# Quantum cosmology as a lattice in a box

Mustafa Saeed<EMAIL_ADDRESS><EMAIL_ADDRESS>Centre College,
Danville, KY, USA 40422  Department of Mathematics and Statistics, University
of New Brunswick, Fredericton, NB, Canada E3B 5A3  Viqar Husain
<EMAIL_ADDRESS>Department of Mathematics and Statistics, University of New
Brunswick, Fredericton, NB, Canada E3B 5A3

###### Abstract

We describe quantization schemes for scalar field cosmology in the metric
variables with fundamental discreteness imposed with a lattice. The variables
chosen for quantization determine the lattice, and each lattice produces
distinct effective equations derived from semiclassical states. We show that
requiring a bounce at the Planck density uniquely selects the volume lattice
and gives the same effective Friedmann equation as that obtained in loop
quantum cosmology. We also present conditions for the validity of the
effective equations.

##  I Introduction

Efforts to quantize gravity date back to the 1930s. The canonical approach
encountered conceptual and technical issues such as the problem of time,
selection of canonical variables, physical Hilbert space, and interpretation
of state vectors of the gravitational field DeWitt (1967). It was realized
that these problems can be more easily addressed in symmetry reduced models
such as Friedmann-Lemaître-Robertson-Walker (FLRW) universe with a scalar
field. Early work on such models used the Arnowitt-Deser-Misner (ADM)
canonical variables DeWitt (1967); Blyth and Isham (1975).

Another method for quantizing cosmological models is derived from loop quantum
gravity (LQG). In LQG, general relativity is expressed in terms of connection-
triad canonical variables and the quantum theory is constructed from a
representation of the so-called holonomy-flux algebra—the holonomy of the
connection on loops and the integral of the triad over surfaces Ashtekar and
Lewandowski (2004). In LQG it is found that the area operator has a discrete
spectrum Rovelli and Smolin (1995). A related quantization of cosmological
models - called loop quantum cosmology (LQC) - uses a particular set of
reduced holonomy-flux variables Ashtekar et al. (2006); Ashtekar and Singh
(2011); this quantization is inequivalent to the Schrödinger quantization
Halvorson (2004); Ashtekar et al. (2003) that is usually deployed. A notable
success of LQC is the resolution of the big bang singularity via a bounce
where matter reaches the Planck scale density.

The bounce also arises in an effective classical theory Taveras (2008);
Rovelli and Wilson-Ewing (2014) where the Hamiltonian constraint opertator is
replaced by its expectation value in a suitably defined semiclassical state
Husain and Winkler (2007). This effective constraint is then used to derive
quantum corrected equations of motion. The effective theory is characterized
by the fiducial coordinate volume, a discreteness scale proportional to the
Planck scale, and a particular choice of reduced holonomy-flux algebra that is
argued to descend from discreteness of area in full LQG.

In this note we quantize ADM variables for cosmology after discretization on a
lattice. Our purpose is to compare the results with LQC, specifically the
choice of variables for quantization, the derivation of the effective
Friedmann equation, and the role of fiducial volume Rovelli and Wilson-Ewing
(2014). We use scale-invariant and dimensionless variables that remove the
fiducial volume at the outset, and use an initially undetermined power of the
scale factor as the configuration variable. This facilitates identifying the
key features necessary to obtain physically acceptable effective equations in
the ADM variables. To derive the effective equations we use the semiclassical
states defined in Husain and Winkler (2007), and note similarities with work
where a direct “polymerization” is applied to the non-scale invariant ADM
variables Montani et al. (2019); Giovannetti et al. (2022).

In Section II we summarize classical cosmology in terms of scale-invariant
dimensionless variables; in Sec. III we give the general lattice quantization
for any function of the scale factor as the configuration variable; in Sec. IV
we derive the effective equations from semiclassical states, and the
conditions for their validity; lastly in Sec. V we give a brief discussion of
the main result — that the effective theory on a volume lattice in the ADM
variables is equivalent to the LQC effective theory.

##  II Hamiltonian Cosmology

The canonical action of the FLRW universe minimally coupled to a massless
scalar field in the ADM canonical variables is

$\begin{split}S=&V_{0}\int dt\left[\left(\frac{1}{8\pi
G}\right)\dot{a}p_{a}+\dot{\phi}p_{\phi}-N\mathcal{H}\right];\\\
\mathcal{H}=&-\left(\frac{1}{8\pi
G}\right)\left(\frac{p_{a}{}^{2}}{24a}\right)+\frac{p_{\phi}{}^{2}}{2a^{3}}\approx
0.\end{split}$ (II.1)

$V_{0}=\int{d^{3}x}$ is the volume of the cubical fiducial cell (the
coordinate box) to which the calculations are restricted, $(a,p_{a})$ and
$(\phi,p_{\phi})$ are the phase space variables for gravity and scalar field
respectively and $N$ is the lapse function, and overdots indicate derivatives
with respect to coordinate time $t$ with metric

$ds^{2}=-N^{2}dt^{2}+a^{2}(t)(dx^{2}+dy^{2}+dz^{2}).$ (II.2)

The Poisson bracket identified from the action

$\\{a,p_{a}\\}=\frac{8\pi G}{V_{0}}$ (II.3)

depends on the fiducial volume but the equations of motion do not; these along
with the Hamiltonian constraint lead to the Friedmann equation

$\left(\frac{\dot{a}}{a}\right)^{2}=\left(\frac{4\pi G}{3}\right)\rho,$ (II.4)

where $\rho=p_{\phi}^{2}/2a^{6}$ is scalar energy density.

In these canonical variables it appears that the quantum theory will depend on
the coordinate volume $V_{0}$ if the usual quantization rule of converting the
Poisson bracket (II.3) into a commutator is followed. Although this would be
acceptable for quantum theory on a fixed background where physical box size is
an infrared cutoff, here the physical volume is $a^{3}V_{0}$, which is a
dynamical variable. Furthermore, the metric is invariant under the scale
factor and coordinate rescalings

$a\longrightarrow a/\lambda,\quad x\rightarrow\lambda x$ (II.5)

and the reduced action is invariant provided

$p_{a}\rightarrow p_{a}/\lambda^{2}.$ (II.6)

Therefore it is natural to introduce phase space variables that are invariant
under these rescalings. These are

$A=V_{0}\,^{1/3}a,\quad\quad p_{A}=V_{0}\,^{2/3}p_{a};$ (II.7)

the Poisson bracket is then

$\\{A,p_{A}\\}=8\pi G,$ (II.8)

and removes the dependence of the Poisson bracket on the arbitrary coordinate
box volume $V_{0}$. Since this rescaling gives the scale factor the dimension
of length, appropriate factors of $\sqrt{G}$ (or alternatively the Planck
length $l_{P}$) may be used to obtain scale invariant and dimensionless phase
space variables, and the time parameter. The final variables we use are

$\displaystyle\bar{a}$ $\displaystyle=$
$\displaystyle\left(\frac{V_{0}}{8\pi}\right)^{1/3}\frac{a}{\sqrt{G}},\quad\bar{p}_{\bar{a}}=\left(\frac{V_{0}}{8\pi}\right)^{2/3}\frac{p_{a}}{\sqrt{G}}$
$\displaystyle\bar{\phi}$ $\displaystyle=$ $\displaystyle\sqrt{8\pi
G}\phi,\quad\quad\quad\bar{p}_{\bar{\phi}}=\left(\frac{V_{0}}{\sqrt{8\pi
G}}\right)p_{\phi};$ (II.9)

these lead to the Hamiltonian constraint

$\displaystyle\mathcal{H}=-\left(\frac{\bar{p}^{2}_{\bar{a}}}{24\bar{a}}\right)+\frac{\bar{p}^{2}_{\bar{\phi}}}{\bar{a}^{3}}\approx
0$ (II.10)

and Friedmann equation

$\left(\frac{\dot{\bar{a}}}{\bar{a}}\right)^{2}=\frac{\bar{\rho}}{6},\quad\bar{\rho}=\frac{\bar{p}_{\bar{\phi}}^{2}}{2\bar{a}^{6}}.$
(II.11)

In the following we drop the over bars.

Our goal is to use appropriate functions of the variables (II) for a lattice
quantization. To that end we note that the canonical pair

$\displaystyle q=f(a),\quad p=\frac{p_{a}}{f^{\prime}(a)},$ (II.12)

where $f(a)$ is any once differentiable function, satisfies $\\{q,p\\}=1$.
Since only finite translation and not momentum operators are definable for
quantization on a lattice, it is useful to consider the alternative Poisson
bracket

$\left\\{q,\exp\left(i\lambda p\right)\right\\}=i\lambda\exp\left(i\lambda
p\right)$ (II.13)

for quantization; $q$ then determines the type of “adaptive mesh”
configuration lattice with spacing parameter $\lambda$; each such lattice
leads to different physics in the effective quantum theory, as we see below.
For the particular choice

$q\equiv f(a)=a^{n},\quad p\equiv\frac{p_{a}}{na^{n-1}}\quad(n>0)$ (II.14)

the Hamiltonian constraint takes the form

${\mathcal{H}}=-\left(\frac{n^{2}}{24}\right)q^{\frac{2n-3}{n}}p^{2}+\frac{p_{\phi}^{2}}{2q^{\frac{3}{n}}}\approx
0.$ (II.15)

This is the case we consider below for deriving effective equations.

##  III Lattice Quantization

With the variables $(q,p)$ defined above, the quantization of the Poisson
bracket

$\displaystyle\\{q,U_{\lambda}(p)\\}=i\lambda U_{\lambda}(p),\quad
U_{\lambda}(p)=e^{i\lambda p}$ (III.1)

is realized on the Hilbert with basis $|q\rangle$ and inner product

$\langle
q|q^{\prime}\rangle=\lim_{T\rightarrow\infty}\frac{1}{2T}\int_{-T}^{T}d\mu\
e^{-iq\mu}e^{iq^{\prime}\mu}=\delta_{q,q^{\prime}}$ (III.2)

by defining the operators Halvorson (2004); Ashtekar et al. (2003)

$\hat{q}|q\rangle=q|q\rangle,\quad\hat{U}_{\lambda}|q\rangle=|q-\lambda\rangle.$
(III.3)

The resulting commutator algebra is

$[\hat{q},\hat{U}_{\lambda}]=-\lambda\hat{U}_{\lambda}.$ (III.4)

$\hat{U}_{\lambda}\,^{\dagger}=\hat{U}_{-\lambda}$; the parameter $\lambda$ is
an arbitrary real number and any particular choice fixes a lattice.

Defining the Hamiltonian constraint (II.15) as an operator requires defining
operators for $p^{2}$ and for powers of $q$, including inverse powers
depending on the choice of $n$. The former must be defined using
$U_{\lambda}$; one choice is squaring the momentum operator

$\hat{p}_{\lambda}=\frac{1}{2i\lambda}\left(\hat{U}_{\lambda}-\hat{U}_{-\lambda}\right).$
(III.5)

Negative powers of $q$ may be defined using the type of identity given in
Thiemann (1998):

$\widehat{\frac{1}{|q|}}=\frac{1}{\lambda^{2}}\left(\hat{U}_{\lambda}\sqrt{|\hat{q}|}\,\hat{U}_{-\lambda}-\hat{U}_{-\lambda}\sqrt{|\hat{q}|}\,\hat{U}_{\lambda}\right)^{2}.$
(III.6)

The standard Schrödinger quantization is used for the matter sector since we
want to introduce fundamental discreteness for the gravity sector only.

###  III.1 Semiclassical states

We summarize here an approach used to define effective equations for cosmology
that makes use of semiclassical states Husain and Winkler (2007); Taveras
(2008). These are Gaussian states of the form
$\psi_{\tilde{q},\tilde{p}}(q)\sim e^{-(q-\tilde{q})^{2}+i\tilde{p}q}$ where
$(\tilde{q},\tilde{p})$ is the point in phase space (defined for the
representation given above) on which the state is peaked. With such states
suitably defined, the expectation value of the Hamiltonian constraint operator
is computed. This gives the effective constraint ${\cal H}_{\rm
eff}(\tilde{q},\tilde{p})$, which is then used to derive effective equations
by imposing a Poisson bracket on the peaking values $\tilde{q}$ and
$\tilde{p}$.

The first step in this procedure is to take the lattice states
$|q_{m}\rangle=|m\lambda\rangle\equiv|m\rangle_{\lambda}$, $m\in\mathbb{Z}$.
Semiclassical states are defined as in Husain and Winkler (2007):

$\displaystyle\ket{\tilde{q},\tilde{p};{t,\lambda}}=\frac{1}{C}\sum^{\infty}_{m=-\infty}\exp\left[-\frac{t}{2}\left(m\lambda-\tilde{q}\right)^{2}+im\lambda\tilde{p}\right]\ket{m}_{\lambda},$
(III.7)

where the normalization constant is

$\displaystyle
C^{2}=\exp\left(-t\tilde{q}^{2}\right)\sum_{m=-\infty}^{\infty}\exp\left[-t\left(m\lambda\right)^{2}+2m\lambda\tilde{q}t\right],$
(III.8)

and $t$ is the width of the state and $\left(\tilde{q},\tilde{p}\right)$ are
the classical phase space points at which the state is peaked.

Expectation values of the elementary operators in these states are the
following:

$\displaystyle\lim_{t\rightarrow 0}\big{<}\hat{q}\big{>}_{t,\lambda}$
$\displaystyle=$ $\displaystyle\tilde{q}$ $\displaystyle\lim_{t\rightarrow
0}\big{<}\hat{U}_{\lambda}\big{>}_{t,\lambda}$ $\displaystyle=$
$\displaystyle\exp\left(i\lambda\tilde{p}\right)$
$\displaystyle\lim_{t\rightarrow 0}\big{<}\hat{p}\big{>}_{t,\lambda}$
$\displaystyle=$ $\displaystyle\,\frac{\sin(\lambda\tilde{p})}{\lambda}.$
(III.9)

These are derived using the Poisson resummation formula Husain and Winkler
(2007). These results are useful for what follows since it bounds the momentum
expectation value in the semiclassical state. This is the analog of the
“holonomy correction” in LQC.

Lastly the semiclassical state for the scalar field in the $\phi$
representation is taken to be

$\displaystyle\psi_{\tilde{\phi},\tilde{p}_{\tilde{\phi}}}(\phi)=\left(\frac{1}{\pi\bar{t}}\right)^{\frac{1}{4}}e^{-\left(\phi-\tilde{\phi}\right)^{2}/2\bar{t}+i\phi\tilde{p}_{\tilde{\phi}}},$
(III.10)

where again $\bar{t}$ is the width of the Gaussian state and
$(\tilde{\phi},\tilde{p}_{\tilde{\phi}})$ are the phase space points on which
the state is peaked. This is not defined on a lattice since the idea is to
introduce discreteness only for the gravitational variables.

##  IV Effective cosmology

We define the effective Hamiltonian constraint as the expectation value of the
Hamiltonian constraint operator in the semiclassical state. With a symmetric
ordering this gives an expression of the type

$\displaystyle\hskip-14.22636pt{\cal H}_{\rm
eff}(\tilde{q},\tilde{p},\tilde{\phi},\tilde{p}_{\phi};\lambda,t)$
$\displaystyle\hskip-14.22636pt=-\left(\frac{n^{2}}{24}\right)\langle\widehat{q^{\frac{2n-3}{2n}}}\widehat{p}^{2}_{\lambda}\widehat{{q}^{\frac{2n-3}{2n}}}\rangle+\frac{\langle\hat{p}_{\phi}^{2}\rangle}{2}\langle\widehat{q^{-\frac{3}{n}}}\rangle.$
(IV.1)

This is tedious but straightforward to calculate. A simplification based on
the results (III.1) that captures the fact that
$\langle\widehat{p}_{\lambda}\,^{2}\rangle$ is bounded above is given by the
following expression for the effective Hamiltonian constraint in the limit
$t\rightarrow 0$:

$\displaystyle\mathcal{H}_{\rm eff}$ $\displaystyle=$
$\displaystyle-\left(\frac{n^{2}}{24}\right)q^{\frac{2n-3}{n}}\left[\frac{\sin(\lambda
p)}{\lambda}\right]^{2}+\frac{p_{\phi}^{2}}{2q^{\frac{3}{n}}}$ (IV.2)
$\displaystyle\approx$ $\displaystyle 0,$

where we have dropped the tilde on the phase space point; the only difference
between this expression and the classical one (II.15) is the sine term
replacing $p^{2}$; the arguments used in deriving this are based on the
results of Husain and Winkler (2007), which are similar to those used in the
derivation of the LQC effective equation Taveras (2008).

###  IV.1 Effective Friedmann equation

Treating the effective Hamiltonian constraint (IV.2) as a classical expression
we can derive the corresponding canonical equations of motion with lapse
$N=1$:

$\displaystyle\begin{split}\dot{q}=&-\frac{n^{2}}{12\lambda}\,q^{\frac{2n-3}{n}}\sin(\lambda
p)\cos(\lambda p),\\\
\dot{p}=&\,\frac{n\left(3-2n\right)}{24}q^{\frac{n-3}{n}}\left[\frac{\sin(\lambda
p)}{\lambda}\right]^{2}\\\
&-\frac{3}{2n}\,p^{2}_{\phi}\,q^{\frac{-(3+n)}{n}},\\\
\dot{\phi}=&\,p_{\phi}\,q^{-\frac{3}{n}},\quad\quad\dot{p}_{\phi}=\,0.\end{split}$
(IV.3)

The square of the Hubble parameter is

$\displaystyle H^{2}$ $\displaystyle=$
$\displaystyle\left(\frac{\dot{q}}{nq}\right)^{2}$ (IV.4) $\displaystyle=$
$\displaystyle\left(\frac{n}{12\,\lambda}\right)^{2}q^{\frac{2(n-3)}{n}}\sin^{2}\left(\lambda
p\right)$ $\displaystyle\times\left[1-\sin^{2}\left(\lambda p\right)\right].$

This is simplified by noting that the Hamiltonian constraint (IV.2) gives

$\displaystyle\sin^{2}\left(\lambda
p\right)=\left(\frac{12\,\lambda}{n}\right)^{2}q^{-\frac{2(n-3)}{n}}\left(\frac{\rho}{6}\right),$
(IV.5)

where $\rho=p^{2}_{\phi}/2q^{6/n}$ is the scalar field energy density.
Substituting this expression into (IV.4) gives the effective Friedmann
equation

$\displaystyle
H^{2}=\left(\frac{\rho}{6}\right)\left[1-q^{-\frac{2(n-3)}{n}}\left(\frac{12\,\lambda}{n}\right)^{2}\left(\frac{\rho}{6}\right)\right].$
(IV.6)

In summary so far, we have shown that starting with the phase space variables
(II.14) in the ADM formalism, preforming a lattice (or polymer) quantization
of the gravity variables and the standard quantization of the scalar field,
and lastly, using the semiclassical states (III.7), yields the effective
Friedmann equation (IV.6). Since the l.h.s. is positive, this equation is
physically meaningful provided

$\rho\leq 6\left(\frac{n}{12\lambda}\right)^{2}q^{\frac{2(n-3)}{n}},$ (IV.7)

and it predicts a bounce when equality holds, i.e. there is a maximum density
which depends on the scale factor.

But this last feature does not make physical sense since a maximum density in
quantum gravity should be independent of the scale factor, and should depend
only on the Planck density up to an overall dimensionless constant. Hence the
effective Friedmann equation is physically meaningful only if $n=3$, i.e. the
scale invariant (dimensionless) physical volume $v=V_{0}\,a^{3}/(8\pi
l_{P}\,^{3})$ is used as the configuration variable. This gives the critical
density

$\rho_{c}\equiv\frac{3}{8\lambda^{2}}$ (IV.8)

in Planck units. Recalling that $\lambda$ is dimensionless lattice parameter,
the effective Friedmann equation becomes

$H^{2}=\left(\frac{\rho}{6}\right)\left[1-\left(\frac{\rho}{\rho_{c}}\right)\right].$
(IV.9)

This result is exactly the same in form as that derived in LQC using the
argument that area is quantized Agullo and Singh (2017) for the elementary
plaquette used to define the curvature of the Ashtekar-Barbero connection
using holonomies in the Hamiltonian constraint. Here the steps and argument
are different, but lead to the same physical result.

###  IV.2 Validity of the effective equation

The effective equations derived above are valid provided quantum fluctuations
are small; this is because the regime of large quantum fluctuations should not
be describable by effective classical equations composed of a few expectation
values. The natural variable for thinking about quantum fluctuations in the
$n=3$ case is the physical volume.

The relevant question is whether volume quantum fluctuations are large at the
bounce point, where the critical density is given by (IV.8) and the
corresponding physical volume (obtained by setting $\rho=\rho_{c}$) is

$v_{\rm bounce}=\frac{p_{\phi}}{\sqrt{2\rho_{c}}}=\frac{2\lambda}{\sqrt{3}}\
p_{\Phi};$ (IV.10)

the scalar field momentum $p_{\phi}$ is a constant of the motion. For the
physically relevant case $n=3$, the phase space variables are the volume and
its conjugate momentum $(v,p_{v})$. For these variables the momentum operator
as defined in (III.5) and the commutator
$\displaystyle\left[\widehat{v},\widehat{U}_{\lambda}\right]=-\lambda\,\widehat{U}_{\lambda}$
lead to the uncertainty relation

$\Delta v\,\Delta\left[\frac{\sin\left(\lambda
p_{v}\right)}{\lambda}\right]\geq\frac{1}{2}\big{|}\big{<}\cos\left(\lambda
p_{v}\right)\big{>}\big{|}.$ (IV.11)

The effective Hamiltonian constraint (IV.2) (with $n=3$) gives

$\frac{\sin\lambda p_{v}}{\lambda}=\frac{2}{\sqrt{3}}\frac{p_{\phi}}{v};$
(IV.12)

substituting this into (IV.11) gives the quantum fluctuation relation

$\left(\frac{\Delta
v}{v}\right)^{2}\geq\frac{\sqrt{3}}{4p_{\phi}}\,\,\big{|}\cos\left(\lambda
p_{v}\right)\big{|}.$ (IV.13)

This shows that minimum uncertainty in the physical volume is controlled by
the value of (the constant of motion) $p_{\phi}$. Therefore for minimum
uncertainty states, $\Delta v/v\ll 1$ provided

$p_{\phi}\gg\frac{\sqrt{3}}{4}\big{|}\cos\left(\lambda p_{v}\right)\big{|}$
(IV.14)

(in Planck units), a result independent of the physical volume $v$. This is
the condition of validity of the semiclassical FLRW-scalar equations as
derived here, and it is apparently independent of the fiducial volume if the
scale invariant scalar momentum is used. However, if we rewrite (IV.14) using
the non-scale invariant scalar momentum $\tilde{p}_{\phi}\equiv
p_{\phi}/V_{0}$, then we find using (II), that

$\tilde{p}_{\phi}\gg\frac{\sqrt{3}}{4V_{0}}\big{|}\cos\left(\lambda
p_{v}\right)\big{|};$ (IV.15)

this means that it is possible to use a sufficiently large $V_{0}$ to satisfy
the inequality; i.e. $\tilde{p}_{\phi}$ need not be at the Planck scale. This
latter inequality is consistent with the condition derived in Rovelli and
Wilson-Ewing (2014) in the connection-triad variables using a similar
analysis.

##  V Summary and discussion

Our main goal in this paper was to show that the FLRW model quantized on a
lattice in a box using the ADM variables gives the same result for the
effective Friedmann dynamics as that derived in LQC. This was done in several
steps: using scale invariant variables for the classical theory that removes
box size (the fiducial volume $V_{0}$), quantizing using an arbitrary power of
the scale factor and its conjugate momentum, using semiclassical states to
derive an effective dynamics, and lastly noting that the effective Friedmann
equation makes physical sense only for a particular choice of configuration
variable: the physical volume of the universe. This means that quantizing on a
volume lattice leads to a natural bounce, with the lattice spacing determining
the critical density; the box coordinate volume does not appear, and neither
does the Immirzi-Barbero parameter of LQG. Despite these differences, our
analysis is comparable to that in the connection-triad variables Corichi and
Singh (2008) with the same final result: the volume and its conjugate momentum
are the preferred phase space variables if the bounce is to occur at the
Planck density.

We also discussed the regime of validity of the effective equations by
deriving a condition for volume fluctuations to be small; this requires the
scale invariant scalar field momentum to be larger than unity in Planck units,
a condition independent of the fiducial volume $V_{0}$. This may be compared
to the analysis in Rovelli and Wilson-Ewing (2014), where the result requires
large $V_{0}$ if the scale non-invariant momentum is used. However our result
matches that of Rovelli and Wilson-Ewing (2014) for scale invariant variables.
We think that the true physical description is provided by variables that are
invariant under coordinate rescalings (which are physically unimportant); thus
the scale invariant variables are the natural ones to use when quantizing FLRW
models in geometrodynamics or connection dynamics.

As noted above, the selection of volume as the relevant configuration variable
arises here by imposing the physical requirement that the bounce occurs at a
critical density that is independent of any function of the scale factor, and
instead is of the order of the Planck density. In LQC the same choice arises
by importing the quantization of area from LQG, and deriving an effective
Hamiltonian constraint by realizing the curvature of the Ashtekar-Barbero
connection as a holonomy around a minimal physical area loop; thus, there one
could argue that “quantization of area leads to the bounce,” whereas here it
comes from requiring that the bounce occur at Planck density.

We conclude by noting that similar lattice quantizations in the ADM variables
may be applied with appropriate physical inputs to other cosmological models
discussed in LQC, and possibly also to models in spherical symmetry Husain and
Winkler (2005); Husain and Terno (2010).

Acknowledgements This work was supported by the NSERC of Canada and the FDC
award of Centre College. We thank Edward Wilson-Ewing for helpful comments on
the manuscript and for pointing out reference Corichi and Singh (2008).

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