University of Wrocław
Department of Physics and Astronomy
Institute of Theoretical Physics

Maciej Kowalczyk
Consequences of regularization ambiguities in Loop Quantum Cosmology

Konsekwencje niejednoznacznej regularyzacji w Pętlowej Kosmologii Kwantowej

Supervisor of master thesis

dr hab. Tomasz Pawłowski

Wrocław 2021

###### Abstract

Ambiguities of the so-called Thiemann regularization in Loop Quantum Cosmology
lead to freedom in how to construct a particular quantization prescription. So
far three distinct examples of such have been proposed in the literature. For
two of them, detailed analysis has been already performed in the literature.
In this thesis, the methodology developed for these is applied to study in
detail the third one, which will be referred to as mLQC-II. In particular, the
mathematical properties of the operator of the quantum version of full
Hamiltonian constraint are examined. These properties indicate that the
evolution of the system is uniquely determined. Furthermore, an investigation
of dynamics is performed by finding the expectation value of the volume as a
function of the scalar field and constants of motion. This result is next
compared to the trajectories predicted with the so-called effective dynamics.
Finally, the main properties of studied prescription are compared against
those of other (already investigated) ones.

Niejednoznaczności wynikające z tak zwanej regularyzacji Thiemana w Pętlowej
Kosmologii Kwantowej prowadzą do swobody w procedurze kwantowania. Do tej pory
wyróżniono trzy konkretne wybory, z których dwa zostały już poddane
szczegółowej analizie. W niniejszej dysertacji użyjemy wypracowanej już
metodologii w szczegółowej analizie trzeciego wyboru, do którego będziemy
odnosić się jako mLQC-II. W szczególności analizujemy własności operatora
kwantowego wiązu hamiltonowskiego, które wskazują na jednoznacznie
zdeterminowaną ewolucję systemu. Ponadto badanie dynamiki przeprowadzono
poprzez obliczenie wartości oczekiwanej objętości jako funkcji zależnej od
pola skalarnego oraz stałe ruchu. Wynik ten jest następnie porównywany z tymi
otrzymanymi już w literaturze dla innych recept regularyzacji.

Key words: General Relativity, Polymer Quantization, Quantum Cosmology, Loop
Quantum Cosmology

Słowa kluczowe: Ogólna Teoria Względności, Kwantyzacja Polimerowa, Kosmologia
Kwantowa, Pętlowa Kosmologia Kwantowa

###### Contents

  1. 1 Introduction
  2. 2 Formalism
    1. 2.1 Loop Quantum Gravity
    2. 2.2 Loop Quantum Cosmology
    3. 2.3 Strict Thiemann regularization of Loop Quantum Cosmology
  3. 3 Second Modification of Loop Quantum Cosmology
    1. 3.1 Regularization and $\hat{\Theta}$ in volume representation
    2. 3.2 $\hat{\Theta}$ in momentum $b$ representation
    3. 3.3 Dynamics
    4. 3.4 Effective Dynamics
  4. 4 Conclusion
  5. 5 Plots
  6. A Constraints
  7. B Deficiency Subspaces

##  1 Introduction

The variety of gathered knowledge forces us to organize it into fields in
order not to get lost in the richness of information. The one dedicated to
finding the most fundamental rules governing the observed reality is Physics.
When trying to comprehend the fundamental interactions within matter, the most
successful ,,component” up to date is Quantum Physics: initialized by a theory
that started growing in the last century - Quantum Mechanic and subsequently
expanded into Quantum Field Theory and in particular the Standard Model neatly
systematizing the richness of forms of observable matter (on the level of
particle physics) and known non-gravitational interactions. On the other hand,
if gravity influence on the matter is the main course of consideration, then
General Relativity (provided that quantum effects are abandoned) delivers,
with great precision, answers about the Universe. Descriptions of nature
provided by QM and GR have been confirmed by numerous experiments in both
cases but they arise from principles that are in partially in contradiction to
each other. In the quantum approach, we operate only with probabilities (thus
expectation values) rather than exact (classical) trajectories like in the
theory of relativity. Departure becomes even more explicit if we take look at
spacetime: in QM it is fixed (mainly Minkowski) space which serves as a
background to describe interaction - one can still consider Quantum Field
Theory in the curve spacetime. While GR metric evolves with matter and cannot
be simply fixed. So it is impossible to build one description of nature by
just simply adding these two theories. In addition, we still encounter several
open problems that are not covered alone by any of them. For example,
observation shows that most of the matter in our universe cannot be described
by the SM. The other one is the absence of a theory that describes the quantum
behaviour of gravity along with experimental confirmation. But it doesn’t mean
that any attempt forward such theory wasn’t made. For example, if one chooses
as a starting point particle physics one ends up with an elegant theory with
extra dimensions - the String Theory. Because of its mathematical elegance and
the fact that this theory combines efforts in the quantization of gravity with
hopes of unification, it was the most examined course at the end of the
previous century [27].
But nature not necessarily has to been ,,enclosed” by a single theory. One of
alternatives branch of propositions for String Theory are attempts at a non-
perturbative and background independent quantization of General Relativity.
Examples of them are quantization of Geometrodynamics which to this day is not
finished due to technical obstacles, or one that performs quantization of GR
in loop variables - Loop Quantum Gravity. To point out the beginning of this
branch, it is sufficient to go back to the XX century, exactly to the 30s to
the work of Matvei Bronstein. He was a young soviet researcher who encouraged
by the debate between Bohr, Rosenfeld and Landau, Peierls about the
application of the uncertainty principle to the electromagnetic field, carried
out similar consideration with respect to the gravitational field [18, 31].
What he found out is that when GR and QM play a role, it is not possible to
localize anything with a precision better than the Planck length
($10^{-35}$m). It means that at some scales to describe the world we need to
use both quantum and gravitational properties. This claim is supported by a
conclusion of similar consideration made by Hawking who applied statistical
quantum mechanics to black holes. What was obtained is a form of black body
radiation which cause black hole evaporation. It confirmed that black holes
obey the second law of thermodynamic and are encode (alongside information
about Planck length) in the Bekenstein-Hawking entropy formula. Several years
after this revelation, namely in 1986 General Relativity has been re-
formulated in connection variables. This moment is usually identified as a
starting point in the development of LQG. In the following months, the
solution of the Wheeler-DeWitt equation was found in the form of Wilson Loops.
From that moment, LQG started to gain shape with the establishment of loop
representation in 1987 and the introduction of the scalar product in 1994. In
the same year, one of the most important results of LQG was shown -
discreteness of area and volume eigenvalues [30]. As it is a relatively young
theory it is still under vigorous development. Its flaw is, as in the case of
String theory, that still no experiment that could confirm or discard it has
been performed [28, 27]. But during the development of LQG subfield by the
name Loop Quantum Cosmology emerge which is an application of methods of LQG
to a cosmological setting. It was first introduced in 2002 in the work of
Martin Bojwald [11] and in the next year, its mathematical structure has been
discussed in detail in [2]. In 2006 first solid calculation of dynamics for
homogeneous isotropic models has been performed by Ashtekar, Pawłowski and
Singh in [6, 7, 8] and in the following years to anisotropic systems (for
example to Bianchi I spacetime in the vacuum [22]). Some focus as well has
been directed to inhomogeneous models [23]. Results obtained in these and many
other papers confirms that this application can predict physically interesting
scenarios.

Because LQC, as well as LQG (and GR for that matter), is a constraint theory,
in order to probe the dynamics within it one has to build and study
Hamiltonian constraint. The matter part of the constraint is not problematic
but the construction of the gravity part is not straightforward [9, 1]. To do
so one first promote elementary variables (in LQC holonomy and densitized
triads) to the operators and then by a procedure known as Thiemann
regularization build the constraint, which procedure is not unique. This
ambiguity in consequence can provide different models. One originally
presented in [6] is referred to as a mainstream LQC. It utilizes the fact that
in the spatially flat Friedmann Lemaitre Robertson Walker spacetime after
symmetry reduction Euclidean and Lorentzian terms of gravity part of
hamiltonian become proportional. Since that, the Lorentzian term can be
rewritten in the form of the Euclidean one on the classical level and then
merge to the Euclidean part. The next step is to quantize this term. The
evolution of this system leads to a single quantum bounce at near-Planckian
energy densities, connecting two epochs of a large semiclassical Universe. But
at it was first shown in [34] when Euclidean and Lorentzian parts are treated
separately, two modifications of LQC can be obtained. One referred, in the
literature, as mLQC-I is obtained by quantizing Euclidean and Lorentzian terms
independently, while one referred as mLQC-II additionally using fact that
extrinsic curvature in k=0 FLRW is proportional to Ashtekar-Barbero
connection. mLQC-I has been already well examined in the literature (eg. [10])
with various novelties with respect to LQC (for example rather than one bounce
two bounces occur with two emergent DeSitter epochs between them) so the goal
of this thesis is to examine mLQC-II to check properties of the operator of
full Hamiltonian constraint and to obtain expectation value of volume operator
and it semiclassical approximation.

This thesis is organized as follows. In the following section, basic
information about formalism needed further will be introduced via a very
general outline of LQG and through examples of LQC and mLQC-I. The content of
the main section will be fully dedicated to the mLQC-II - to building
Hamiltonian constraint, to finding the evolution operator $\theta$ and scalar
product in the different representations and most important to finding
expectation values of volume operator. The last chapter will serve to discuss
and compare obtained results.

##  2 Formalism

Let us start with introducing briefly the formalism applied in this
dissertation. First, in section 2.1 we will briefly present the main ideas and
components of Loop Quantum Gravity as the most general but not yet complete
framework. Next, in sec. 2.2 we will introduce the core elements of Loop
Quantum Cosmology as the symmetry reduced version of LQG, discussing how the
ideas behind LQG are realised in the context of cosmological models. Finally,
a version of LQC born of applying directly the regularization prescription
introduced by Thiemann in LQG will be presented in sec. 2.3.

###  2.1 Loop Quantum Gravity

Since Loop Quantum Gravity implements strictly the main principles of General
Relativity it treats spacetime itself and the gravitational field as the
aspects of the same physical entity. Dividing the spacetime metric into a
fixed background metric and fluctuating quantum field is thus forbidden. So it
is a non-perturbative approach. Moreover, it can be split into two branches -
covariant (the so-called spin foam models) and the canonical one.
Here we focus on the latter, where one performs foliation of the space-time
described by a 4-dimensional manifold with some metric into a set of
3-dimensional space-like surfaces $\Sigma_{t}$ ($t$ denotes the global time
function) [20, 31]. The spacetime $4$-metric spits then onto spatial
$3$-metric (on $\Sigma_{t}$), a lapse function and a shift vector. In such
case, elementary variables are taken to be 3-metric and its conjugate
momentum. But 3-metric can be expressed in terms of orthonormal co-triad. Thus
as a set of canonically conjugate variables, one can chooses a Sen-Ashtekar-
Barbero variables: $su(2)$ valued connection $A_{a}^{i}$ (indices $a=1,2,3$
are spatial indices and $i=1,2,3$ enumerating vectors) and densitized triads
$E_{i}^{a}$. Now, the starting point - like in ADM formulation - is the
Einstein-Hilbert action written in terms of new variables and with appropriate
boundary term (Holst term). Its canonical splitting and Legendre transform
leads to a standard set of constraints of ADM theory: Hamiltonian and
diffeomorphism one. However in this Hamiltonian formulation emerge additional
constraint, which ensures that the state belonging in the system is invariant
under gauge transformation - it is called Gauss constraint. Since it is a
system with constraint, its quantization requires special care. Here the
systematic method known as the Dirac program is used.
At first quantization is performed only on the kinematical level - ignoring
the constraints. But presented variables are not however direct candidates for
quantization. Instead, one takes the holonomies of connection along curves and
fluxes of $E_{i}^{a}$ across surfaces. They form a (so called) holonomy flux
$*$-algebra, which allows to use the machinery of the Gellfand-Neumark-Segal
method to build a quantum theory. It produces non-separable kinematical
Hilbert space spanned by spin-networks states represented by cylindrical
functions [19] living on the piecewise-analytic graphs (embedded in 3D
manifold). In addition state living on an empty graph is distinguished as a
vacuum.

The next step is to implement the constraints and for each of them, a
different method is used. Gauss constraint implies certain restrictions on the
spin labels of edges intersecting particular vertex and allows to distinguish
a sub-basis of the chosen basis of the original kinematical Hilbert space. The
diffeomorphisms are implemented as finite transformations not via
infinitesimal generators (the constraint). The diffeomorphism invariant states
are identified by the procedure of the so-called group averaging [5] \- are
literary averages (integrals) over the action of all the spatial
diffeomorphisms. At this subspace manifest one of the most important results
of LQG - the discreteness of the geometry. Now all the properties of
continuous geometry (for example volumes or angles) are represented by quantum
operators. In particular, most relevant for LQG is the area one which features
purely discrete spectra [33, 30]. Its minimum nonzero eigenvalue is
$A_{1}=2\sqrt{3}\pi\gamma l_{p}^{2}$, where: $l_{p}$ is a Planck length and
$\gamma$ is the Barbero-Immirzi parameter that in these paper will be treated
(as suggested by the black hole entropy calculations [4, 16, 24]) as a real
number close to $0,2375$ (in first consideration of Ashtekar it was a pure
imaginary number). At this point, one needs to be aware that neither of the
above operators is physical observable.

Finally, the Hamiltonian constraint is implemented by promoting the classical
constraint to a quantum operator. It is done by reformulating it - through a
procedure called Thiemann regularization - in terms of elementary variables.
What is important here is that such an outcome is not unique. The physical
Hilbert space will then be identified as a kernel of that operator which is
well defined mathematically, but due to the complicated nature of the operator
extremely difficult technically. But one can reformulate the given theory as a
free theory by introducing matter fields - which serves as an evolution
parameter - coupled to gravity [15, 19, 17]. Thanks to that the Hamiltonian
constraint operator becomes an evolution equation with the gravitational part
becoming a true Hamiltonian. According to the Dirac program, it is an
appropriate time to construct physical observables. As such one uses the so-
called partial observables [29, 14] measuring relevant quantities in relation
to other ones. For example, if the choice of matter field allows identifying
kinematical diffeomorphism invariant Hilbert space as a physical one,
mentioned earlier operators can become partial observables measuring
geometrical quantities with respect to the clock field.

It is worth pointing out, that consideration of above-described the theory is
the gravitational field itself. Unfortunately, so far the only examples for
which the quantum dynamics has been evaluated correspond to textbook examples
of simplest spin networks [35], not corresponding to any relevant physical
system. But one can use the above tools and steps to focus on the entire
universe as a quantum system without an external observer. Such an approach
simplify consideration and can provide useful information - for example in the
regime of critical energy.

###  2.2 Loop Quantum Cosmology

Despite being called the symmetric sector of LQG, this approach is not so in
the strict sense. It is not derived from full theory. Instead, it is an
independent simplified approach constructed by applying the quantization
program of LQG to cosmological (highly) symmetric models in a step-by-step
basis. Here we are going to consider it in its simplest setting: applied to a
model of a flat FLRW universe, which metric in polar coordinates has the form:

$ds^{2}=dt^{2}-a^{2}(t)\Big{(}dr^{2}+r^{2}d\Omega^{2}\Big{)}$ (2.1)

In particular, our calculations will be performed for such a universe
admitting the massless scalar field.

At this point we have the first departure from the diffeomorphism-invariant
formalism of LQG: because homogeneous spacetimes admit additional
distinguished structure, we partially fix the spatial gauge [4]. In particular
as the foliation by spatial surfaces, we choose the one distinguished symmetry
- foliation by homogeneous surfaces. Furthermore, the isotropy allows to
partially fix the gauge when selecting the orthonormal triad, in particular
allowing to set the Ashtekar variables defined in LQG to take the form:

$\displaystyle A_{a}^{i}\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\ cV_{0}^{-\frac{1}{3}}\delta^{i}_{a}$
$\displaystyle E_{i}^{a}\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\ pV_{0}^{-\frac{2}{3}}\delta^{a}_{i}\ .$
(2.2)

where $c$ and $p$ are constant on the spatial slice. The coefficient $V_{0}$
is a volume of the fiducial cell which define as the finite region of the
Universe that remains constant in comoving coordinates. Choosing such a cell
is necessary as all the momenta calculated by integrals of densities over the
whole spatial slice would be infinite due to the homogeneity of spacetime. It
means that all quantities evaluated in the process of dynamical evolution will
refer only to that finite region.

The Poisson Bracket of connection and densities triad (A.6) becomes:

$\displaystyle\\{c,p\\}=\frac{\kappa\gamma}{3}$ (2.3)

where $\kappa$ is constant equal to $8\pi G$.

As in LQG, we proceed with choosing as our basic variables holonomies and
fluxes. Here however the symmetry isotropy allows us to greatly simplify the
holonomy flux algebra we work with. Instead of the full one, we can choose
only the holonomies along the straight edges - in the direction $k$ \- in
$\Sigma_{t}$:

$h_{k}^{(\lambda)}=\cos\frac{\lambda c}{2}\mathds{1}+2\sin\frac{\lambda
c}{2}\tau_{k}$ (2.4)

where $\lambda$ is any real number and $\tau_{k}$ are Pauli matrices
(multiplied by $-\frac{i}{2}$) and the fluxes across unit square surfaces here
also denoted as $p$ [2].

Having selected our basic variables we now proceed with the Dirac quantization
program. In the first step (kinematic) we apply the GNS method to our reduced
holonomy algebra. The full space of the cylindrical functions now becomes just
space spanned by almost periodic functions and can be identified with the
space of square summable functions on the Bohr compactification of the real
line with the Haar measure $d\mu$.

$\mathcal{H}_{\bf kin}=L^{2}(\bar{\mathbb{R}},d\mu)$ (2.5)

As a basis, eigenstates of the unit flux operator $\hat{p}$ can be selected.
Such state can be labelled by a real number $\mu$ and the scalar product
between its basis elements is discrete, namely:

$\braket{\mu_{1}}{\mu_{2}}=\delta_{\mu_{1},\mu_{2}}$ (2.6)

Thanks to that quantum vacuum is distinguished as $\delta_{\mu,0}\mathbb{I}$.
Note that despite the discrete inner product $\mu$ can take any real value,
which means that our Hilbert space is non-separable. State in that base is
expressed as: $\ket{\Psi}=\sum_{i}c^{(i)}\ket{\mu_{i}}$ with inner product:
$\braket{\Psi_{1}}{\Psi_{2}}=\sum_{i}\bar{c}^{(i)}_{1}c^{(i)}_{2}$. As the
basic operators alongside $\hat{p}$, the $U(1)$ component of the holonomy has
been chosen:

$\displaystyle\hat{p}\ket{\mu}=\frac{8\pi\gamma
l_{p}^{2}}{6}\mu\ket{\mu}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \hat{e^{\frac{i\lambda
c}{2}}}\ket{\mu}=\ket{\mu+\lambda}$ (2.7)

However, it is more convenient to work with slightly differently relabelled
base and specifically chosen holonomy components. Namely $\mu$ is exchanged to
$v$ and restrict $\lambda$ to $\bar{\mu}(p)$ such that:

$\displaystyle\hat{V}\ket{v}$ $\displaystyle=\alpha|v|\ket{v}$
$\displaystyle\hat{N}\ket{v}:=\widehat{e^{\frac{i\bar{\mu}c}{2}}}\ket{v}=\ket{v+1}$
(2.8)

where $\braket{v}{v^{\prime}}=\delta_{v,v^{\prime}}$ and $\alpha=2\pi
G\hbar\gamma\sqrt{\Delta}$. The meaning of this choice will become clear
later. This choice is tailored to the Hamiltonian constraint regularization in
the so-called _improved dynamics_ scheme [7].

In the next step of the Dirac program, the constraints should be implemented.
Due to symmetries and partial gauge fixing the Gauss and Diffeonorphism one
are already satisfied. The only remaining is Hamiltonian one:

$H_{g}=H^{E}-2(1+\gamma^{2})T$ (2.9)

with Euclidean Hamiltonian and Lorentzian Hamiltonian constraint define as:

$\displaystyle H^{E}$
$\displaystyle=\frac{1}{2\kappa}\int_{\mathcal{V}}d^{3}x\epsilon_{ijk}\frac{E^{ai}E^{bj}}{\sqrt{\det(h)}}F^{k}_{ab}$
$\displaystyle T$
$\displaystyle=\frac{1}{2\kappa}\int_{\mathcal{V}}d^{3}x\frac{E^{ai}E^{bj}}{\sqrt{\det(h)}}K^{j}_{[a}K^{i}_{b]}$
(2.10)

where $h$ denotes metric on one 3-dimensional space-like surface $\Sigma_{t}$,
$F^{k}_{ab}$ is curvature of connection $A^{k}_{a}$ and $K^{i}_{a}$ is
extrinsic curvature 1-form. To be allowed to quantize constraint it must be
re-expressed in terms of holonomies and densitized triads. As suggested in [6,
7], the term involving triads is re-expressed as:

$\epsilon_{ijk}\frac{E^{ai}E^{bj}}{\sqrt{det(h)}}=\sum_{k}\frac{4sgn(p)}{\kappa\lambda
V_{0}^{\frac{1}{3}}}\epsilon^{abc}\delta^{k}_{c}Tr\Big{(}h_{i}^{(\lambda)}\\{(h_{i}^{(\lambda)})^{-1},V\\}\tau_{i}\Big{)}$
(2.11)

and if we consider square loop $\square_{ij}$ (parallel to one of the faces of
the elementary cell and with sides of length $\lambda V_{0}^{\frac{1}{3}}$
each) the curvature of connection becomes:

$F^{k}_{ab}=-2\lim_{Ar_{\square}\rightarrow
0}Tr\Big{(}\frac{h^{(\lambda)}_{\square_{ij}}-1}{\lambda^{2}V_{0}^{\frac{2}{3}}}\Big{)}\tau^{k}\delta^{i}_{a}\delta^{j}_{b}$
(2.12)

where $h^{(\lambda)}_{\square_{ij}}$ is holonomy around square loop
$\square_{ij}$, defined as:

$h^{(\lambda)}_{\square_{ij}}=h_{i}^{(\lambda)}h_{j}^{(\lambda)}(h_{i}^{(\lambda)})^{-1}(h_{j}^{(\lambda)})^{-1}$
(2.13)

However classical limit $Ar_{\square}\rightarrow 0$ is problematic in quantum
theory - it means that loops area shrinks to be zero which, due to
discontinuity, is forbidden in LQG. Rather than this, we set the loop till its
area will be equal to twice the smallest non-zero eigenvalue of the area
operator from LQG. In LQC it is called area gap $\Delta=2A_{1}$:

$\Delta=\bar{\mu}^{2}|p|$ (2.14)

One can now see that the new label $v$ introduced in (2.8) corresponds to
’would be’ affine parameter generated by the holonomy component
$e^{i\frac{\bar{\mu}}{2}c}$ such that its quantum counterpart becomes a unit
shift.

So by taking this into account and by plugging formula (2.11) and (2.12) to
Euclidean part of Hamiltonian constraint, it becomes in the desired form:

$H^{E}=-\frac{2sgn(p)}{\kappa\gamma\bar{\mu}^{3}}\sum_{ijk}\epsilon^{ijk}Tr(h_{i}^{(\bar{\mu})}h_{j}^{(\bar{\mu})}(h_{i}^{(\bar{\mu})})^{-1}(h_{j}^{(\bar{\mu})})^{-1}h_{k}^{(\bar{\mu})}\\{(h_{k}^{(\bar{\mu})})^{-1},V\\})$
(2.15)

Now we wish to reformulate the Lorentzian part of the constraint. And we will
do it in a particular way by pointing out that on classical level $K^{i}_{a}$
can be rewritten, according to equation (A.5), as
$K^{i}_{a}=\frac{1}{\gamma}(A^{k}_{a}-\Gamma^{i}_{a})$ to obtain:

$E^{ai}E^{bj}K^{j}_{[a}K^{i}_{b]}=\frac{1}{2\gamma^{2}}\epsilon_{ijk}E^{ai}E^{bj}(F^{k}_{ab}-\Omega^{k}_{ab})$
(2.16)

with $\Omega^{k}_{ab}$ define as curvature of spin connection
$\Gamma^{i}_{a}$. Thanks to that, that Lorentzian part is constituted by two
terms: one proportional to the Euclidean part and one corresponds to Ricci
curvature. But for flat model $\Omega^{k}_{ab}=0$ thus Lotentzian part is only
proportional to the Euclidean part and can be subsumed into it. Such procuder
leads to the standard LQC presented in [6, 7]. However such a choice is not
unique - different choices of expressing $K^{i}_{a}$ cause different operators
in quantum theory - we will use that fact in the next sections.

Thus Hamiltonian constraint for standard LQC, takes the form:

$H_{g}=-\frac{4sgn(p)}{\kappa\gamma\bar{\mu}^{3}}\sum_{ijk}\epsilon^{ijk}Tr(h_{i}^{(\bar{\mu})}h_{j}^{(\bar{\mu})}(h_{i}^{(\bar{\mu})})^{-1}(h_{j}^{(\bar{\mu})})^{-1}h_{k}^{(\bar{\mu})}\\{(h_{k}^{(\bar{\mu})})^{-1},V\\})$
(2.17)

which after quantization (for improved dynamics scheme [7]) becomes:

$\hat{H_{g}}=\frac{24i}{\kappa^{2}\gamma\hbar\sqrt{\Delta^{3}}}\hat{\sin}(b)\hat{A}\hat{\sin}(b)$
(2.18)

where:

$\displaystyle\hat{A}=\hat{sgn}(v)\hat{V}\Big{(}\hat{\sin}(\frac{b}{2})\hat{V}\hat{\cos}(\frac{b}{2})-\hat{\cos}(\frac{b}{2})\hat{V}\hat{\sin}(\frac{b}{2})\Big{)}\leavevmode\nobreak\
\leavevmode\nobreak\ ,$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\
\hat{\sin}(\frac{b}{2})=\frac{1}{2i}(\hat{N}-\hat{N}^{-1})\leavevmode\nobreak\
\leavevmode\nobreak\ ,$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\
\hat{\cos}(\frac{b}{2})=\frac{1}{2}(\hat{N}+\hat{N}^{-1})$

and $b:=c\bar{\mu}$. This operator is selfadjoint and positive define on
$\mathcal{H}_{\bf kin}=L^{2}(\bar{\mathbb{R}},d\mu)$. Its action on the state
in volume basis returns:

$\hat{H_{g}}\Psi(v)=f_{+}\Psi(v+4)+f_{0}\Psi(v)+f_{-}\Psi(v-4)$ (2.19)

As a regular difference operator it divides its domain onto subspaces of
functions supported on the lattices:

$\mathcal{L}_{\epsilon}=\\{\epsilon+4n;\leavevmode\nobreak\
\leavevmode\nobreak\ n\in\mathds{Z}\\}$ (2.20)

where the value of $\epsilon{\color[rgb]{0.33,0.42,0.18}\in[0,4]}$ can be
fixed. Consideration of $\hat{H}_{g}$ can then be restricted only to Hilbert
space supported on a chosen subset (lattice). Such subspaces, unlike the full
Hilbert space, are separable. In our calculations, we will focus on a
particular lattice $\epsilon=0$. In such case the subsets of
$\mathcal{L}_{\epsilon}$ corresponding to $n\in\\{0\\},\mathbb{Z}^{\pm}$ are
not mixed by the action of $\hat{H}_{g}$ and we can work with just one semi-
lattice.

Lets now focus on the matter content. We consider free massless scalar field
$\phi$ that is minimally coupled to gravity. So total Hilbert space given by:

$\mathcal{H}=\mathcal{H}_{\bf kin}=L^{2}(\bar{\mathbb{R}},d\mu)\otimes
L^{2}(\mathds{R},d\phi)$ (2.21)

is build of solutions to total Hamiltonian constraint define as a tensor
product of presented earlier (gravitational) Hamiltonian constraint (2.18) and
constraint of the matter field which is provided by:

$C^{\prime}_{\phi}=|p|^{-\frac{2}{3}}\frac{p^{2}_{\phi}}{2}$ (2.22)

Its basic Poisson bracket is given by $\\{\phi,p_{\phi}\\}=1$ and
Hamiltonian’s equations are:

$\displaystyle\dot{p_{\phi}}=\\{p_{\phi},C^{\prime}_{\phi}\\}=0\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $
$\displaystyle\dot{\phi}=\\{\phi,C^{\prime}_{\phi}\\}=|p|^{-\frac{2}{3}}p_{\phi}$
(2.23)

What can be read from them is that, $p_{\phi}$ is a constant of motion, thus
$\phi$ is a monotonic function of the time and it will serve as a parameter
with respect to which the whole system will be evolving. The procedure to
quantize this constraint will be, to begin with, $|p|^{-\frac{2}{3}}$ then (as
in the standard quantum mechanic) $p_{\phi}$. But total Hamiltonian constraint
can be multiplied by lapse function $N=2V$ (because it is a Lagrange
multiplier it will not change physics). Thus in
$C_{\phi}:=2VC^{\prime}_{\phi}$ only momentum of $\phi$ is quantized and
constraint will be promoted to the following operator:

$\hat{C}_{\phi}=\mathds{1}\otimes(-\hbar^{2}\partial_{\phi}^{2})$ (2.24)

But rather than simply multiply the gravity part of the constraint with $N$,
symmetric ordering with respect to the volume operator should be applied. It
leads to the operator:

$\hat{\Theta}=\frac{48i}{\kappa^{2}\gamma\hbar\sqrt{\Delta^{3}}}\sqrt{\hat{V}}\hat{\sin}(b)\hat{A}\hat{\sin}(b)\sqrt{\hat{V}}\otimes\mathds{1}$
(2.25)

Thanks to all of that, full quantum constraint takes the form of Klein-Gordon
equation:

$\hbar^{2}\partial_{\phi}^{2}\Psi(v,\phi)=-\hat{\Theta}\Psi(v,\phi)$ (2.26)

and allows to apply the standard quantum mechanical methods for finding
solution (to a particular lattice $\epsilon=4$):

$\Psi(v,\phi)=\int_{0}^{\infty}dk\Big{(}\Psi_{+}(k)e_{k}(v)e^{i\omega\phi}+\Psi_{-}(k)\bar{e}_{k}(v)e^{-i\omega\phi}\Big{)}$
(2.27)

where $e_{k}(v)$ is eigenfunction for the operator $\Theta$ which corresponds
to eigenvalue $\omega$ of operator of the matter field and $k$ is connected to
it by $\omega^{2}=12\pi Gk^{2}$. Such structure leads to another splitting on
$\mathcal{H}_{\epsilon}$ corresponding to positive and negative energies. If
we restrict consideration, for example for a sector of positive energies
evolution of the system is provided by:

$\Psi(v,\phi)=e^{i(\phi-\phi_{0})\sqrt{\Theta}}\Psi(v,\phi_{0})$ (2.28)

and the physical inner product for any given $\phi_{0}$ is:

$\braket{\Psi_{1}}{\Psi_{2}}_{\epsilon}=\sum_{v\in\mathcal{L}_{\epsilon}}\Psi_{1}(v,\phi_{0})\bar{\Psi}_{2}(v,\phi_{0})$
(2.29)

To complete the Dirac program, one more step is required to be done - the
construction of physical observables. In the presented model they are found to
be an operator of the momentum of the field $\phi$ and volume operator for
fixed $\phi_{0}$ [6, 7], which action on the state $\Psi$ returns:

$\displaystyle\hat{V}_{\phi_{0}}\Psi(v,\phi)$ $\displaystyle=\alpha
e^{i(\phi-\phi_{0})\sqrt{\Theta}}|v|\Psi(v,\phi_{0})$
$\displaystyle\hat{p}_{\phi}\Psi(v,\phi)=-i\hbar\partial_{\phi}\Psi(v,\phi)$
(2.30)

and they expectation values are:

$\displaystyle\braket{\hat{V}_{\phi_{0}}}$
$\displaystyle=\alpha\sum_{v\in\mathcal{L}_{\epsilon}}|v||\Psi(v,\phi_{0})|^{2}$
$\displaystyle\braket{\hat{p}_{\phi}}=-i\hbar\sum_{v\in\mathcal{L}_{\epsilon}}\bar{\Psi}(v,\phi)\partial_{\phi}\Psi(v,\phi)$
(2.31)

To interpret the evolution of the model let’s recall results from [7] for the
expectation value of the volume operator. Because General Relativity is
recovered from LQC for a $\phi\rightarrow\pm\infty$ expectation values of
volume (by taking into account dispersion) approach classical trajectories
asymptotically. Departure becomes significant near-Planckian energy densities
- classical trajectories collapsing into a singularity, while effects of
quantum geometry cause repulsive behaviour of gravitational force leading to
quantum bounce at critical energy density
$\rho_{c}=\frac{3}{\kappa\gamma^{2}\Delta}$.

Similar results are obtained by examination of the effective dynamic of the
system. Consideration begins with a regularized but classical version of the
full constraint [6, 21] given by:

$H_{eff}=-\frac{3v\sin^{2}(b)}{\kappa\gamma^{2}\Delta}+\frac{p^{2}_{\phi}}{2v}$

and because $H_{eff}=0$, the energy density $\rho=\frac{p^{2}_{\phi}}{2v^{2}}$
is:

$\rho=\rho_{c}\sin^{2}(b)$ (2.32)

Evolution of this system is provided with Hamilton’s equation for $b$ and $v$
which takes form:

$\displaystyle\dot{v}=$
$\displaystyle\\{H_{eff},v\\}=\frac{3v}{\gamma\Delta}\sin(2b)$ (2.33a)
$\displaystyle\dot{b}=$
$\displaystyle\\{H_{eff},b\\}=-\frac{3v}{\gamma\Delta}\sin^{2}(b)+4\pi G\rho$
(2.33b)

and $\phi$ and it conjugate momentum as in (2.23). By combining (2.33a) with
(2.23) and by using relation (2.32) evolution of volume with respect to scalar
field is expressed as:

$\frac{dv}{d\phi}=\pm\sqrt{6\kappa(1-\frac{\rho}{\rho_{c}}})v$ (2.34)

which is satisfied, according to [3], by hyperbolic function expressed as:

$V=\frac{p_{\phi}}{\sqrt{2\rho_{c}}}\cosh(\beta\phi-\beta\phi_{0})$ (2.35)

This solution is in good agreement with quantum predictions.

###  2.3 Strict Thiemann regularization of Loop Quantum Cosmology

In Loop Quantum Cosmology we chose to at first, reformulate the Lorentzian
part to form, where it can be subsumed into the Euclidean part and then to
perform quantization of the Hamiltonian Constraint given by formula (2.17).
But as has been already pointed we are allowed to use different regularization
formula to obtain constraint in elementary variables. For example in [34] two-
approach were proposed, to which by following [32] we will refer as mLQC-I and
mLQC-II. The first follows directly the algorithm proposed originally by
Thiemann [33] for full LQG namely, we obtain $K^{i}_{a}$ by a Poisson bracket:

$K^{i}_{a}=\frac{1}{\kappa\gamma^{3}}\\{A^{i}_{a},\\{H^{E},V\\}\\}$ (2.36)

which in cosmological setings reduce to:

$K^{i}_{a}=-\frac{1}{\kappa\gamma^{3}\lambda}h_{i}^{(\lambda)}\\{(h_{i}^{(\lambda)})^{-1},\\{H^{E},V\\}\\}$
(2.37)

To formulate above expresion in improved dynamics scheme [7] trivial swap from
$\lambda$ to $\bar{\mu}$ is not sufficient because of dependence of
$\bar{\mu}$ on triads [34] \- formula (2.14). This leads to the following:

$K^{i}_{a}=-\frac{2}{3\kappa\gamma^{3}\bar{\mu}}h_{i}^{(\bar{\mu})}\\{(h_{i}^{(\bar{\mu})})^{-1},\\{H^{E},V\\}\\}$
(2.38)

and by plugging it - alongside with formula (2.11) - into Lorentzian
constraint, $T$ can be expressed in required form [34]:

$T=\frac{8sgn(p)}{\kappa^{4}\gamma^{7}\bar{\mu}^{3}}\sum_{ijk}\epsilon^{ijk}\operatorname*{Tr}\Big{(}h_{i}^{(\bar{\mu})}\\{(h_{i}^{(\bar{\mu})})^{-1},\\{H^{E},V\\}\\}h_{j}^{(\bar{\mu})}\\{(h_{j}^{(\bar{\mu})})^{-1},\\{H^{E},V\\}\\}h_{k}^{(\bar{\mu})}\\{(h_{k}^{(\bar{\mu})})^{-1},V\\}\Big{)}$
(2.39)

The implementation of the Dirac program is the same as in the previous
section. To be exact - quantization on the kinematical level as well as the
introduction of the matter field is the same as for mainstream LQC. The point
of divergence is a slightly different form of the quantum Hamiltonian
constriant which Lorentzian part is:

$\hat{T}=-\frac{96i}{9\kappa^{4}\gamma^{7}\hbar^{5}}\Big{(}\hat{\sin}(\frac{b}{2})\hat{B}\hat{\cos}(\frac{b}{2})-\hat{\cos}(\frac{b}{2})\hat{B}\hat{\sin}(\frac{b}{2})\Big{)}\hat{A}\Big{(}\hat{\sin}(\frac{b}{2})\hat{B}\hat{\cos}(\frac{b}{2})-\hat{\cos}(\frac{b}{2})\hat{B}\hat{\sin}(\frac{b}{2})\Big{)}$
(2.40)

where:

$\displaystyle\hat{\sin}(\frac{b}{2})=\frac{1}{2i}(\hat{N}-\hat{N}^{-1})\leavevmode\nobreak\
\leavevmode\nobreak\ $ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\hat{\cos}(\frac{b}{2})=\frac{1}{2}(\hat{N}+\hat{N}^{-1})$ (2.41)

and $\hat{B}$ is commutator between $\hat{H}^{E}$ and $\hat{V}$, while
Euclidean term is as in equation (2.15). Full Hamiltonian constraint again
takes the form of the Klein-Gordon equation as in 2.26 but with new $\Theta$
formulated in shift and volume operators as:

$\hat{\Theta}=\frac{3}{2\kappa\hbar^{2}\Delta}\sqrt{\hat{V}}\Big{(}-s^{\prime}\hat{N}^{4}\hat{V}\hat{N}^{4}+\hat{N}^{2}\hat{V}\hat{N}^{2}+2(s^{\prime}-1)\hat{V}+\hat{N}^{-2}\hat{V}\hat{N}^{-2}-s^{\prime}\hat{N}^{-4}\hat{V}\hat{N}^{-4}\Big{)}\sqrt{\hat{V}}\otimes\mathds{1}$
(2.42)

where $s^{\prime}=\frac{1+\gamma^{2}}{4\gamma^{2}}$. Thanks to that, it is
straightforward to see that, as in mainstream LQC, $\Theta$ acts on states in
volume representation as a difference operator - but now it is of the 4th
order. So we again make a restriction to the superselection sector for lattice
$\epsilon=4$. But by defining Fourier transformation of the state in volume
base to momentum $b$ base as:

$\psi(b)=\sum_{v\in\mathcal{L}_{\epsilon}}\frac{1}{\sqrt{|v|}}\psi(v)e^{i\frac{vb}{2}}$
(2.43)

with $b\in\left[0,\pi\right]$, $\hat{\Theta}$ becomes differential operator of
the second-order [10], given by the formula:

$\hat{\Theta}=12\pi
G\hbar^{2}\gamma^{2}\Big{(}(\sin(b)\partial_{b})^{2}-s^{\prime}(\sin(\frac{b}{2})\partial_{b})^{2}\Big{)}$
(2.44)

which is in agreement with the order of the difference operator in mainstream
LQC. In addition, $\hat{\Theta}$ is symmetric but not essentially self-
adjoint. By investigating its deficiency subspace which has both dimensions
equal to one (it was shown in [10]), this operator admits a family of self-
adjoint extensions. It indicates that evolution is not unique and we can
restrict ourselves to work on a subspace of $\mathcal{H}$ corresponding to a
particular extension of $\hat{\Theta}$. To do so in addition some boundary
data (corresponding to our choice) needs to be added. Then physical states are
constructed by analogy to mainstream LQC but with basis build out of
eigenfunctions evaluated for particular extension (it was done analytically in
b-representation in [10]). Moreover Klein Gordon equation for mLQC-II for
large volumes of the The universe is stable (in the sense of Von Neumann
stability [32]). It means that eigenfunction can be numerically evaluated from
initial data in $v$.

To check the dynamics physical observables must be built. In LQC they are the
momentum of the field $\phi$ and volume at any given value of $\phi$. In this
model, momentum can be again promoted to the operator, but
$\hat{V}_{\phi_{0}}$ could act outside of the physical Hilbert space [10, 25].
Solution her is to simply replace $\hat{V}_{\phi_{0}}$ with a bounded function
of this operator. Thus we recall from [10], as a second Dirac observable -
compactified volume:

$\hat{\theta}_{K}=\arctan\Big{(}\frac{\hat{V}}{\alpha K}\Big{)}$ (2.45)

where $K$ is any positive real number. This operator is bounded by
$\frac{\pi}{2}$ which corresponds to infinite volume obtained in finite value
of scalar field - no such thing happens in considered earlier model. But to
extract physics and not exactly qualitative solution from it we can examine
effective dynamic:

$H_{eff}=\frac{3v}{\kappa\Delta}\Big{(}\sin^{2}(b)-s^{\prime}\sin^{2}(2b)\Big{)}$
(2.46)

Now vanishing of constraint leds to two solutions:

$\sin^{2}(b_{\pm})=\frac{1\pm\sqrt{1-\frac{\rho}{\rho_{c}^{I}}}}{2(\gamma^{2}+1)}$
(2.47)

where critical energy density in mLQC-I is:

$\rho_{c}^{I}=\frac{\rho_{c}}{4(\gamma^{2}+1)}$ (2.48)

Solutions to $b_{+}$ and $b_{-}$ are both physically achievable and indicate
asymmetric evolution of the universe with respect to the bounce. But more
detailed analysis [10], shows that in fact for $\phi\rightarrow\pm\infty$
General Relativity is recovered and the expectation value of volume, classical
and effective trajectories will coincide. But in the near Planck energy regime
classical trajectory will collapse into singularity while in quantum theory
and effective dynamics bounce occurs. But after the bounce in the mLQC-I
Universe will rapidly expand (as in GR in presence of cosmological constant)
to a moment when the finite value of scalar field volume will be infinity.
This point was identified as a transition from expanding Universe (in its
conformal future) to contracting Universe (explicitly to its conformal past)
where second bounce occurs [10]. Thanks to that we can identify solutions to
$b_{+}$ and $b_{-}$ (from effective dynamics) as corresponding to emerging
DeSiter epochs (between bounces) and for trajectories for
$\phi\rightarrow\pm\infty$ respectively.

##  3 Second Modification of Loop Quantum Cosmology

In this chapter, we describe the next version of the LQC as a result of
Thieman’s regularization, which (unlike the previous two) has not been yet
investigated in detail. It was first proposed (together with mLQC-I) in [34]
and its effective dynamic has ben probed for example in [21]. Because Dirac
program of quantization is as in mainstream LQC and mLQC-I, in the section 3.1
we start at the point of divergence - construction of the constraint. To be
exact we will present regularization and action of operator $\Theta$ on the
state in volume representation and study those properties that can be probed
in this representation. Since for mLQC-I analysis in the momentum
representation has been much more useful, in the next section - 3.2 we will
perform a transformation to this representation from volume representation and
check properties of $\Theta$. In the section 3.3 we will attempt to probe the
quantum dynamics via finding the expectation value of the volume as a function
of the scalar field and constant of motion. Finally in sec. 3.4 we will
compare our results against obtained with effective dynamics.

###  3.1 Regularization and $\hat{\Theta}$ in volume representation

As mLQC-I quantization arises from taking the general form of $K^{i}_{a}$ in
The Lorentzian term, the second modification of LQC will be applied only to
spatially flat models because in that case extrinsic curvature 1-form, can be
expressed as:

$K^{i}_{a}=\frac{1}{\gamma}A^{i}_{a}$ (3.1)

and constraint becomes:

$T=\frac{1}{2\kappa\gamma^{2}}\int_{\mathcal{V}}d^{3}x\frac{E^{ai}E^{bj}}{\sqrt{det(h)}}A^{j}_{[a}A^{i}_{b]}$
(3.2)

where at clasical level it is equivalent to $T$ formulated in equation (2.10).
In cosmological setings, it reduce to:

$T=-\frac{2sgn(p)}{\kappa^{2}\gamma^{3}}\epsilon^{ijk}\operatorname*{Tr}\Big{(}c\tau_{i}c\tau_{j}\\{c\tau_{k},V\\}\Big{)}$
(3.3)

where to obtain it in terms of holonomy we will use the following two
properties from [34]:

$\displaystyle c\tau_{i}=\lim_{\bar{\mu}\rightarrow
0}\frac{1}{2\bar{\mu}}\Big{(}h_{i}^{(\bar{\mu})}-(h_{i}^{(\bar{\mu})})^{-1}\Big{)}$
(3.4a)
$\displaystyle\\{c\tau_{k},V\\}=h_{k}^{(\bar{\mu})}\\{(h_{k}^{(\bar{\mu})})^{-1},V\\}$
(3.4b)

and thus Lorentzian part becomes:

$T=\frac{sgn(p)}{2\kappa^{2}\gamma^{3}\bar{\mu}^{3}}\lim_{\bar{\mu}\rightarrow
0}\sum_{ijk}\epsilon^{ijk}\operatorname*{Tr}\Big{(}h_{i}^{(\bar{\mu})}-(h_{i}^{(\bar{\mu})})^{-1})(h_{j}^{(\bar{\mu})}-(h_{j}^{(\bar{\mu})})^{-1})h_{k}^{(\bar{\mu})}\\{(h_{k}^{(\bar{\mu})})^{-1},V\\}\Big{)}$
(3.5)

And again as in mLQC-I quantization on the kinematical level is the same as in
LQC thus we encounter the very same problem in the above expression - ill-
definiteness of the limit. To resolve this we proceed as already been
mentioned - by setting the holonomy length the same way as when quantizing the
Euclidean part of the Hamiltonian constraint. By this, the operator of the
Lorentzian part becomes:

$\hat{T}=-\frac{24i}{\gamma^{3}\kappa^{2}\hbar}\hat{\sin}(\frac{b}{2})\hat{A}\hat{\sin}(\frac{b}{2})$
(3.6)

where $\hat{\sin}(\frac{b}{2})$ is defined as in (2.41) and full operator of
gravity part of Hamiltonian constraint in volume representation can be defined
as:

$\hat{\Theta}=-3\pi
G\hbar^{2}\gamma^{2}\sqrt{|\hat{v}|}\Big{(}\hat{N}^{2}|\hat{v}|\hat{N}^{2}-s\hat{N}|\hat{v}|\hat{N}+2(s-1)|\hat{v}|-s\hat{N}^{-1}|\hat{v}|\hat{N}^{-1}+\hat{N}^{-2}|\hat{v}|\hat{N}^{-2}\Big{)}\sqrt{|\hat{v}|}$
(3.7)

where $s=\frac{4(1+\gamma^{2})}{\gamma^{2}}$. In volume representation
$\hat{\Theta}$ act on the state as in LQC as difference operator and as in
mLQC-I it is of 4th order operator. If we consider its eigenvalue problem
(corresponding to an eigenvalue of matter part operator $\omega$), this an
equation can be rewritten as recurrence relation:

$f_{-4}e_{\omega}(v+4)=f_{-2}e_{\omega}(v+2)+(\omega^{2}-f_{0})e_{\omega}(v)-f_{+4}e_{\omega}(v-4)+f_{+2}e_{\omega}(v-2)$
(3.8)

where:

$\displaystyle f_{+4}$ $\displaystyle=\sqrt{v}\sqrt{v-4}|v-2|$ $\displaystyle
f_{-4}$ $\displaystyle=\sqrt{v}\sqrt{v+4}|v+2|$ (3.9a) $\displaystyle f_{+2}$
$\displaystyle=s\sqrt{v}\sqrt{v-2}|v-1|$ $\displaystyle f_{-2}$
$\displaystyle=s\sqrt{v}\sqrt{v+2}|v+1|$ (3.9b) $\displaystyle f_{0}$
$\displaystyle=2(s-1)v^{2}$ (3.9c)

and solutions are obtained by fixing for $e_{\omega}(v=2)$ and
$e_{\omega}(v=4)$. This indicates that the space of solutions has dimension 2.
We performed a numerical analysis of this problem. What we have found is that
solution is unstable - they will grow exponentially as it is shown (for
particular eigenvalue $\omega=10$) in Fig.1. It means that unlike in mLQC-I,
eigenfunction in volume representation cannot be numerically evaluated from
initial data. The instability of these solutions was first established via
VonNeuman stability analysis in [32] where the large volume behavior of the
equation has been studied. Here we presented the roots of this equation:

$\displaystyle 1,\leavevmode\nobreak\ \leavevmode\nobreak\ $
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ 1$ (3.10a)
$\displaystyle\frac{2+\gamma^{2}-2\sqrt{1+\gamma^{2}}}{\gamma^{2}},\leavevmode\nobreak\
\leavevmode\nobreak\ $ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\frac{2+\gamma^{2}+2\sqrt{1+\gamma^{2}}}{\gamma^{2}}$ (3.10b)

where first is made of two identical roots equal to one but in the second pair
at least one is clearly greater than unity. It not necessary means that in
large volumes of the Universe, GR will not be recovered. Rather it points out
the possibility of the occurrence of not valid physical roots of the equation.
It was confirmed by investigation of the effective dynamics in [21], where one
of the solutions to the vanishing of effective constraint was identified as
nonphysical. Now to check properties of $\Theta$ we switch as in [10] to
momentum $b$ representation.

###  3.2 $\hat{\Theta}$ in momentum $b$ representation

We begin with defining a type of a Fourier transformation of state in volume
base to momentum $b$ base as:

$\psi(b)=\sum_{v\in\mathcal{L}_{\epsilon}}\frac{1}{\sqrt{|v|}}\psi(v)e^{i\frac{vb}{2}}$
(3.11)

wherewith respect to formula (2.43), the novelty lies in the domain of $b$ \-
now is a circle of unit radius rather than half unit. We also write down,
formula for the inverse of the above transformation:

$\psi(v)=\frac{1}{\pi}\sqrt{|v|}\int_{0}^{2\pi}db\psi(b)e^{-i\frac{vb}{2}}$
(3.12)

and by plugging it in (2.29) scalar product in this representation is:

$\braket{\psi_{1}}{\psi_{2}}=\frac{1}{\pi^{2}}\sum_{v\in\mathcal{L}_{\epsilon}}|v|\int
dbdb^{\prime}\bar{\psi}_{1}(b^{\prime})e^{i\frac{vb^{\prime}}{2}}\psi_{2}(b)e^{-i\frac{vb}{2}}$
(3.13)

where because of the absolute value of volume, $\braket{\psi_{1}}{\psi_{2}}$
cannot be formulated as a single integral on the whole Hilbert space. But if
we consider projections $P^{\pm}$ such that $[P^{\pm}\psi](v)=\theta(\pm
v)\psi(v)$ products on subspaces of positive and negative valuse of $v$ can be
presented as:

$\displaystyle\braket{\psi_{1}}{\psi_{2}}_{\pm}$
$\displaystyle=\pm\frac{1}{\pi^{2}}\sum_{v\in\mathcal{L}_{\epsilon}}\int
dbdb^{\prime}\bar{\psi}_{1}(b^{\prime})v\psi_{2}(b)e^{-i\frac{v(b-b^{\prime})}{2}}=\pm\frac{1}{\pi}\int
dbdb^{\prime}\bar{\psi}_{1}(b^{\prime})(-2i\partial_{b})\psi_{2}(b)\delta(b-b^{\prime})$
$\displaystyle=\mp\frac{2i}{\pi}\int
db\bar{\psi}_{1}(b)\partial_{b}\psi_{2}(b)$ (3.14)

However, because we work with states symmetric in $v$ (thus need only positive
semilattice) we will restrict ourselves to work with non-negative and grater
than 2 values of $v$ so we will be using integral with the minus sign. Thanks
to that, it is straightforward to find action of volume operator as:

$\displaystyle\alpha|\hat{v}|\psi(b)=-2i\alpha\partial_{b}\psi(b)$ (3.15)

We now, by inspecting:

$\displaystyle\sqrt{|\hat{v}|}\hat{N}\frac{1}{\sqrt{|\hat{v}|}}\psi(b)$
$\displaystyle=\sum_{v\in\mathcal{L}_{\epsilon}}\frac{1}{\sqrt{|v|}}\sqrt{|\hat{v}|}\hat{N}\frac{1}{\sqrt{|\hat{v}|}}\psi(v)e^{i\frac{vb}{2}}=\sum_{v\in\mathcal{L}_{\epsilon}}\frac{1}{\sqrt{|v|}}\frac{\sqrt{|v|}}{\sqrt{|v-1|}}\psi(v-1)e^{i\frac{vb}{2}}$
$\displaystyle=\sum_{v^{\prime}\in\mathcal{L}_{\epsilon}}\frac{1}{\sqrt{|v^{\prime}|}}\psi(v^{\prime})e^{i\frac{v^{\prime}b}{2}}e^{i\frac{b}{2}}=e^{i\frac{b}{2}}\psi(b)$
(3.16)

find action of the shift operator to be in the following form:

$\hat{N}\psi(b)=\frac{1}{\sqrt{|\hat{v}|}}\exp(\frac{ib}{2})\sqrt{|\hat{v}|}\psi(b)$
(3.17)

and thus:

$\hat{\sin}(b)\psi(b)=\frac{1}{\sqrt{|\hat{v}|}}\sin(b)\sqrt{|\hat{v}|}\psi(b)$
(3.18)

In such case, $\hat{\Theta}$ takes the form:

$\displaystyle\hat{\Theta}$ $\displaystyle=12\pi
G\hbar^{2}\gamma^{2}\Big{(}(\sin(b)\partial_{b})^{2}-\frac{4(1+\gamma^{2})}{\gamma^{2}}(\sin(\frac{b}{2})\partial_{b})^{2}\Big{)}$
$\displaystyle=-12\pi
G\hbar^{2}\gamma^{2}\Big{(}f(b)\partial_{b}^{2}+h(b)\partial_{b}\Big{)}$
(3.19)

which is the differential operator of the second order where:

$\displaystyle
f(b)=4\sin^{2}(\frac{b}{2})\Big{(}\frac{1+\gamma^{2}}{\gamma^{2}}-\cos^{2}(\frac{b}{2})\Big{)}\leavevmode\nobreak\
\leavevmode\nobreak\ $ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
h(b)=\sin(b)\Big{(}\frac{1+\gamma^{2}}{\gamma^{2}}-\cos(b)\Big{)}$ (3.20)

Because $f(b)$ is always non-negative, unlike in [10], the signature of the
Klein-Gordon equation - with $\Theta$ defined as in (3.19) - will not be mixed
and constraint will be hyperbolic. Now, we wish to introduce a new variable as
a function of $b$ so that the equation for evolution takes the strict K-G
form. After application of chain rule, alongside with condition

$f(b)(\frac{dx}{db})^{2}=1$ (3.21)

which has been reformulated to

$\frac{dx}{db}=\frac{1}{\sqrt{f(b)}}$ (3.22)

$x(b)$ has been be integrated out analytically, to be:

$x(b)=\tanh^{-1}\Big{(}\frac{\sqrt{2}\cos(\frac{b}{2})}{\sqrt{2+\gamma^{2}-\gamma^{2}\cos(b)}}\Big{)}$
(3.23)

with $x(b=\pi)=0$ and appropriate limits:

$\displaystyle\lim_{b\rightarrow 0}x(b)=-\infty\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\lim_{b\rightarrow 2\pi}x(b)=\infty$

so, the range of this function is the whole real line. Moreover this function
is well defined (continuous and differentiable at every point) on the whole
domain. Operator in new coordinates becomes:

$\hat{\Theta}=-12\pi G\hbar^{2}\partial_{x}^{2}$ (3.24)

and sclar product in $x$ takes form:

$\braket{\psi_{1}}{\psi_{2}}=-\frac{2i}{\pi}\int^{\infty}_{-\infty}dx\bar{\psi}_{1}(x)\partial_{x}\psi_{2}(x)$
(3.25)

The operator $|\hat{v}|$ on the subdomain supported on $v>0$ has been found
after expressing $\frac{dx}{db}$ as a function of $x$ variable:

$|\hat{v}|=-i\frac{1}{\sqrt{\gamma^{2}+1}}f(x)\partial_{x}$ (3.26)

where $f(x)=(\gamma^{2}\tanh^{2}(x)+1)\cosh(x)$.
Now we will check whether $\hat{\Theta}$ is self-adjoint operator. To do so
lets inspect the deficiency subspaces $\mathcal{K}_{\pm}$ defined [26] as the
spaces of (kinematically) normalizable solutions $\psi^{\pm}$ to the equation:

$12\pi G\hbar^{2}\partial_{x}^{2}\psi^{\pm}=\pm i\psi^{\pm}$ (3.27)

where both solutions have been found to be as plain weave type:

$\displaystyle\psi^{+}$
$\displaystyle=\exp(\frac{(-1)^{\frac{1}{4}}x}{\sqrt{12\pi
G}\hbar})+\exp(-\frac{(-1)^{\frac{1}{4}}x}{\sqrt{12\pi G}\hbar})$
$\displaystyle\psi^{-}$
$\displaystyle=\exp(\frac{(-1)^{\frac{3}{4}}x}{\sqrt{12\pi
G}\hbar})+\exp(-\frac{(-1)^{\frac{3}{4}}x}{\sqrt{12\pi G}\hbar})$ (3.28)

We now, as it was defined in (3.25) computed $\braket{\psi^{+}}{\psi^{+}}$ and
$\braket{\psi^{-}}{\psi^{-}}$ to find out, that both of these scalar products
are not finite. It means that both solutions (3.28) are not normalizable and
by that only zero will satisfy equation 3.27. So both deficiency subspaces are
spaned only by a zeroth vector so:

$\displaystyle\dim(\mathcal{K}_{+})=0\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\dim(\mathcal{K}_{-})=0$

and by using the definition from [26] which was quoted in Appendix B, we
conclude that $\hat{\Theta}$ is essentially self-adjoint.

###  3.3 Dynamics

Because $\Theta$ turns out to be essentially self-adjoint evolution of the
system is uniquely determined. As well physical Hilbert space can be
identified as a solution to full quantum constraint. Thus we start probing the
dynamic by writing it in $x$ representation:

$\partial_{\phi}^{2}\Psi(x,\phi)=12\pi G\partial_{x}^{2}\Psi(x,\phi)$ (3.29)

Eigenfunctions of $\partial_{\phi}^{2}$ and $\partial_{x}^{2}$ are $e^{ikx}$,
$e^{i\omega\phi}$ respectively where $\omega$ is ,,frequency” and it is
connected to $k$ by $\omega=\sqrt{12\pi G}|k|$. Solutions to equation (3.29),
are find by using standard quantum mechanical methods to be:

$\Psi(x,\phi)=\int_{0}^{\infty}dk\psi(k)e^{ikx}e^{i\beta|k|\phi}$ (3.30)

where $\beta=\sqrt{12\pi G}$. Finally by complete analogous to formulas
(2.30), Dirac observables are:

$\displaystyle\hat{V}_{\phi_{0}}\Psi(x,\phi)=\alpha
e^{i(\phi-\phi_{0})\sqrt{\Theta}}|v|\Psi(x,\phi_{0})\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\hat{p}_{\phi}\Psi(x,\phi)=-i\hbar\partial_{\phi}\Psi(x,\phi)$ (3.31)

Now we wish to find the expectation value of $V$ in $x$ representation. So as
mentioned earlier, we restricted ourselves to act only on subspace where $v$
is nonnegative and larger than $2$. It leads to:

$\braket{\psi_{1}}{\hat{V}\psi_{2}}=-\frac{2\alpha}{\pi}\frac{1}{\sqrt{\gamma^{2}+1}}\int^{\infty}_{-\infty}dx\bar{\psi}_{1}(x)(f(x)\partial^{2}_{x}+f^{\prime}(x)\partial_{x})\psi_{2}(x)$
(3.32)

Now by using form (3.30) of the wave function in the above expression we have:

$\displaystyle=-\frac{2\alpha}{\pi}\frac{1}{\sqrt{\gamma^{2}+1}}\int^{\infty}_{-\infty}\int^{\infty}_{0}\int^{\infty}_{0}dxdkdk^{\prime}\bar{\psi}_{1}(k^{\prime})e^{-ik^{\prime}(x+\beta\phi)}(f(x)\partial^{2}_{x}+f^{\prime}(x)\partial_{x})\psi_{2}(k)\bar{\psi}_{2}(k)e^{ik(x+\beta\phi)}$
$\displaystyle=-\frac{2\alpha}{\pi}\frac{1}{\sqrt{\gamma^{2}+1}}\int^{\infty}_{-\infty}\int^{\infty}_{0}\int^{\infty}_{0}dxdkdk^{\prime}\bar{\psi}_{1}(k^{\prime})(-k^{2}f(x)+ikf^{\prime}(x))\psi_{2}(k)e^{i(k-k^{\prime})x}e^{i(k-k^{\prime})\beta\phi)}$
$\displaystyle=-\frac{2\alpha}{\pi}\frac{1}{\sqrt{\gamma^{2}+1}}\int^{\infty}_{0}dk\bar{\psi}_{1}(k)(-k^{2}f(x)+ikf^{\prime}(x))\psi_{2}(k)$
(3.33)

To proceed further we need function $f$ and $f^{\prime}$ in energy $k$
variable. At first we expand them in to Taylor series:

$f(x)=\sum_{n=0}^{\infty}\frac{1}{n!}f^{n}(0)x^{n}$ (3.34)

then by investigation of

$\displaystyle\hat{x}\Psi(x,\phi)$
$\displaystyle=\int_{0}^{\infty}dkx\psi(k)e^{ikx}e^{i\beta|k|\phi}$
$\displaystyle=\int_{0}^{\infty}dk\psi(k)(i\partial_{k}-\beta\phi)e^{ikx}e^{i\beta|k|\phi}$
(3.35)

we find that $x=i\partial_{k}-\beta\phi$ and thus by using binomial formula:

$x^{n}=(i\partial_{k}-\beta\phi)^{n}=\sum_{l=0}^{n}{n\choose
l}(-\beta\phi)^{n-l}(i\partial_{k})^{l}$ (3.36)

so, finally, by detailed analysis function, $f$ can be re-expressed as:

$f(x)=\sum_{i=0}^{\infty}\frac{1}{l!}\partial^{l}_{\beta\phi}f(\beta\phi)(i\partial_{k})^{l}$
(3.37)

By plugging this form of functions into formula (3.33) we ended with:

$\displaystyle\braket{\hat{V}}$
$\displaystyle=\frac{2\alpha}{\pi}\frac{1}{\sqrt{\gamma^{2}+1}}\sum_{i=0}^{\infty}\frac{1}{l!}\partial^{l}_{\beta\phi}f(\beta\phi)\int^{\infty}_{0}dk\bar{\psi}_{1}(k)k^{2}(i\partial_{k})^{l}\psi_{2}(k)$
$\displaystyle-\frac{2\alpha}{\pi}\frac{1}{\sqrt{\gamma^{2}+1}}\sum_{i=0}^{\infty}\frac{1}{l!}\partial^{l}_{\beta\phi}f^{\prime}(\beta\phi)\int^{\infty}_{0}dk\bar{\psi}_{1}(k)ik(i\partial_{k})^{l}\psi_{2}(k)$
(3.38)

Before going further, we find form of the scalar product in $k$
representation, we start from formula (3.14):

$\displaystyle\braket{\psi_{1}}{\psi_{2}}$
$\displaystyle=-\frac{2i}{\pi}\int_{0}^{2\pi}db\bar{\psi}_{1}(b)\partial_{b}\psi_{2}(b)$
$\displaystyle=-\frac{2i}{\pi}\int_{0}^{2\pi}\int_{0}^{\infty}\int_{0}^{\infty}dbdkdk^{\prime}\bar{\psi}_{1}(k^{\prime})e^{-i\beta
k^{\prime}\phi}e^{-ik^{\prime}b}\partial_{b}\psi_{2}(k)e^{i\beta
k\phi}e^{ikb}$
$\displaystyle=-\frac{2i}{\pi}\int_{0}^{2\pi}\int_{0}^{\infty}\int_{0}^{\infty}dbdkdk^{\prime}\bar{\psi}_{1}(k^{\prime})ik\psi_{2}(k)e^{i\beta(k-k^{\prime})\phi}e^{ik(b-b^{\prime})}$
$\displaystyle=\frac{2}{\pi}\int_{0}^{\infty}dk\bar{\psi}_{1}(k)k\psi_{2}(k)$
(3.39)

thus formula (3.38) becomes:

$\braket{\hat{V}}=\alpha\frac{1}{\sqrt{\gamma^{2}+1}}\sum_{l=0}^{\infty}\frac{1}{l!}\Big{(}\partial^{l}_{\beta\phi}f(\beta\phi)\braket{:k(i\partial_{k})^{l}:}-i\partial^{l}_{\beta\phi}f^{\prime}(\beta\phi)\braket{(i\partial_{k})^{l}}\Big{)}$
(3.40)

where $:\cdot:$ denotes some symmetric ordering.

Now let us consider a system that is completely described by the position $X$
and its conjugate momentum $P$. During quantization, these variables become
operators and physics is extracted through their expectation values. But
because the expectation value of the product of operators, in general, is not
equal to products of expectation values, to describe the system completely
after quantization we need moments [12, 13], defined as:

$\displaystyle G^{a,b}$
$\displaystyle:=\braket{(\hat{P}-P)^{a}(\hat{X}-X)^{b}}_{Weyl}$
$\displaystyle=\sum_{k=0}^{a}\sum_{n=0}^{b}(-1)^{a+b-k-n}{a\choose k}{b\choose
n}P^{a-k}X^{b-n}\braket{\hat{P}^{k}\hat{X}^{n}}_{Weyl}$ (3.41)

where index on expectation value means totally symmetric ordering. After
inverting relation 3.41 to obtain the formula for
$\braket{\hat{P}^{k}\hat{X}^{n}}_{Weyl}$, we can apply it to find:

$\displaystyle\braket{k(i\partial_{k})^{l}}$
$\displaystyle=\sum_{n=0}^{1}\sum_{i=0}^{l}{l\choose
i}\braket{k}^{1-n}\braket{i\partial_{k}}^{l-i}G^{ni}$
$\displaystyle=\braket{k}\braket{i\partial_{k}}^{l}+\braket{i\partial_{k}}^{l}G^{1,0}+\sum_{i=1}^{l}{l\choose
i}\braket{i\partial_{k}}^{l-i}\Big{(}\braket{k}G^{0,i}+G^{1,1}\Big{)}$ (3.42)

where on the left-hand side of the equation (3.42), $\braket{...}$ denotes
expectation value while on the right-hand side, it represents classical
trajectories. Form of (3.42) indicates that leading term is
$\braket{k}\braket{i\partial_{k}}^{l}$ and terms with $G^{a,b}$ are quantum
corections. If they will be neglected:

$\braket{k(i\partial_{k})^{l}}\cong\braket{k}\braket{i\partial_{k}}^{l}$
(3.43)

and by plugging it to equation (3.40), we found semi classical trajectory of
volume to be:

$\braket{\hat{V}}\cong\frac{\alpha}{\sqrt{\gamma^{2}+1}}\Big{(}\braket{k}f(\beta\phi+\braket{i\partial_{k}})-if^{\prime}(\beta\phi+\braket{i\partial_{k}})\Big{)}$
(3.44)

But it is straightforward to see an alarming property of this solution -
because the second term is pure imaginary, the whole solution will return a
complex value. To resolve this problem one more comment on (3.42) should be
done - in this formula sum of all terms must be real but each of them could be
complex. It means if we take the whole expression for
$\braket{k(i\partial_{k})^{l}}$ imaginary part will cancel out. So we can just
delete the problematic part. Thanks to that:

$\braket{\hat{V}}\cong\frac{\alpha}{\sqrt{\gamma^{2}+1}}\braket{k}\Big{(}\gamma^{2}\tanh^{2}(\beta\phi+\braket{i\partial_{k}})+1\Big{)}\cosh(\beta\phi+\braket{i\partial_{k}})$
(3.45)

which is symmetric function.

###  3.4 Effective Dynamics

Now we will check if our result is consistent with the prediction of effective
dynamics, which with good approximation provides quantitative prediction about
quantum system [9]. To do so, we will be treating full Hamiltonian constraint
- after regularization - as a classical constraint. In the following
consideration we are inspecting form of $H_{eff}$ from [21] given by:

$H_{eff}=-\frac{3v}{\kappa\Delta\gamma^{2}}\sin^{2}(\frac{b}{2})[1+\gamma^{2}\sin^{2}(\frac{b}{2})]+\frac{p^{2}_{\phi}}{2v}$

which condition that it must vanish leads to:

$\rho=4\rho_{c}\sin^{2}(\frac{b}{2})\Big{(}1+\gamma^{2}\sin^{2}(\frac{b}{2})\Big{)}$
(3.46)

with equation of motion [21] given by:

$\displaystyle\dot{b}$
$\displaystyle=\\{b,H_{eff}\\}=-\frac{6v}{\gamma\sqrt{\Delta}}\sin^{2}(\frac{b}{2})(1+\gamma^{2}\sin^{2}(\frac{b}{2}))+4\pi
G\rho$ (3.47a) $\displaystyle\dot{v}$
$\displaystyle=\\{v,H_{eff}\\}=\frac{3v}{\gamma\sqrt{\Delta}}\sin(b)(1+2\gamma^{2}\sin^{2}(\frac{b}{2}))$
(3.47b)

But derivative in above expressions are with respect to conformal time and to
compare the evolution of $v$ with one obtained in (3.45) we need to formulate
it, as a function dependent on the scalar field. To do so we introduce an
auxiliary function define as:

$y:=\sin^{2}(\frac{b}{2})$ (3.48)

and by using formula (3.47a) we have found:

$\frac{db}{d\phi}=\frac{db}{dt}(\frac{d\phi}{dt})^{-1}=-2\beta
y(1+\gamma^{2}y)$ (3.49)

Now we calculate:

$(\frac{dy}{d\phi})^{2}=\frac{1}{4}y(1-y)(\frac{db}{d\phi})^{2}=\beta^{2}y^{2}(1-y)(1+\gamma^{2}y)$
(3.50)

what allow us to get $y$ as a function of scalar field:

$y(\phi)=\frac{1-\tanh^{2}(\beta\phi-\beta\phi_{0})}{\gamma^{2}\tanh^{2}(\beta\phi-\beta\phi_{0})+1}$
(3.51)

and reformulated $\rho$ from (3.46) as:

$\rho=4\rho_{c}\frac{1+\gamma^{2}}{(\gamma^{2}\tanh^{2}(\beta\phi-\beta\phi_{0})+1)^{2}\cosh^{2}(\beta\phi-\beta\phi_{0})}$
(3.52)

and by thus we obtain:

$\displaystyle V$
$\displaystyle=\frac{|p_{\phi}|}{\sqrt{2\rho}}=\frac{\hbar\sqrt{12\pi
G}|k|}{\sqrt{8\rho_{c}}\sqrt{(1+\gamma^{2})}}(\gamma^{2}\tanh^{2}(\beta\phi-\beta\phi_{0})+1)\cosh(\beta\phi-\beta\phi_{0})$
$\displaystyle=\frac{\alpha}{\sqrt{(1+\gamma^{2})}}|k|(\gamma^{2}\tanh^{2}(\beta\phi-\beta\phi_{0})+1)\cosh(\beta\phi-\beta\phi_{0})$
(3.53)

which, by defing $\phi_{0}=-\frac{1}{\beta}\braket{i\partial_{k}}$ is exactly
what we have acquired in formula (3.45). It means that our solution is
consistent with one obtained in [21]. Morover it indicates that:

  * •

evolution of the Universe will be symetric with respect to the bounce that
will ocure for:

$\rho_{c}^{II}=4(1+\gamma^{2})\rho_{c}$

  * •

for $\phi\rightarrow\pm\infty$ General Relativity will be recovered.

##  4 Conclusion

In this thesis, we recollected the most important information about two
regularization prescriptions studied in detail in literature - the so called
(mainstream) LQC and mLQC-I. As well, we provided our detailed analysis of
prescription first introduced (alongside with mLQC-I) in [34] to which we
referred as mLQC-II. Because quantization on the kinematical level is the same
as in the prescription mentioned earlier, we have begun by building the
gravity part of the constraint. To be exact we started with the Lorentzian
part (because Euclidean is as in formula (2.15)) and pointing out that because
we chose to work in k=0 FLRW model externistic curvature 1-form can be
expressed as a term proportional to Ashtekar-Barbero connection (3.1). Then we
expressed this constraint in terms of elementary variables and performed
quantization as in [34]. It has been done to obtain the full operator of
gravity part as in (3.7). Because we chose to introduce the matter field as in
mainstream LQC full quantum constraint forms a type of Klein-Gordon equation
where the scalar field $\phi$ play the role of the evolution parameter. By
investigation of properties of $\Theta$ in volume representation, we
identified it as difference operator of the fourth-order and by reformulating
its eigenvalue problem to a recurrence relation, we find out that its solution
cannot be evaluated numerically for given initial conditions in $v$. It means
that solution is unstable which confirmed the conclusion from [32]. Because of
that, we chose to examine the properties of the operator of the gravitational
part of constraint in a more convenient representation. Thus, as suggested in
[10], we switched to momentum $b$ representation in which $\Theta$ has been
found to act as a differential operator of the second order. At this point, we
inspected its deficiency subspace to determined its self-adjointness. What we
have found is that both of these subspaces are built only of zeroth vector
which means that the operator is essentially self-adjoint. By this, evolution
can be uniquely determined. Then we find physical Hilbert space constituted of
states satisfying condition (3.29) which has a form of standard Klein-Gordon
equation. As well we have found Dirac observables given by formula (3.31).
Following that, we attempted to probe the dynamic of that system by evaluating
the expectation values of the volume as a function of $\phi$ for semiclassical
states. We found out that this function is symmetric and built out of
hyperbolic functions. Moreover, our result is in agreement with results
obtained in [21]. It means that for $\phi\to\pm\infty$ classical behavior
predicted by General Relativity will be recovered. Departures from classical
theory, become at near-Planckian energy density regime, where for critical
energy density $\rho_{c}^{II}\approx 1,73\rho_{Pl}$ bounce will occur.

We can now compare obtained results of mLQC-II against other known
regularizations. We can point out the following observations:

  * •

In $\phi\to\pm\infty$ trajectories obtained through both effective dynamics
and quantum methods for all three models approximate well provided than by
General Relativity. It means that in the asymptotic future/past every known
regularization of the Loop Quantum Cosmology will recreate classical theory.
Effective trajectories has been gathered in Fig2 \- for LQC it is described by
hyperbolic cosine of $\phi$ (2.35), for mLQC-II it is again by hyperbolic
cosine but multiplied by $(\gamma^{2}\tanh^{2}(\phi)+1)$ (3.45) and for mLQC-
II it has been obtained numerically.

  * •

In all models, the singularity of the Big Bang of General Relativity has been
resolved. At near-Planckian energy densities regime, we encounter bounce that
occurs at a critical density which differs for each prescription and they
satisfy:

$\rho_{c}^{I}<\rho_{c}<\rho_{c}^{II}$

  * •

As was shown for mLQC-I modifications are not necessarily restricted to
Planckian density regions - rather than one bounce two occurs. Between them
emerging two DeSitter epochs separated by point were for a finite value of the
scalar field, the volume of the Universe becomes infinite.

  * •

A different approach in the regularization of the gravity part of Hamiltonian
constraint cause different forms and properties of operator $\Theta$. In
mainstream LQC it is a self-adjoint (thus evolution can be uniquely
determined) operator which acts on the state in volume representation as
difference operator and in momentum $b$ representation as a differential
operator. Both are of the second order. In mLQC-I differential, an operator is
again of the second-order but the order of the difference one is four. That
discrepancy is accounted for by admitting multiple self-adjoint extensions,
each generating independent evolution. For mLQC-II we have a similar situation
(fourth for $v$, second for $b$), but both deficiency subspaces are spaned
only by a zeroth vector so the operator is essentially self-adjoint. This
apparent paradox is resolved by the fact that only some subspace of solutions
to the eigenvalue problem spans physical Hilbert space.

### Acknowledgements

At first, I would like to express my gratitude to my supervisor prof. Tomasz
Pawłowski without whom this work couldn’t be done. Thank you for guiding me
and helping to solve an encountered problems. I also want to thank all my
lecturers and fellow students from Physics Department for encouraging me to
continue study.

##  5 Plots

Figure 1: Values of the eigenfunction $e_{\omega}$ to $\Theta$ -in logarytmic
scale, for $\omega=10$ Figure 2: Effective Dynamical Trajectories for
$p_{\phi}=5000$, Source: prof. Tomasz Pawłowski

##  Appendix A Constraints

Einstein equations (without cosmological constant) are obtained by variation
(with respect to the metric tensor) of Einstein-Hilbert action:

$S_{EH}=\frac{1}{2\kappa}\int_{\mathcal{M}}d^{4}xR\sqrt{-g}$ (A.1)

where: $R$ is 4-dimensional Ricci scalar, $\mathcal{M}$ represents Manifold of
spacetime and $g$ is determinant of metric tensor with signature (-,+,+,+).
$\mathcal{M}$ can be foliated in $\mathcal{M}\cong\Sigma\times\mathcal{R}$
thus the action (A.1) can be re-express by the Arnowitt-Deser-Misner action:

$S_{EH}=\frac{c^{4}}{2\kappa}\int_{\Sigma\times\mathcal{R}}dtd^{3}xN\sqrt{h}(R+K^{ij}K_{ij}+K^{2})$
(A.2)

where: $h$ is a determinant of metric on one 3-dimensional hypersurface
$\Sigma_{t}$, $N$ is the lapse function which expresses the proper time
elapsed between the event belonging in $\Sigma_{t}$ and $\Sigma_{t+dt}$ (with
shift vector denoted by $N^{i}$ they serve as transformation to obtain every
event on the spacetime) and $R$ is now 3-dimensional Ricci scalar [20].
$K^{ij}$ is extrinsic curvature that can be understood as the rate of change
of $h_{ij}$ with respect to the global time.
By obtaining Hamiltonian density by standard Legendre transformaition, (A.2)
takes form:

$S_{EH}=\frac{1}{2\kappa}\int_{\Sigma\times\mathcal{R}}dtd^{3}x(p^{ij}\dot{h}_{ij}-N\mathcal{H}_{g}-N^{i}\mathcal{H}_{i})$
(A.3)

where: $p^{ij}$ is conjugate momentum to $h_{ij}$. They Poisson Bracket
satisfy:

$\displaystyle\\{p^{ij}(x),h_{kl}(x^{\prime})\\}=\delta^{i}_{k}\delta^{j}_{l}\delta(x-x^{\prime})$
(A.4)

What is left is gravitational Hamiltonian constraint $\mathcal{H}_{g}$ and
diffeomorphism constraint $\mathcal{H}_{i}$ [20] that, alongside the addition
of Hamiltonian equation of motion, recreates physics.
When triads formalism is taken into consideration 3 metric takes form

$h_{ij}=e^{a}_{i}e^{b}_{j}\delta_{ab}$

and new variables can be introduce (a,b = 1, 2, 3 are space index and i,j = 1,
2, 3 are internal indices enumering the vectors). Playing the role of $h_{ij}$
is not triad itself but it densitize version:

$E^{a}_{i}=\sqrt{h}e^{a}_{i}$

and it canonically conjugate momentum is given by connection:

$A^{i}_{a}=\Gamma^{i}_{a}+\gamma K^{i}_{a}$ (A.5)

where $K^{i}_{a}$ is externistic curveture 1-form, $\Gamma^{i}_{a}$ encodes
information about the spin connection. Corresponding to A.4, Poisson Bracket
of these new variables is:

$\\{A^{i}_{a}(x),E^{b}_{j}(x^{\prime})\\}=\kappa\gamma\delta^{i}_{j}\delta^{b}_{a}\delta(x-x^{\prime})$
(A.6)

But changing $(h,p)$ to $(A,E)$ is not enough to obtain the same physics,
because the considered system obtains additional degrees of freedom. To fix
that, the addition of a new constraint is needed. Thanks to that, A.3 is re-
expressed as:

$S_{EH}=\frac{c^{4}}{2\kappa}\int_{\Sigma\times\mathcal{R}}dtd^{3}x(E^{a}_{i}\dot{A}_{a}^{i}-N\mathcal{H}_{g}-N^{a}\mathcal{H}_{a}-\mathcal{N}^{i}\mathcal{G}_{i})$
(A.7)

where $\mathcal{N}^{i}$ is Lagrange multiplier of the additional constraint
$G_{i}$. Such constraint referred to in literature [20] as Gaus Constraint.
Now constraints in connection variables takes explicit form:

$\displaystyle\mathcal{G}_{i}$
$\displaystyle=\frac{1}{2\kappa\gamma}\int_{\mathcal{V}}d^{3}x(\partial_{a}E^{a}_{i}+\epsilon_{ijk}A^{j}_{a}E^{a}_{k})$
(A.8.a) $\displaystyle\mathcal{H}_{a}$
$\displaystyle=\frac{1}{\kappa\gamma}\int_{\mathcal{V}}d^{3}xF^{i}_{ab}E^{b}_{i}$
(A.8.b) $\displaystyle\mathcal{H}_{g}$
$\displaystyle=\frac{1}{2\kappa}\int_{\mathcal{V}}d^{3}x\frac{E^{ai}E^{bj}}{\sqrt{det(h)}}(\epsilon_{ijk}F^{k}_{ab}-2(1+\gamma^{2})K^{j}_{[a}K^{i}_{b]})$
(A.8.c)

where:

$H^{E}=\frac{1}{2\kappa}\int_{\mathcal{V}}d^{3}x\epsilon_{ijk}\frac{E^{ai}E^{bj}}{\sqrt{det(h)}}F^{k}_{ab}$
(A.9)
$T=\frac{1}{2\kappa}\int_{\mathcal{V}}d^{3}x\frac{E^{ai}E^{bj}}{\sqrt{det(h)}}K^{j}_{[a}K^{i}_{b]}$
(A.10)

are Euclidean Hamiltonian constraint and Lorentzian Hamiltonian constraint
respective [9, 33]. Integration here is limited to an elementary cell
$\mathcal{V}$ , and $F^{k}_{ab}$ is curvature of connection $A^{i}_{a}$ define
as:

$F^{k}_{ab}=\partial_{a}A^{k}_{b}-\partial_{b}A^{k}_{a}+\epsilon_{kij}A^{i}_{a}A^{j}_{b}$

##  Appendix B Deficiency Subspaces

Here we recal Corollary from [26]. If we consider $T$ to be a symmetric
operator on a Hilbert space $\mathcal{H}$ then:

$\displaystyle\mathcal{K}_{+}=Ker(i-T^{\star})$
$\displaystyle\mathcal{K}_{-}=Ker(i+T^{\star})$

are deficiency subspaces of $T$. In addition lets define a pair of numbers:

$\displaystyle n_{+}(T)=dim(\mathcal{K}_{+})$ $\displaystyle
n_{-}(t)=dim(\mathcal{K}_{-})$

which are called the deficiency indices of $T$. If:

  * (i)

$n_{+}(T)=n_{-}(t)=0$, $T$ is self-adjoint.

  * (ii)

$n_{+}(T)=n_{-}(t)$, $T$ has self-adjoint extensions. There is a one-to-one
correspondence between self-adjoint extensions of $T$ and unitary maps from
$\mathcal{K}_{+}$ onto $\mathcal{K}_{-}$.

  * (iii)

$n_{+}(T)\neq n_{-}(t)$, $T$ has no nontrivial symmetric extensions.

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