††thanks: Corresponding author

# Alternative $k=-1$ loop quantum cosmology

Jinsong Yang<EMAIL_ADDRESS>School of Physics, Guizhou University, Guiyang
550025, China  Cong Zhang<EMAIL_ADDRESS>Institut für
Quantengravitation, Friedrich-Alexander-Universität Erlangen-Nürnberg,
Staudtstr. 7/B2, 91058 Erlangen, Germany  Faculty of Physics, University of
Warsaw, Pasteura 5, 02-093 Warsaw, Poland  Xiangdong Zhang
<EMAIL_ADDRESS>Department of Physics, South China University of
Technology, Guangzhou 510641, China

###### Abstract

An alternative quantization of the gravitational Hamiltonian constraint of the
$k=-1$ Friedmann-Robertson-Walker model is proposed by treating the Euclidean
term and the Lorentzian term independently, mimicking the treatment of full
loop quantum gravity. The resulting Hamiltonian constraint operator for the
$k=-1$ model with a massless scalar field is successfully constructed, and is
shown to have the corrected classical limit. Compared to the former
quantization schemes in the literature where only the Euclidean term is
quantized, the new quantum dynamics of the $k=-1$ model with a massless scalar
field indicates that the classical big-bang singularity is replaced by an
asymmetric quantum bounce.

##  I Introduction

How to quantize general relativity (GR) in a consistent manner is a great
challenge to theoretical physics. One of the promising candidates is the so-
called loop quantum gravity (LQG) which is a nonperturbative approach to
quantum GR Rovelli:2004tv ; Thiemann:2007pyv ; Ashtekar:2004eh ; Han:2005km .
In the past three decades, LQG has made remarkable progress, such as making
the natural predictions of the discretized geometries and providing the
microscopic interpretation of BH entropy Rovelli:1994ge ; Ashtekar:1996eg ;
Ashtekar:1997fb ; Yang:2016kia ; Thiemann:1996at ; Ma:2010fy ; Ashtekar:1997yu
; Song:2020arr . The nonperturbative quantization procedure of LQG has been
successfully applied to the metric $f(R)$ theories Zhang:2011vi ; Zhang:2011qq
, scalar-tensor theories Zhang:2011gn ; Zhang:2011vg , higher-dimensional
gravity Bodendorfer:2011nv , and so on Zhang:2020smo . Despite these
achievements, the dynamics of full LQG is still an unsolved issue. To gain a
certain level of understanding of the dynamics, the quantization ideas and
technologies developed in LQG have also be applied to its symmetry-reduced
models, such as the Friedmann-Robertson-Walker (FRW) models and the
spherically symmetric black hole models, leading to loop quantum cosmology
(LQC) and loop quantum black hole models Ashtekar:2003hd ; Ashtekar:2005qt .
The most successful feature of LQC is that it can resolve the classical big
bang singularity by a quantum bounce due to the quantum geometry effects. We
refer to Ashtekar:2003hd ; Bojowald:2005epg ; Ashtekar:2006wn ;
Ashtekar:2011ni  for more complete reviews on LQC.

In full LQG, the gravitational Hamiltonian constraint is a combination of the
so-called Euclidean term and the Lorentzian term. In the spatially flat, $k=0$
FRW model, the Lorentzian term and the Euclidean term are proportional to each
other. Thus one often combines these two terms into one term proportional to
the Euclidean term, and then quantizes the Euclidean term to obtain the well-
defined gravitational Hamiltonian constraint operator Ashtekar:2003hd ;
Ashtekar:2006wn . It turns out that in this quantization scheme the classical
big-bang singularity is replaced by a symmetric quantum bounce for the $k=0$
FRW model with a massless scalar field in the framework of LQC Ashtekar:2006wn
. Note that in full LQG, the Lorentzian term is quantized independently by
employing the Thiemann’s trick Thiemann:1996aw . Thus, to mimic the full LQG
quantization procedure in the $k=0$ model of LQC, the Euclidean term and the
Lorentzian term were treated independently Bojowald:2002gz ; Henriques:2006qb
; Yang:2009fp . This alternative quantization scheme leads to an asymmetric
quantum bounce, which relates the spatially flat FRW model with an asymptotic
de Sitter universe, and thus an effective cosmological constant and an
effective Newtonian constant can be obtained Assanioussi:2018hee ; Li:2018opr
; Zhang:2021zfp .

As in the $k=0$ model, the quantization technologies for the gravitational
Hamiltonian constraint developed in LQG have been extended to the $k=-1,+1$
models Vandersloot:2006ws ; Szulc:2007uk ; Szulc:2006ep ; Ashtekar:2006es ;
Mielczarek:2009kh ; Corichi:2011pg ; Corichi:2013usa ; Dupuy:2016upu .
Compared to the $k=0$ model where the spin connection vanishes and hence the
Ashtekar connection equals to the extrinsic curvature multiplied by the
Immirzi parameter, the Lorentzian term is not proportional to the total
Euclidean term, but is proportional to the part of the Euclidean term
involving the extrinsic curvature due to the nonvanishing spin connection for
both the $k=-1$ model and the $k=+1$ model. Hence in the literatures one often
absorbs the Lorentzian term into a part of the Euclidean term, and then
quantize the two parts of the Euclidean term, respectively. It turns out that,
as the $k=0$ model with the similar treatment, the resulting $k=-1$ LQC model
also predicts a vacuum repulsion in the high curvature regime that would lead
to a symmetric bounce Vandersloot:2006ws . Moreover, the $k=-1$ model of LQC
also possesses some new features that never appears in the $k=0$ model, for
example, due to a vacuum repulsion in the high curvature regime, the scale
factor has the minimum value as $a_{\rm min}=\gamma\sqrt{\Delta}$
Vandersloot:2006ws . It is natural to ask whether the treatment of the
Lorentzian term independently, mimicking the treatment in the full theory, can
be directly carried to the $k=-1$ model, and whether an asymmetric bounce can
still be held for the $k=-1$ model. This is the main motivation of the present
paper. In this paper, we consider an alternative quantization of the
gravitational Hamiltonian constraint in the $k=-1$ model by treating the
Lorentzian term independently.

This paper is organized as follows. The canonical formulation of the $k=-1$
model is briefly recalled in Sec. II. Then we propose an alternative
gravitational Hamiltonian constraint operator by treating the Lorentzian term
independently, and provide a new quantum dynamics for the $k=-1$ model in Sec.
III. The effective theory of the new quantum dynamics and its asymptotic
behavior are studied in Sec. IV. Summary is included in the last section.

##  II canonical formulation of the $k=-1$ model

According to the cosmological principle, the line elements of the homogenous
isotropic cosmological models take as

$\displaystyle{\rm d}s^{2}=-{\rm d}t^{2}+a^{2}(t)\left[\frac{1}{1-kr^{2}}{\rm
d}r^{2}+r^{2}\left({\rm d}\theta^{2}+\sin^{2}\theta{\rm
d}\phi^{2}\right)\right],$ (1)

where $a(t)$ is the scale factor, and $k=-1,0,1$ for the open, flat, and
closed FRW models, respectively.

In what follows, we present the canonical formulation of the $k=-1$ model
following Ref. Vandersloot:2006ws . For the spatially noncompact $k=0,-1$
models with topology homeomorphic to $\mathbb{R}^{3}$, one introduces an
“elemental cell” ${\cal V}$ on the homogeneous spatial manifold
$\mathbb{R}^{3}$ and restrict all integrals to this elemental cell. Then one
chooses a fiducial metric
${}^{o}\\!q_{ab}={}^{o}\\!\omega^{i}_{a}{}^{o}\\!\omega^{j}_{b}\delta_{ij}$ on
$\mathbb{R}^{3}$ with ${}^{o}\\!\omega^{i}_{a}$ being the left- and right-
invariant fiducial one-forms in the $k=0$ model, and only the left-invariant
fiducial one-forms in the $k=-1$ model. Here $a,b,\cdots$ denote the spatial
indices while $i,j,\cdots=1,2,3$. Denote by $V_{o}$ the volume of ${\cal V}$
measured by the fiducial metric ${}^{o}\\!q_{ab}$. The left-invariant one-
forms ${}^{o}\\!\omega^{i}_{a}$ satisfy the Maurer-Cartan equation

$\displaystyle{\rm d}\ {}^{o}\\!\omega^{i}+\frac{1}{2}{C^{i}}_{jk}\
{}^{o}\\!\omega^{j}\wedge{}^{o}\\!\omega^{k}=0,$ (2)

where for the $k=-1$ model the structure constants read

$\displaystyle{C^{i}}_{jk}=\delta^{i}_{j}\delta_{k1}-\delta^{i}_{k}\delta_{j1},$
(3)

while for the $k=0$ model they take zero. The corresponding left-invariant
vector fields ${}^{o}\\!e^{a}_{i}$ are dual to ${}^{o}\\!\omega^{i}_{a}$,
satisfying ${}^{o}\\!e^{a}_{i}{}^{o}\\!\omega^{j}_{a}=\delta^{j}_{i}$ and
${}^{o}\\!e^{a}_{i}{}^{o}\\!\omega^{i}_{b}=\delta^{a}_{b}$. The commutators
between the left-invariant vector fields read

$\displaystyle[{}^{o}\\!e_{i},{}^{o}\\!e_{j}]={C^{k}}_{ij}{}^{o}\\!e_{k}.$ (4)

Classically, the dynamical variables of LQC are obtained by symmetrically
reducing those of full LQG. In the full theory, the dynamical variables
consist of the $su(2)$-valued connection $A_{a}^{i}$ and the densitized triad
$\tilde{E}^{b}_{j}$ with the nontrivial Poisson bracket

$\displaystyle\\{A^{i}_{a}(x),\tilde{E}_{j}^{b}(y)\\}$
$\displaystyle=\kappa\gamma\delta^{b}_{a}\delta^{i}_{j}\delta(x,y),$ (5)

where $\kappa=8\pi G$ with $G$ being the Newtonian constant, and $\gamma$ is
the Immirzi parameter Barbero:1994ap ; Immirzi:1996dr . The connection
$A^{i}_{a}$ is related to the spin connection $\Gamma^{i}_{a}$ and the
extrinsic curvature $K^{i}_{a}$ by $A^{i}_{a}=\Gamma^{i}_{a}+\gamma
K^{i}_{a}$. It turns out that the symmetry-reduced extrinsic curvature
$K^{i}_{a}$ is diagonal in the basis of left-invariant one-forms for the
$k=0,-1$ models. While, unlike the $k=0$ model where $\Gamma^{i}_{a}$
vanishes, the symmetry-reduced spin connection $\Gamma^{i}_{a}$ in the $k=-1$
model takes the form Vandersloot:2006ws

$\displaystyle\Gamma^{i}_{a}=-\epsilon^{1ij}\ {}^{o}\\!\omega^{j}_{a},$ (6)

and thus it is nondiagonal. Hence the symmetry-reduced connection and
densitized triad for the $k=-1$ model read Vandersloot:2006ws

$\displaystyle A_{a}^{i}$ $\displaystyle=-\epsilon^{1ij}\
{}^{o}\\!\omega^{j}_{a}+cV_{o}^{-\frac{1}{3}}\ {}^{o}\\!\omega^{i}_{a}\equiv
A^{i}_{j}V_{o}^{-\frac{1}{3}}\ {}^{o}\\!\omega^{j}_{a},$ (7)
$\displaystyle\tilde{E}^{a}_{i}$
$\displaystyle=pV_{o}^{-\frac{2}{3}}\sqrt{\det({}^{o}\\!q)}\
{}^{o}\\!e^{a}_{i},$ (8)

where

$\displaystyle A^{i}_{j}=\begin{pmatrix}c&0&0\\\ 0&c&-V_{o}^{\frac{1}{3}}\\\
0&V_{o}^{\frac{1}{3}}&c\end{pmatrix},$ (9)

the variables $c$ and $p$ are only functions of $t$, and $\det({}^{o}\\!q)$
denotes the determinant of ${}^{o}\\!q_{ab}$. Hence the gravitational phase
space of the $k=-1$ model consists of conjugate pairs $(c,p)$. The nontrivial
Poisson bracket reads

$\displaystyle\\{c,p\\}$ $\displaystyle=\frac{\kappa}{3}\gamma.$ (10)

Note that the variables $c$ and $p$ are related to the scale factor $a$ by
$|p|=a^{2}V_{o}^{\frac{2}{3}}$ and $c=\gamma\dot{a}V_{o}^{\frac{1}{3}}$. The
physical volume $V$ of the elemental cell ${\cal V}$ measured by the spatial
(physical) metric $q_{ab}=|p|V_{o}^{-\frac{2}{3}}{}^{o}\\!q_{ab}$ is related
to $p$ via $V={|p|}^{3/2}$. In the improved scheme, it is convenient to choose
the following variables to simplify the dynamics Ding:2008tq

$\displaystyle b$ $\displaystyle:=\frac{\bar{\mu}c}{2},\qquad v:=\frac{{\rm
sgn}(p)|p|^{3/2}}{2\pi\gamma\ell^{2}_{\rm p}\sqrt{\Delta}},$ (11)

where $\ell_{\rm p}\equiv\sqrt{G\hbar}$ denotes the Planck length, ${\rm
sgn}(p)$ is the signature of $p$, $\Delta\equiv
4\sqrt{3}\,\pi\gamma\ell^{2}_{\rm p}$ is the minimum nonzero eigenvalue of the
area operator in full LQG Ashtekar:2008zu , and
$\bar{\mu}\equiv\sqrt{\Delta/|p|}$. The Poisson bracket between $b$ and $v$ is
given by

$\displaystyle\\{b,v\\}=\frac{1}{\hbar}.$ (12)

As in the $k=0$ model, the Gauss and diffeomorphism constraints of the
gravitational part are automatically satisfied for the symmetry-reduced
variables in Eqs. (7) and (8) in the $k=-1$ model, and thus the classical
dynamics is encoded in the Hamiltonian constraint. The gravitational
Hamiltonian constraint of the $k=-1$ model reads

$\displaystyle{\cal H}^{k=-1}_{\rm grav}$ $\displaystyle:=\int_{\cal V}{\rm
d}^{3}x\,\frac{\tilde{E}^{a}_{i}\tilde{E}^{b}_{j}}{2\kappa\sqrt{\textrm{det}(q)}}\left[{\epsilon^{ij}}_{k}{}^{\scriptscriptstyle(A)}\\!F^{k}_{ab}-2(1+\gamma^{2})K^{i}_{[a}K^{j}_{b]}\right]$
$\displaystyle=\int_{\cal V}{\rm
d}^{3}x\,\frac{\tilde{E}^{a}_{i}\tilde{E}^{b}_{j}}{2\kappa\sqrt{\textrm{det}(q)}}\left[{\epsilon^{ij}}_{k}{}^{\scriptscriptstyle(\gamma
K)}\\!F^{k}_{ab}-2(1+\gamma^{2})K^{i}_{[a}K^{j}_{b]}\right]+\int_{\cal V}{\rm
d}^{3}x\,\frac{\tilde{E}^{a}_{i}\tilde{E}^{b}_{j}}{2\kappa\sqrt{\textrm{det}(q)}}{\epsilon^{ij}}_{k}{}^{(\Gamma)}\\!F^{k}_{ab}$
$\displaystyle\equiv\tilde{\cal H}^{{\rm E},k=0}_{\rm
grav}-2(1+\gamma^{2})\tilde{\cal H}^{{\rm L},k=0}_{\rm grav}+{\cal H}_{\rm
grav}^{\Gamma,k=-1},$ (13)

where $\det(q)$ denotes the determinant of $q_{ab}$, and

$\displaystyle{}^{\scriptscriptstyle(x)}\\!F^{k}_{ab}:=2\partial_{[a}x^{k}_{b]}+{\epsilon^{k}}_{lm}x^{l}_{a}x^{m}_{b}.$
(14)

Hence, in the $k=-1$ model, the Euclidean term consists of $\tilde{\cal
H}^{{\rm E},k=0}_{\rm grav}$ and ${\cal H}_{\rm grav}^{\Gamma,k=-1}$, while
the Lorentzian term is $\tilde{\cal H}^{{\rm L},k=0}_{\rm grav}$. Seen from
the above formulation, the two terms $\tilde{\cal H}^{{\rm E},k=0}_{\rm grav}$
and $\tilde{\cal H}^{{\rm L},k=0}_{\rm grav}$ have the same formulations as
the Euclidean term ${\cal H}^{{\rm E},k=0}_{\rm grav}$ and the Lorentzian term
${\cal H}^{{\rm E},k=0}_{\rm grav}$ in the $k=0$ model, respectively. Thus the
gravitational Hamiltonian constraint of the $k=-1$ model differs from that of
the $k=0$ model by the third term ${\cal H}_{\rm grav}^{\Gamma,k=-1}$ due to
the nonvanishing $\Gamma^{i}_{a}$ in the $k=-1$ model. A straightforward
calculation shows that the three terms in Eq. (II) can be expressed by the
variables $(b,v)$ as

$\displaystyle\tilde{\cal H}_{\rm grav}^{{\rm E},k=0}$
$\displaystyle=\frac{3\gamma\hbar}{\sqrt{\Delta}}\,b^{2}|v|,$ (15)
$\displaystyle\tilde{\cal H}_{\rm grav}^{{\rm L},k=0}$
$\displaystyle=\frac{3\hbar}{2\gamma\sqrt{\Delta}}\,b^{2}|v|,$ (16)
$\displaystyle{\cal H}^{\Gamma,k=-1}_{\rm grav}$
$\displaystyle=\frac{3\left(\gamma\sqrt{\Delta}\,\hbar\right)^{\frac{1}{3}}V_{o}^{\frac{2}{3}}}{4\left(2\pi
G\right)^{\frac{2}{3}}}|v|^{\frac{1}{3}}.$ (17)

Hence the gravitational Hamiltonian constraint (II) of the $k=-1$ model
reduces to

$\displaystyle{\cal H}^{k=-1}_{\rm grav}$
$\displaystyle=-\frac{3\hbar|v|}{\gamma\sqrt{\Delta}}\left[b^{2}-V_{o}^{\frac{2}{3}}\left(\frac{\gamma^{2}\Delta}{16\pi
G\hbar|v|}\right)^{\frac{2}{3}}\right]$
$\displaystyle\equiv-\frac{3\hbar|v|}{\gamma\sqrt{\Delta}}g(b,v).$ (18)

At the classical level, we assume that the universe is filled by a massless
scalar field $\phi$. The Hamiltonian of the scalar field $\phi$ is given by

$\displaystyle{\cal
H}_{\phi}=\frac{p^{2}_{\phi}}{2V}=\frac{p_{\phi}^{2}}{4\pi\gamma\ell_{\rm
P}^{2}\sqrt{\Delta}\,|v|},$ (19)

where $p_{\phi}$ denotes the conjugate momentum of $\phi$. The Poisson bracket
between $\phi$ and $p_{\phi}$ is $\\{\phi,p_{\phi}\\}=1$. Hence the total
Hamiltonian constraint of gravity coupled to a massless scalar field reads

$\displaystyle{\cal H}_{\rm tot}^{k=-1}$ $\displaystyle={\cal H}^{k=-1}_{\rm
grav}+{\cal H}_{\phi}$
$\displaystyle=-\frac{3\hbar|v|}{\gamma\sqrt{\Delta}}g(b,v)+\frac{p_{\phi}^{2}}{4\pi\gamma\ell_{\rm
P}^{2}\sqrt{\Delta}\,|v|}.$ (20)

By the total Hamiltonian constraint equation

$\displaystyle{\cal H}_{\rm tot}^{k=-1}=0,$ (21)

the classical Friedmann equation can be obtained as

$\displaystyle H^{2}_{k=-1}$
$\displaystyle=\left(\frac{\dot{v}}{3v}\right)^{2}=\left(\frac{\\{v,{\cal
H}_{\rm tot}^{k=-1}\\}}{3v}\right)^{2}$ $\displaystyle=\frac{8\pi
G}{3}\rho_{\phi}+\frac{V_{o}^{2/3}}{V^{2/3}}$ $\displaystyle=\frac{8\pi
G}{3}\rho_{\phi}+\frac{1}{a^{2}},$ (22)

where $\cdot{}$ denotes a derivative with respect to the time determined by
${\cal H}_{\rm tot}^{k=-1}$, and $\rho_{\phi}=\frac{p_{\phi}^{2}}{2V^{2}}$ is
the energy density of the scalar field $\phi$.

##  III Loop quantization of the $k=-1$ model

To pass the classical theory of $k=-1$ model to its quantum theory, one needs
to construct the kinematical Hilbert space. In the $k=0$ model, the vanishing
$\Gamma^{i}_{a}$ enables us to identify $A^{i}_{a}$ with $\gamma K^{i}_{a}$,
leading to the identification of the holonomies of the connection and those of
the extrinsic curvature (mutiplied by $\gamma$). The resulting holonomies of
the connection $A^{i}_{a}$, equal to $\gamma K^{i}_{a}$ in the $k=0$ model,
along edges generated by the left- and right-invariant vector fields
${}^{o}\\!e^{a}_{i}$ with physical length $\lambda V^{1/3}$ take the form
$h^{(\lambda)}_{i}=\cos\left(\frac{\lambda
c}{2}\right)\mathbb{I}+2\tau_{i}\sin\left(\frac{\lambda c}{2}\right)$, where
$\tau_{i}:=-\frac{{\rm i}}{2}\sigma_{i}$ with $\sigma_{i}$ being the Pauli
matrices. Hence the related algebra is that of the almost periodic functions,
and thus the kinematical Hilbert space for the gravitational part can be
defined as ${\mathscr{H}}^{{\rm gr},k=0}_{\rm
kin}=L^{2}(\mathbb{R}_{\mathrm{Bohr}},{\rm d}\mu_{\mathrm{Bohr}})$, where
${\mathbb{R}}_{\rm Bohr}$ and ${\rm d}\mu_{\rm Bohr}$ are respectively the
Bohr compactification of the real line ${\mathbb{R}}$ and the Haar measure on
it Ashtekar:2003hd . However, in the $k=-1$ case, the spin connection
$\Gamma^{i}_{a}$ takes the nonvanishing expression (6), resulting in a
difference between the holonomy of the connection and the one of the extrinsic
curvature. Moreover, due to the nondiagonal form (7) of the connection, the
holonomies of the connection take complicated forms in the $k=-1$ model,
leading to the algebra generated is no longer that of the almost periodic
function Vandersloot:2006ws . Instead, one often considers the holonomies of
the extrinsic curvature $\gamma K^{i}_{a}$ in the $k=-1$ model, which take the
same forms as those in the $k=0$ model. More precisely, considering an edge
$e_{i}$ starting from the basepoint of the elemental cell ${\cal V}$, with
tangent vector parallel to the vector ${}^{o}\\!e^{a}_{i}$ and taking length
$\lambda$, following Refs. Vandersloot:2006ws ; Szulc:2007uk , we define the
“holonomy” of $\gamma K^{i}_{a}=cV_{o}^{-\frac{1}{3}}\
{}^{o}\\!\omega^{i}_{a}$ as

$\displaystyle h^{(\lambda)}_{i}$ $\displaystyle:={\cal P}\exp\int_{e_{i}}{\rm
d}t\,\gamma K^{j}_{a}\tau_{j}{}^{o}\\!e^{a}_{i}$ $\displaystyle=e^{\lambda
c\tau_{i}}$ $\displaystyle=\cos\left(\frac{\lambda
c}{2}\right)\mathbb{I}+2\sin\left(\frac{\lambda c}{2}\right)\tau_{i}.$ (23)

Here ${\cal P}$ denotes the path ordering which orders the smallest path
parameter to the left Thiemann:2007pyv , and it takes the trivial action in
our model as in the works Thiemann:2007pyv ; Ashtekar:2003hd ;
Vandersloot:2006ws ; Szulc:2007uk . Clearly, these holonomies (III) generate
the algebra of almost periodic functions, and thus result in the kinematical
Hilbert space for the $k=-1$ model being ${\mathscr{H}}^{\rm gr}_{\rm
kin}\equiv{\mathscr{H}}^{{\rm gr},k=-1}_{\rm kin}={\mathscr{H}}^{{\rm
gr},k=0}_{\rm kin}$ Vandersloot:2006ws . As in the $k=0$ model, we will employ
the $\bar{\mu}$-scheme to define the Hamiltonian operator. This requires us to
consider the holonomis along the edges taking physical length $\sqrt{\Delta}$
Thiemann:2007pyv ; Vandersloot:2006ws ; Szulc:2007uk , which are given by

$\displaystyle h^{(\bar{\mu})}_{i}$ $\displaystyle:={\cal
P}\exp\left(\int_{0}^{\bar{\mu}V_{o}^{1/3}}{\rm d}t\,\gamma
K^{j}_{a}\tau_{j}{}^{o}\\!e^{a}_{i}\right)$
$\displaystyle=e^{\bar{\mu}c\tau_{i}}$
$\displaystyle=\cos\left(\frac{\bar{\mu}c}{2}\right)\,\mathbb{I}+2\sin\left(\frac{\bar{\mu}c}{2}\right)\tau_{i}$
$\displaystyle=\cos(b)\,\mathbb{I}+2\sin(b)\,\tau_{i}\,,$ (24)

and their inverse take the forms:

$\displaystyle{h_{i}^{(\bar{\mu})}}^{-1}=\cos(b)\,\mathbb{I}-2\sin(b)\,\tau_{i}.$
(25)

In the $v$-representation for both the $k=0$ model and the $k=-1$ model, the
two elementary operators, $\widehat{e^{{\rm i}b}}$ and $\hat{v}$, act on the
basis $|v\rangle$ of ${\mathscr{H}}^{\rm gr}_{\rm kin}$ as

$\displaystyle\widehat{e^{{\rm i}b}}\,|v\rangle$
$\displaystyle=|v+1\rangle\,,\qquad\hat{v}\,|v\rangle=v\,|v\rangle\,.$ (26)

Thus one can easily write down the action of the operators

$\displaystyle\widehat{h^{(\bar{\mu})}_{i}}=\widehat{\cos(b)}\,\mathbb{I}+2\,\widehat{\sin(b)}\,\tau_{i}$
(27)

corresponding to the holonomies $h^{(\bar{\mu})}_{i}$ of the extrinsic
curvature $\gamma K^{i}_{a}$ on $|v\rangle$ in terms of $\widehat{e^{{\rm
i}b}}$. For the scalar field, it is convenient to choose the Schrödinger
representation Ashtekar:2011ni . Thus the kinematical Hilbert space for the
scalar field part can be chosen as ${\mathscr{H}}_{{\rm kin}}^{\rm
sc}:=L^{2}({\mathbb{R}},d\phi)$. Hence the total kinematical Hilbert space of
the $k=-1$ model with a scalar field is ${\mathscr{H}}_{\rm kin}^{\rm
tot}={\mathscr{H}}^{\rm gr}_{\rm kin}\otimes{\mathscr{H}}^{\rm sc}_{\rm kin}$.

We now consider an alternative regularization of the gravitational Hamiltonian
constraint of the $k=-1$ model in Eq. (II), such that it is closer to that in
the $k=0$ model as well as to that in full LQG. As mentioned previously, the
two terms $\tilde{\cal H}^{{\rm E},k=0}_{\rm grav}$ and $\tilde{\cal H}^{{\rm
L},k=0}_{\rm grav}$ in Eq. (II) have the same forms as the Euclidean and
Lorentzian terms in the $k=0$ model, respectively. Hence it is natural to
expect that the two terms in the $k=-1$ model can be regularized as the
corresponding forms in the $k=0$ model. To realize explicitly this idea, some
subtle issues should be clarified. Firstly, we consider the first term

$\displaystyle\tilde{\cal H}^{{\rm E},k=0}_{\rm grav}$
$\displaystyle=\int_{\cal V}{\rm
d}^{3}x\,\frac{\tilde{E}^{a}_{i}\tilde{E}^{b}_{j}}{2\kappa\sqrt{\textrm{det}(q)}}{\epsilon^{ij}}_{k}{}^{\scriptscriptstyle(\gamma
K)}\\!F^{k}_{ab}$ $\displaystyle=\int_{\cal V}{\rm
d}^{3}x\,\frac{\tilde{E}^{a}_{i}\tilde{E}^{b}_{j}}{2\kappa\sqrt{\textrm{det}(q)}}{\epsilon^{ij}}_{k}\left[2\partial_{[a}\gamma
K^{k}_{b]}\right.$ $\displaystyle\hskip
113.81102pt\left.+{\epsilon^{k}}_{lm}\gamma K^{l}_{a}\gamma K^{m}_{b}\right]$
$\displaystyle=\int_{\cal V}{\rm
d}^{3}x\,\frac{\tilde{E}^{a}_{i}\tilde{E}^{b}_{j}{\epsilon^{ij}}_{k}}{2\kappa\sqrt{\textrm{det}(q)}}{\epsilon^{k}}_{lm}\gamma
K^{l}_{a}\gamma K^{m}_{b},$ (28)

where in the third step we used the fact that the first term in the integral
of the second line vanishes due to

$\displaystyle{}^{o}\\!e^{a}_{i}{}^{o}\\!e^{b}_{j}{\epsilon^{ij}}_{k}2\partial_{[a}\gamma
K^{k}_{b]}$
$\displaystyle=cV_{o}^{-\frac{1}{3}}{}^{o}\\!e^{a}_{i}{}^{o}\\!e^{b}_{j}{\epsilon^{ij}}_{k}2\partial_{[a}{}^{o}\\!\omega^{k}_{b]}$
$\displaystyle=-cV_{o}^{-\frac{1}{3}}{}^{o}\\!e^{a}_{i}{}^{o}\\!e^{b}_{j}{\epsilon^{ij}}_{k}{C^{k}}_{lm}{}^{o}\\!\omega^{l}_{a}{}^{o}\\!\omega^{m}_{b}$
$\displaystyle=-cV_{o}^{-\frac{1}{3}}{\epsilon^{ij}}_{k}{C^{k}}_{ij}$
$\displaystyle=0.$ (29)

To regularize $\tilde{\cal H}^{{\rm E},k=0}_{\rm grav}$ in Eq. (III), one
needs to use the Thiemann’s trick

$\displaystyle\frac{\tilde{E}^{a}_{i}\tilde{E}^{b}_{j}}{\sqrt{\textrm{det}(q)}}{\epsilon^{ij}}_{k}$
$\displaystyle=\frac{2}{\kappa\gamma}\tilde{\epsilon}^{abc}\\{A^{k}_{c},V\\}=\frac{2}{\kappa\gamma}\tilde{\epsilon}^{abc}\\{\gamma
K^{k}_{c},V\\},$ (30)

where $\tilde{\epsilon}^{abc}$ is the Levi-Civita density, and then to express
the curvature ${}^{\scriptscriptstyle(\gamma K)}\\!F^{k}_{ab}$ of $\gamma
K^{i}_{a}$ in terms of holonomies of $\gamma K^{i}_{a}$. In the $k=0$ case,
since the left- and right-invariant vector fields ${}^{o}\\!e^{a}_{i}$ commute
to each other, the integral curvatures of ${}^{o}\\!e^{a}_{i}$ and
${}^{o}\\!e^{b}_{j}$ can form closed loops $\Box_{ij}$, around which the
curvature ${}^{\scriptscriptstyle(\gamma K)}\\!F^{k}_{ab}$ can be recast as
the holonomies $h^{(\bar{\mu})}_{\Box_{ij}}$ of $\gamma K^{i}_{a}$. Compared
to the $k=0$ case, due to the noncommutativity of the left-invariant vector
fields ${}^{o}\\!e^{a}_{i}$ in the $k=-1$ case, the integral curves of
${}^{o}\\!e^{a}_{i}$ and ${}^{o}\\!e^{a}_{j}$ can not provide closed loops. In
Vandersloot:2006ws , holonomies of $\gamma K^{i}_{a}$ based on the open curves
generated by ${}^{o}\\!e^{a}_{i}$ and ${}^{o}\\!e^{a}_{j}$ were adopted to
regularize the curvature ${}^{\scriptscriptstyle(\gamma K)}\\!F^{k}_{ab}$. In
Szulc:2007uk , the author proposed closed loops $\Box_{ij}$ generated by the
integral curves of the left-invariant vector fields ${}^{o}\\!e^{a}_{i}$ and
the right-invariant vector fields ${}^{o}\\!\eta^{b}_{j}$ commuting with
${}^{o}\\!e^{a}_{i}$. In the present paper, we will consider open holonnomies
to represent extrinsic curvature following Refs. Yang:2009fp ; Ashtekar:2009um
; Corichi:2011pg ; Dupuy:2016upu . Inputting Eq. (30) into Eq. (III), one
obtains

$\displaystyle\tilde{\cal H}^{{\rm E},k=0}_{\rm grav}$
$\displaystyle=\frac{1}{\kappa^{2}\gamma}\int_{\cal V}{\rm
d}^{3}x\,\tilde{\epsilon}^{abc}{\epsilon^{k}}_{lm}\gamma K^{l}_{a}\gamma
K^{m}_{b}\\{\gamma K^{k}_{c},V\\}$
$\displaystyle=-\frac{4}{\kappa^{2}\gamma}\int_{\cal V}{\rm
d}^{3}x\,\tilde{\epsilon}^{abc}{\rm Tr}\left(\gamma K_{a}\gamma K_{b}\\{\gamma
K_{c},V\\}\right),$ (31)

where the identity ${\rm
Tr}(\tau_{i}\tau_{j}\tau_{k})=-\frac{1}{4}\epsilon_{ijk}$ was used. In
cosmology the known identities Yang:2009fp ; Ashtekar:2009um ; Corichi:2011pg
; Dupuy:2016upu  take the forms:

$\displaystyle\gamma K_{a}$
$\displaystyle=\frac{h^{(2\bar{\mu})}_{i}-{h_{i}^{(2\bar{\mu})}}^{-1}}{4\bar{\mu}V_{o}^{1/3}}{}^{o}\\!\omega^{i}_{a},$
(32) $\displaystyle\\{\gamma K_{c},V\\}$
$\displaystyle=-\frac{1}{\bar{\mu}V_{o}^{1/3}}\sum_{k}h^{(\bar{\mu})}_{k}\left\\{{h_{k}^{(\bar{\mu})}}^{-1},V\right\\}{}^{o}\\!\omega^{k}_{c},$
(33)

where $h_{i}^{(\bar{\mu})}$ (or $h^{(2\bar{\mu})}_{i}$) is defined by (III).
It should be noticed that Eq. (32), which is precisely valid in the limit
$\bar{\mu}\rightarrow 0$, should be understood as a regularized expression in
the ${\bar{\mu}}$-scheme with $\bar{\mu}=\sqrt{\Delta/|p|}$. Substituting Eqs.
(32) and (33) into Eq. (III) and assuming for simplicity that the holonomies
of $k=-1$ can be approximated with the holonomies of $k=0$, we arrive at

$\displaystyle\tilde{\cal H}^{{\rm E},k=0,{\rm reg}}_{\rm grav}$
$\displaystyle=\frac{{\rm
sgn}(p)}{4\kappa^{2}\gamma\bar{\mu}^{3}}\sum_{i,j,k}\epsilon^{ijk}{\rm
Tr}\left[\left(h^{(2\bar{\mu})}_{i}-{h_{i}^{(2\bar{\mu})}}^{-1}\right)\left(h^{(2\bar{\mu})}_{j}-{h_{j}^{(2\bar{\mu})}}^{-1}\right)h^{(\bar{\mu})}_{k}\left\\{{h_{k}^{(\bar{\mu})}}^{-1},V\right\\}\right]$
$\displaystyle=\frac{\hbar^{2}\gamma}{4\sqrt{\Delta}}\sin(2b)\left[v\sum_{k}{\rm
Tr}\left(\tau_{k}h^{(\bar{\mu})}_{k}\left\\{{h_{k}^{(\bar{\mu})}}^{-1},|v|\right\\}\right)\right]\sin(2b),$
(34)

where the identity
$\tau_{i}\tau_{j}=\frac{1}{2}\epsilon_{ijm}\tau^{m}-\frac{1}{4}\delta_{ij}$
was used. The resulting regularized expression $\tilde{\cal H}^{{\rm
E},k=0,{\rm reg}}_{\rm grav}$ in Eq. (III) is the same as the regularized
Euclidean Hamiltonian constraint ${\cal H}^{{\rm E},k=0,{\rm reg}}_{\rm grav}$
in the $k=0$ model Ashtekar:2006wn . We now consider the second term
$\tilde{\cal H}^{{\rm L},k=0}_{\rm grav}$ in Eq. (II). Classically, the term
$\tilde{\cal H}^{{\rm L},k=0}_{\rm grav}$ is proportional to the term
$\tilde{\cal H}^{{\rm E},k=0}_{\rm grav}$, and hence the term $\tilde{\cal
H}^{{\rm L},k=0}_{\rm grav}$ does not need to be quantized independently. This
approach to quantization of $\tilde{\cal H}^{{\rm L},k=0}_{\rm grav}$ in the
$k=-1$ has been adopted in Vandersloot:2006ws ; Szulc:2007uk , similar to the
$k=0$ case in Ashtekar:2006wn . Alternatively, the Lorentzian term in the
$k=0$ case can be regularized independently in Yang:2009fp , mimicking the
treatment of full LQG. It is natural to ask whether the treatment for the
Lorentzian term in the $k=0$ case can be directly carried to that for
$\tilde{\cal H}^{{\rm L},k=0}_{\rm grav}$ in the $k=-1$ model, and the
resulting operator is the same as that in the $k=0$ model. The answer is in
the affirmative. To this end, let us recall the key identities for
regularizing the Lorentzian term of the gravitational Hamiltonian constraint
in the full theory, study their symmetry-reduced forms in the $k=-1$ model,
and then compare them with those in the $k=0$ model. The first classical
identity reads

$\displaystyle K_{a}^{i}\tau_{i}$
$\displaystyle=\frac{1}{\kappa\gamma}\\{A^{i}_{a}\tau_{i},K\\}=\frac{1}{\kappa\gamma}\\{\Gamma^{i}_{a}\tau_{i}+\gamma
K^{i}_{a}\tau_{i},K\\}$ $\displaystyle=\frac{1}{\kappa\gamma}\\{\gamma
K^{i}_{a}\tau_{i},K\\}$
$\displaystyle=-\frac{2}{3\kappa\gamma}\frac{1}{\bar{\mu}V_{o}^{1/3}}\sum_{i}h^{(\bar{\mu})}_{i}\left\\{{h_{i}^{(\bar{\mu})}}^{-1},K\right\\}{}^{o}\\!\omega^{i}_{a},$
(35)

where $K:=\int{\rm d}^{3}xK_{a}^{i}\tilde{E}^{a}_{i}$, the former steps hold
for the full theory and thus hold for its symmetry-reduced models, while the
third step holds due to the fact that the spin connection $\Gamma^{i}_{a}$ in
Eq. (6) is proportional to ${}^{o}\\!\omega^{j}_{a}$ up to a constant for the
$k=-1$ case, and the vanishing connection for the $k=0$ model. In the last
step, we have used the relation Yang:2009fp

$\displaystyle\\{c\tau_{i},K\\}$
$\displaystyle=-\frac{2}{3\bar{\mu}}h^{(\bar{\mu})}_{i}\left\\{{h_{i}^{(\bar{\mu})}}^{-1},K\right\\}.$
(36)

Here, it is worth noting that the above equation is satisfied only in the
$\bar{\mu}$ scheme where $\bar{\mu}$ is a function of $p$, rather that a
certain constant in the $\mu_{o}$ scheme. In the $\bar{\mu}$ scheme,
$\bar{\mu}$ depending on $p$ does not commute with $K$, leading to a factor
$2/3$ on the right-hand side of Eq. (36). The second identity is

$\displaystyle K$ $\displaystyle=\frac{1}{\gamma^{2}}\\{{}^{(A)}{\cal H}^{\rm
E},V\\}=\frac{1}{\gamma^{2}}\\{{}^{(\Gamma)}{\cal H}^{\rm E}+{}^{(\gamma
K)}{\cal H}^{\rm E},V\\}$ $\displaystyle=\frac{1}{\gamma^{2}}\\{{}^{(\gamma
K)}{\cal H}^{\rm E},V\\}=\frac{1}{\gamma^{2}}\\{\tilde{\cal H}^{{\rm
E},k=0}_{\rm grav},V\\}$ $\displaystyle=\frac{1}{\gamma^{2}}\\{{\cal H}^{{\rm
E},k=0}_{\rm grav},V\\},$ (37)

where ${}^{(x)}{\cal H}^{\rm E}:=\int{\rm
d}^{3}x\,\frac{\tilde{E}^{a}_{i}\tilde{E}^{b}_{j}}{2\kappa\sqrt{\textrm{det}(q)}}{\epsilon^{ij}}_{k}{}^{\scriptscriptstyle(x)}\\!F^{k}_{ab}$.
Hence, to regularize $K$, one just replaces $\tilde{\cal H}^{{\rm E},k=0}_{\rm
grav}$ by its regularized version $\tilde{\cal H}^{{\rm E},k=0,{\rm reg}}_{\rm
grav}$ in Eq. (III). It should also be noted here that both $\bar{\mu}$ and
$V$ depend only on $p$. As a result, the Poisson bracket between $\tilde{\cal
H}^{{\rm E},k=0,{\rm reg}}_{\rm grav}$ and $V$ has the same form for both the
$\bar{\mu}$ and $\mu_{o}$ schemes. Thus the above two classical identities
which play key roles in the regularization of the Lorentzian term hold in the
$k=-1$ model, and take the same forms as those in the $k=0$ model. Therefore,
to regularize $\tilde{\cal H}^{{\rm L},k=0}_{\rm grav}$ independently in the
$k=-1$ model, one can follow directly the treatment of the Lorentzian term
${\cal H}^{{\rm L},k=0}_{\rm grav}$ in the $k=0$ model, mimicking the
treatment of the full theory. To this end, we first re-express $\tilde{\cal
H}^{{\rm L},k=0}_{\rm grav}$ in the form

$\displaystyle\tilde{\cal H}^{{\rm L},k=0}_{\rm grav}$
$\displaystyle=\int_{\cal V}{\rm
d}^{3}x\,\frac{\tilde{E}^{a}_{i}\tilde{E}^{b}_{j}}{2\kappa\sqrt{\textrm{det}(q)}}K^{i}_{[a}K^{j}_{b]}$
$\displaystyle=\int_{\cal V}{\rm
d}^{3}x\,\frac{\tilde{E}^{a}_{i}\tilde{E}^{b}_{j}}{2\kappa\sqrt{\textrm{det}(q)}}{\epsilon^{ij}}_{k}\frac{1}{2}{\epsilon^{k}}_{lm}K^{l}_{a}K^{m}_{b}$
$\displaystyle=-\frac{2}{\kappa^{2}\gamma}\int_{\cal V}{\rm
d}^{3}x\,\tilde{\epsilon}^{abc}{\rm Tr}\left(K_{a}K_{b}\\{\gamma
K_{c},V\\}\right).$ (38)

Combining Eq. (III) with Eqs. (III) and (33) and replacing $\tilde{\cal
H}^{{\rm E},k=0}_{\rm grav}$ by its regularized version $\tilde{\cal H}^{{\rm
E},k=0,{\rm reg}}_{\rm grav}$, one obtains the regularized expression of
$\tilde{\cal H}^{{\rm L},k=0}_{\rm grav}$ as Yang:2009fp

$\displaystyle\tilde{\cal H}^{{\rm L},k=0,{\rm reg}}_{\rm grav}$
$\displaystyle=\frac{8\,{\rm
sgn}(p)}{9\kappa^{4}\gamma^{7}\bar{\mu}^{3}}\sum_{i,j,k}\epsilon^{ijk}{\rm
Tr}\left(h^{(\bar{\mu})}_{i}\left\\{{h_{i}^{(\bar{\mu})}}^{-1},\left\\{\tilde{\cal
H}^{{\rm E},k=0,{\rm reg}}_{\rm
grav},V\right\\}\right\\}h^{(\bar{\mu})}_{j}\left\\{{h_{j}^{(\bar{\mu})}}^{-1},\left\\{\tilde{\cal
H}^{{\rm E},k=0,{\rm reg}}_{\rm
grav},V\right\\}\right\\}h^{(\bar{\mu})}_{k}\left\\{{h_{k}^{(\bar{\mu})}}^{-1},V\right\\}\right)$
$\displaystyle=\frac{8\,{\rm
sgn}(p)}{9\kappa^{4}\gamma^{7}\bar{\mu}^{3}}\sum_{i,j,k}\epsilon^{ijk}{\rm
Tr}\left(h^{(\bar{\mu})}_{j}\left\\{{h_{j}^{(\bar{\mu})}}^{-1},\left\\{\tilde{\cal
H}^{{\rm E},k=0,{\rm reg}}_{\rm
grav},V\right\\}\right\\}h^{(\bar{\mu})}_{k}\left\\{{h_{k}^{(\bar{\mu})}}^{-1},V\right\\}h^{(\bar{\mu})}_{i}\left\\{{h_{i}^{(\bar{\mu})}}^{-1},\left\\{\tilde{\cal
H}^{{\rm E},k=0,{\rm reg}}_{\rm grav},V\right\\}\right\\}\right)$
$\displaystyle=\frac{\hbar^{4}\sqrt{\Delta}}{288\gamma^{3}}\sum_{i,j,k}\epsilon^{ijk}{\rm
Tr}\left(h^{(\bar{\mu})}_{i}\left\\{{h_{i}^{(\bar{\mu})}}^{-1},\left\\{\tilde{\cal
H}^{{\rm E},k=0,{\rm reg}}_{\rm
grav},|v|\right\\}\right\\}v\,h^{(\bar{\mu})}_{j}\left\\{{h_{j}^{(\bar{\mu})}}^{-1},|v|\right\\}h^{(\bar{\mu})}_{k}\left\\{{h_{k}^{(\bar{\mu})}}^{-1},\left\\{\tilde{\cal
H}^{{\rm E},k=0,{\rm reg}}_{\rm grav},|v|\right\\}\right\\}\right).$ (39)

Similarly, the last term in Eq. (II) can be regularized as Vandersloot:2006ws
; Szulc:2007uk

$\displaystyle{\cal H}^{\Gamma,k=-1,{\rm reg}}_{\rm grav}$
$\displaystyle=\frac{{\rm sgn}(p)V_{o}^{\frac{2}{3}}}{2\kappa\pi
G\gamma\bar{\mu}}\sum_{k}{\rm
Tr}\left(\tau_{k}h^{(\bar{\mu})}_{k}\left\\{{h_{k}^{(\bar{\mu})}}^{-1},V\right\\}\right)$
$\displaystyle=\frac{\hbar(\hbar\gamma\sqrt{\Delta})^{\frac{1}{3}}V_{o}^{\frac{2}{3}}}{4(2\pi
G)^{\frac{2}{3}}}{\rm sgn}(v)|v|^{\frac{1}{3}}\sum_{k}{\rm
Tr}\left(\tau_{k}h^{(\bar{\mu})}_{k}\left\\{{h_{k}^{(\bar{\mu})}}^{-1},|v|\right\\}\right).$
(40)

Up to now, the three parts of the gravitational Hamiltonian constraint in Eq.
(II) have been regularized as the expressions in Eqs. (III), (III) and (III),
which can be directly promoted to quantum operators by replacing the functions
by the corresponding operators and replacing $\\{\cdot,\cdot\\}$ by
$[\cdot,\cdot]/({\rm i}\hbar)$. Then $\tilde{\cal H}^{{\rm E},k=0,{\rm
reg}}_{\rm grav}$ in (III) can be quantized as

$\displaystyle\hat{\cal H}^{{\rm E},k=0}_{\rm grav}$
$\displaystyle=-\frac{{\rm
i}\hbar\gamma}{4\sqrt{\Delta}}\widehat{\sin(2b)}\left(\hat{v}\sum_{k}{\rm
Tr}\left(\tau_{k}\widehat{h^{(\bar{\mu})}_{k}}\left[\widehat{{h_{k}^{(\bar{\mu})}}^{-1}},|\hat{v}|\right]\right)\right)\widehat{\sin(2b)}$
$\displaystyle=\frac{{\rm
i}\hbar\gamma}{2\sqrt{\Delta}}\widehat{\sin(2b)}\left(\hat{v}\,\hat{O}_{|\hat{v}|}\right)\widehat{\sin(2b)}\sum_{k}{\rm
Tr}(\tau_{k}\tau_{k})$ $\displaystyle=-\frac{{\rm
i}3\hbar\gamma}{4\sqrt{\Delta}}\,\widehat{\sin(2b)}\left(\hat{v}\,\hat{O}_{|\hat{v}|}\right)\widehat{\sin(2b)},$
(41)

where in the second step we have used

$\displaystyle\widehat{h^{(\bar{\mu})}_{k}}\left[\widehat{{h_{k}^{(\bar{\mu})}}^{-1}},\hat{B}\right]$
$\displaystyle=\hat{B}\,\mathbb{I}-\widehat{h^{(\bar{\mu})}_{k}}\hat{B}\,\widehat{{h_{k}^{(\bar{\mu})}}^{-1}}$
$\displaystyle=\hat{B}\,\mathbb{I}-\left(\widehat{\cos(b)}\,\mathbb{I}+2\,\widehat{\sin(b)}\,\tau_{k}\right)\hat{B}\left(\widehat{\cos(b)}\,\mathbb{I}-2\,\widehat{\sin(b)}\,\tau_{k}\right)$
$\displaystyle=\hat{B}\,\mathbb{I}-\left[\widehat{\cos(b)}\hat{B}\,\widehat{\cos(b)}\,\mathbb{I}-4\,\widehat{\sin(b)}\hat{B}\,\widehat{\sin(b)}\,\tau_{k}\tau_{k}+2\left(\widehat{\sin(b)}\hat{B}\,\widehat{\cos(b)}-\widehat{\cos(b)}\hat{B}\,\widehat{\sin(b)}\right)\tau_{k}\right]$
$\displaystyle=\left(\hat{B}-\widehat{\sin(b)}\hat{B}\,\widehat{\sin(b)}-\widehat{\cos(b)}\hat{B}\,\widehat{\cos(b)}\right)\mathbb{I}-2\,\hat{O}_{\hat{B}}\,\tau_{k},$
(42)

here in the fourth step the identity
$\tau_{k}\tau_{k}=-\frac{1}{4}\,\mathbb{I}$ for $k=1,2,3$ was used, the
operator $\hat{O}_{\hat{B}}$, depending on the operator $\hat{B}$, is defined
by

$\displaystyle\hat{O}_{\hat{B}}$
$\displaystyle:=\widehat{\sin(b)}\hat{B}\,\widehat{\cos(b)}-\widehat{\cos(b)}\hat{B}\,\widehat{\sin(b)},$
(43)

and ${\rm Tr}(\tau_{k})=0$. In the last step in Eq. (III) we have used
$\sum_{k}{\rm Tr}(\tau_{k}\tau_{k})=-\frac{3}{2}$. Similarly, the regularized
expression (III) can be quantized as

$\displaystyle\hat{\cal H}^{{\rm L},k=0}_{\rm grav}$
$\displaystyle=-\frac{{\rm
i}\sqrt{\Delta}}{288\hbar\gamma^{3}}\sum_{i,j,k}\epsilon^{ijk}{\rm
Tr}\left(\widehat{h^{(\bar{\mu})}_{i}}\left[\widehat{{h_{i}^{(\bar{\mu})}}^{-1}},\left[\hat{\cal
H}^{{\rm E},k=0}_{\rm
grav},|\hat{v}|\right]\right]\hat{v}\,\widehat{h^{(\bar{\mu})}_{j}}\left[\widehat{{h_{j}^{(\bar{\mu})}}^{-1}},|\hat{v}|\right]\widehat{h^{(\bar{\mu})}_{k}}\left[\widehat{{h_{k}^{(\bar{\mu})}}^{-1}},\left[\hat{\cal
H}^{{\rm E},k=0}_{\rm grav},|\hat{v}|\right]\right]\right)$
$\displaystyle=\frac{{\rm
i}\sqrt{\Delta}}{36\hbar\gamma^{3}}\,\hat{O}_{[\hat{\cal H}^{{\rm E},k=0}_{\rm
grav},|\hat{v}|]}\left(\hat{v}\,\hat{O}_{|\hat{v}|}\right)\hat{O}_{[\hat{\cal
H}^{{\rm E},k=0}_{\rm grav},|\hat{v}|]}\sum_{i,j,k}\epsilon^{ijk}{\rm
Tr}(\tau_{i}\tau_{j}\tau_{k})$ $\displaystyle=-\frac{{\rm
i}\sqrt{\Delta}}{24\hbar\gamma^{3}}\,\hat{O}_{[\hat{\cal H}^{{\rm E},k=0}_{\rm
grav},|\hat{v}|]}\left(\hat{v}\,\hat{O}_{|\hat{v}|}\right)\hat{O}_{[\hat{\cal
H}^{{\rm E},k=0}_{\rm grav},|\hat{v}|]},$ (44)

where in the second step we have used Eq. (III) and used that the terms
involving zero, one and two $\tau$ in the trace vanish due to
$\sum_{ijk}\epsilon^{ijk}=0$, $\sum_{ijk}\epsilon^{ijk}{\rm Tr}(\tau_{k})=0$,
and $\sum_{ijk}\epsilon^{ijk}{\rm
Tr}(\tau_{i}\tau_{j})=-\frac{1}{2}\sum_{ijk}\epsilon^{ijk}\delta_{ij}=0$. The
operators $\hat{O}_{|\hat{v}|}$ and $\hat{O}_{[\hat{\cal H}^{{\rm E},k=0}_{\rm
grav},|\hat{v}|]}$ are defined according to Eq. (43) with $\hat{B}=|\hat{v}|$
and $\hat{B}=[\hat{\cal H}^{{\rm E},k=0}_{\rm grav},|\hat{v}|]$, respectively.
Moreover, in the last step in Eq. (III), we have used
$\sum_{i,j,k}\epsilon^{ijk}{\rm
Tr}(\tau_{i}\tau_{j}\tau_{k})=-\frac{1}{4}\sum_{i,j,k}\epsilon^{ijk}\epsilon_{ijk}=-\frac{3}{2}$.
Finally, the operator corresponding to ${\cal H}^{\Gamma,k=-1,{\rm reg}}_{\rm
grav}$ in Eq. (III) reads if assuming again for simplicity that the holonomies
of $k=-1$ can be approximated with the holonomies of $k=0$

$\displaystyle\hat{\cal H}^{\Gamma,k=-1}_{\rm grav}$
$\displaystyle=\frac{(\hbar\gamma\sqrt{\Delta})^{\frac{1}{3}}V_{o}^{\frac{2}{3}}}{{\rm
i}4(2\pi G)^{\frac{2}{3}}}{\rm
sgn}(\hat{v})|\hat{v}|^{\frac{1}{3}}\sum_{k}{\rm
Tr}\left(\tau_{k}\widehat{h^{(\bar{\mu})}_{k}}\left[\widehat{{h_{k}^{(\bar{\mu})}}^{-1}},|\hat{v}|\right]\right)$
$\displaystyle=-\frac{(\hbar\gamma\sqrt{\Delta})^{\frac{1}{3}}V_{o}^{\frac{2}{3}}}{{\rm
i}2(2\pi G)^{\frac{2}{3}}}{\rm
sgn}(\hat{v})|\hat{v}|^{\frac{1}{3}}\hat{O}_{|\hat{v}|}\sum_{k}{\rm
Tr}(\tau_{k}\tau_{k})$
$\displaystyle=\frac{3(\hbar\gamma\sqrt{\Delta})^{\frac{1}{3}}V_{o}^{\frac{2}{3}}}{{\rm
i}4(2\pi G)^{\frac{2}{3}}}{\rm
sgn}(\hat{v})|\hat{v}|^{\frac{1}{3}}\hat{O}_{|\hat{v}|}.$ (45)

The actions of the operators $\hat{\cal H}^{{\rm E},k=0}_{\rm grav}$,
$\hat{\cal H}^{{\rm L},k=0}_{\rm grav}$, and $\hat{\cal H}^{\Gamma,k=-1}_{\rm
grav}$ on $|v\rangle$ read

$\displaystyle\hat{\cal H}^{{\rm E},k=0}_{\rm grav}|v\rangle$
$\displaystyle=E_{+}(v)|v+4\rangle+E_{0}(v)|v\rangle+E_{-}(v)|v-4\rangle,$
(46) $\displaystyle\hat{\cal H}^{{\rm L},k=0}_{\rm grav}|v\rangle$
$\displaystyle=L_{+}(v)|v+8\rangle+L_{0}(v)|v\rangle+L_{-}(v)|v-8\rangle,$
(47) $\displaystyle\hat{\cal H}^{\Gamma,k=-1}_{\rm grav}|v\rangle$
$\displaystyle=\Gamma(v)|v\rangle,$ (48)

where

$\displaystyle E_{+}(v)$
$\displaystyle=\frac{3\gamma\hbar}{32\sqrt{\Delta}}(v+2)M_{1,3}(v),$ (49)
$\displaystyle E_{-}(v)$ $\displaystyle=E_{+}(v-4),$ (50) $\displaystyle
E_{0}(v)$ $\displaystyle=-E_{+}(v)-E_{-}(v),$ (51) $\displaystyle L_{+}(v)$
$\displaystyle=-\frac{\sqrt{\Delta}}{192\gamma^{3}\hbar}(v+4)M_{-1,1}(v+4)$
$\displaystyle\hskip 56.9055pt\times G_{-}(v+4)G_{+}(v+4),$ (52)
$\displaystyle L_{-}(v)$ $\displaystyle=L_{+}(v-8),$ (53) $\displaystyle
L_{0}(v)$
$\displaystyle=-\frac{\sqrt{\Delta}}{192\gamma^{3}\hbar}\left\\{(v+4)M_{-1,1}(v+4)[G_{+}(v)]^{2}\right.$
$\displaystyle\hskip
36.98866pt\left.+(v-4)M_{-1,1}(v-4)[G_{-}(v)]^{2}\right\\},$ (54)
$\displaystyle\Gamma(v)$
$\displaystyle=\frac{3\left(\gamma\sqrt{\Delta}\,\hbar\right)^{\frac{1}{3}}V_{o}^{\frac{2}{3}}}{8\left(2\pi
G\right)^{\frac{2}{3}}}{\rm sgn}(v)|v|^{\frac{1}{3}}M_{1,-1}(v).$ (55)

Here

$\displaystyle M_{a,b}(v)$ $\displaystyle:=|v+a|-|v+b|,$ (56) $\displaystyle
G_{\pm}(v)$ $\displaystyle:=E_{\pm}(v-1)M_{0,\pm 4}(v-1)$ $\displaystyle\hskip
28.45274pt-E_{\pm}(v+1)M_{0,\pm 4}(v+1).$ (57)

The function $M_{a,b}(v)$ satisfies

$\displaystyle M_{0,|k|}(v)=-M_{0,-|k|}(v+|k|).$ (58)

Hence the action of the $k=-1$ gravitational Hamiltonian constraint operator
$\hat{\cal H}^{k=-1}_{\rm grav}$ on $|v\rangle$ reads

$\displaystyle\hat{\cal H}^{k=-1}_{\rm grav}|v\rangle=$
$\displaystyle\tilde{L}_{+}(v)|v+8\rangle+E_{+}(v)|v+4\rangle$
$\displaystyle+\left[E_{0}(v)+\tilde{L}_{0}(v)+\Gamma(v)\right]|v\rangle$
$\displaystyle+E_{-}(v)|v-4\rangle+\tilde{L}_{-}(v)|v-8\rangle,$ (59)

where $\tilde{L}_{*}:=-2(1+\gamma^{2})L_{*}$, here $*=+,-,0$.

On the other hand, the Hamiltonian constraint ${\cal H}_{\phi}$ for the scalar
field can be quantized as a well-defined operator $\hat{\cal H}_{\phi}$ in
${\mathscr{H}}^{\rm tot}_{\rm kin}$, and the action of $\hat{\cal H}_{\phi}$
on a quantum state $|\psi\rangle=\psi(v,\phi)|v,\phi\rangle$ with
$|v,\phi\rangle\equiv|v\rangle\otimes|\phi\rangle\in{\mathscr{H}}^{\rm
gr}_{\rm kin}\otimes{\mathscr{H}}^{\rm sc}_{\rm kin}$ is given by
Ashtekar:2006wn

$\displaystyle\hat{\cal
H}_{\phi}\cdot\psi(v,\phi)=-\frac{\hbar^{2}}{4\pi\gamma\sqrt{\Delta}\ell_{\mathrm{p}}^{2}}C(v)\partial^{2}_{\phi}\psi(v,\phi),$
(60)

where

$\displaystyle
C(v)\equiv\left(\frac{3}{2}\right)^{3}|v|\left|{|v+1|^{{1}/{3}}}-{|v-1|^{{1}/{3}}}\right|^{3}.$
(61)

Combining equations above, one can write down the resulting quantum
Hamiltonian constraint equation corresponding to its classical one (21) as

$\displaystyle\hat{\cal H}_{\rm tot}^{k=-1}\cdot\psi(v,\phi)=\left(\hat{\cal
H}^{k=-1}_{{\rm grav}}+\hat{\cal H}_{\phi}\right)\cdot\psi(v,\phi)=0,$ (62)

which describes the quantum evolution of the coupled system with the scalar
field $\phi$ as an emergent time.

##  IV Effective theory and its asymptotic behavior of the alternative $k=-1$
LQC

By constructing certain coherent states peaked at points of the classical
phase space and computing the expectation value of the Hamiltonian constraint
operator under the coherent states, one can obtain the corresponding effective
Hamiltonian constraint. To this end, we first note that the symmetry-reduced
phase space of the $k=-1$ model coincides with that of the $k=0$ model. Hence
certain coherent states constructed for the $k=0$ model can be directly
carried to the $k=-1$ model. A Gaussian coherent state peaked at a point
$(b_{o},v_{o},\phi_{o},p_{\phi})$ in the classical phase space with spreads
$\epsilon$ and $\sigma$ in the gravitational sector and scalar field sector
takes the form Ding:2008tq

$\displaystyle\left(\Psi_{(b_{o},v_{o},\phi_{o},p_{\phi})}\right|$
$\displaystyle:=\int{\rm
d}\phi\sum_{v\in\mathbb{R}}e^{-\frac{\epsilon^{2}}{2}(v-v_{o})^{2}}e^{{\rm
i}b_{o}(v-v_{o})}$ $\displaystyle\hskip 28.45274pt\times
e^{-\frac{\sigma^{2}}{2}(\phi-\phi_{o})^{2}}e^{\frac{\rm
i}{\hbar}p_{\phi}(\phi-\phi_{o})}(v|\otimes(\phi|,$ (63)

and its shadow on the regular lattice with spacing one reads

$\displaystyle\left|\Psi\right\rangle$ $\displaystyle:=\int{\rm
d}\phi\sum_{n\in\mathbb{Z}}e^{-\frac{\epsilon^{2}}{2}(n-v_{o})^{2}}e^{-{\rm
i}b_{o}(n-v_{o})}$ $\displaystyle\hskip 28.45274pt\times
e^{-\frac{\sigma^{2}}{2}(\phi-\phi_{o})^{2}}e^{-\frac{\rm
i}{\hbar}p_{\phi}(\phi-\phi_{o})}\,|n\rangle\otimes|\phi\rangle$
$\displaystyle\equiv\left|\Psi_{\rm
grav}\right\rangle\otimes\left|\Psi_{\phi}\right\rangle.$ (64)

To make the state be sharply peaked in the classical phase space of the
universe with large volume, one should require that $\epsilon\ll b_{o}$,
$v_{o}\epsilon\gg 1$, $\sigma\ll\phi_{o}$ and $p_{\phi}\sigma\gg 1$. Denote by
$\langle\hat{O}\rangle:=\frac{\langle\Psi|\hat{O}|\Psi\rangle}{\langle\Psi|\Psi\rangle}$
the expectation value of an operator $\hat{O}$ under the coherent states (IV).
By using the Poisson resummation on the sum over $n$ and the steepest descent
approximation, the expectation value of each term of the gravitational
Hamiltonian constraint operator $\hat{\cal H}^{k=-1}_{\rm grav}$ can be
calculated, and thus the resulting expectation value of $\hat{\cal
H}^{k=-1}_{\rm grav}$ can be obtained. For brevity, in the remainder of this
paper, we will suppress the label $o$ appearing in $b_{o},v_{o}$ and
$\phi_{o}$. A straightforward calculation reveals that (see Appendix A for a
derivation) Yang:2009fp ; Assanioussi:2018hee ; Assanioussi:2019iye

$\displaystyle{\cal H}^{{\rm E},k=0}_{\rm grav,eff}$
$\displaystyle:=\langle\hat{\cal H}^{{\rm E},k=0}_{\rm
grav}\rangle=\frac{3\hbar\gamma
v}{4\sqrt{\Delta}}\left[\sin^{2}(2b)+O(\epsilon^{2})\right]\left\\{1+O(e^{-\pi^{2}/\epsilon^{2}})+O\left[1/(v\epsilon)^{2}\right]\right\\},$
(65) $\displaystyle{\cal H}^{{\rm L},k=0}_{\rm grav,eff}$
$\displaystyle:=\langle\hat{\cal H}^{{\rm L},k=0}_{\rm
grav}\rangle=\frac{3\hbar
v}{32\gamma\sqrt{\Delta}}\left[\sin^{2}(4b)+O(\epsilon^{2})\right]\left\\{1+O(e^{-\pi^{2}/\epsilon^{2}})+O\left[1/(v\epsilon)^{2}\right]\right\\},$
(66) $\displaystyle{\cal H}^{\Gamma,k=-1}_{\rm grav,eff}$
$\displaystyle:=\langle\hat{\cal H}^{\Gamma,k=-1}_{\rm
grav}\rangle=\frac{3\left(\gamma\sqrt{\Delta}\,\hbar\right)^{\frac{1}{3}}V_{o}^{\frac{2}{3}}}{4\left(2\pi
G\right)^{\frac{2}{3}}}{v}^{\frac{1}{3}}\left\\{1+O(e^{-\pi^{2}/\epsilon^{2}})+O\left[1/(v\epsilon)^{2}\right]\right\\}.$
(67)

Hence, in the region with $\epsilon\ll b$, $v\epsilon\gg 1$, $\sigma\ll\phi$
and $p_{\phi}\sigma\gg 1$ the higher-order corrections can be omitted. In what
follows, we focus on the leading terms. Hence the effective Hamiltonian
constraint of gravitational part for the $k=-1$ model reads

$\displaystyle{\cal H}^{k=-1}_{\rm grav,eff}$ $\displaystyle={\cal H}^{{\rm
E},k=0}_{\rm grav,eff}-2(1+\gamma^{2}){\cal H}^{{\rm L},k=0}_{\rm
grav,eff}+{\cal H}^{\Gamma,k=-1}_{\rm grav,eff}$ $\displaystyle=-\frac{3\hbar
v}{\gamma\sqrt{\Delta}}\left\\{\frac{1}{4}\sin^{2}(2b)\left[1-(1+\gamma^{2})\sin^{2}(2b)\right]\right.$
$\displaystyle\hskip
85.35826pt\left.-\frac{\gamma^{2}\Delta}{4}\frac{V_{o}^{2/3}}{V^{2/3}}\right\\}$
$\displaystyle\equiv-\frac{3\hbar v}{\gamma\sqrt{\Delta}}g_{\rm eff}(b,v).$
(68)

Taking into account the result for the scalar field in Yang:2009fp , the total
effective Hamiltonian constraint of the gravity coupled with a massless scalar
field reads

$\displaystyle{\cal H}^{k=-1}_{\rm tot,eff}$ $\displaystyle=-\frac{3\hbar
v}{\gamma\sqrt{\Delta}}g_{\rm eff}(b,v)+\frac{p_{\phi}^{2}}{4\pi\gamma
G\hbar\sqrt{\Delta}\,v}.$ (69)

Before calculating the dynamics of this effective Hamiltonian, we clarify some
subtle issues. First, we expect here that the evolution of, saying,
$\langle\hat{v}\rangle$ up to $O(\hbar)$ order coincides with the dynamics
determined by the effective Hamiltonian constraint (69). In other words, if we
compute the quantum dynamics of the coherent state (IV) and investigate the
evolution of the expectation value of $\hat{v}$, we conjecture that the result
coincides with the dynamics of $v$ obtained by solving the Hamilton’s equation
concerning the effective Hamiltonian (69). Second, as claimed before, we
consider the region with $\epsilon\ll b$ so that the higher-order corrections
are omitted. However, as shown later, in the FRW phase of the evolution given
by the effective Hamiltonian, $b$ approaches $0$ asymptotically. We thus
obtain a tension that, on the one hand, we require $b\gg\epsilon$ but, on the
other hand, $b$ goes to $0$ along the evolution. To resolve this tension, we
still have to compute the quantum dynamics to see how the spread
$\epsilon=1/\sqrt{2|\langle\hat{v}\rangle^{2}-\langle\hat{v}^{2}\rangle|}$
evolves. Even though the quantum dynamics, which will be left as our future
work, has not been investigated yet, the previous results in the $k=0$ model
Ashtekar:2006wn ; Zhang:2019dgi  make us expect $\epsilon\sim
1/\langle\hat{v}\rangle$ which would make $\epsilon\ll b$ true along the
evolution. Indeed, coherent states with the phase-space dependent spread have
been considered in the regular LQC Taveras:2008ke . Moreover, another approach
to understand the effective dynamics is to apply the path integral formulation
to study transition amplitude $A(v_{f},\phi_{f};v_{i},\phi_{i})=\langle
v_{f},\phi_{f}|v_{i},\phi_{i}\rangle_{\rm phy}$ with
$\langle\cdot|\cdot\rangle_{\rm phy}$ denoting the physical inner products
Ashtekar:2010ve ; Qin:2012gaa . Since the Hamiltonian constraint operator
$\hat{\cal H}^{k=-1}_{\rm grav}$ is the same as that of the $k=0$ model up to
the term $\hat{\cal H}^{\Gamma,k=-1}_{\rm grav,eff}$ which takes $|v\rangle$
as its eigenstate, one can simply generalize the results in Qin:2012gaa  to
conclude that the classical path resulting from dynamics of the effective
Hamiltonian dominates $A(v_{f},\phi_{f};v_{i},\phi_{i})$. Finally, in the
region with $b\rightarrow 0$ and $v\gg 1$, it is easy to see that, as
$b\rightarrow 0$, $g_{\rm eff}(b,v)$ goes to $g(b,v)$, and thus the total
effective Hamiltonian constraint ${\cal H}^{k=-1}_{\rm tot,eff}$ in Eq. (69)
reduces to its classical expression ${\cal H}_{\rm tot}^{k=-1}$ in Eq. (II).
Hence the new alternative quantum dynamics has the corrected classical limit.

It is easy to see that, in the effective theory, $p_{\phi}$ is a constant of
motion due to $\dot{p}_{\phi}=\\{p_{\phi},{\cal H}^{k=-1}_{\rm tot,eff}\\}=0$,
and $\phi$ can be regarded as an internal clock because of $\dot{\phi}>0$,
similar to the classical theory. The effective Hamiltonian constraint equation

$\displaystyle{\cal H}^{k=-1}_{\rm tot,eff}=0$ (70)

can determine the evolutions of $v$ with respect to $b$ for some given
$p_{\phi}$, which is plotted in Fig. 1.

Figure 1: Plots of $v$ with respect to $b$ determined by the total Hamiltonian
constraint equation ${\cal H}^{k=-1}_{\rm tot,eff}=0$ for different values of
$p_{\phi}$, with $V_{0}=G=\hbar=1$ and $\gamma=0.2375$.

Figure 1 depicts that $v=0$ can never be a solution to Eq. (70). It indicates
that the classical singularity at $v=0$ can be avoided in the effective
theory. By Eq. (70), the matter density can be expressed as

$\displaystyle\rho_{\phi}(v)$
$\displaystyle=\frac{p_{\phi}^{2}}{2V^{2}}=-\frac{{\cal H}^{k=-1}_{\rm
grav,eff}}{V}$ $\displaystyle=\frac{3}{2\pi G\gamma^{2}\Delta}g_{\rm
eff}(b,v)\equiv\rho_{\phi}^{\rm eff}(b,v).$ (71)

The effective Hubble parameter is determined by the total effective
Hamiltonian constraint (69), and reads

$\displaystyle H^{2}_{{\rm eff},k=-1}$
$\displaystyle=\left(\frac{\dot{v}}{3v}\right)^{2}=\left(\frac{\left\\{v,{\cal
H}^{k=-1}_{\rm tot,eff}\right\\}}{3v}\right)^{2}$
$\displaystyle=\frac{1}{\gamma^{2}\Delta}[g_{\rm eff}^{\prime}(b,v)]^{2},$
(72)

where ′ denotes the first-order derivative with respect to $b$. In what
follows, we focus on the region $b\in[0,\pi/4]$ where we live. A bounce
appears when $H^{2}_{{\rm eff},k=-1}=0$, i.e.,

$\displaystyle g_{\rm eff}^{\prime}(b,v)=0\quad\Leftrightarrow\quad
b=\frac{1}{2}\arcsin\sqrt{\frac{1}{2(1+\gamma^{2})}},$ (73)

at which the energy density $\rho_{\phi}$ takes the maximal value, namely the
critical energy density, as

$\displaystyle\rho^{k=-1}_{\rm crit}$ $\displaystyle=\rho_{F}-\frac{3}{8\pi
G}\frac{V_{o}^{2/3}}{V^{2/3}}=\rho_{F}-\frac{3}{8\pi G}\frac{1}{a^{2}},$ (74)

where

$\displaystyle\rho_{F}$ $\displaystyle:=\frac{3}{32\pi
G\gamma^{2}(1+\gamma^{2})\Delta}.$ (75)

In comparison with the effective $k=0$ model proposed in Yang:2009fp  where
the critical density is given by the first term of Eq. (74), the effective
$k=-1$ model contributes an additional term, the second term of Eq. (74), to
the critical energy density. It is worth mentioning that the critical energy
density $\rho^{k=-1}_{\rm crit}$ in the effective $k=-1$ model depends on the
value of $v$ (or the scale factor $a$) at the bounce point. The value $v_{\rm
bounce}$ of $v$ at the bounce point can be determined by

$\displaystyle\rho_{\phi}(v_{\rm bounce})=\rho^{k=-1}_{\rm crit}(v_{\rm
bounce}).$ (76)

In Fig. 2, the value $v_{\rm bounce}$ as a function of $p_{\phi}$ is plotted.
It is shown that as $p_{\phi}$ increase, the value $v_{\rm bounce}$ increase
monotonically. Moreover the condition $\rho_{\phi}(v_{\rm bounce})\geq 0$
implies that

$\displaystyle a\geq\sqrt{4\gamma^{2}(1+\gamma^{2})\Delta},$ (77)

and thus the effective theory predicts that the scale factor $a$ is bounced
below $a_{\rm min}=\sqrt{4\gamma^{2}(1+\gamma^{2})\Delta}$.

Figure 2: Plot of $v_{\rm bounce}$ with respect to $p_{\phi}$, with
$V_{0}=G=\hbar=1$ and $\gamma=0.2375$.

Solving the total effective Hamiltonian constraint equation ${\cal
H}^{k=-1}_{\rm tot,eff}=0$ for $b$ yields

$\displaystyle b=\begin{cases}b_{\rm
I}\equiv\frac{1}{2}\arcsin\sqrt{\frac{1-\sqrt{1-\frac{\rho_{\phi}+\frac{3}{8\pi
G}\frac{1}{a^{2}}}{\rho_{F}}}}{2(1+\gamma^{2})}}\\\ b_{\rm
II}\equiv\frac{1}{2}\arcsin\sqrt{\frac{1+\sqrt{1-\frac{\rho_{\phi}+\frac{3}{8\pi
G}\frac{1}{a^{2}}}{\rho_{F}}}}{2(1+\gamma^{2})}}\end{cases}.$ (78)

Hence there exists two types of classical universe, namely the type-I universe
and the type-II universe. The two universes are connected by a quantum bounce.
The effective Hubble parameter is determined by

$\displaystyle H_{{\rm eff},k=-1,{\rm I/II}}^{2}$
$\displaystyle=\left.\left(\frac{\dot{v}}{3v}\right)^{2}\right|_{b=b_{\rm
I/II}}$ $\displaystyle=\frac{8\pi
G}{3}\frac{\rho_{F}}{1+\gamma^{2}}\left(1-\frac{\rho_{\phi}+\frac{3}{8\pi
G}\frac{1}{a^{2}}}{\rho_{F}}\right)\left(1\mp\sqrt{1-\frac{\rho_{\phi}+\frac{3}{8\pi
G}\frac{1}{a^{2}}}{\rho_{F}}}\right)\left(1+2\gamma^{2}\pm\sqrt{1-\frac{\rho_{\phi}+\frac{3}{8\pi
G}\frac{1}{a^{2}}}{\rho_{F}}}\right),$ (79)

which can be expressed as the following more convenient forms

$\displaystyle H_{{\rm eff},k=-1,{\rm I}}^{2}$ $\displaystyle=\frac{8\pi
G}{3}\left(\rho_{\phi}+\frac{3}{8\pi
G}\frac{1}{a^{2}}\right)\left(1-\frac{\rho_{\phi}+\frac{3}{8\pi
G}\frac{1}{a^{2}}}{\rho_{F}}\right)\left[1+\frac{\gamma^{2}}{1+\gamma^{2}}\left(\frac{\sqrt{\frac{\rho_{\phi}+\frac{3}{8\pi
G}\frac{1}{a^{2}}}{\rho_{F}}}}{1+\sqrt{1-\frac{\rho_{\phi}+\frac{3}{8\pi
G}\frac{1}{a^{2}}}{\rho_{F}}}}\right)^{2}\right],$ (80)

and

$\displaystyle H_{{\rm eff},k=-1,{\rm II}}^{2}$ $\displaystyle=\frac{8\pi
G}{3}\rho_{\Lambda_{\rm eff}}\left(1-\frac{\rho_{\phi}+\frac{3}{8\pi
G}\frac{1}{a^{2}}}{\rho_{F}}\right)\left[1+\frac{1-2\gamma^{2}+\sqrt{1-\frac{\rho_{\phi}+\frac{3}{8\pi
G}\frac{1}{a^{2}}}{\rho_{F}}}}{4\gamma^{2}\left(1+\sqrt{1-\frac{\rho_{\phi}+\frac{3}{8\pi
G}\frac{1}{a^{2}}}{\rho_{F}}}\right)}\frac{\rho_{\phi}+\frac{3}{8\pi
G}\frac{1}{a^{2}}}{\rho_{F}}\right],$ (81)

where

$\displaystyle\rho_{\Lambda_{\rm eff}}:=\frac{\Lambda_{\rm eff}}{8\pi
G},\qquad{\text{with}}\quad\Lambda_{\rm
eff}:=\frac{3}{(1+\gamma^{2})^{2}\Delta}.$ (82)

Now let us study the asymptotic behavior of the effective dynamics at the
large $v$ limit. For $v\rightarrow\infty$, the matter density $\rho_{\phi}(v)$
in Eq. (IV) goes to zero, and thus

$\displaystyle\sin^{2}(2b)\left[1-(1+\gamma^{2})\sin^{2}(2b)\right]\rightarrow
0,$ (83)

which implies

$\displaystyle b\rightarrow b_{0}=\begin{cases}b_{\rm I,c}\equiv 0,\\\ b_{\rm
II,c}\equiv\frac{1}{2}\arcsin\left(\frac{1}{\sqrt{1+\gamma^{2}}}\right).\end{cases}$
(84)

Expanding ${\cal H}^{k=-1}_{\rm tot,eff}$ at $b_{0}$ up to the second order
yields the classical behavior of the effective Hamiltonian constraint as

$\displaystyle{\cal H}^{k=-1}_{\rm tot,eff}\;\rightarrow\;$
$\displaystyle-\frac{3\hbar v}{\gamma\sqrt{\Delta}}\Big{[}g_{\rm
eff}(b_{0},v)+g_{\rm eff}^{\prime}(b_{0})(b-b_{0})$ $\displaystyle\hskip
28.45274pt+\frac{g_{\rm
eff}^{\prime\prime}(b_{0})}{2}(b-b_{0})^{2}\Big{]}+\frac{p_{\phi}^{2}}{4\pi\gamma
G\hbar\sqrt{\Delta}\,v}\,,$ (85)

where ′′ denotes the second-order derivative with respective to $b$, and

$\displaystyle g_{\rm eff}^{\prime}(b_{0})=\begin{cases}0,&b_{0}=b_{\rm
I,c},\\\ -\frac{\gamma}{1+\gamma^{2}},&b_{0}=b_{\rm II,c},\end{cases}$ (86)
$\displaystyle g_{\rm eff}^{\prime\prime}(b_{0})=\begin{cases}2,&b_{0}=b_{\rm
I,c},\\\ \frac{2-10\gamma^{2}}{1+\gamma^{2}},&b_{0}=b_{\rm II,c}.\end{cases}$
(87)

Plugging these asymptotic expressions into Eq. (IV), we have

$\displaystyle H_{{\rm eff},k=-1}^{2}$ $\displaystyle\rightarrow\frac{8\pi
G}{3}\left(\frac{g^{\prime\prime}_{\rm
eff}(b_{0})}{2}\,\rho_{\phi}+\frac{3\left[g^{\prime}_{\rm
eff}(b_{0})\right]^{2}}{8\pi G\gamma^{2}\Delta}+\frac{3g^{\prime\prime}_{\rm
eff}(b_{0})}{16\pi G}\frac{V_{0}^{2/3}}{V^{2/3}}\right)=\frac{8\pi
G}{3}\left[\frac{g^{\prime\prime}_{\rm
eff}(b_{0})}{2}\,\left(\rho_{\phi}+\frac{3}{8\pi
G}\frac{1}{a^{2}}\right)+\frac{3\left[g^{\prime}_{\rm
eff}(b_{0})\right]^{2}}{8\pi G\gamma^{2}\Delta}\right]$
$\displaystyle=\begin{cases}\frac{8\pi
G}{3}\rho_{\phi}+\frac{1}{a^{2}},&b_{0}=b_{\rm I,c},\\\ \frac{8\pi
G}{3}\left(\frac{1-5\gamma^{2}}{1+\gamma^{2}}\right)\rho_{\phi}+\left(\frac{1-5\gamma^{2}}{1+\gamma^{2}}\right)\frac{1}{a^{2}}+\frac{\Lambda_{\rm
eff}}{3},&b_{0}=b_{\rm II,c}.\end{cases}$ (88)

The above asymptotic behavior (IV) of the effective Hubble parameter can be
also obtained directly from Eqs. (80) and (81). Equation (IV) implies that the
type-II universe is an asymptotic de Sitter universe with a positive effective
cosmological constant $\Lambda_{\rm eff}$. Therefore the asymptotical $k=-1$
FRW universe (the type-I universe) will be bounced to an asymptotic de Sitter
universe (the type-II universe) coupled to a scalar field.

We now numerically study the the effective dynamical evolution of $v$ with
$\phi$. To this end, we can firstly solve the effective Hamiltonian constraint
equation ${\cal H}^{k=-1}_{\rm tot,eff}=0$ to yield

$\displaystyle v=v(b,p_{\phi}).$ (89)

Secondly, we consider the evolution equation of $\phi$ with respect to $b$,
namely

$\displaystyle\frac{{\rm d}\phi}{{\rm d}b}=\frac{\\{\phi,{\cal H}^{k=-1}_{\rm
tot,eff}\\}}{\\{b,{\cal H}^{k=-1}_{\rm tot,eff}\\}}=f(b,p_{\phi}),$ (90)

where Eq. (89) was inserted in the second step. Solving Eq. (90) yields

$\displaystyle\phi=\phi(b,p_{\phi}).$ (91)

By combining Eq. (89) with Eq. (91) and then eliminating $b$, we arrive at
$v=v(\phi,p_{\phi})$. In Fig. 3, the effective dynamical evolution of $v$ with
respect to $\phi$ for given $p_{\phi}$ is plotted. It indicates that an
asymmetric bounce appears in the backward evolution of the universe sourced by
a massless scalar field $\phi$, and the classical big-bang singularity is
resolved.

Figure 3: Plots of the effective dynamical evolution of $v$ with respect to
$\phi$, in reverse direction of the cosmological time, determined by the
effective Hamiltonian constraint for different values of $p_{\phi}$ with
initial data $v(b_{\rm start},p_{\phi})=8p_{\phi}$ and $\phi(b_{\rm
start},p_{\phi})=1$, with $V_{0}=G=\hbar=1$ and $\gamma=0.2375$.

##  V Summary

The quantization ambiguities often exist in constructing the gravitational
Hamiltonian constraint operator of full LQG as well as of LQC. It has been
shown that in LQC different quantizations of the gravitational Hamiltonian
constraint may lead to different quantum dynamics, and thus affect the fate of
the universe. Hence the study of the quantization ambiguities of the
gravitational Hamiltonian constraint plays an important role in the quantum
dynamics of LQC. In present paper, we have studied an alternative quantization
of the gravitational Hamiltonian constraint in the $k=-1$ model of LQC closely
following that in the $k=0$ model proposed in Yang:2009fp , mimicking the
treatment of full LQG.

Classically, the connection $A^{i}_{a}$ in the $k=-1$ model takes the
nondiagonal expression (7) on the left-invariant one-forms
${}^{o}\\!\omega^{i}_{a}$, while it takes the diagonal form on the left- and
right-invariant one-forms ${}^{o}\\!\omega^{i}_{a}$ in the $k=0$ model. In the
$k=-1$ model, the nondiagonal expression of $A^{i}_{a}$ leads to the
complicated forms of the resulting holonomies of the connection. Instead, one
often considers the holonomies of the extrinsic curvature $K^{i}_{a}$
multiplied by $\gamma$ in the $k=-1$ model, and thus they have the same
expressions as the holonomies of the connection $A^{i}_{a}=\gamma K^{i}_{a}$
in the $k=0$ model. Hence both the $k=0$ model and the $k=-1$ model have the
same Hilbert space ${\mathscr{H}}^{\rm gr}_{\rm
kin}=L^{2}(\mathbb{R}_{\mathrm{Bohr}},{\rm d}\mu_{\mathrm{Bohr}})$.

To study the quantum dynamics of the $k=0,-1$ models in the framework of LQC,
one needs to promote the gravitational Hamiltonian constraint into a well-
defined operator in ${\mathscr{H}}^{\rm gr}_{\rm kin}$. At the classical
level, the gravitational Hamiltonian constraint of the $k=-1$ model can be
expressed as the three terms in Eq. (II). On one hand, the former two terms,
$\tilde{\cal H}^{{\rm E},k=0}_{\rm grav}$ and $\tilde{\cal H}^{{\rm L},k=0}$,
in Eq. (II) have the same expressions as the Euclidean and Lorentzian terms,
${\cal H}^{{\rm E},k=0}_{\rm grav}$ and ${\cal H}^{{\rm L},k=0}$, in the $k=0$
model, respectively, although involving different fiducial one-forms
${}^{o}\\!\omega^{i}_{a}$ (the only left-invariant one-forms v.s. the left-
and right-invariant one-forms). From the symmetry-reduced expressions in Eqs.
(15) and (16), the former two terms are proportional to each other, and thus
one can firstly combine the two terms and then quantize the first and third
terms to obtain the gravitational Hamiltonian constraint Vandersloot:2006ws .
On the other hand, from the viewpoint of full LQG, the sum of both the first
and last terms forms the Euclidean term, while the middle term represents the
Lorentzian term. Hence, alternatively, we can quantize the Euclidean term and
the Lorentzian term respectively, mimicking the treatment in full LQG. We have
shown that the former two terms $\tilde{\cal H}^{{\rm E},k=0}_{\rm grav}$ and
$\tilde{\cal H}^{{\rm L},k=0}$ in Eq. (II) can be quantized as the operators,
$\hat{\cal H}^{{\rm E},k=0}_{\rm grav}$ and $\hat{\cal H}^{{\rm L},k=0}_{\rm
grav}$ in Eqs. (III) and (III), corresponding precisely to the Euclidean and
Lorentzian Hamiltonian operators of the $k=0$ model proposed in Yang:2009fp ,
while the third term has been quantized as $\hat{\cal H}^{\Gamma,k=-1}_{\rm
grav}$ in Eq. (III). The resulting gravitational Hamiltonian constraint
operator $\hat{\cal H}^{k=-1}_{\rm grav}$ is symmetric, which has the action
(III) on $|v\rangle$. Moreover, we have shown that the new quantum dynamics
determined by the alternative Hamiltonian constraint operator $\hat{\cal
H}_{\rm tot}^{k=-1}$ in Eq. (62) for the $k=-1$ model coupled to a massless
scalar field has the corrected classical limit, and obtained its effective
Hamiltonian constraint (69), by semi-classical analysis. The effective
Friedmann equation for the $k=-1$ model was derived in Eq. (IV), which shows
that it has two branches (80) and (81) relating to two types of universes,
similar to the $k=0$ LQC proposed in Yang:2009fp . It turns out that the
asymptotical $k=-1$ FRW universe (the type-I universe) will be bounced to an
asymptotic de Sitter universe (the type-II universe) coupled to a scalar
field. Last but not least, by requiring the condition $\rho_{\phi}(v_{\rm
bounce})\geq 0$, the effective theory predicts that the scale factor $a$ is
bounced below $a_{\rm min}=\sqrt{4\gamma^{2}(1+\gamma^{2})\Delta}$, which is
different from that in previous $k=-1$ LQC model Vandersloot:2006ws .

So far, the Thiemann’s trick for regularizing the gravitational Hamiltonian
constraint in full LQG, by treating the Euclidean term and the Lorentzian term
independently, has been successfully applied to the $k=0,-1$ LQC models.
However, to our knowledge, a similar treatment for the $k=+1$ model in the
framework of LQC has not been carried out. In spite of the Thiemann-
regularization of the Hamiltonian constraint on the hyperspherical lattice for
the $k=+1$ model has been studied from the viewpoint of full LQG in
Liegener:2019out . The expectation value of the Hamiltonian constraint under
certain coherent states was computed, and an effective Hamiltonian constraint
was obtained in the $\mu_{o}$ scheme rather than the $\bar{\mu}$ scheme
Liegener:2019out . By numerical simulations of the dynamical evolution, an
asymmetric bounce replacing the classical big bang was also obtained in the
model Liegener:2019out .

It should be noted that there are still many aspects of the loop quantum
$k=-1$ model that deserves further investigation. Recently, some works have
focused on the relation between LQG and the $k=0$ LQC by calculating the
expectation value of the Hamiltonian in LQG under certain coherent state
peaked at some point in the classical phase space Dapor:2017rwv ; Han:2019vpw
; Han:2019feb ; Liegener:2020dcg ; Zhang:2020mld ; Zhang:2021qul . How to
generate these works to the $k=-1$ case will be interesting. Moreover, except
the alternative regularization from the Thiemann’s trick adopted in the
present paper following directly that in the $k=0$ case, the other alternative
regularizations employed in the $k=0$ model Liegener:2019zgw ; Yang:2019ujs ;
Yang:2020eby ; Zhang:2021zfp  can also be in principle extended to the $k=-1$
model.

###### Acknowledgements.

This work is supported in part by NSFC Grants No. 12165005, No. 11961131013,
No.12275087, No. 11775082, and “the Fundamental Research Funds for the Central
Universities”.

##  Appendix A Derivation of the expectation values

In this appendix, we present the calculations of expectation values of the
Hamiltonian constraint operator on the coherent states. To this end, let us
firstly consider an operator $\hat{H}$ with the action on $|v\rangle$

$\displaystyle\hat{H}|v\rangle$
$\displaystyle=f_{+}(v)|v+l\rangle+f_{0}(v)|v\rangle+f_{-}(v)|v-l\rangle,$
(92)

where $l$ is a positive even number, and

$\displaystyle f_{-}(v)=f_{+}(v-l).$ (93)

Then the expectation value of $\hat{H}$ on the coherent states (IV) reads

$\displaystyle\langle\Psi_{\rm grav}|\hat{H}|\Psi_{\rm grav}\rangle$
$\displaystyle=\sum_{n,n^{\prime}\in\mathbb{Z}}e^{-\frac{\epsilon^{2}}{2}(n^{\prime}-v_{o})^{2}}e^{{\rm
i}b_{o}(n^{\prime}-v_{o})}e^{-\frac{\epsilon^{2}}{2}(n-v_{o})^{2}}e^{-{\rm
i}b_{o}(n-v_{o})}$ $\displaystyle\hskip 56.9055pt\times\left[f_{+}(n)\langle
n^{\prime}|n+l\rangle+f_{-}(n)\langle n^{\prime}|n-l\rangle+f_{0}(n)\langle
n^{\prime}|n\rangle\right]$
$\displaystyle=\sum_{n\in\mathbb{Z}}e^{-\frac{\epsilon^{2}}{2}(n+l-v_{o})^{2}}e^{{\rm
i}b_{o}(n+l-v_{o})}e^{-\frac{\epsilon^{2}}{2}(n-v_{o})^{2}}e^{-{\rm
i}b_{o}(n-v_{o})}f_{+}(n)$
$\displaystyle\quad+\sum_{n\in\mathbb{Z}}e^{-\frac{\epsilon^{2}}{2}(n-l-
v_{o})^{2}}e^{{\rm i}b_{o}(n-l-
v_{o})}e^{-\frac{\epsilon^{2}}{2}(n-v_{o})^{2}}e^{-{\rm
i}b_{o}(n-v_{o})}f_{-}(n)$
$\displaystyle\quad+\sum_{n\in\mathbb{Z}}e^{-\epsilon^{2}(n-v_{o})^{2}}f_{0}(n)$
$\displaystyle=\sum_{n\in\mathbb{Z}}e^{-\frac{\epsilon^{2}}{2}(n+l-v_{o})^{2}}e^{{\rm
i}b_{o}(n+l-v_{o})}e^{-\frac{\epsilon^{2}}{2}(n-v_{o})^{2}}e^{-{\rm
i}b_{o}(n-v_{o})}f_{+}(n)$
$\displaystyle\quad+\sum_{n^{\prime}\in\mathbb{Z}}e^{-\frac{\epsilon^{2}}{2}(n^{\prime}-v_{o})^{2}}e^{{\rm
i}b_{o}(n^{\prime}-v_{o})}e^{-\frac{\epsilon^{2}}{2}(n^{\prime}+l-v_{o})^{2}}e^{-{\rm
i}b_{o}(n^{\prime}+l-v_{o})}f_{+}(n^{\prime})$
$\displaystyle\quad+\sum_{n\in\mathbb{Z}}e^{-\epsilon^{2}(n-v_{o})^{2}}f_{0}(n)$
$\displaystyle=(e^{{\rm i}lb_{o}}+e^{-{\rm
i}lb_{o}})\sum_{n^{\prime}\in\mathbb{Z}}e^{-\frac{\epsilon^{2}}{2}\left[(n^{\prime}-v_{o})^{2}+(n^{\prime}+l-v_{o})^{2}\right]}f_{+}(n^{\prime})+\sum_{n\in\mathbb{Z}}e^{-\epsilon^{2}(n-v_{o})^{2}}f_{0}(n)$
$\displaystyle=2\cos(lb_{o})e^{-\frac{l^{2}}{4}\epsilon^{2}}\sum_{n\in\mathbb{Z}}e^{-\epsilon^{2}\left(n-v_{o}\right)^{2}}f_{+}\left(n-\frac{l}{2}\right)+\sum_{n\in\mathbb{Z}}e^{-\epsilon^{2}(n-v_{o})^{2}}f_{0}(n)$
$\displaystyle=\left[\sin^{2}\left(\frac{l}{2}b_{o}\right)-\frac{1}{2}\right]e^{-\frac{l^{2}}{4}\epsilon^{2}}\left[-4\sum_{n\in\mathbb{Z}}e^{-\epsilon^{2}\left(n-v_{o}\right)^{2}}f_{+}\left(n-\frac{l}{2}\right)\right]+\frac{1}{2}\left[2\sum_{n\in\mathbb{Z}}e^{-\epsilon^{2}(n-v_{o})^{2}}f_{0}(n)\right],$
(94)

where in the third step we have used Eq. (93) and relabeled $n-l$ by
$n^{\prime}$. Applying the Possion resummation formula and the steepest decent
method, for an arbitrary analytic function $g(n)$, one has Ashtekar:2003hd

$\displaystyle\sum_{n\in\mathbb{Z}}e^{-\epsilon^{2}\left(n-v_{o}\right)^{2}}g(n)=\frac{\sqrt{\pi}}{\epsilon}g(v_{o})\left\\{1+O(e^{-\pi^{2}/\epsilon^{2}})+O\left[1/(v_{o}\epsilon)^{2}\right]\right\\}.$
(95)

Then one gets

$\displaystyle\langle\Psi_{\rm grav}|\Psi_{\rm grav}\rangle$
$\displaystyle=\sum_{n,n^{\prime}\in\mathbb{Z}}e^{-\frac{\epsilon^{2}}{2}(n^{\prime}-v_{o})^{2}}e^{{\rm
i}b_{o}(n^{\prime}-v_{o})}e^{-\frac{\epsilon^{2}}{2}(n-v_{o})^{2}}e^{-{\rm
i}b_{o}(n-v_{o})}\langle n^{\prime}|n\rangle$
$\displaystyle=\sum_{n\in\mathbb{Z}}e^{\epsilon^{2}(n-v_{o})^{2}}$
$\displaystyle=\frac{\sqrt{\pi}}{\epsilon}\left\\{1+O(e^{-\pi^{2}/\epsilon^{2}})+O\left[1/(v_{o}\epsilon)^{2}\right]\right\\},$
(96)

and

$\displaystyle\langle\Psi_{\rm grav}|\hat{H}|\Psi_{\rm grav}\rangle$
$\displaystyle=\frac{\sqrt{\pi}}{\epsilon}\left\\{e^{-\frac{l^{2}}{4}\epsilon^{2}}\left[\sin^{2}\left(\frac{l}{2}b_{o}\right)-\frac{1}{2}\right]\left[-4f_{+}\left(v_{o}-\frac{l}{2}\right)\right]+\frac{1}{2}\left[2f_{0}(v_{o})\right]\right\\}\left\\{1+O(e^{-\pi^{2}/\epsilon^{2}})+O\left[1/(v_{o}\epsilon)^{2}\right]\right\\},$
(97)

for analytic functions $f_{+}(y)$ and $f_{0}(y)$. If $f_{+}(y)$ [or
$f_{0}(y)$] is not analytic function which is the case under considerations
due to the involved absolute value, one can replace it with its analytic
extention $\bar{f}_{+}(y)$ [or $\bar{f}_{0}(y)$] (for example, omitting the
absolute value symbol). It turns out that the error
$\sum_{n}\exp(-\epsilon^{2}(n-v_{o}))[f_{+}(n)-\bar{f}_{+}(n)]$ can be shown
to be the order $O\left(e^{-v_{o}^{2}\epsilon^{2}}\right)$, which is
negligible compared to the corrections derived above Ashtekar:2003hd . Hence
the resulting normalized expectation value of $\hat{H}$ reads

$\displaystyle\langle\hat{H}\rangle$
$\displaystyle=\frac{\langle\Psi|\hat{H}|\Psi\rangle}{\langle\Psi|\Psi\rangle}=\frac{\langle\Psi_{\rm
grav}|\hat{H}|\Psi_{\rm grav}\rangle}{\langle\Psi_{\rm grav}|\Psi_{\rm
grav}\rangle}$
$\displaystyle=\left\\{e^{-\frac{l^{2}}{4}\epsilon^{2}}\left[\sin^{2}\left(\frac{l}{2}b_{o}\right)-\frac{1}{2}\right]\left[-4f_{+}\left(v_{o}-\frac{l}{2}\right)\right]+\frac{1}{2}\left[2f_{0}(v_{o})\right]\right\\}\left\\{1+O(e^{-\pi^{2}/\epsilon^{2}})+O\left[1/(v_{o}\epsilon)^{2}\right]\right\\}.$
(98)

Now we turn to the three parts of the gravitational Hamiltonian constraint
operator $\hat{\cal H}^{k=-1}_{\rm grav}$. Applying the result in Eq. (A) to
the first two terms $\hat{\cal H}^{{\rm E},k=0}_{\rm grav}$ and $\hat{\cal
H}^{{\rm L},k=0}_{\rm grav}$ with $l=4$ and $l=8$, respectively, we have

$\displaystyle\langle\hat{\cal H}^{{\rm E},k=0}_{\rm grav}\rangle$
$\displaystyle=\frac{3\hbar\gamma
v_{o}}{4\sqrt{\Delta}}\left[\sin^{2}\left(2b_{o}\right)e^{-4\epsilon^{2}}+\frac{1}{2}\left(1-e^{-4\epsilon^{2}}\right)\right]\left\\{1+O(e^{-\pi^{2}/\epsilon^{2}})+O\left[1/(v_{o}\epsilon)^{2}\right]\right\\}$
$\displaystyle=\frac{3\hbar\gamma
v_{o}}{4\sqrt{\Delta}}\left[\sin^{2}\left(2b_{o}\right)+O(\epsilon^{2})\right]\left\\{1+O(e^{-\pi^{2}/\epsilon^{2}})+O\left[1/(v_{o}\epsilon)^{2}\right]\right\\},$
(99) $\displaystyle\langle\hat{\cal H}^{{\rm L},k=0}_{\rm grav}\rangle$
$\displaystyle=\frac{3\hbar
v_{o}}{32\gamma\sqrt{\Delta}}\left[\sin^{2}\left(4b_{o}\right)e^{-16\epsilon^{2}}+\frac{1}{2}\left(1-e^{-16\epsilon^{2}}\right)\right]\left\\{1+O(e^{-\pi^{2}/\epsilon^{2}})+O\left[1/(v_{o}\epsilon)^{2}\right]\right\\}$
$\displaystyle=\frac{3\hbar
v_{o}}{32\gamma\sqrt{\Delta}}\left[\sin^{2}\left(4b_{o}\right)+O(\epsilon^{2})\right]\left\\{1+O(e^{-\pi^{2}/\epsilon^{2}})+O\left[1/(v_{o}\epsilon)^{2}\right]\right\\},$
(100)

where we have used the results

$\displaystyle-4\bar{E}_{+}(v_{o}-2)$
$\displaystyle=\frac{3\hbar\gamma}{4\sqrt{\Delta}}v_{o},\qquad
2\bar{E}_{0}(v_{o})=\frac{3\hbar\gamma}{4\sqrt{\Delta}}v_{o},$ (101)
$\displaystyle-4\bar{L}_{+}(v_{o}-4)$
$\displaystyle=\frac{3\hbar}{32\gamma\sqrt{\Delta}}v_{o},\qquad
2\bar{L}_{0}(v_{o})=\frac{3\hbar}{32\gamma\sqrt{\Delta}}v_{o}.$ (102)

To get the expectation value $\langle\hat{\cal H}^{\Gamma,k=-1}_{\rm
grav}\rangle$ from Eq. (A), we drop the term involving $f_{+}(v)$ since
$\hat{\cal H}^{\Gamma,k=-1}_{\rm grav}$ has $|v\rangle$ as its eigenstate, and
obtain

$\displaystyle\langle\hat{\cal H}^{\Gamma,k=-1}_{\rm grav}\rangle$
$\displaystyle=\frac{3\left(\gamma\sqrt{\Delta}\,\hbar\right)^{\frac{1}{3}}V_{o}^{\frac{2}{3}}}{4\left(2\pi
G\right)^{\frac{2}{3}}}v_{o}^{\frac{1}{3}}\left\\{1+O(e^{-\pi^{2}/\epsilon^{2}})+O\left[1/(v_{o}\epsilon)^{2}\right]\right\\},$
(103)

where we have used

$\displaystyle\bar{\Gamma}(v_{o})$
$\displaystyle=\frac{3\left(\gamma\sqrt{\Delta}\,\hbar\right)^{\frac{1}{3}}V_{o}^{\frac{2}{3}}}{4\left(2\pi
G\right)^{\frac{2}{3}}}v_{o}^{\frac{1}{3}}.$ (104)

## References

  * (1) C. Rovelli, Quantum Gravity (Cambridge University Press, Cambridge, England, 2004).
  * (2) T. Thiemann, Modern Canonical Quantum General Relativity (Cambridge University Press, Cambridge, England, 2007).
  * (3) A. Ashtekar and J. Lewandowski, Background independent quantum gravity: A status report, Class. Quant. Grav. 21, R53 (2004).
  * (4) M. Han, Y. Ma, and W. Huang, Fundamental structure of loop quantum gravity, Int. J. Mod. Phys. D 16, 1397 (2007).
  * (5) C. Rovelli and L. Smolin, Discreteness of area and volume in quantum gravity, Nucl. Phys. B 442, 593 (1995).
  * (6) A. Ashtekar and J. Lewandowski, Quantum theory of geometry: I. Area operators, Class. Quant. Grav. 14, A55 (1997).
  * (7) A. Ashtekar and J. Lewandowski, Quantum theory of geometry II: Volume operators, Adv. Theor. Math. Phys. 1, 388 (1997).
  * (8) J. Yang and Y. Ma, New volume and inverse volume operators for loop quantum gravity, Phys. Rev. D 94, 044003 (2016).
  * (9) T. Thiemann, A length operator for canonical quantum gravity, J. Math. Phys. 39, 3372 (1998).
  * (10) Y. Ma, C. Soo, and J. Yang, New length operator for loop quantum gravity, Phys. Rev. D 81, 124026 (2010).
  * (11) A. Ashtekar, J. Baez, A. Corichi, and K. Krasnov, Quantum Geometry and Black Hole Entropy, Phys. Rev. Lett. 80, 904 (1998).
  * (12) S. Song, H. Li, Y. Ma, and C. Zhang, Entropy of black holes with arbitrary shapes in loop quantum gravity, Sci. China Phys. Mech. Astron. 64, 120411 (2021).
  * (13) X. Zhang and Y. Ma, Extension of Loop Quantum Gravity to $f(R)$ Theories, Phys. Rev. Lett. 106, 171301 (2011).
  * (14) X. Zhang and Y. Ma, Loop quantum $f(R)$ theories, Phys. Rev. D 84, 064040 (2011).
  * (15) X. Zhang and Y. Ma, Loop quantum Brans-Dicke theory, J. Phys. Conf. Ser. 360, 012055 (2012).
  * (16) X. Zhang and Y. Ma, Nonperturbative loop quantization of scalar-tensor theories of gravity, Phys. Rev. D 84, 104045 (2011).
  * (17) N. Bodendorfer, T. Thiemann, and A. Thurn, New variables for classical and quantum gravity in all dimensions: I. Hamiltonian Analysis, Class. Quant. Grav. 30, 045001 (2013).
  * (18) X. Zhang, J. Yang, and Y. Ma, Canonical loop quantization of the lowest-order projectable Horava gravity, Phys. Rev. D 102, 124060 (2020).
  * (19) A. Ashtekar, M. Bojowald, and J. Lewandowski, Mathematical structure of loop quantum cosmology, Adv. Theor. Math. Phys. 7, 233 (2003).
  * (20) A. Ashtekar and M. Bojowald, Quantum geometry and the Schwarzschild singularity, Class. Quant. Grav. 23, 391 (2006).
  * (21) M. Bojowald, Loop quantum cosmology, Living Rev. Rel. 8, 11 (2005).
  * (22) A. Ashtekar, T. Pawlowski, and P. Singh, Quantum nature of the big bang: Improved dynamics, Phys. Rev. D 74, 084003 (2006).
  * (23) A. Ashtekar and P. Singh, Loop quantum cosmology: A status report, Class. Quant. Grav. 28, 213001 (2011).
  * (24) T. Thiemann, Quantum spin dynamics (QSD), Class. Quant. Grav. 15, 839 (1998).
  * (25) M. Bojowald, Isotropic loop quantum cosmology, Class. Quant. Grav. 19, 2717 (2002).
  * (26) A. B. Henriques, Loop quantum cosmology and the Wheeler-De Witt equation, Gen. Rel. Grav. 38, 1645 (2006).
  * (27) J. Yang, Y. Ding, and Y. Ma, Alternative quantization of the Hamiltonian in loop quantum cosmology, Phys. Lett. B 682, 1 (2009).
  * (28) M. Assanioussi, A. Dapor, K. Liegener, and T. Pawłowski, Emergent de Sitter Epoch of the Quantum Cosmos from Loop Quantum Cosmology, Phys. Rev. Lett. 121, 081303 (2018).
  * (29) B.-F. Li, P. Singh, and A. Wang, Towards cosmological dynamics from loop quantum gravity, Phys. Rev. D 97, 084029 (2018).
  * (30) X. Zhang, G. Long, and Y. Ma, Loop quantum gravity and cosmological constant, Phys. Lett. B 823, 136770 (2021).
  * (31) K. Vandersloot, Loop quantum cosmology and the $k=-1$ Robertson-Walker model, Phys. Rev. D 75, 023523 (2007).
  * (32) L. Szulc, An open FRW model in loop quantum cosmology, Class. Quant. Grav. 24, 6191 (2007).
  * (33) L. Szulc, W. Kaminski, and J. Lewandowski, Closed FRW model in loop quantum cosmology, Class. Quant. Grav. 24, 2621 (2007).
  * (34) A. Ashtekar, T. Pawlowski, P. Singh, and K. Vandersloot, Loop quantum cosmology of $k=1$ FRW models, Phys. Rev. D 75, 024035 (2007).
  * (35) J. Mielczarek, O. Hrycyna, and M. Szydlowski, Effective dynamics of the closed loop quantum cosmology, JCAP 11, 014 (2009).
  * (36) A. Corichi and A. Karami, Loop quantum cosmology of $k=1$ FRW: A tale of two bounces, Phys. Rev. D 84, 044003 (2011).
  * (37) A. Corichi and A. Karami, Loop quantum cosmology of $k=1$ FLRW: Effects of inverse volume corrections, Class. Quant. Grav. 31, 035008 (2014).
  * (38) J. L. Dupuy and P. Singh, Implications of quantum ambiguities in $k=1$ loop quantum cosmology: Distinct quantum turnarounds and the super-Planckian regime, Phys. Rev. D 95, 023510 (2017).
  * (39) J. F. Barbero, Real Ashtekar variables for Lorentzian signature space-times, Phys. Rev. D 51, 5507 (1995).
  * (40) G. Immirzi, Quantum gravity and Regge calculus, Nucl. Phys. B - Proc. Suppl. 57, 65 (1997).
  * (41) Y. Ding, Y. Ma, and J. Yang, Effective scenario of loop quantum cosmology, Phys. Rev. Lett. 102, 051301 (2009).
  * (42) A. Ashtekar, Loop quantum cosmology: An overview, Gen. Rel. Grav. 41, 707 (2009).
  * (43) A. Ashtekar and E. Wilson-Ewing, Loop quantum cosmology of Bianchi type II models, Phys. Rev. D 80, 123532 (2009).
  * (44) M. Assanioussi, A. Dapor, K. Liegener, and T. Pawłowski, Emergent de Sitter epoch of the loop quantum cosmos: A detailed analysis, Phys. Rev. D 100, 084003 (2019).
  * (45) C. Zhang, J. Lewandowski, H. Li, and Y. Ma, Bouncing evolution in a model of loop quantum gravity, Phys. Rev. D 99, 124012 (2019).
  * (46) V. Taveras, Corrections to the Friedmann equations from loop quantum gravity for a universe with a free scalar field, Phys. Rev. D 78, 064072 (2008).
  * (47) A. Ashtekar, M. Campiglia, and A. Henderson, Casting loop quantum cosmology in the spin foam paradigm, Class. Quant. Grav. 27, 135020 (2010).
  * (48) L. Qin, G. Deng, and Y. Ma, Path integrals and alternative effective dynamics in loop quantum cosmology, Commun. Theor. Phys. 57, 326 (2012).
  * (49) K. Liegener and S. A. Weigl, Effective LQC model for $k$ = +1 isotropic cosmologies from spatial discretizations, Phys. Rev. D 102, 106014 (2020).
  * (50) A. Dapor and K. Liegener, Cosmological effective Hamiltonian from full loop quantum gravity dynamics, Phys. Lett. B 785, 506 (2018).
  * (51) M. Han and H. Liu, Effective dynamics from coherent state path integral of full loop quantum gravity, Phys. Rev. D 101, 046003 (2020).
  * (52) M. Han and H. Liu, Improved $\bar{\mu}$-scheme effective dynamics of full loop quantum gravity, Phys. Rev. D 102, 064061 (2020).
  * (53) K. Liegener and L. Rudnicki, Algorithmic approach to cosmological coherent state expectation values in loop quantum gravity, Class. Quant. Grav. 38, 205001 (2021).
  * (54) C. Zhang, S. Song, and M. Han, First-order quantum correction in coherent state expectation value of loop-quantum-gravity Hamiltonian: Overview and results, arXiv:2012.14242.
  * (55) C. Zhang, S. Song, and M. Han, First-order quantum correction in coherent state expectation value of loop-quantum-gravity Hamiltonian, Phys. Rev. D 105, 064008 (2022).
  * (56) K. Liegener and P. Singh, Gauge-invariant bounce from loop quantum gravity, Class. Quant. Grav. 37, 085015 (2020).
  * (57) J. Yang, C. Zhang, and Y. Ma, Loop quantum cosmology from an alternative Hamiltonian, Phys. Rev. D 100, 064026 (2019).
  * (58) J. Yang, C. Zhang, and Y. Ma, Loop quantum cosmology from an alternative Hamiltonian. II. Including the Lorentzian term, Phys. Rev. D 102, 084018 (2020).