Omega(t)$. As a consequence, the existence of the
singular Big Bang grid-point limit (42) is rendered possible.

In the language of mathematics the innovation lies in the enhancement of the
flexibility of the dynamical laws. One arrives at the less usual, non-
Hermitian NIP formulation of quantum mechanics. In its framework the unitary
and closed quantum systems may be defined via structures using more than one
inner product, i.e., strictly speaking, more than one Hilbert space.

One of the most welcome consequences is an enhancement of the flexibility
while on dark side one may find terminological misunderstandings. In the
phenomenologically oriented literature several nicknames denote more or less
the same theory. Thus, in different papers one encounters, e.g., the reference
to the quasi-Hermitian quantum theory [22, 59], to the pseudo-Hermitian
quantum theory [7], to the non-Hermitian but ${\cal PT}-$symmetric quantum
theory (usually also with ${\cal P}$ = parity and with ${\cal T}$ = time
reversal [38]), to the three-Hilbert-space quantum theory [11] or to the
crypto-Hermitian quantum theory [60], etc.

###  6.3 The danger of an over-determination of the dynamical input

All of the latter approaches lead to a perceivable gain in flexibility of the
realistic models of various quantum systems. This is to be paid merely by the
necessity of keeping trace of the more sophisticated forms of Hermitian
conjugations. One can conclude that the subject is still hot. On a model-
independent level of discussion it is worth adding that the consistence of the
dynamical input need not in fact be easily guaranteed. In review [22], for
example, the authors stressed that in the over-determined cases the necessary
Hilbert space metric (and, hence, the theory itself) need not exist at all. In
[61], such a non-existence has been shown to occur even in some fairly popular
realistic models. An abstract analysis of such an unpleasant possibility was
presented in [62]. Only recently, more encouraging results were obtained in
[25] offering a certain systematic guide to the construction of the mutually
compatible non-Hermitian observables.

Once we restrict attention to the applicability of the NIP approach in
cosmology, an encouragement may be sought in the progress and simplifications
of the canonical quantization [3, 5]. The latter two reviews of the state of
art differ by the language, with the former one being more mathematically
oriented. Still, both of these monographs share the traditional philosophy
interpreting the quantum theory as a result of a modification of its classical
predecessor. In our final remark we would like to point out that one could
also try to weaken our dependence on the classical-physics-based intuition by
treating, as primary, the tentative quantum hypotheses in a way defended,
e.g., by Brody and Hughston [63].

##  7 Summary

The core of our present message is that the consistency of the quantum-
mechanical interpretation of the non-stationary WDW systems requires that
Schrödinger equation ceases to be perceived as offering a complete picture of
the evolution. In this sense, it is necessary to add a parallel and full-
fledged description of the evolution of the operators of observables using the
Heisenberg-like evolution equations. In the natural physical quantum-gravity
context, the unitarity of the WDW-controlled evolution can be then guaranteed.
The apparently non-unitary evolution of the left and right wavefunctions
(controlled by the respective two Schrödinger-type equation) is precisely
compensated by the apparent non-unitarity of the evolution of the operators
representing the observables (controlled, in parallel, by non-Hermitian
Heisenberg-type equations).

Having accepted such a philosophy, our present paper can be read as a more or
less purely methodical return to the question whether, in the framework of
quantum cosmology, the birth of our Universe should be perceived as a point-
like Big Bang or as a smeared Big Bounce. In essence, we presented here a few
arguments supporting our persuasion that in the purely theoretical NIP
framework such a question remains, at present, open.

## Appendix: Two Hilbert spaces in quantum mechanics.

In the conventional quantum mechanics of textbooks [21] the predictions of the
results of experiments have their mathematical background, in NSP, in the
evaluation of matrix elements

$\prec\\!\psi^{(NSP)}(t_{f})\,|\mathfrak{q}_{(NSP)}(t_{f})\,|\psi^{(NSP)}(t_{f})\\!\\!\succ\,\,\,.$
(56)

The symbol $\mathfrak{q}_{(NSP)}$ denotes here a self-adjoint operator of the
observable in question: Usually, this operator is time-independent,
$\mathfrak{q}_{(NSP)}\neq\mathfrak{q}_{(NSP)}(t)$. All information about the
evolution of the system in time is carried, in the pure state regime, by a
ket-vector element $|\psi^{(NSP)}(t)\\!\\!\succ\,\,$ of a Hilbert space of
states ${\cal H}^{(textbook)}$. This state is assumed prepared at
$t_{initial}=0$ and measured at $t=t_{final}=t_{f}$. Prediction (56) is
probabilistic and contains just the NSP wave-ket solutions
$|\psi^{(NSP)}(t)\\!\\!\succ\,\,$ of Schrödinger equation

${\rm i}\frac{\partial}{\partial
t}\,|\psi(t)\\!\\!\succ\,\,=\mathfrak{h}_{(NSP)}\,|\psi(t)\\!\\!\succ\,\,\,,\
\ \ \ \ \ \ \ \ \ |\psi(t)\\!\\!\succ\,\,\in{\cal H}^{(textbook)}\,.$ (57)

Due to the Stone theorem the evolution is unitary if and only if the
Hamiltonian is self-adjoint in ${\cal H}^{(textbook)}$,
$\mathfrak{h}_{(NSP)}=\mathfrak{h}_{(NSP)}^{\dagger}\,$ [64].

One of many efficient simplifications of the practical solution of Eq. (57) is
due to Dyson [65]. He revealed that in many cases one has to work with a
technically unfriendly Hamiltonian which can be perceivably simplified via a
suitable isospectral preconditioning $\mathfrak{h}_{(NSP)}\to
H_{(NSP)}=\Omega^{-1}\,\mathfrak{h}_{(NSP)}\,\Omega$. This is formally
equivalent to the transformation of the ket-vector wave functions,

$|\psi_{n}^{(textbook)}\\!\\!\succ\,\,=\Omega\,|\psi_{n}^{(auxiliary)}\rangle\,,\
\ \ \ n=0,1,\ldots\,.$ (58)

Operator $\Omega$ has to be $\,n-$independent and stationary
($\Omega\neq\Omega(t)$). Dyson also recommended to make the choice of $\Omega$
non-unitary ($\Omega^{\dagger}\Omega=\Theta\neq I$). In analogy with the so
called coupled-cluster method based on a similar idea [66], one may also treat
the simpler partner of the Hilbert space ${\cal H}^{(textbook)}$ as formally
different, denoted by a different dedicated symbol, say, ${\cal
H}^{(friendlier)}$.

Schrödinger Eq. (57) becomes replaced, in the majority of applications of the
Dyson-recommended and $\Omega-$mediated change of space ${\cal
H}^{(textbook)}\to{\cal H}^{(friendlier)}$, by a friendlier equation

${\rm i}\frac{\partial}{\partial
t}\,|\psi^{(Dyson)}(t)\rangle=H_{(Dyson)}\,|\psi^{(Dyson)}(t)\rangle\,,\ \ \ \
\ \ |\psi^{(Dyson)}(t)\rangle\in{\cal H}^{(friendlier)}\,.$ (59)

The transformed Hamiltonian is de-Hermitized since
$H=\Omega^{-1}\,\mathfrak{h}\,\Omega\neq H^{\dagger}$ in ${\cal
H}^{(friendlier)}$. In the early review [22] of the procedure a change of the
philosophy has been proposed, resulting in a reformulation of the textbook NSP
approach called, in the spirit of the mathematician’s terminology [59], quasi-
Hermitian quantum mechanics. In this framework the model-building process has
to start directly from Eq. (59) and from a guarantee of the user-friendliness
of the preconditioned Hamiltonian $H$. Whenever necessary, one may, after all,
re-Hermitize the model, say, via a reconstruction of $\Omega$ [7].

The non-unitarity of the map $\Omega$ implies, for the manifestly auxiliary
Hilbert space ${\cal H}^{(friendlier)}$, the loss of its physical-space
status. Fortunately, it appeared sufficient to amend the inner product and to
convert ${\cal H}^{(friendlier)}$ into a fully acceptable and physical Hilbert
space ${\cal H}^{(standard)}$. By construction, the latter space has to be
unitary equivalent to ${\cal H}^{(textbook)}$, with the most straightforward
method being the reconstruction of the so called metric
$\Theta=\Omega^{\dagger}\,\Omega$. The mathematical details can be found in
reviews [22] or [7]. The essence of the trick is that the correct space ${\cal
H}^{(standard)}$ can be represented via the mere amendment of the bra vectors
in ${\cal H}^{(friendlier)}$,

$\langle\psi|\ \to\ \langle\psi|\Theta\ \equiv\ \langle\psi_{\Theta}|\ \ \ \ \
{\rm for}\ \ \ \ \ {\cal H}^{(friendlier)}\to{\cal H}^{(standard)}\,.$ (60)

In the terminology of functional analysis the definition of the dual alias
bra-vector space of the linear functionals is merely amended and transferred
back, from ${\cal H}^{(standard)}$ to ${\cal H}^{(friendlier)}$, via formula
${\cal V}^{\prime}\to{\cal V}^{\prime}_{\Theta}\,$. In other words, one just
converts the conventional, unphysical bra-ket inner product
$\langle\psi|\chi\rangle$ into its physical alternative,

$\langle\psi|\chi\rangle\to\langle\psi|\Theta|\chi\rangle\ \equiv\
\langle\psi_{\Theta}|\chi\rangle\,.$ (61)

In the light of this relation it is possible to perform all calculations in
${\cal H}^{(friendlier)}$. Still, in practice, the redundancy of the
introduction of the manifestly unphysical Hilbert space ${\cal
H}^{(friendlier)}$ must be well motivated. The expense must be more than
compensated by the simplification of the evaluation of the experimental
predictions. Also the loss of the direct connection with ${\cal
H}^{(textbook)}$ has to be taken into account because in this space we usually
define the operators of observables using the principle of correspondence
[21].

One can often pull at least some of the necessary operators from ${\cal
H}^{(textbook)}$ up to the auxiliary Hilbert space ${\cal H}^{(friendlier)}$
(see, e.g., [58]). E.g., whenever one knows the Dyson map, one can define the
necessary operators in ${\cal H}^{(friendlier)}$ using formula

$Q_{(Dyson)}=\Omega_{(Dyson)}^{(-1)}\,\mathfrak{q}_{(NSP)}\,\Omega_{(Dyson)}\neq
Q_{(Dyson)}^{\dagger}\,.$ (62)

The experiment-predicting NSP formulae (56) then acquire the upgraded forms,

$\prec\\!\psi^{(NSP)}(t_{f})\,|\mathfrak{q}_{(NSP)}(t_{f})\,|\psi^{(NSP)}(t_{f})\\!\\!\succ\,\,=\langle\psi^{(Dyson)}_{\Theta}(t_{f})\,|Q_{(Dyson)}(t_{f})\,|\psi^{(Dyson)}(t_{f})\rangle\,$
(63)

in which one can use, at worst, just some reasonably precise approximate forms
of the physical Hilbert space metric $\Theta=\Omega^{\dagger}\,\Omega$ in Eq.
(60) (cf. [22, 67]).

.

Funding:  This work is supported by the Faculty of Science of the University
of Hradec Králové and by the Nuclear Physics Institute in Řež.

.

Conflicts of Interest:  The author declares no conflict of interest.

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