Semiclassical states, effective dynamics, and classical emergence in loop quantum cosmology
Parampreet Singh
1,
*
and Kevin Vandersloot
1,2,†
1
Institute for Gravitational Physics and Geometry, The Pennsylvania State University, 104 Davey Lab, University Park,
Pennsylvania 16802, USA
2
Max-Planck-Institut fu
¨
r Gravitationsphysik, Albert-Einstein-Institut, Am Mu
¨
hlenberg 1, D-14476 Golm, Germany
(Received 15 July 2005; published 6 October 2005)
We construct physical semiclassical states annihilated by the Hamiltonian constraint operator in the
framework of loop quantum cosmology as a method of systematically determining the regime and validity
of the semiclassical limit of the quantum theory. Our results indicate that the evolution can be effectively
described using continuous classical equations of motion with nonperturbative corrections down to near
the Planck scale below which the Universe can only be described by the discrete quantum constraint.
These results, for the first time, provide concrete evidence of the emergence of classicality in loop
quantum cosmology and also clearly demarcate the domain of validity of different effective theories. We
prove the validity of modified Friedmann dynamics incorporating discrete quantum geometry effects
which can lead to various new phenomenological applications. Furthermore the understanding of
semiclassical states allows for a framework for interpreting the quantum wave functions and under-
standing questions of a semiclassical nature within the quantum theory of loop quantum cosmology.
DOI: 10.1103/PhysRevD.72.084004 PACS numbers: 04.60.Pp, 04.60.Kz, 98.80.Qc
I. INTRODUCTION
A fundamental input of loop quantum gravity (LQG) [1]
to our understanding of quantum spacetime is that it is
inherently discrete and spatial geometry is quantized.
Quantum features of spacetime become evident in the
regime of very high curvature whereas continuous space-
time emerges as a large eigenvalue limit of quantum ge-
ometry. Perhaps the most interesting avenue to explore this
idea is in cosmological Friedmann-Robertson-Walker
(FRW) spacetimes in the regime of high curvature and
small volume such that quantum gravitational effects are
expected to be dominant with the possibility of potentially
observable signatures. It is a fundamental question in any
viable model of LQG as to the scale at which the classical
picture can be recovered and where precisely we expect to
see modifications to the classical Friedmann dynamics.
Since tools to address these issues are still under develop-
ment [2], it is an open question whether the picture sug-
gested by LQG holds for our Universe.
To construct a FRW model within the field theoretic
framework of LQG would be quite difficult. Thus much
progress has been made by restricting the model to a mini-
superspace quantization known as loop quantum cosmol-
ogy (LQC). In this simplified setting fundamental ques-
tions can be answered directly with explicit calculations
and the physical consequences can be explored. As in
LQG, the underlying geometry in LQC is discrete and the
scale factor operator has discrete eigenvalues. Quantum
dynamics is governed by a discrete difference equation
which leads to a nonsingular evolution through the classi-
cal big bang singularity [3,4]. This important result can be
traced directly to the discrete nature of quantum geometry.
A related and important feature of LQC is the modification
to the behavior of the eigenvalues of the inverse scale
factor below a critical scale factor a
. Unlike in the clas-
sical regime, eigenvalues of the inverse scale factor opera-
tor become proportional to positive powers of the scale
factor for a<a
. These considerations have led to an
effective description of the evolution of the Universe,
where the standard Friedmann dynamics receives modifi-
cations from the inverse scale factor eigenvalues below a
.
Hence it has been assumed that classical emergence occurs
above a
and that the Universe can still be described in
terms of continuum dynamics below this scale.
The effective modified Friedmann dynamics leads to
various interesting phenomenological effects (see
Ref. [5] for a recent review). For example, it has been
demonstrated that effective dynamics leads to a phase of
kinetic dominated superinflation [6] which can provide
correct initial conditions for conventional chaotic inflation
[7–10], thus making inflation more natural. The loop quan-
tum phase has been shown to alleviate the problem of
inflation in closed models [11] and inflation without scalar
field [12]. It is interesting to note that effects of a super-
inflationary phase prior to the conventional potential driven
inflation may leave observable signatures at large scales in
the cosmic microwave background [8]. Apart from effects
in the early Universe, the effective dynamics promises to
resolve various cosmological singularities, for example,
the big crunch [13–16] and brane collision singularity
[17]. Application of LQC techniques has also shown to
yield nonsingular gravitational collapse scenarios with
associated observable signatures [18,19].
Although LQC phenomenology leads to various poten-
tially observable effects, it should be noted that strictly
speaking these investigations are not based directly on the
*
Electronic address: singh@gravity.psu.edu
†
Electronic address: kfvander@gravity.psu.edu
PHYSICAL REVIEW D 72, 084004 (2005)
1550-7998=2005=72(8)=084004(8)$23.00 084004-1 © 2005 The American Physical Society
quantum theory of LQC, but rather on heuristically moti-
vated effective continuous equations of motion. Beyond
heuristic ideas it is not clear as to the domain of validity of
the effective theory and where the continuous effective
description breaks down. Thus it merits a more careful
derivation of the effective picture directly from the quan-
tum theory and difference equation. In the absence of any
such quantitative proof, the above assumptions on the
existence of the modified dynamics below a
appear ad
hoc. These issues have been noted before and in fact
phenomenological constraints on the domain of validity
have also been discussed [9,20], but in the absence of any
comparison with the underlying quantum evolution, such
investigations are far from being complete.
A further issue less emphasized previously pertains to
the fact that LQC predicts a discrete difference equation as
opposed to the continuous Wheeler-DeWitt equation of
standard quantum cosmology. It is thus important to de-
termine under what conditions the discreteness can modify
the dynamics and play an important role phenomenologi-
cally. An important question is whether the continuum
picture breaks down in the regime where discreteness
effects are important and if not how can the dynamics be
described in terms of an effective continuous picture.
The aim of this paper is to answer these questions
systematically through the application of the full quantum
features of LQC. To this end we will study the model of a
homogeneous and isotropic universe with matter coupled
in the form of a massless scalar field. The main difficulty in
describing dynamics within LQC pertains to the oft men-
tioned problem of time in quantum cosmology [21]. The
problem can however be overcome by treating the scale
factor as a clock variable and considering the evolution of
the scalar field. Thus the semiclassical states constructed
will consist of sharply peaked wave packets centered
around a value of the scalar field at a particular scale factor.
The states can be evolved forward (or backward) using the
difference equation and the trajectory can be compared
with that from classical Friedmann or effective dynamics.
All of the physics is captured in this picture without
resorting to an external time. This allows us to directly
verify the semiclassical limit of LQC and determine where
the continuum picture breaks down. The results will vali-
date the effective continuous picture incorporating the
inverse volume modifications. Furthermore, we will also
test and verify previously proposed effective continuous
equations that include modifications associated with the
discreteness effects. We will indicate with the quantum
evolution the conditions under which the discreteness ef-
fects play an important role.
II. CLASSICAL THEORY AND THE QUANTUM
CONSTRAINT
The model we will investigate is composed of a homo-
geneous and isotropic universe with zero spatial curvature
and matter in the form of a massless scalar field. The
classical phase space is parametrized by four quantities:
the connection c, the triad p, the scalar field , and the
scalar field momentum P
which satisfy the following
Poisson bracket relations:
fc; pg
1
3
; f; P
g1: (1)
Here is the Barbero-Immirzi parameter whose value can
be fixed to 0.2375 by black hole thermodynamics [22] and
8G. We will work with c @ 1 and Planck length
l
P
G
p
. Note that formulas in LQC typically are written
in terms of the modified Planck length l
P
8G
p
, hence
our formulas will appear slightly modified from previous
works.
The gravitational variables c and p encode the curvature
and geometry, respectively, of the gravitational field which
can be seen in their relation to the standard metric variables
jpja
2
;c
_
a: (2)
Governing the dynamics is the Hamiltonian constraint
which, in terms of c and p variables, is given by
H
3
2
sgnp
jpj
q
c
2
1
2
a
3
P
2
: (3)
The equations of motion are derived through the vanishing
of the Hamiltonian constraint H 0, and the Hamiltonian
equations (
_
x fx; Hg for any phase space variable x). For
the scalar field momentum, the Hamiltonian equations
immediately imply that P
is constant in time. The Hamil-
tonian equations for p and along with the vanishing of
the Hamiltonian constraint give their time dependence
_
p
2
3
s
P
a
;
_
a
3
P
(4)
which can be integrated to give the time evolution of both p
and . The standard Friedmann equation can be derived by
combining these equations, eliminating P
, and using the
relation between p and a to get
_
a
a
2
6
_
2
: (5)
Our goal is to compare the trajectories of quantum wave
packets in LQC with the classical equations of motion (4).
The difficulty lies in the lack of any @=@t term in the
difference equation that governs the behavior of the wave
packets. Thus it is impossible within LQC to consider a
wave packet peaked around some value of p and and
evolve it forward in time to compare with the classical
trajectory. This is the ‘‘problem of time’’ in quantum
cosmology [21]. The origin of this difficulty can be under-
stood even at the classical level; namely, that the classical
equations of motion are not unique since the lapse is a
freely specifiable function which we have implicitly fixed
PARAMPREET SINGH AND KEVIN VANDERSLOOT PHYSICAL REVIEW D 72, 084004 (2005)
084004-2
to one to arrive at (4). The trajectory pt itself has no
physical meaning since we can reparametrize t to get a
different trajectory. Physically, an observer could never
measure the value of t by measuring p, c, , and P
.
A solution to the problem, as noted by various authors
[23], is to notice that while, for instance, pt and t by
themselves have no physical meaning, the correlation be-
tween p and for a given value of t is a physically
meaningful statement. The correlations are invariant under
reparametrizations of time. These correlations can be de-
termined by deparametrizing the classical equations to
remove the reference to t. For the model under considera-
tion, we arrive at a differential equation governing the
evolution of the scalar field as a function of the scale factor
by noting d=da
_
=
_
a which from the Friedmann equa-
tion gives
d
da
6
a
2
s
: (6)
Integrating this we find that the scalar field evolves as
cl
a
6
s
logaC (7)
where C is a constant. The time parameter t showing up in
the classical equations of motion can now be seen as an
arbitrary parametrization of the correlations between the
physical quantities given in (7).
To recover a notion of dynamics without any explicit
reference to time in our model we can choose the scale
factor to play the role of a physical clock since it is a
monotonically increasing function as indicated in Eq. (4).
While on general grounds it is not necessary to interpret the
quantum theory by singling out a clock variable, in our
model it proves useful for interpreting our results in terms
of a ‘‘time evolution’’ of the scalar field given in Eq. (7).
Thus we can consider wave packets peaked around a value
of the scalar field at a given scale factor and then the
difference equation of LQC will determine the trajectory
of the wave packet for different values of the scale factor
from which we can determine how the scalar field evolves
with respect to the scale factor.
With an understanding of the classical framework we
can now turn to the loop quantization of the model. In the
framework of Dirac quantization used in LQG and LQC
the physical quantum states are annihilated by the
Hamiltonian constraint (3) represented as an operator.
The gravitational side of the constraint when quantized
using loop techniques leads to a partial difference equation
[4]. The matter term is quantized as
^
H
1
2
^
a
3
^
P
2
1
2
d
J
@
2
@
2
(8)
with d
J
being the LQC quantized eigenvalues of the in-
verse volume operator. The key feature of the d
J
is that it is
a bounded function with maxima at a characteristic scale
factor a
8J
0
=3
p
l
P
. The inverse volume eigen-
values are labeled with the quantum ambiguity parameter J
which arises from the fact that the inverse volume operator
is computed by tracing over SU(2) holonomies in an
irreducible spin J representation [24]. Physically, below
the scale factor a
determined by J, the inverse volume
eigenvalues are suppressed in contrast to the classical
inverse volume which diverges for small a.
0
is an addi-
tional quantum ambiguity parameter which is heuristically
related to the smallest eigenvalue of area operator in LQG
[4] (which fixes its value to
3
p
=4). The eigenvalues d
J
can
be approximated as d
J
aDaa
3
[24] with
Da8=77
6
q
3=2
f7q 1
11=4
jq 1j
11=4
11qq 1
7=4
sgnq 1jq 1j
7=4
g
6
;
(9)
where q a
2
=a
2
.Fora>a
we have Da1 and we
recover the classical expression for the inverse volume
eigenvalues d
J
aa
3
.
The constraint equation satisfied by the physical states
becomes
3
4
2
2
0
s 4
0
4
0
2s
s 4
0
4
0
1
2
d
J
@
2
@
2
0 (10)
where the parameter is an eigenvalue of the scale factor
operator and volume operator given by
a
8jj
6
s
l
P
;V
8jj
6
3=2
l
3
P
(11)
and the function s2=8
0
l
2
P
V
0
V
0
can be shown to be equal to
p
p
a for large volumes
[20]. We note that this difference equation is derived from a
non-self-adjoint constraint operator. One can construct a
self-adjoint constraint [25,26] and derive the correspond-
ing difference equation, however we do not expect our
results to be modified significantly. The difference equa-
tion can also be seen as a discrete approximation of the
second order hyperbolic Wheeler-DeWitt equation
3
@
2
@p
2
p
p
p;
1
2
d
J
@
2
p;
@
2
(12)
obtained by quantizing
^
c i
1
3
@=@p. It has been shown
in Ref. [27] that wave packets which satisfy the Wheeler-
DeWitt equation (though for a different factor ordering
than that given here and without d
J
corrections) follow
the classical trajectory given in Eq. (7). The constraint
equation (10) differs radically from the Wheeler-DeWitt
equation by the fact that it is a discrete difference equation
and by the presence of the inverse volume eigenvalues d
J
.
SEMICLASSICAL STATES, EFFECTIVE DYNAMICS, ... PHYSICAL REVIEW D 72, 084004 (2005)
084004-3
It is our goal now to determine the precise effects of these
differences in the trajectory of the wave packets.
Attempts have been made to describe the new features of
the quantum theory of LQC within an effective continuous
theory. The very first attempts simply replaced the inverse
volume in the classical equations of motion with the ei-
genvalues d
J
[6]. Many of the phenomenological investi-
gations are based on this effective framework. The
effective equations have been generalized both through a
path integral framework [20] and a WKB analysis [28,29]
to include effects arising from the fundamental discrete-
ness of the theory. The effective framework can be de-
scribed in terms of an effective Hamiltonian constraint
given by (additional terms arise in WKB analysis, which
are not considered here)
H
eff
3
2
2
0
p
p
sin
2
0
c
1
2
d
J
aP
2
(13)
where the sin
2
0
c modifications can be understood as
discreteness corrections from the difference equation.
Most phenomenological investigations so far have ignored
the discreteness corrections by assuming
0
c 1 and
sin
0
c
0
c whence the modified Friedmann equation
becomes
_
a
a
2
3
m
3
_
2
eff
2Da
(14)
where
m
1
2
d
J
P
2
=a
3
1
2
DaP
2
and we have used
_
eff
d
J
P
acquired from the Hamiltonian equations. It
is then straightforward to deparametrize the equations of
motion to obtain
d
eff
da
6ad
J
a
s
: (15)
For a<a
, d
J
a is proportional to positive powers of the
scale factor which leads to radical modifications of the
eff
a trajectory compared to the one obtained classically.
For a larger than a
, we have d
J
aa
3
and we recover
the classical behavior of the scalar field given in Eq. (7).
In our analysis we would also like to consider the dis-
creteness corrections from the gravitational side of the
constraint. From the vanishing of the effective constraint
(13) it is clear that the discreteness corrections become
relevant when the matter term becomes large. More pre-
cisely when the matter density
m
is on the order of a
critical density
crit
3=
2
0
2
a
2
, the corrections are
appreciable [20]. The equations of motion can be calcu-
lated from the effective constraint (13) and deparametrized
to give
d
eff
da
6ad
J
1
1
2
2
0
d
J
P
2
=6a
v
u
u
t
6ad
J
1
1
m
=
crit
s
(16)
whence it is clear that the modified equations of motion in
Eq. (15) are recovered when
m
crit
. The Friedmann
equation with discrete quantum corrections can be ob-
tained from the above equation and is given by [20]
_
a
a
2
3
m
1
9
2
2
2
0
a
2
2
m
: (17)
For simplicity we will refer to effective theories described
by Eqs. (15) and (16) as ET-I and ET-II, respectively.
As emphasized earlier, the use of these effective con-
tinuous equations requires a more careful consideration.
Most phenomenological investigations have so far as-
sumed that the effective continuous equations remain valid
even near the Planck regime, in particular, until the funda-
mental step size of the difference equation 4
0
which
corresponds to a scale factor of a
0
16
0
=3
p
l
P
.
There-
fore it is crucial to determine under what conditions the
effective equations hold and the phenomenological predic-
tions can be trusted. It is important to determine the scale at
which discrete quantum geometric effects play a prominent
role and influence dynamics. We can then answer the
question as to what scale the continuum spacetime arises
and classicality emerges. We can also determine if further
corrections arise from pure quantum effects.
III. COHERENT STATE EVOLUTION AND
QUANTUM DYNAMICS
Our aim is to compare the classical and effective theo-
ries with the evolution from the difference equation, and
thus for simplicity we do not consider the evolution beyond
the classical singularity. The difference equation (10) is
sufficiently complicated such that an analytic solution is
not available. We will thus compute the solutions numeri-
cally. In our method we consider a semiclassical state at a
large initial scale factor and evolve it backward toward the
singularity using the difference equation. As a semiclassi-
cal state we consider a Gaussian wave packet sharply
peaked around 0 and some classical P
at the scale
factor a
init
a
,
a
i
expiP
exp
2
=2
2
(18)
with a spread . Since the difference equation is
second order in , to find a physical solution we must
specify initial conditions at
init
[determined from a
init
using Eq. (11)] and
init
4
0
. The difference equation
then gives us the wave function at
init
8
0
which
serves to determine the wave function at the next step
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and so on, thus yielding us evolution of the initial
Gaussian. The scalar field trajectory a will then be
obtained from the peak of the semiclassical state (we will
comment on the validity of this in the discussion).
Given the Gaussian initial condition, exact solutions can
be computed numerically by assuming the form
expiP
exp
2
=2
2
X
1
n0
C
n
n
: (19)
With the above ansatz, the partial difference equation is
reduced to a difference equation for the coefficients C
n
which can be solved numerically (the ansatz avoids com-
putation of finite differences of @
2
=@
2
). The initial
condition is then simply C
n
init
n0
for some large
init
. We are left to specify the wave function at
init
4
0
. This choice does not affect our results appreciably.
We specify the initial condition by analogy with the Klein-
Gordon equation where an arbitrary solution is the sum of
positive and negative frequency solutions [which corre-
spond to the solutions of Eqs. (7), (15), and (16)]. We
can then tune the initial conditions to pick out one of the
two solutions.
For the given initial coherent state peaked at 0 and
P
we can also evaluate the classical and effective trajec-
tories from ET-I and ET-II and compare them with the
trajectory of the peak of the coherent state. For larger
values of J, a
increases and we thus expect to see devia-
tions from the classical dynamics at larger scales (yet still
below a
). The deviations from ET-II are expected when
the matter density approaches the critical value. We can
test these corrections by choosing a large initial value of
P
. Furthermore, since the fundamental discrete step is of
size 4
0
we do not expect classical or effective theories to
be valid for & 4
0
. We now discuss some representative
cases from our numerics:
(i) J 1=2, P
10l
p
: This case corresponds to the
smallest value of J and a small value of P
with
init
200. The matter density remains small compared to the
critical value which ensures that differences between ET-I
and ET-II are negligible throughout the evolution. Since
J 1=2, we have a
<a
0
which implies that the effective
dynamics agrees with the classical dynamics for all scale
factors. The evolution of the coherent state via the quantum
difference equation is shown in Fig. 1. The coherent state is
sharply peaked at
init
and evolves toward 0 without
losing its semiclassical character and retaining its sharp
peak. The trajectory of the peak is compared to the classi-
cal and effective theories in Fig. 2. The classical and
effective theories are in very good agreement with quantum
theory until the smallest nonzero value of . It is clear that
for this choice of parameters, the classical evolution can be
trusted until the first step in the quantum evolution before
the classical singularity, i.e. until a
0
. Below a
0
the classical
evolution would lead to a blow up of resulting in a
singularity whereas the evolution is nonsingular with the
quantum difference equation. For the chosen value of
parameters, classicality and continuum thus emerge as
soon as we consider a scale factor greater than a
0
.
(ii) J 500, P
100l
p
: Here we start the evolution at
init
350 which corresponds to an initial scale factor
twice a
. In this case a
>a
0
and we expect a region where
the classical theory breaks down and dynamics can be
approximated by the effective theory. We have chosen
P
and
i
in such a way that differences between ET-I
and ET-II are negligible even for very small . The results
are plotted in Fig. 3. It is evident that the classical theory
departs from the quantum evolution at a larger scale factor
as compared to the case of J 1=2. The scale at which
−1.5
−1
−0.5
0
0.25
0
5
10
a
φ
FIG. 1 (color online). Evolution of the coherent state for J
1=2 and P
10l
p
. The coherent state remains sharply peaked
and follows the classical trajectory given by Eq. (7).
Amplification at scales close to a 0 are due to discrete
quantum effects.
0 2 4 6 8 10 12 14
−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
a
φ
FIG. 2 (color online). The trajectory of the peak of the coher-
ent state ( line) is compared with classical theory (solid line)
for J 1=2 and P
10l
p
. The classical trajectory agrees
extremely well with the quantum curve until the smallest dis-
crete step in the scale factor.
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