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Quantum nature of the big bang: An analytical and numerical investigation
Abhay Ashtekar,
1,2,3,
*
Tomasz Pawlowski,
1,†
and Parampreet Singh
1,2,‡
1
Institute for Gravitational Physics and Geometry, Physics Department, Pennsylvania State University, University Park,
Pennsylvania 16802, USA
2
Inter-University Centre for Astronomy and Astrophysics, post bag 4, Ganeshkhind, Pune 411 017, India
3
Isaac Newton Institute for Mathematical Sciences, 20 Clarkson Road, Cambridge CB3 0EH, United Kingdom
(Received 13 April 2006; published 29 June 2006)
Analytical and numerical methods are developed to analyze the quantum nature of the big bang in the
setting of loop quantum cosmology. They enable one to explore the effects of quantum geometry both on
the gravitational and matter sectors and significantly extend the known results on the resolution of the big
bang singularity. Specifically, the following results are established for the homogeneous isotropic model
with a massless scalar field: (i) the scalar field is shown to serve as an internal clock, thereby providing a
detailed realization of the ‘‘emergent time’’ idea; (ii) the physical Hilbert space, Dirac observables, and
semiclassical states are constructed rigorously; (iii) the Hamiltonian constraint is solved numerically to
show that the big bang is replaced by a big bounce. Thanks to the nonperturbative, background
independent methods, unlike in other approaches the quantum evolution is deterministic across the
deep Planck regime. Our constructions also provide a conceptual framework and technical tools which can
be used in more general models. In this sense, they provide foundations for analyzing physical issues
associated with the Planck regime of loop quantum cosmology as a whole.
DOI: 10.1103/PhysRevD.73.124038 PACS numbers: 04.60.Kz, 04.60.Pp, 98.80.Qc
I. INTRODUCTION
Loop quantum gravity (LQG) is a background indepen-
dent, nonperturbative approach to quantum gravity [1–3].
It is therefore well suited for the analysis of certain long
standing questions on the quantum nature of the big bang.
Examples of such questions are
(i) How close to the big bang does a smooth space-
time of general relativity make sense? In particular,
can one show from first principles that this approxi-
mation is valid at the onset of inflation?
(ii) Is the big bang singularity naturally resolved by
quantum gravity? Or, is some external input such as
a new principle or a boundary condition at the big
bang essential?
(iii) Is the quantum evolution across the ‘‘singularity’’
deterministic? Since one needs a fully nonpertur-
bative framework to answer this question in the
affirmative, in the pre-big bang [4] and Ekpyrotic/
Cyclic [5,6] scenarios, for example, so far the
answer has been in the negative [7,8].
(iv) If the singularity is resolved, what is on the ‘‘other
side’’? Is there just a ‘‘quantum foam,’’ far removed
from any classical space-time (as suggested, e.g., in
[9]), or, is there another large, classical universe (as
suggested, e.g., in [4 –6])?
Over the years, these and related issues had been generally
relegated to the ‘‘wish list’’ of what one would like the
future, satisfactory quantum gravity theory to eventually
address. However, over the past five years, thanks to the
seminal ideas introduced by Bojowald and others, notable
progress was made on such questions in the context of
symmetry reduced, minisuperspaces [10]. In particular, it
was found that Riemannian quantum geometry, which
comes on its own in the Planck regime, has just the right
features to resolve the big bang singularity [11] in a precise
manner [12]. However, the physical ramifications of this
resolution—in particular the answer to what is on the other
side—have not been worked out. It is therefore natural to
ask whether one can complete that analysis and systemati-
cally address the questions listed above, at least in the
limited context of simple cosmological models. It turns
out that the answer is in fact in the affirmative. A brief
summary of arguments leading to this conclusion appeared
in [13]. The purpose of this paper is to provide the detailed
constructions and numerical simulations that underlie
those results.
Let us begin with a brief summary of the main results of
loop quantum cosmology (LQC). (For a comprehensive
survey, see, e.g., [10].) They can be divided into two broad
classes:
(i) singularity resolution based on exact quantum equa-
tions (see e.g. [11,12,14–19]), and,
(ii) phenomenological predictions based on ‘‘effective’’
equations (see e.g. [20 –32]).
Results in the first category make a crucial use of the
effects of quantum geometry on the gravitational part of
the Hamiltonian constraint. Because of these effects, the
quantum ‘‘evolution’’ is now dictated by a second order
difference equation rather than the second order differen-
tial equation of the Wheeler-DeWitt (WDW) theory.
Nonetheless, the intuitive idea of regarding the scale factor
a as ‘‘internal time’’ was maintained. The difference is that
the evolution now occurs in discrete steps. As explained in
*
Electronic address: ashtekar@gravity.psu.edu
†
Electronic address: pawlowsk@gravity.psu.edu
‡
Electronic address: singh@gravity.psu.edu
PHYSICAL REVIEW D 73, 124038 (2006)
1550-7998=2006=73(12)=124038(33) 124038-1 © 2006 The American Physical Society
detail in Sec. II B, this discreteness descends directly from
the quantum nature of geometry in LQG; in particular, the
step size is dictated by the lowest nonzero eigenvalue of the
area operator. When the Universe is large, the WDW
differential equation is an excellent approximation to the
‘‘more fundamental’’ difference equation. However, in the
Planck regime, there are major deviations. In particular,
while the classical singularity generically persists in the
WDW theory without additional inputs, this is not the case
in LQG. This difference does not arise simply because the
discrete evolution enables one to ‘‘jump over’’ the classical
singularity. Indeed, even when the discrete evolution
passes through the point a 0, the difference equation
remains well defined and enables one to ‘‘evolve’’ any
initial data across this classically singular point. It is in
this sense that the singularity was said to be resolved.
In the second category of results, by and large the focus
was on quantum geometry modifications of the matter part
of the Hamiltonian constraint. The main idea is to work
with approximation schemes which encode the idea of
semiclassicality and to incorporate quantum geometry ef-
fects on the matter Hamiltonian by adding suitable effec-
tive terms to the classical Hamiltonian constraint. The hope
is that dynamics generated by the ‘‘quantum-corrected’’
classical Hamiltonian would have a significant domain of
validity and provide a physical understanding of certain
aspects of the full quantum evolution. This strategy has led
to a number of interesting insights. For example, using
effective equations, it was argued that the singularity reso-
lution is generic for all homogeneous models; that there is
an early phase of inflation driven by quantum gravity
effects; that this phase leads to a reduction of power
spectrum in CMB at large angular scales; and that quantum
geometry effects suppress the classical chaotic behavior of
Bianchi IX models near classical singularity.
Attractive as these results are, important limitations have
persisted. Let us begin with the results on singularity
resolution. As in most nontrivially constrained systems,
solutions to the quantum constraint fail to be normalizable
in the kinematical Hilbert space on which the constraint
operators are well defined. Therefore one has to endow
physical states (i.e. solutions to the Hamiltonian constraint)
with a new, physical inner product. Systematic strat-
egies—e.g., the powerful ‘‘group averaging procedure’’
[33,34]—typically require that the constraint be repre-
sented by a self-adjoint operator on the kinematic Hilbert
space while, so far, most of the detailed discussions of
singularity resolution are based on Hamiltonian constraints
which do not have this property. Consequently, the space of
solutions was not endowed with a Hilbert space structure.
This in turn meant that one could not introduce Dirac
observables nor physical semiclassical states. Therefore,
as pointed out, e.g. in [35], the physical meaning of singu-
larity resolution remained somewhat obscure. In particular,
even in simple models, there was no clear-cut answer as to
what the Universe did ‘‘before’’ the big bang. Was there a
genuine quantum foam or was the quantum state peaked at
a large classical universe on the other side? In absence of a
physical Hilbert space, the second and the third questions
posed in the beginning of this section could only be an-
swered partially and the first and the fourth remained
unanswered.
The phenomenological predictions have physically in-
teresting features and serve as valuable guides for future
research. However, the current form of this analysis also
faces some important limitations. Many of these discus-
sions focus only on the quantum geometry modifications of
the matter Hamiltonian. This strategy has provided new
insights and does serve as a useful starting point. However,
there is no a priori reason to believe that it is consistent to
ignore modifications of the gravitational part of the
Hamiltonian and retain only the matter modifications.
Conclusions drawn from such analysis can be taken as
attractive suggestions, calling for more careful investiga-
tions, rather than firm predictions of LQC, to be compared
with observations. On the conceptual side, a number of
semiclassical approximations are made while deriving the
effective equations. Many of them are largely violated in
the Planck regime. Therefore it is difficult to regard con-
clusions drawn from effective equations on singularity
avoidance, and on the fate of the Universe beyond, as
reliable. It does happen surprisingly often in physics that
approximation schemes turn out to work even outside the
regimes for which they were originally intended. But there
is no a priori reason to think that this must happen. To
develop intuition on the validity of approximations, it is
essential to make, in at least a few simple models, detailed
comparisons of predictions of effective equations with
those of quantum equations that are being approximated.
To summarize, while LQC has led to significant progress
by opening new avenues and by indicating how qualita-
tively new and physically desirable results can arise, the
program has remained incomplete even within the realm of
symmetry reduced, mini-superspace models.
The goal of this paper is to complete the program in the
simplest of models by using a combination of analytical
and numerical methods. The resulting theory will enable us
to answer, in the context of these models, the questions
raised in the beginning of this section. Specifically, we will
show from first principles that: (i) a classical space-time
continuum is an excellent approximation till very early
times; (ii) the singularity is resolved in the sense that a
complete set of Dirac observables on the physical Hilbert
space remains well defined throughout the evolution;
(iii) the big bang is replaced by a big bounce in the
quantum theory; (iv) there is a large classical universe on
the other side, and, (v) the evolution bridging the two
classical branches is deterministic, thanks to the back-
ground independence and nonperturbative methods.
While the paper is primarily concerned with basic concep-
ASHTEKAR, PAWLOWSKI, AND SINGH PHYSICAL REVIEW D 73, 124038 (2006)
124038-2
tual and computational issues, our constructions also pro-
vide some tools for a more systematic analysis of phe-
nomenological questions. Finally, our approach can be
used in more general models. In particular, our construc-
tions can be used also for anisotropic models and for
models in which the scalar field has a potential, although
certain conceptual subtleties have to be handled carefully
and, more importantly, the subsequent numerical analysis
is likely to be significantly more complicated. Nonetheless,
in a rather well-defined sense, these constructions provide
a foundation from which one can systematically analyze
the Planck regime in LQC well beyond the specific model
discussed in detail.
The main ideas of our analysis can be summarized as
follows: First, our Hamiltonian constraint is self-adjoint on
the kinematical (more precisely, auxiliary) Hilbert space.
Second, we use the scalar field as internal time. In the
classical theory of k 0 models with a massless scalar
field , the scale factor a as well as are monotonic
functions of time in any given solution. In the k 1 case,
on the other hand, since the Universe recollapses, only
has this property. Therefore, it seems more natural to use
as internal time, which does not refer to space-time coor-
dinates or any other auxiliary structure. It turns out that
is well suited to be emergent time also in the quantum
theory. Indeed, our self-adjoint Hamiltonian constraint is
of the form
@
2
@
2
; (1.1)
where does not involve and is a positive, self-adjoint,
difference operator on the auxiliary Hilbert space of quan-
tum geometry. Hence, the quantum Hamiltonian constraint
can be readily regarded as a means to evolve the wave
function with respect to . Moreover, this interpretation
makes the group averaging procedure similar to that used
in the quantization of a ‘‘free’’ particle in a static space-
time, and therefore conceptually more transparent. Third,
we can carry out the group averaging and arrive at the
physical inner product. Fourth, we identify complete sets
of Dirac observables on the physical Hilbert space. One
such observable is provided by the momentum
^
p
con-
jugate to the scalar field and a set of them by
^
aj
o
, the
scale factor at the ‘‘instants of emergent time’’
o
.
1
Fifth,
we construct semiclassical states which are peaked at
values of these observables at late times, when the
Universe is large. Finally we evolve them backwards using
the Hamiltonian constraint using the adaptive step, fourth
order Runge-Kutta method. The numerical tools are ade-
quate to keep track not only of how the peak of the wave
function evolves but also of fluctuations in the Dirac ob-
servables in the course of evolution. A variety of numerical
simulations have been performed and the existence of the
bounce is robust.
The paper is organized as follows: In Sec. II we sum-
marize the framework of LQC in the homogeneous, iso-
tropic setting, keeping the matter field generic. We first
summarize the kinematical structure [12], highlighting the
origin of the qualitative difference between the WDW
theory and LQC already at this level. We then provide a
self-contained and systematic derivation of the quantum
Hamiltonian constraint, spelling out the underlying as-
sumptions. In Sec. III we restrict the matter field to be a
massless scalar field, present the detailed WDW theory and
show that the singularity is not resolved. The choice of a
massless scalar field in the detailed analysis was motivated
by two considerations. First, it makes the basic construc-
tions easier to present and the numerical simulations are
substantially simpler. More importantly, in the massless
case every classical solution is singular, whence the singu-
larity resolution by LQC is perhaps the clearest illustration
of the effects of quantum geometry. In Sec. IV we discuss
this model in detail within the LQC framework, emphasiz-
ing the dynamical differences from the WDW theory.
Specifically, we present the general solution to the quan-
tum constraint, construct the physical Hilbert space and
introduce operators corresponding to Dirac observables. In
Sec. V we present the results of our numerical simulations
in some detail. Solutions to the Hamiltonian constraint are
constructed using two different methods, one using a fast
Fourier transform and another by evolving initial data
using the adaptive step, fourth order Runge-Kutta method.
To further ensure the robustness of conclusions, the initial
data (at late ‘‘times’’ when the Universe is large) is speci-
fied in three different ways, reflecting three natural choices
in the construction of semiclassical states. In all cases, the
classical big bang is replaced by a quantum big bounce and
the two ‘‘classical branches’’ are joined by a deterministic
quantum evolution. Section VI compares and contrasts the
main results with those in the literature.
Issues which are closely related to (but are not an
integral part of) the main results are discussed in three
appendices. Appendix A is devoted to certain heuristics on
effective equations and uncertainty relations which provide
a physical intuition for ‘‘mechanisms’’ underlying certain
constructions and results. In Appendix B we discuss tech-
nical aspects of numerical simulations which are important
but whose inclusion in the main text would have broken the
flow of the argument. Finally in Appendix C we present an
alternate physical Hilbert space which can be constructed
by exploiting certain special features of LQC which are not
found in the general setting of constrained systems. This
space is more closely related to the WDW theory and could
1
As we will see in Sec. II, to construct a Hamiltonian frame-
work in the open model, one has to fix a fiducial cell. The scale
factor a (and the momentum p conjugate to the gravitational
connection introduced later) refers to the volume of this cell.
Alternatively, one can avoid the reference to the fiducial cell by
fixing a
o
and considering the ratios aj
=a
o
as Dirac observ-
ables. However, for simplicity of presentation we will not follow
this route.
QUANTUM NATURE OF THE BIG BANG: AN ... PHYSICAL REVIEW D 73, 124038 (2006)
124038-3
exist also in more general contexts. Its existence illustrates
an alternate way to address semiclassical issues which may
well be useful in full LQG.
II. LQC IN THE k 0 HOMOGENEOUS,
ISOTROPIC SETTING
This section is divided into three parts. To make the
paper self-contained, in the first we provide a brief sum-
mary of the kinematical framework, emphasizing the con-
ceptual structure that distinguishes LQC from the WDW
theory. In the second, we introduce the self-adjoint
Hamiltonian constraints and their WDW limits. By spell-
ing out the underlying assumptions clearly, we also pave
the way for the construction of a more satisfactory
Hamiltonian constraint in [36].
2
In the third part, we first
list issues that must be addressed to extract physics from
this general program and then spell out the model consid-
ered in the rest of the paper.
A. Kinematics
1. Classical phase space
In the k 0 case, the spatial 3-manifold M is topologi-
cally R
3
, endowed with the action of the Euclidean group.
This action can be used to introduce on M a fiducial, flat
metric
o
q
ab
and an associated constant orthonormal triad
o
e
a
i
and cotriad
o
!
i
a
. In full general relativity, the gravita-
tional phase space consists of pairs A
i
a
;E
a
i
of fields on M,
where A
i
a
is a SU(2) connection and E
a
i
an orthonormal
triad (or moving frame) with density weight 1 [1]. As one
would expect, in the present homogeneous, isotropic con-
text, one can fix the gauge and spatial diffeomorphism
freedom in such a way that A
i
a
has only one independent
component,
~
c, and E
a
i
, only one independent component,
~
p:
A
~
c
o
!
i
i
;E
~
p
o
q
p
o
e
i
i
; (2.1)
where
~
c and
~
p are constants and the density weight of E has
been absorbed in the determinant of the fiducial metric.
3
Denote by
S
grav
the subspace of the gravitational phase
space
grav
defined by (2.1). Because M is noncompact and
our fields are spatially homogeneous, various integrals
featuring in the Hamiltonian framework of the full theory
diverge. However, the presence of spatial homogeneity
enables us to bypass this problem in a natural fashion:
Fix a ‘‘cell’’ V adapted to the fiducial triad and restrict
all integrations to this cell. (For simplicity, we will assume
that this cell is cubical with respect to
o
q
ab
.) Then the
gravitational symplectic structure
grav
on
grav
and fun-
damental Poisson brackets are given by [12]:
S
grav
3V
o
8G
d
~
c ^ d
~
p and f
~
c;
~
pg
8G
3V
o
; (2.2)
where V
o
is the volume of V with respect to the auxiliary
metric
o
q
ab
. Finally, there is a freedom to rescale the
fiducial metric
o
q
ab
by a constant and the canonical vari-
ables
~
c,
~
p fail to be invariant under this rescaling. But one
can exploit the availability of the elementary cell V to
eliminate this additional ‘‘gauge’’ freedom. For,
c V
1=3
o
~
c and p V
2=3
o
~
p (2.3)
are independent of the choice of the fiducial metric
o
q
ab
.
Using c; p, the symplectic structure and the fundamental
Poisson bracket can be expressed as
S
grav
3
8G
dc ^ dp and fc; pg
8G
3
: (2.4)
Since these expressions are now independent of the volume
V
o
of the cell V and make no reference to the fiducial
metric
o
q
ab
, it is natural to regard the pair c; p as the
basic canonical variables on
S
grav
. In terms of p, the
physical triad and cotriad are given by
e
a
i
sgnpjpj
1=2
V
1=3
o
o
e
a
i
and
!
i
a
sgnpjpj
1=2
V
1=3
o
o
!
i
a
:
(2.5)
The function sgnp arises because in connection dynamics
the phase space contains triads with both orientations, and
since we have fixed a fiducial triad
o
e
a
i
, the orientation of
the physical triad e
a
i
is captured in the sign of p. (As in the
full theory, we also allow degenerate cotriads which now
correspond to p 0, for which the triad vanishes.)
Finally, note that although we have introduced an ele-
mentary cell V and restricted all integrals to this cell, the
spatial topology is still R
3
and not T
3
. Had the topology
been toroidal, connections with nontrivial holonomy
around the three circles would have enlarged the configu-
ration space and the phase space would then have inherited
additional components.
2. Quantum kinematics
To construct quantum kinematics, one has to select a set
of elementary observables which are to have unambiguous
operator analogs. In nonrelativistic quantum mechanics
they are taken to be x, p. One might first imagine using
c, p in their place. This would be analogous to the proce-
2
Our conventions are somewhat different from those in the
literature, especially [12]. First, we follow the standard quantum
gravity convention and set ‘
2
Pl
G@ (rather than 8G@).
Second, we follow the general convention in geometry and set
the volume element e on M to be e
:
jdetEj
p
(rather than e
:
jdetEj
p
sgndetE). This gives rise to some differences in factors
of sgnp in various terms in the expression of the Hamiltonian
constraint. Finally, the role of the minimum nonzero eigenvalue
of area is spelled out in detail, and the typographical error in the
expression of
o
that features in the Hamiltonian constraint is
corrected.
3
Our conventions are such that
i
j
1
2
"
ijk
k
1
4
ij
. Thus,
2i
k
k
, where
i
are the Pauli matrices.
ASHTEKAR, PAWLOWSKI, AND SINGH PHYSICAL REVIEW D 73, 124038 (2006)
124038-4
dure adopted in the WDW theory. However, unlike in
geometrodynamics, LQG now provides a well-defined ki-
nematical framework for full general relativity, without
any symmetry reduction. Therefore, in the passage to the
quantum theory, we do not wish to treat the reduced theory
as a system in its own right but follow the procedure used
in the full theory. There, the elementary variables are
(i) holonomies h
e
defined by the connection A
i
a
along
edges e, and, (ii) the fluxes of triads E
a
i
across 2-surfaces
S. In the present case, this naturally suggests that we use
(i) the holonomies h
k
along straight edges (
o
e
a
k
) defined
by the connection A c=
V
o
3
p
o
!
i
i
, and (ii) the momen-
tum p itself [12]. Now, the holonomy along the kth edge is
given by
h
k
cos
c
2
I 2 sin
c
2
k
; (2.6)
where I is the identity 2 2 matrix. Therefore, the ele-
mentary configuration variables can be taken to be
expic=2 of c. These are called almost periodic func-
tions of c because is an arbitrary real number (positive if
the edge is oriented along the fiducial triad vector
o
e
a
i
and
negative if it is oriented in the opposite direction). The
theory of these functions was first analyzed by the mathe-
matician Harold Bohr (who was Niels’s brother).
Thus, in LQC one takes e
ic=2
and p as the elementary
classical variables which are to have unambiguous operator
analogs. They are closed under the Poisson bracket on
S
grav
. Therefore, as in quantum mechanics, one can con-
struct an abstract ?-algebra a generated by e
ic=2
and p,
subject to the canonical commutation relations. The main
task in quantum kinematics is to find the appropriate
representation of a.
In this task, one again follows the full theory. There,
surprisingly powerful theorems are now available: By ap-
pealing to the background independence of the theory, one
can select an irreducible representation of the holonomy-
flux algebra uniquely [37,38]. The unique representation
was constructed already in the mid-nineties and is there-
fore well understood [34,39– 43]. In this representation,
there are well-defined operators corresponding to the triad
fluxes and holonomies, but the connection itself does not
lead to a well-defined operator. Since one follows the full
theory in LQC, the resulting representation of a also has
well-defined operators corresponding to p and almost pe-
riodic functions of c (and hence, holonomies), but there is
no operator corresponding to c itself.
The Hilbert space underlying this representation is
H
grav
kin
L
2
R
Bohr
;d
Bohr
, where R
Bohr
is the Bohr com-
pactification of the real line and
Bohr
is the Haar measure
on it. It can be characterized as the Cauchy completion of
the space of continuous functions fc on the real line with
finite norm, defined by
kfk
2
lim
D!1
1
2D
Z
D
D
ffdc: (2.7)
An orthonormal basis in H
grav
kin
is given by the almost
periodic functions of the connection, N
c
:
e
ic=2
.
(The N
c are the LQC analogs of the spin network
functions in full LQG [44,45]). They satisfy the relation
hN
jN
0
ihe
ic=2
je
i
0
c=2
i
;
0
: (2.8)
Note that, although the basis is of the plane wave type, the
right side has a Kronecker delta, rather than the Dirac
distribution. Therefore a generic element of H
grav
kin
can
be expanded as a countable sum c
P
k
k
N
k
where
the complex coefficients
k
are subject to
P
k
j
k
j
2
< 1.
Consequently, the intersection between H
grav
kin
and the
more familiar Hilbert space L
2
R;dcof quantum mechan-
ics (or of the WDW theory) consists only of the zero
function. Thus, already at the kinematic level, the LQC
Hilbert space is very different from that used in the WDW
theory.
The action of the fundamental operators, however, is the
familiar one. The configuration operator acts by multi-
plication and the momentum by differentiation:
^
N
cexp
ic
2
c and
^
pci
8‘
2
Pl
3
d
dc
;
(2.9)
where, as usual, ‘
2
Pl
G@. The first of these provides a 1-
parameter family of unitary operators on H
grav
kin
while the
second is self-adjoint.
It is often convenient to use the Dirac bra-ket notation
and set e
ic=2
hcji. In this notation, the eigenstates of
^
p
are simply the basis vectors ji:
^
pji
8‘
2
Pl
6
ji: (2.10)
Finally, since the operator
^
V representing the volume of the
elementary cell V is given by
^
V j
^
pj
3=2
, the basis vectors
are also eigenstates of
^
V:
^
Vji
8
6
jj
3=2
‘
3
Pl
ji: (2.11)
The algebra a, of course, also admits the familiar rep-
resentation on L
2
R;dc. Indeed, as we will see in Sec. III,
this is precisely the ‘‘Schro
¨
dinger representation’’ under-
lying the WDW theory. The LQC representation outlined
above is unitarily inequivalent. This may seem surprising
at first in the light of the von-Neumann uniqueness theorem
of quantum mechanics. The LQG representation evades
that theorem because there is no operator
^
c corresponding
to the connection component c itself. Put differently, the
theorem requires that the unitary operators
^
N
be weakly
continuous in , while our operators on H
grav
kin
are not.
(For further discussion, see [46].)
QUANTUM NATURE OF THE BIG BANG: AN ... PHYSICAL REVIEW D 73, 124038 (2006)
124038-5
B. The Hamiltonian constraint
1. Strategy
Because of spatial flatness, the gravitational part of the
Hamiltonian constraint of full general relativity simplifies
and assumes the form
C
grav
2
Z
V
d
3
xN"
ijk
F
i
ab
e
1
E
aj
E
bk
; (2.12)
where e
:
jdetEj
p
, and where we have restricted the
integral to our elementary cell V . Because of spatial
homogeneity the lapse N is constant and we will set it to
one.
To obtain the corresponding constraint operator, we
need to first express the integrand in terms of our elemen-
tary phase space functions h
e
, p and their Poisson brackets.
The term involving triad can be treated using the Thiemann
strategy [3,47]:
"
ijk
e
1
E
aj
E
bk
X
k
sgnp
2G
o
V
1=3
o
o
"
abco
!
k
c
Trh
o
k
fh
o
1
k
;Vg
i
; (2.13)
where h
0
k
is the holonomy along the edge parallel to the
kth basis vector of length
o
V
1=3
o
with respect to
o
q
ab
, and
V jpj
3=2
is the volume function on the phase space.
While the right side of (2.13) involves
o
, it provides an
exact expression for the left side which is independent of
the value of
o
.
For the field strength F
i
ab
, we use the standard strategy
used in gauge theories. Consider a square 䊐
ij
in the i-j
plane spanned by two of the triad vectors
o
e
a
i
, each of
whose sides has length
o
V
1=3
o
with respect to the fiducial
metric
o
q
ab
. Then, ‘‘the ab component’’ of the curvature is
given by
F
k
ab
2 lim
o
!0
Tr
h
o
䊐
ij
1
2
o
V
2=3
o
ko
!
i
a
o
!
j
b
; (2.14)
where the holonomy h
o
䊐
ij
around the square 䊐
ij
is just the
product of holonomies (2.6) along the four edges of 䊐
ij
:
h
o
ij
h
o
i
h
o
j
h
o
i
1
h
o
j
1
: (2.15)
By adding the two terms and simplifying, C
grav
can be
expressed as
C
grav
lim
o
!0
C
o
grav
; where
C
o
grav
4sgnp
8
3
3
o
G
X
ijk
"
ijk
Trh
o
i
h
o
j
h
o
1
i
h
o
1
j
h
o
k
fh
o
1
k
;Vg:
(2.16)
Since C
o
grav
is now expressed entirely in terms of holono-
mies and p and their Poisson bracket, it is straightforward
to write the corresponding quantum operator on H
grav
kin
:
^
C
o
grav
4isgnp
8
3
3
o
‘
2
Pl
X
ijk
"
ijk
Tr
^
h
o
i
^
h
o
j
^
h
o
1
i
^
h
o
1
j
^
h
o
k
^
h
o
1
k
;
^
V
24isgnp
8
3
3
o
‘
2
Pl
sin
2
o
c
sin
o
c
2
^
V cos
o
c
2
cos
o
c
2
^
V sin
o
c
2
: (2.17)
However, the limit
o
! 0 of this operator does not
exist. This is not accidental; had the limit existed, there
would be a well-defined operator directly corresponding to
the curvature F
i
ab
and we know that even in full LQG,
while holonomy operators are well defined, there are no
operators corresponding to connections and curvatures.
This feature is intimately intertwined with the quantum
nature of Riemannian geometry of LQG. The viewpoint in
LQC is that the failure of the limit to exist is a reminder
that there is an underlying quantum geometry where ei-
genvalues of the area operator are discrete, whence there is
a smallest nonzero eigenvalue, , i.e., an area gap [48,49].
Thus, quantum geometry is telling us that it is physically
incorrect to let
o
go to zero because that limit corre-
sponds to shrinking the area enclosed by loops 䊐
ij
to zero.
Rather, the ‘‘correct’’ field strength operator in the quan-
tum theory should be in fact nonlocal, given by setting
o
in (2.14) to a nonzero value, appropriately related to .In
quantum theory, we simply cannot force locality by shrink-
ing the loops 䊐
ij
to zero area. In the classical limit,
however, we are led to ignore quantum geometry, and
recover the usual, local field F
ab
[12,23].
There are two ways of implementing this strategy. In this
paper, we will discuss the one that has been used in the
literature [10,12,18]. The second strategy will be discussed
in [36]; it has a more direct motivation and is more sat-
isfactory in semiclassical considerations especially in pres-
ence of a nonzero cosmological constant.
In the above discussion,
o
enters through holonomies
h
o
k
. Now, it is straightforward to verify that (every matrix
element of) h
o
k
is an eigenstate of the area operator
c
Ar
c
jpj (associated with the face of the elementary cell V
orthogonal to the kth direction):
c
Arh
o
k
c
8
o
6
‘
2
Pl
h
o
k
c: (2.18)
In the first strategy, one fixes
o
by demanding that this
eigenvalue be 2
3
p
‘
2
Pl
, the area gap [1,49], so that
o
3
3
p
=2.
To summarize, by making use of some key physical
features of quantum geometry in LQG, we have arrived
at a ‘‘quantization’’ of the classical constraint C
grav
:
ASHTEKAR, PAWLOWSKI, AND SINGH PHYSICAL REVIEW D 73, 124038 (2006)
124038-6
^
C
grav
^
C
o
grav
j
o
3
3
p
=2
: (2.19)
There are still factor ordering ambiguities which will be
fixed in Sec. II B 2.
We will conclude our summary of the quantization
strategy by comparing this construction with that used in
full LQG [1,3,47]. As one would expect, the curvature
operator there is completely analogous to (2.14). How-
ever, in the subsequent discussion of the Hamiltonian con-
straint certain differences arise: While the full theory is
diffeomorphism invariant, the symmetry reduced theory is
not because of gauge fixing carried out in the beginning of
Sec. II A 1. (These differences are discussed in detail in
Secs. 4 and 5 of [12].) As a result, in the dynamics of the
reduced theory, we have to ‘‘parachute’’ the area gap
from the full theory. From physical considerations, this
new input seems natural and the strategy is clearly viable
since the constraint operator has the correct classical limit.
However, so far there is no systematic procedure which
leads us to the dynamics of the symmetry reduced theory
from that of the full theory. This is not surprising because
the Hamiltonian constraint in the full theory has many
ambiguities and there is not a canonical candidate that
stands out as being the most satisfactory. The viewpoint
in LQC is rather that, at this stage, it would be more fruitful
to study properties of LQC constraints, such as the one
introduced in this section and in [36], and use the most
successful of them as a guide to narrow down the choices in
the full theory.
2. Constraint operators and their properties
It is easy to verify that, although the classical constraint
function C
grav
we began with is real on the phase space
S
grav
, the operator
^
C
o
grav
is not self-adjoint on H
grav
kin
. This
came about just because of the standard factor ordering
ambiguities of quantum mechanics and there are two natu-
ral reorderings that can rectify this situation.
First, we can simply take the self-adjoint part of (2.19)
on H
grav
kin
and use it as the gravitational part of the con-
straint:
^
C
0
grav
1
2
^
C
grav
^
C
grav
y
: (2.20)
It is convenient to express its action on states
:
hji in the —or the triad/geometry —representation.
The action is given by
^
C
0
grav
f
0
4
o
f
0
o
f
0
4
o
; (2.21)
where the coefficients f
0
, f
0
o
are functions of :
f
0
o
4
3
s
‘
Pl
2
o
3=2
jj
o
j
3=2
j
o
j
3=2
j;
f
0
1
4
f
0
o
f
0
o
4
o
;
f
0
1
4
f
0
o
f
0
o
4
o
:
(2.22)
By construction, this operator is self-adjoint and one
can show that it is also bounded above (in par-
ticular, h;
^
C
0
grav
i <
=3
p
‘
Pl
o
3=2
5
3=2
3
3=2
hji). Next, consider the ‘‘parity’’ operator defined
by
:
. It corresponds to the flip of the
orientation of the triad and thus represents a large gauge
transformation. It will play an important role in Sec. IV.
Here we note that the functions f
0
, f
0
o
are such that the
constraint operator
^
C
0
grav
commutes with :
^
C
0
grav
; 0: (2.23)
Finally, for
o
3
3
p
=2, this difference operator
can be approximated, in a well-defined sense, by a second
order differential operator (a connection dynamics analog
of the WDW operator of geometrodynamics). Since
^
C
0
grav
is
just the self-adjoint part of the operator used in [12], we
can directly use results of Sec. 4.2 of that paper. Set p
8‘
2
Pl
=6 and consider functions p which, together
with their first four derivatives are bounded. On these
functions, we have
^
C
0
grav
p
64
2
3
‘
4
Pl
p
p
d
2
dp
2
d
2
dp
2
p
p
p
:
^
C
0WDW
grav
p; (2.24)
where stands for equality modulo terms of the order
O
o
. That is, had we left
o
as a free parameter, the
equality would hold in the limit
o
! 0. This is the limit in
which the area gap goes to zero, i.e., quantum geometry
effects can be neglected.
Let us now turn to the second natural factor ordering.
The form of the expression (2.17) of
^
C
grav
suggests [50]
that we simply ‘‘redistribute’’ the sin
2
o
c term in a sym-
metric fashion. Then this factor ordering leads to the
following self-adjoint gravitational constraint:
^
C
grav
24isgnp
8
3
3
o
‘
2
Pl
sin
o
c
sin
o
c
2
^
V cos
o
c
2
cos
o
c
2
^
V sin
o
c
2
sin
o
c
o
3
3
p
=2
:
sin
o
c
^
A sin
o
c
o
3
3
p
=2
: (2.25)
For concreteness in most of this paper we will work with
this form of the constraint and, for notational simplicity,
unless otherwise stated set
o
3
3
p
=2. However, nu-
QUANTUM NATURE OF THE BIG BANG: AN ... PHYSICAL REVIEW D 73, 124038 (2006)
124038-7
merical simulations have been performed also using the
constraint
^
C
0
grav
of Eq. (2.20) and the results are robust.
Properties of
^
C
grav
which we will need in this paper can
be summarized as follows. First, the eigenbasis ji of
^
p
diagonalizes the operator
^
A. Therefore in the p represen-
tation,
^
A acts simply by multiplication. It is easy to verify
that
^
A2
8
6
s
‘
Pl
2
o
3=2
jj
o
j
3=2
j
o
j
3=2
j: (2.26)
By inspection,
^
A is self-adjoint and negative definite on
H
grav
kin
. The form of
^
C
grav
now implies that it is also self-
adjoint and negative definite. Its action on states is
given by
^
C
grav
f
4
o
f
o
f
4
o
; (2.27)
where the coefficients f
, f
o
are again functions of :
f
1
2
8
6
s
‘
Pl
2
o
3=2
jj 3
o
j
3=2
j
o
j
3=2
j;
f
f
4
o
;f
o
f
f
:
(2.28)
It is clear from Eqs. (2.25) and (2.26) that
^
C
grav
com-
mutes with the parity operator which flips the orienta-
tions of triads:
^
C
grav
; 0: (2.29)
Finally, the ‘‘continuum limit’’ of the difference operator
^
C
grav
yields a second order differential operator. Let us first
set
f
2
o
2
o
2
o
:
(2.30)
Then,
^
C
grav
2
o
2
o
: (2.31)
Therefore, if we again set p 8‘
2
Pl
=6 and consider
functions p which, together with their first four deriva-
tives are bounded we have
^
C
grav
p
128
2
3
‘
4
Pl
d
dp
p
p
d
dp
:
^
C
WDW
grav
p;
(2.32)
where again stands for equality modulo terms of the
order O
o
. That is, in the limit in which the area gap goes
to zero, i.e., quantum geometry effects can be neglected,
the difference operator reduces to a WDW type differential
operator. (As one might expect, the limiting WDW opera-
tor is independent of the Barbero-Immirzi parameter .)
We will use this operator extensively in Sec. III.
In much of computational physics, especially in numeri-
cal general relativity, the fundamental objects are differen-
tial equations and discrete equations are introduced to
approximate them. In LQC the situation is just the oppo-
site. The physical fundamental object is now the discrete
Eq. (2.27) with
o
3
3
p
=2. The differential equation is
the approximation. The leading contribution to the differ-
ence between the two—i.e., to the error—is of the form
O
o
2
0000
where the second term depends on the wave
function under consideration. Therefore, the approxi-
mation is not uniform. For semiclassical states,
0000
can be
large, of the order of 10
16
=
4
in the examples considered
in Sec. V. In this case, the continuum approximation can
break down already at 10
4
.
As one might expect the two differential operators,
^
C
0WDW
grav
and
^
C
WDW
grav
, differ only by a factor ordering:
^
C
0WDW
grav
^
C
WDW
grav
p
16
2
3
‘
4
Pl
jpj
3=2
p:
(2.33)
Remark.—As noted in Sec. II B 1, the continuum limit
o
! 0 of any of the quantum constraint operators of LQC
does not exist on H
grav
kin
because of the quantum nature
of underlying geometry. To take this limit, one has to work
in the setting in which the quantum geometry effects are
neglected, i.e., on the WDW type Hilbert space L
2
R;dc.
On this space, operators
^
C
0
grav
,
^
C
grav
,
^
C
WDW
grav
, and
^
C
0WDW
grav
are
all densely defined and the limit can be taken on a suitable
dense domain.
C. Open issues and the model
In Sec. II A we recalled the kinematical framework used
in LQC and in II B we extended the existing results by
analyzing two self-adjoint Hamiltonian constraint opera-
tors in some detail. Physical states ; can now be
constructed as solutions of the Hamiltonian constraint:
^
C
grav
^
C
matt
; 0; (2.34)
where stands for matter fields. Given any matter model,
one could solve this equation numerically. However, ge-
nerically, the solutions would not be normalizable in the
total kinematic Hilbert space H
total
kin
of gravity plus matter.
Therefore, although Sec. II B goes beyond the existing
literature in LQC, one still cannot calculate expectation
values, fluctuations, and probabilities —i.e., extract phys-
ics—knowing only these solutions.
To extract physics, then, we still have to complete the
following tasks:
(i) In the classical theory, seek a dynamical variable
which is monotonically increasing on all solutions
(or at least ‘‘large’’ portions of solutions). Attempt
to interpret the Hamiltonian constraint (2.34) as an
ASHTEKAR, PAWLOWSKI, AND SINGH PHYSICAL REVIEW D 73, 124038 (2006)
124038-8
‘‘evolution equation’’ with respect to this internal
time. If successful, this strategy would provide an
‘‘emergent time’’ in the background independent
quantum theory. Although it is possible to extract
physics under more general conditions, physical
interpretations are easier and more direct if one
can locate such an emergent time.
(ii) Introduce an inner product on the space of solutions
to (2.34) to obtain the physical Hilbert space H
phy
.
The fact that the orientation reversal induced by
is a large gauge transformation will have to be
handled appropriately.
(iii) Isolate suitable Dirac observables in the classical
theory and represent them by self-adjoint operators
on H
phy
.
(iv) Use these observables to construct physical states
which are semiclassical at ‘‘late times,’’ sharply
peaked at a point on the classical trajectory repre-
senting a large classical universe.
(v) Evolve these states using (2.34). Monitor the mean
values and fluctuations of Dirac observables. Do
the mean values follow a classical trajectory?
Investigate if there is a drastic departure from the
classical behavior. If there is, analyze what replaces
the big bang.
In the rest of the paper, we will carry out these tasks in
the case when matter consists of a zero rest mass scalar
field. In the classical theory, the phase space
S
grav
is now
four dimensional, coordinatized by c; p; ; p
. The basic
(nonvanishing) Poisson brackets are given by
fc; pg
8G
3
and f; p
g1: (2.35)
The symmetry reduction of the classical Hamiltonian con-
straint is of the form
C
grav
C
matt
6
2
c
2
jpj
q
8G
p
2
jpj
3=2
0: (2.36)
Using this constraint, one can solve for c in terms of p
and p
. Furthermore, since does not enter the expression
of the constraint, p
is a constant of motion. Therefore,
each dynamical trajectory can be specified on the two-
dimensional p; plane. Typical trajectories are shown
in Fig. 1. Because the phase space allows triads with both
orientations, the variable p can take both positive and
negative values. At p 0 the physical volume of the
Universe goes to zero and, if the point lies on any dynami-
cal trajectory, it is an end point of that trajectory, depicting
a curvature singularity. As the figure shows, for each fixed
value of p
, there are four types of trajectories, two in the
p 0 half plane and two in the p 0 half plane.
Analytically they are given by
3
16G
s
ln
jpj
jp
?
j
?
(2.37)
where p
?
,
?
are integration constants. These trajectories
are related by a ‘‘parity transformation’’ on the phase space
which simply reverses the orientation of the physical triad.
As noted before, since the metric and the scalar field are
unaffected, it represents a large gauge transformation.
Therefore, it suffices to focus just on the portion p 0
of Fig. 1. Then for each fixed value of p
, there are two
solutions passing through any given point
?
;p
?
. In one,
the Universe begins with the big bang and then expands
and in the other the Universe contracts into a big crunch.
Thus in this model, every classical solution meets the
singularity.
Finally, we can introduce a natural set of Dirac observ-
ables. Since p
is a constant of motion, it is obviously
one. To introduce others, we note that is a monotonic
function on each classical trajectory. Furthermore, in each
solution, the space-time metric takes the form ds
2
dt
2
V
2=3
o
jpjtdS
2
o
and the time dependence of the
scalar field is given by
d
dt
16Gp
?
jp
?
j
3=2
exp
12G
p
?
; (2.38)
where p
?
,
?
, and p
?
are constants. Thus, in every solu-
tion is a monotonic function of time and can therefore
serve as a good ‘‘internal clock.’’ This interpretation sug-
gests the existence of a natural family of Dirac observ-
ables: pj
o
, the value of p at the ‘‘instant’’
o
. The set
p
;pj
o
constitutes a complete set of Dirac observables
since their specification uniquely determines a classical
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
-20000 -10000 0 10000 20000
expanding
contracting
µ
φ
FIG. 1 (color online). Classical phase space trajectories are
plotted in the , p plane. For 0, there is a branch
which starts with a big bang (at 0) and expands out and a
branch which contracts into a big crunch (at 0). Their
mirror images appear in the 0 half plane.
QUANTUM NATURE OF THE BIG BANG: AN ... PHYSICAL REVIEW D 73, 124038 (2006)
124038-9
trajectory on the symmetry reduced phase space
S
, i.e., a
point in the reduced phase space
~
S
. While the interpreta-
tion of as internal time motivates this construction and,
more generally, makes physics more transparent, it is not
essential. One can do all of physics on
~
S
: Since physical
states are represented by points in
~
S
and a complete set of
observables is given by p
;pj
o
, one can work just with
this structure.
Remark.—In the open i.e., k 0 model now under
consideration, since p is also monotonic along any classi-
cal trajectory,
p
o
is also a Dirac observable and p could
also be used as an internal time. However, in LQC the
expression of the gravitational part of the constraint opera-
tor makes it difficult to regard p as the emergent time in
quantum theory. Moreover, even in the classical theory,
since the Universe expands and then recollapses in the k
1 case, p fails to be monotonic along solutions and cannot
serve as internal time globally.
III. WDW THEORY
The WDW theory of the model has been analyzed in
some detail within geometrodynamics (see especially
[51]). However, that analysis was primarily in the context
of a WKB approximation. More recently, the group aver-
aging technique was used to construct the physical Hilbert
space in a general cosmological context [33,52], an ele-
mentary example of which is provided by the present
model. However, to our knowledge a systematic comple-
tion of the program outlined in Sec. II C has not appeared
in the literature.
In this section we will construct the WDW type quantum
theory in the connection dynamics. This construction will
serve two purposes. First it will enable us to introduce the
key notions required for the completion of the program in a
familiar and simpler context. Second, we will be able to
compare and contrast the results of the WDW theory and
LQC in detail, thereby bringing out the role played by
quantum geometry in quantum dynamics.
A. Emergent time and the general solution
to the WDW equation
Recall that the phase space of the model is four dimen-
sional, coordinatized by c; p; ; p
and the fundamental
nonvanishing Poisson brackets are given by (2.35). The
Hamiltonian constraint has the following form:
C
grav
C
6
2
c
2
jpj
q
8G
p
2
jpj
3=2
0: (3.1)
To make comparison with the standard geometrodynami-
cal WDW theory, it is most convenient to work in the p,
representation. Then, the kinematic Hilbert space is given
by H
wdw
kin
L
2
R
2
; dpd. Operators
^
p,
^
operate by
multiplication while
^
c and
^
p
are represented as
^
c i@
8G
3
@
@p
and
^
p
i@
@
@
: (3.2)
Note that, while in geometrodynamics the scale factor is
restricted to be nonnegative, here p ranges over the entire
real line, making the specification of the Hilbert space and
operators easier.
To write down the quantum constraint operator, we have
to make a choice of factor ordering. Since our primary
motivation behind the introduction of the WDW theory is
to compare it with LQC, it is most convenient to use the
factor ordering that comes from the continuum limit (2.32)
of the constraint operator of LQC. Then, the WDW equa-
tion becomes
2
3
8G@
2
@
@p
p
p
@
@p
8G@
2
d
jpj
3=2
@
2
@
2
:
8G@
2
Bp
@
2
@
2
; (3.3)
where we have denoted the eigenvalue jpj
3=2
of
d
jpj
3=2
by Bp to facilitate later comparison with LQC.
4
The
operator on the left side of this equation is self-adjoint on
L
2
R;dp and the equation commutes with the orientation
reversal operator p; p; representing a
large gauge transformation. Thus, if is a solution to
Eq. (3.3), so is p; .
For a direct comparison with LQC, it is convenient to
replace p with defined by p 8G@=6. Then,
(3.3) becomes
@
2
@
2
16G
3
B
1
@
@
p
@
@
:
; (3.4)
where we have recast the equation in such a way that
operators involving only appear on the left side and
operators involving only appear on the right. The
WDW equation now has the same form as the Klein-
Gordon equation in a static space-time, playing the
role of time and
of the (elliptic operator constructed
from the norm of the Killing field and the) spatial
Laplacian. Thus, the form of the quantum Hamiltonian
constraint is such that can be interpreted as emergent
time in the quantum theory. In this factor ordering of the
constraint operator, which emerged in the continuum limit
(2.32) of LQC, it is not as convenient to regard p as
emergent time.
Physical states will be suitably regular solutions to (3.4).
Since is a large gauge transformation, we can divide
physical states into eigenspaces of
^
P. Physical observables
will preserve each eigenspace. Since
2
1, there are
only two eigenspaces, one representing the symmetric
sector and the other, antisymmetric. Since the standard
4
Here, and in what follows, quantities with an underbar will
refer to the WDW theory.
ASHTEKAR, PAWLOWSKI, AND SINGH PHYSICAL REVIEW D 73, 124038 (2006)
124038-10
WDW theory deals with metrics, it is completely insensi-
tive to the orientation of the triad. Therefore, it is natural to
work with the symmetric sector. Thus, the physical Hilbert
space will consist of suitably regular solutions ; to
(3.4) which are symmetric under !.
The mathematical similarity with the Klein-Gordon
equation in static space-times immediately suggests a
strategy to obtain the general solution of (3.4). We first
note that d=d
p
d=d is a negative definite, self-
adjoint operator on L
2
S
R;d, the symmetric sector of
L
2
R;d. Therefore, is a positive definite, self-adjoint
operator on L
2
S
R; Bd. Its eigenfunctions provide us
with an orthonormal basis. It is easy to verify that the
eigenvectors
e
k
can be labeled by k 2 R and are given
by
e
k
:
jj
1=4
4
e
ik lnjj
: (3.5)
Their eigenvalues are given by
e
k
!
2
e
k
; with !
2
16G
3
k
2
1
16
;
(3.6)
(where the factor of 1=16 is an artifact of the factor order-
ing choice which we were led to from LQC). The eigen-
functions satisfy the orthonormality relations:
Z
1
1
dBe
k
e
k
0
k; k
0
(3.7)
(where the right side is the standard Dirac distribution, not
the Kronecker symbol as on L
2
R
Bohr
;d
Bohr
); and the
completeness relation:
Z
1
1
dB
e
k
0 8 k iff 0;
(3.8)
for any 2L
2
S
R; Bd.
With these eigenfunctions at hand, we can now write
down a ‘‘general’’ symmetric solution to (3.3). Any solu-
tion, whose initial data at
o
is such that
1=4
;
o
and
1=4
_
;
o
are symmetric and
lie in the Schwartz space of rapidly decreasing functions,
has the form
;
Z
1
1
dk
~
ke
k
e
i!
~
k
e
k
e
i!
; (3.9)
for some
~
k in the Schwartz space. Following the
terminology used in the Klein-Gordon theory, if
~
k
have support on the negative k axis, we will say the
solution is ‘‘outgoing’’ (or ‘‘expanding’’) while if it has
support on the positive k axis, it is ‘‘incoming’’ (or ‘‘con-
tracting’’). If
~
k vanishes, the solution will be said to
be of positive frequency and if
~
k vanishes, it will be
said to be of negative frequency. Thus, every solution (3.9)
admits a natural decomposition into positive and negative
frequency parts. Finally we note that positive (respectively,
negative) frequency solutions satisfy a first order (in )
equation which can be regarded as the square root of (3.4):
i
@
@
p
; (3.10)
where
p
is the positive, self-adjoint operator defined via
spectral decomposition of
on L
2
R; Bd.
Regarding as time, this is just a first order Schro
¨
dinger
equation with a nonlocal Hamiltonian
p
. Therefore, a
general ‘‘initial datum’’ f
at
o
can be evolved
to obtain a solution to (3.10) via
; e
i
p
o
f
;
o
: (3.11)
B. The physical sector
Solutions (3.9) are not normalizable in H
wdw
kin
(because
zero is in the continuous part of the spectrum of the WDW
operator). Our first task is to endow the space of these
physical states with a Hilbert space structure. There are
several possible avenues. We will begin with one that is
somewhat heuristic but has direct physical motivation. The
idea [53,54] is to introduce operators corresponding to a
complete set of Dirac observables and select the required
inner product by demanding that they be self-adjoint. In the
classical theory, such a set is given by p
and j
o
. Since
^
p
commutes with the WDW operator in (3.4), given a
(symmetric) solution ; to (3.4),
^
p
;
:
i@
@
@
(3.12)
is again a (symmetric) solution. So, we can just retain this
definition of
^
p
from H
wdw
kin
. The Schro
¨
dinger type evo-
lutions (3.11) enable us to define the other Dirac observable
d
jj
o
, where the absolute value suffices because the states
are symmetric under . Given a (symmetric) solution
; to (3.4), we can first decompose it into positive
and negative frequency parts
; , freeze them at
o
, multiply this initial datum by jj, and evolve
via (3.11):
d
jj
o
; e
i
p
o
jj
;
o
e
i
p
o
jj
;
o
: (3.13)
The result is again a (symmetric) solution to (3.4). Now, we
see that both these operators have the further property that
they preserve the positive and negative frequency subspa-
ces. Since they constitute a complete family of Dirac
observables, we have superselection. In quantum theory
we can restrict ourselves to one superselected sector. In
QUANTUM NATURE OF THE BIG BANG: AN ... PHYSICAL REVIEW D 73, 124038 (2006)
124038-11
what follows, for definiteness we will focus on the positive
frequency sector and, from now on, drop the suffix .
We now seek an inner product on positive frequency
solutions ; to (3.4) (invariant under the reflec-
tion) which makes j
^
pj
and j ^j
o
self-adjoint. Each of
these solutions is completely determined by its initial
datum ;
o
and the Dirac observables have the fol-
lowing action on the datum:
d
jj
o
;
o
jj;
o
and
^
p
;
o
@
p
;
o
:
(3.14)
Therefore, it follows that (modulo an overall rescaling,) the
unique inner product which will make these operators self-
adjoint is just
h
1
j
2
i
phy
Z
o
dB
1
2
(3.15)
(see e.g. [53,54]). Note that the inner product is conserved,
i.e., is independent of the choice of the instant
o
.
Thus, the physical Hilbert space H
wdw
phy
is the space of
positive frequency wave functions ; which are sym-
metric under reflection and have a finite norm (3.15).
The procedure has already provided us with a representa-
tion of our complete set of Dirac observables on this
H
wdw
phy
:
d
jj
o
; e
i
p
o
jj;
o
and
^
p
; @
p
; :
(3.16)
We will now show that the same representation of the
algebra of Dirac observables can be obtained by the more
systematic group averaging method [33,52] which also
brings out the mathematical inputs that go in this choice.
(The two methods have been applied and compared for a
nontrivially constrained system in [55]). Here, one first
notes that the total constraint operator is self-adjoint on
an auxiliary Hilbert space H
wdw
aux
:
L
2
S
R
2
; Bdd
(where, as before the subscript S denotes restriction to
functions which are symmetric under reflection). One
must then select an appropriate dense subspace of
H
wdw
aux
. A natural candidate is the Schwartz space of rap-
idly decreasing functions. One then ‘‘averages’’ elements
of under the 1-parameter family group generated by the
constraint operator
^
C @
2
to produce a solution to
(3.4):
f
;
:
Z
1
1
de
i
^
C
f;
Z
1
1
dk
2j!j
~
fk; !
e
k
e
i!
~
fk; !e
k
e
i!
; (3.17)
where to arrive at the second step we expanded f in
the eigenbasis of
and
^
p
with
~
f as the coefficients.
Thus, the group averaging procedure reproduces the solu-
tion (3.9) with
~
k
~
fk; !=2j!j and
~
k
~
fk; !=2j!j. Solutions
f
are regarded as ‘‘distribu-
tions’’ or elements of the dual
?
of and the physical
norm is given by the action of this distribution,
f
, on the
‘‘test function’’ f. However, there is some freedom in the
specification of this action which generally results in seem-
ingly different but unitarily equivalent representations of
the algebra of Dirac observables. For us the most conve-
nient choice is
kk
2
:
f
f
:
Z
1
1
d
Z
1
1
dB
f
;
p
f; ;
(3.18)
where
f
f is the action of the distribution
f
on the test
field f. Then, the inner product coincides with (3.15) and
the representation of the Dirac observables is the same as in
(3.12) and (3.13). Had we chosen to drop the factor of
p
in defining the action of
f
on f, we would have obtained a
unitarily equivalent representation in which the action of
^j
o
is more complicated.
Finally, with the physical Hilbert space and a complete
set of Dirac observables at hand, we can now introduce
semiclassical states and study their evolution. Let us fix a
time
o
and construct a semiclassical state which is
peaked at p
p
?
and jj
o
?
. We would like the
peak to be at a point that lies on a large classical universe.
This implies that we should choose
?
1 and (in the
natural classical units c G 1), p
?
@. In the closed
(k 1) models, for example, the second condition is nec-
essary to ensure that the Universe expands out to a size
much larger than the Planck scale. At time
o
, con-
sider the state
;
o
Z
1
1
dk
~
ke
k
e
i!
o
?
;
where
~
ke
kk
?
2
=2
2
;
(3.19)
where k
?
3=16G@
2
p
p
?
and
?
3=16G
p
lnj
?
j
o
. It is easy to evaluate the inte-
gral in the approximation !
16G=3
p
k (which is
justified because k
?
1) and calculate mean values of
the Dirac observables and their fluctuations. One finds that,
as required, the state is sharply peaked at values
?
, p
?
.
The above construction is closely related to that of coher-
ent states in nonrelativistic quantum mechanics. The main
difference is that the observables of interest are not and
its conjugate momentum but rather and p
—the mo-
mentum conjugate to time, i.e., the analog of the Hamil-
tonian in nonrelativistic quantum mechanics.
We can now ask for the evolution of this state. Does it
remain peaked at the classical trajectory defined by p
p
?
passing through
?
at
o
? This question is easy
ASHTEKAR, PAWLOWSKI, AND SINGH PHYSICAL REVIEW D 73, 124038 (2006)
124038-12
to answer because (3.11) implies that the (positive fre-
quency) solution to ; (3.4) defined by initial data
(3.19) is obtained simply by replacing
o
by in (3.19).
Since the measure of dispersion in (3.19) does not
depend on , it follows that the initial state ;
o
which is the semiclassical, representing a large universe
at time
o
continues to be peaked at a trajectory defined by
3
16G
s
ln
jj
j
?
j
o
: (3.20)
This is precisely the classical trajectory with p
p
?
,
passing through
?
at
o
. This is just as one would
hope during the epoch in which the Universe is large.
However, this holds also in the Planck regime and, in the
backward evolution, the semiclassical state simply follows
the classical trajectory into the big bang singularity. (Had
we worked the positive k
?
, we would have obtained a
contracting solution and then the forward evolution would
have followed the classical trajectory into the big crunch
singularity.) In this sense, the WDW evolution does not
resolve the classical singularity.
Remark.—In the above discussion for simplicity we
restricted ourselves to eigenfunctions
e
k
which are
symmetric under ! from the beginning. Had we
dropped this requirement, we would have found that there
is a 4-fold (rather than 2-fold) degeneracy in the eigen-
functions of
. Indeed, if is the step function [
0 if <0 and 1 if >0], then
e
jkj
, e
jkj
,
e
jkj
, e
jkj
are all continuous functions of
which satisfy the eigenvalue equation (in the distributional
sense) with eigenvalue !
2
16G=3k
2
1=16. This
fact will be relevant in the next section.
IV. ANALYTICAL ISSUES IN LOOP QUANTUM
COSMOLOGY
We will now analyze the model within LQG. We will
first observe that the form of the Hamiltonian constraint is
such that the scalar field can again be used as emergent
time. Since the form of the resulting evolution equation is
very similar in the WDW theory, we will be able to con-
struct the physical Hilbert space and Dirac observables
following the ideas introduced in Sec. III B.
A. Emergent time and the general solution to the LQC
Hamiltonian constraint
The quantum constraint has the form
^
C
grav
^
C
0; (4.1)
where
^
C
grav
is given by (2.27). Since
^
C
8G
d
1=p
3=2
^
p
2
, the constraint becomes
8G
^
p
2
;
~
Bp
1
^
C
grav
; ; (4.2)
where
~
Bp is the eigenvalue of the operator
d
1=jpj
3=2
:
~
Bp
:
6
8‘
2
Pl
3=2
B;
where B
2
3
o
6
j
o
j
3=4
j
o
j
3=4
6
:
(4.3)
Thus, we now have a separation of variables. Both the
classical and the WDW theory suggests that could serve
as emergent time. To implement this idea, let us introduce
an appropriate kinematical Hilbert space for both geometry
and the scalar field: H
total
kin
:
L
2
R
Bohr
;Bd
Bohr
L
2
R; d. Since is to be thought of as time and as
the genuine, physical degree of freedom which evolves
with respect to this time, we chose the standard
Schro
¨
dinger representation for but the ‘‘polymer repre-
sentation’’ for which correctly captures the quantum
geometry effects. This is a conservative approach in that
the results will directly reveal the manifestations of quan-
tum geometry; had we chosen a nonstandard representation
for the scalar field, these effects would have been mixed
with those arising from an unusual representation of ‘‘time
evolution.’’ Comparison with the WDW theory would also
become more complicated. (However, the use of a polymer
representation for may become necessary to treat inho-
mogeneities in an adequate fashion.)
On H
total
kin
, the constraint takes the form:
@
2
@
2
B
1
C
4
o
;
C
o
; C
4
o
;
; ; (4.4)
where the functions C
, C
o
are given by
5
:
C
G
9j
o
j
3
jj 3
o
j
3=2
j
o
j
3=2
j;
C
C
4
o
;
C
o
C
C
:
(4.5)
The form of (4.4) is the same as that of the WDW con-
straint (3.4), the only difference is that the -independent
operator is now a difference operator rather than a
differential operator. Thus, the LQC quantum Hamil-
tonian constraint also can be regarded as an evolution
equation which evolves the quantum state in the emergent
time .
However, since is a difference operator, an important
difference arises from the WDW analysis. For, now the
5
Note that this fundamental evolution equation makes no
reference to the Barbero-Immirzi parameter . If we set ~
=
o
and
~
~; ; , the equation satisfied by
~
~; makes no reference to
o
either. This is the equation
used in numerical simulations. To interpret the results in terms of
scale factor, however, values of and
o
become relevant.
QUANTUM NATURE OF THE BIG BANG: AN ... PHYSICAL REVIEW D 73, 124038 (2006)
124038-13
space of physical states, i.e. of appropriate solutions to the
constraint equation, is naturally divided into sectors each
of which is preserved by the evolution and by the action of
our Dirac observables. Thus, there is superselection. Let
L
j"j
denote the ‘‘lattice’’ of points fj"j4n
o
;n2 Zg on
the axis, L
j"j
the lattice of points fj"j4n
o
;n2
Zg and let L
"
L
j"j
[ L
j"j
. Let H
grav
j"j
, H
grav
j"j
and
H
grav
"
denote the subspaces of L
2
R
Bohr
;Bd
Bohr
with states whose support is restricted to lattices L
j"j
,
L
j"j
, and L
"
. Each of these three subspaces is mapped
to itself by . Since
^
C
grav
is self-adjoint and positive
definite on H
grav
kin
L
2
R
Bohr
; d
Bohr
, it follows that
is self-adjoint and positive definite on all three Hilbert
spaces.
Note, however, that since H
grav
j"j
and H
grav
j"j
are mapped
to each other by the operator , only H
grav
"
is left invari-
ant by . Now, because reverses the triad orientation, it
represents a large gauge transformation. In gauge theories,
we have to restrict ourselves to sectors, each consisting of
an eigenspace of the group of large gauge transformations.
(In QCD, in particular, this leads to the sectors.) The
group generated by is just Z
2
, whence there are only two
eigenspaces, with eigenvalues 1. Since there are no
fermions in our theory, there are no parity violating pro-
cesses whence we are led to choose the symmetric sector
with eigenvalue 1. (Also, in the antisymmetric sector all
states are forced to vanish at the singularity 0 while
there is no such a priori restriction in the symmetric
sector.) Thus, we are primarily interested in the symmetric
subspace of H
grav
"
; the other two Hilbert spaces will be
useful only in the intermediate stages of our discussion.
Our first task is to explore properties of the operator .
Since it is self-adjoint and positive definite, its spectrum is
nonnegative. Therefore, as in the WDW theory, we will
denote its eigenvalues by !
2
. Let us first consider a generic
", i.e., not equal to 0 or 2
o
. Then, on each of the two
Hilbert spaces H
grav
j"j
, we can solve for the eigenvalue
equation e
!
!
2
e
!
, i.e.,
C
e
!
4
o
C
o
e
!
C
e
!
4
o
!
2
Be
!
: (4.6)
Since this equation has the form of a recursion relation and
since the coefficients C
never vanish on the lattices
under consideration, it follows that we will obtain an
eigenfunction by freely specifying, say,
?
and
?
4
o
for any
?
on the lattice L
j"j
or L
j"j
.
Hence the eigenfunctions are 2-fold degenerate on each
of H
grav
j"j
and H
grav
j"j
.OnH
grav
"
therefore, the eigenfunc-
tions are 4-fold degenerate as in the WDW theory. Thus,
H
grav
"
admits an orthonormal basis e
I
!
where the degen-
eracy index I ranges from 1 to 4, such that
he
I
!
je
I
0
!
0
i
I;I
0
!; !
0
: (4.7)
[The Hilbert space H
grav
"
is separable and the spectrum is
equipped with the standard topology of the real line.
Therefore, we have the Dirac distribution !; !
0
rather
than the Kronecker delta
!;!
0
.] As usual, every element
of H
grav
"
can be expanded as
Z
sp
d!
~
I
!e
I
!
where
~
I
!he
I
!
ji;
(4.8)
where the integral is over the spectrum of . The numeri-
cal analysis of Sec. V and comparison with the WDW
theory are facilitated by making a convenient choice of
this basis in H
grav
"
, i.e., by picking specific vectors from
each four-dimensional eigenspace spanned by e
I
!
.Todo
so, note first that, as one might expect, every eigenvector
e
I
!
has the property that it approaches unique eigen-
vectors
e
! of the WDW differential operator as !
1. The precise rate of approach is discussed in Sec. VA.
In general, the two WDW eigenfunctions
e
! are dis-
tinct. Indeed, because of the nature of the WDW operator
, its eigenvectors can be chosen to vanish on the entire
negative (or positive) axis; their behavior on the two half
lines is uncorrelated. (See the remark at the end of
Sec. III B.) Eigenvectors of the LQC on the other hand
are rigid; their values at any two lattice points determine
their values on the entire lattice L
j"j
. Second, recall that
the spectrum of the WDW operator is bounded below by
!
2
G=3, whence e
!
with !
2
< G=3 does not appear
in the spectral decomposition of
. Note, however, that
solutions to the eigenvalue equation
e
!
!
2
e
!
con-
tinue to exist even for !
2
< G=3. But such eigenfunc-
tions diverge so fast as !1or as !1that h
e
!
ji
fails to converge for all 2 L
2
R; Bd, whence they
do not belong to the basis. What is the situation with
eigenvectors of the LQC ? Since eigenvectors e
!
of
approach those of
, he
!
ji again fails to converge for all
2 H
grav
"
if !
2
< G=3. Thus the spectrum of is
again bounded below by G=3.
6
Therefore, to facilitate
comparison with the WDW theory, we will introduce a
variable k via !
2
G=3 16G=3k
2
and use k in
place of ! to label the orthonormal basis. To be specific,
let us
(i) denote by e
jkj
the basis vector in H
grav
j"j
with
eigenvalue !
2
, which is proportional to the WDW
e
jkj
as !1; (i.e., it has only outgoing or ex-
panding component in this limit);
(ii) denote by e
jkj
the basis vector in H
grav
j"j
with
eigenvalue !
2
which is orthogonal to e
jkj
,
[since eigenvectors are 2-fold degenerate in each
of H
grav
j"j
, the vector e
jkj
is uniquely determined
up to a multiplicative phase factor.]
6
A rigorous version of this argument can be constructed e.g. by
using the Gel’fand triplet [56] associated with the operator .
However, this step has not been carried out.
ASHTEKAR, PAWLOWSKI, AND SINGH PHYSICAL REVIEW D 73, 124038 (2006)
124038-14
As we will see in Sec. VA, this basis is well suited for
numerical analysis.
We thus have an orthonormal basis e
k
in H
grav
"
with
k 2 R: he
k
je
k
0
ik; k
0
and he
k
je
k
0
i0. The four ei-
genvectors with eigenvalue !
2
are now e
jkj
, e
jkj
which
have support on the lattice L
j"j
, and e
jkj
, e
jkj
which have
support on the lattice L
j"j
. We will be interested only in
the symmetric combinations:
e
s
k
1
2
e
k
e
k
e
k
e
k
;
(4.9)
which are invariant under . Finally, we note that any
symmetric element of H
grav
"
can be expanded as
Z
1
1
dk
~
ke
s
k
: (4.10)
We can now write down the general symmetric solution
to the quantum constraint (4.4) with initial data in H
grav
"
:
;
Z
1
1
dk
~
ke
s
k
e
i!
~
k
e
s
k
e
i!
(4.11)
where
~
k are in L
2
R;dk.As !1, these ap-
proach solutions (3.9) to the WDW equation. However,
the approach is not uniform in the Hilbert space but varies
from solution to solution. As indicated in Sec. II B, the
LQC solutions to (4.4) which are semiclassical at late times
can start departing from the WDW solutions for relatively
large values of , say 10
4
o
.
As in the WDW theory, if
k vanishes, we will say
that the solution is of positive frequency and if
k
vanishes we will say it is of negative frequency. Thus,
every solution to (4.4) admits a natural positive and nega-
tive frequency decomposition. The positive (respectively,
negative) frequency solutions satisfy a Schro
¨
dinger type
first order differential equation in :
i
@
@
p
(4.12)
but with a Hamiltonian
p
(which is nonlocal in ).
Therefore the solutions with initial datum ;
o
f
are given by
; e
i
p
o
f
; : (4.13)
Remark.—In the above discussion, we considered a
generic ". We now summarize the situation in the special
cases, " 0 and " 2
o
. In these cases, differences arise
because the individual lattices are invariant under the
reflection !, i.e., the lattices L
j"j
and L
j"j
coin-
cide. As before, there is a 2-fold degeneracy in the eigen-
vectors of on any one lattice. For concreteness, let us
label the Hilbert spaces H
grav
j"j
and choose the basis vectors
e
k
, with k 2 R as above. Now, symmetrization can be
performed on each of these Hilbert spaces by itself. So, we
have
e
s
k
1
2
p
e
k
e
k
: (4.14)
However, the vector e
s
jkj
coincides with the vector
e
s
jkj
so there is only one symmetric eigenvector per
eigenvalue. This is not surprising: the original degeneracy
was 2-fold (rather than 4-fold) and so there is one sym-
metric and one antisymmetric eigenvector per eigenvalue.
Nonetheless, it is worth noting that there is a precise sense
in which the Hilbert space of symmetric states is only ‘‘half
as big’’ in these exceptional cases as they are for a generic
".
For " 2
o
, there is a further subtlety because C
vanishes at 2
o
and C
vanishes at 2
o
.
Thus, in this case, as in the WDW theory, there is a
decoupling and the knowledge of the eigenfunction
e
k
on the positive axis does not suffice to determine
it on the negative axis and vice versa. However,
the degeneracy of the eigenvectors does not increase
but remains 2-fold because (4.4) now introduces two
new constraints: C
2
o
e
k
6
o
!
2
B2
o
C
o
2
o
e
k
2
o
0. Conceptually, this difference
is not significant; there is again a single symmetric eigen-
function for each eigenvalue.
B. The physical sector
Results of Sec. IVA show that while the LQC operator
differs from the WDW operator
in interesting ways,
the structural form of the two Hamiltonian constraint
equations is the same. Therefore, apart from the issue of
superselection sectors which arises from the fact that is
discrete, introduction of the Dirac observables and deter-
mination of the inner product, either by demanding that the
Dirac observables be self-adjoint or by carrying out group
averaging, is completely analogous in the two cases.
Therefore, we will not repeat the discussion of Sec. III B
but only summarize the final structure.
The sector of physical Hilbert space H
"
phy
labelled by
" 20; 2
o
consists of positive frequency solutions
; to (4.4) with initial data ;
o
in the sym-
metric sector of H
"
grav
. Equation (4.11) implies that they
have the explicit expression in terms of our eigenvectors
e
s
k
;
Z
1
1
dk
~
ke
s
k
e
i!
; (4.15)
where, as before, !
2
16G=3k
2
1=16 and e
s
k
is given by (4.9) and (4.14). By choosing appropriate
functions
~
k, this expression will be evaluated in
Sec. VA using fast Fourier transforms. The resulting
; will provide, numerically, quantum states which
are semiclassical for large . The physical inner product is
QUANTUM NATURE OF THE BIG BANG: AN ... PHYSICAL REVIEW D 73, 124038 (2006)
124038-15
given by
1
;
o
h
1
j
2
i
"
X
2fj"j4n
o
;n2Zg
B
1
;
o
2
;
o
(4.16)
for any
o
. The action of the Dirac observables is inde-
pendent of " and has the same form as in the WDW theory:
d
jj
o
; e
i
p
o
jj;
o
and
^
p
; i@
@;
@
:
(4.17)
The kinematical Hilbert space H
total
kin
is nonseparable
but, because of superselection, each physical sector H
"
phy
is separable. Eigenvalues of the Dirac observable
d
jj
o
constitute a discrete subset of the real line in each sector. In
the kinematic Hilbert space H
grav
kin
, the spectrum of
^
p is
discrete in a subtler sense: while every real value is al-
lowed, the spectrum has discrete topology, reflecting the
fact that each eigenvector has a finite norm in H
grav
kin
. Thus,
the more delicate discreteness of the spectrum of
^
p on
H
grav
kin
descends to the standard type of discreteness of
Dirac observables. The question is often raised whether
the kinematic discreteness in LQG will have strong im-
prints in the physical sector or if they will be washed away
in the passage to the physical Hilbert space. A broad
answer— illustrated by the area eigenvalues of isolated
horizons [57,58] —is that the discreteness will generically
descend to the physical sector at least in cases where one
can construct Dirac observables directly from the kine-
matical geometrical operators [1]. The present discussion
provides another illustration of this situation.
Note that the eigenvalues of
d
jj
o
in distinct sectors are
distinct. Therefore, which sector actually occurs is a ques-
tion that can be in principle answered experimentally,
provided one has access to microscopic measurements
which can distinguish between values of the scale factor
which differ by 10‘
Pl
. This will not be feasible in the
foreseeable future. Of greater practical interest are the
coarse-grained measurements, where the coarse graining
occurs at significantly greater scales. For these measure-
ments, different sectors would be indistinguishable and one
could work with any one.
The group averaging procedure used in this section is
quite general in the sense that it is applicable for a large
class of systems, including full LQG if, e.g., its dynamics is
formulated using Thiemann’s master constraint program
[59]. In this sense, the physical Hilbert spaces H
"
phy
con-
structed here are natural. However, using the special struc-
tures available in this model, one can also construct an
inequivalent representation which is closer to that used in
the WDW theory. The main results on the bounce also hold
in that representation. Although that construction appears
to have an ad hoc element, it may well admit extensions
and be useful in more general models. Therefore, it is
presented in Appendix C.
V. NUMERICS IN LOOP QUANTUM COSMOLOGY
In this section, we will find physical states of LQG and
analyze their properties numerically. This section is di-
vided into three parts. In the first we study eigenfunctions
e
k
of and then use them to directly evaluate the right
side of (4.15), thereby obtaining a general physical state. In
the second part we solve the initial value problem starting
from initial data at
o
, thereby obtaining a general
solution to the difference Eq. (4.4). In the third we sum-
marize the main results and compare the outcome of the
two methods. Readers who are not interested in the details
of simulations can go directly to the third subsection.
A large number of simulations were performed within
each of the approaches by varying the parameters in the
initial data and working with different lattices L
"
. They
show that the final results are robust. To avoid making the
paper excessively long, we will only show illustrative plots.
A. Direct evaluation of the integral representation
(4.15) of solutions
The goal of this subsection is to evaluate the right side of
(4.15) using suitable momentum profiles
~
k. This calcu-
lation requires the knowledge of eigenfunctions e
s
k
of
. Therefore, we will first have to make a somewhat long
detour to numerically calculate the basis functions e
k
and e
s
k
introduced in Sec. IVA. The integral in (4.15)
will be then evaluated using a fast Fourier transform.
1. General eigenfunctions of the operator: Asymptotics
We will first establish properties of the general eigen-
functions e
!
of that were used in Sec. IVA.
Let us fix a lattice, say L
j"j
. Since the left side of (4.6)
approaches
^
C
WDW
grav
e
!
as jj!1, in this limit one
would expect each e
!
to converge to an eigenfunction
e
!
of with the same eigenvalue. Numerical simula-
tions have shown that this expectation is correct and have
also provided the rate of approach.
Recall that each WDW eigenfunction
e
!
is a linear
combination of basis functions
e
jkj
, e
jkj
defined in
Sec. III A. Therefore, given an e
!
it suffices to calculate
the coefficients of the decomposition of
e
!
with respect
to this basis. The method of finding these coefficients is
presented in detail in Appendix B.
7
Once the limiting
e
!
were found, they were compared with the original
7
If !
2
< G=3, then !
2
is not part of the spectrum of the self-
adjoint operator
. Nonetheless, by directly solving the eigen-
value equation
e
!
!
2
e
!
one can introduce an analogous
decomposition onto fixed eigenfunctions,
e
jk
0
j
:
jj
1=4k
0
,
k
02
1=16 3=16G!
2
and also write the limit of e
!
in
terms of coefficients in this ‘‘basis.’’
ASHTEKAR, PAWLOWSKI, AND SINGH PHYSICAL REVIEW D 73, 124038 (2006)
124038-16
eigenfunction e
!
for a variety of values of !.An
illustrative plot comparing e
!
with its limit is shown
in Fig. 2. In general, each e
!
approaches distinct ei-
genfunctions
e
!;
of in the limits !1. The rate
of approach is given by
1=4
e
!
(
1=4
e
!;
O
1
2
; for >0;
1=4
e
!;
O
1
2
; for <0:
(5.1)
Numerical tests were performed up to jj10
6
o
. The
quantity
7=4
j e
!
e
!;
j was found to be bounded.
The bound decreases with . For the case " 0, ! 20
depicted in Fig. 2 the absolute bound in the interval
10
2
o
; 10
6
o
was less than 90.
Because the eigenfunctions e
!
of are determined
on the entire lattice L
j"j
by their values on (at most) two
points; the WDW limits for positive and negative are not
independent. Thus, if the limits are expressed as
e
!
!
1
Ae
jkj
Be
jkj
;
e
!
!
1
Ce
jkj
De
jkj
;
(5.2)
the coefficients C, D are uniquely determined by values of
A, B (and vice versa). One relation, suggested by analytical
considerations involving the physical inner product, was
verified in detail numerically:
jAj
2
jBj
2
jCj
2
jDj
2
: (5.3)
It will be useful in the analysis of basis functions in the next
two subsections.
2. Construction of the basis e
jkj
In Sec. IVA we introduced a specific basis of H
"
which
is well adapted for comparison with the WDW theory. We
will now use numerical methods to construct this basis and
analyze its properties. Our investigation will be restricted
to the vectors e
jkj
because the physical states ;
of Eq. (4.15) we are interested in will have negligible
projections on the vectors e
jkj
. [Recall that in general
2 H
"
has support on L
j"j
[ L
j"j
. e
jkj
has sup-
port on L
j"j
and e
jkj
on L
j"j
.]
Each of the eigenfunctions e
jkj
is calculated as follows:
To solve (4.6), we need to specify initial conditions at two
points on each of the two lattices. We fix large positive
?
2 L
j"j
and demand that the values of e
jkj
agree
with those of
e
jkj
at the points
?
and
?
4
o
.
Then e
jkj
are evaluated separately on finite domains
L
j"j
\
?
;
?
of each of the two lattices. For large
negative , these eigenfunctions are linear combinations
of the WDW basis functions C
e
jkj
D
e
jkj
. The coef-
ficients C
, D
are evaluated using the method specified in
Appendix B.
Eigenfunctions e
jkj
were calculated for approxi-
mately 2 10
4
different jkj’s in the range 5 ! 10
3
.
They revealed the following properties:
(i) Each e
jkj
is well approximated by a WDW
eigenfunction until one reaches the ‘‘genuinely
quantum region.’’ In this region the absolute value
je
jkj
j grows very quickly as decreases: je
jkj
j/
e
sgn
!jj
p
, where " is a constant on
any given lattice. This property is illustrated by
Fig. 3. This region of rapid growth is symmetric
about 0 and its size depends linearly on ! (the
square root of the eigenvalue of ); its boundary
lies at 0:5!
o
). [However, this region ex-
cludes the interval 4
o
; 4
o
where B de-
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
1 10 100 1000 10000
µ
-1/4
e
k
µ
e
µ/µ
(
)
o
µ/µ
o
( )
ω
µ
−1/4
FIG. 2 (color online). Crosses denote the values of an eigen-
function e
!
of for " 0 and ! 20. The solid curve is
the eigenfunction
e
!
of the WDW to which e
!
approaches at large positive .As increases, the set of points
on L
"
becomes denser and fills the solid curve. For visual clarity
only some of these points are shown for >100.
-15
-10
-5
0
5
10
-30 -20 -10 0 10 20 30
ω = 250
ω = 500
ω = 750
|k|
( )
µ
/
−
ω
1/2
e
ln
|
|
sgn( )
|µ/µ |
1/2
o
µ
+
−
FIG. 3 (color online). The exponential growth of je
jkj
j in
the genuinely quantum region is shown for three different values
of !, where ! is given by e
jkj
!
2
e
jkj
.
QUANTUM NATURE OF THE BIG BANG: AN ... PHYSICAL REVIEW D 73, 124038 (2006)
124038-17
creases and goes to zero, departing significantly
from its WDW analog
B.]
(ii) After leaving this region of growth, the basis func-
tion e
jkj
again approaches some WDW eigen-
function
e
jkj
!
1
C
e
jkj
D
e
jkj
; (5.4)
where the coefficients C
, D
are large. Their
absolute values grow exponentially with !.To
investigate this property qualitatively we defined
an ‘‘amplification factor’’
:
jC
jjD
j: (5.5)
The numerical calculations show that
,so
parts of e
jkj
supported on L
j"j
and L
j"j
are
amplified equally. The dependence of
on !
and " is shown in Fig. 4. Almost everywhere it
can be well approximated by the function
!; "e
a!b
; (5.6)
where a, b are rather complicated functions of ".
The fits of a", b" are presented in Figs. 5 and 6.
The actual simulations were carried out for various
values of p
@!,uptop
10
3
.
(iii) The general relation (5.3) holds in our case with
jBj0. Existence of the tremendous amplification
now implies that the absolute values of C and D are
almost equal (with differences of the order of
1=
)
jC
jjD
j: (5.7)
Thus for negative , eigenfunctions asymptotically
approach WDW eigenfunctions and are almost
equally composed of incoming and outgoing
waves.
3. Basis for the symmetric sector
Once the basis functions e
jkj
are known one can readily
use (4.9) to construct the basis e
s
jkj
for solutions to (4.6)
which are symmetric under !. Because of strong
amplification in the region around 0 the behavior of
e
s
jkj
is dominated by properties of e
jkj
for <0. The
numerical calculations show the following properties:
(i) Each symmetric basis eigenfunction is strongly
suppressed in the genuinely quantum region around
to 0. The behavior of e
jkj
in this region
implies that je
s
jkj
j/cosh
!
p
, where is a
function of " only.
(ii) Outside the genuinely quantum region, e
s
jkj
quickly approaches the WDW eigenfunction al-
most equally composed of incoming and outgoing
‘‘plane waves’’ (
e
jkj
and e
jkj
). In the exceptional
20
30
40
50
60
70
80
90
100
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
10
0
10
5
10
10
10
15
10
20
10
25
10
30
10
35
λ
ε
ω
ε/µ
o
FIG. 4 (color online). The amplification factor
in the
genuinely quantum region is shown as a function of the parame-
ter " labeling the lattice and !.
-8
-6
-4
-2
0
2
4
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
b
ε
ε/µ
o
b
FIG. 6 (color online). The function b" of Eq. (5.6) is plotted
by connecting numerically calculated data points.
0.7
0.72
0.74
0.76
0.78
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
a
ε
ε/µo
a
FIG. 5 (color online). The function a" of Eq. (5.6) is plotted
by connecting numerically calculated data points.
ASHTEKAR, PAWLOWSKI, AND SINGH PHYSICAL REVIEW D 73, 124038 (2006)
124038-18
cases " 0 or 2
o
, the two contributions are ex-
actly equal. To establish this result, note first that
the symmetry requirement and the fact that
C
2
o
C
2
o
0 imply that in both
cases the value of e
s
jkj
at one point already deter-
mines the complete initial data (i.e., values at some
and
4
o
) and hence the eigenfunction on
the entire L
"
:
e
s
jkj
4
o
e
s
jkj
4
o
e
s
jkj
0; for " 0;
(5.8a)
e
s
jkj
6
o
!
2
B2
o
C
o
2
o
C
2
o
e
s
jkj
2
o
;
for " 2
o
: (5.8b)
Because of the reality of coefficients of Eq. (4.6),
this implies that the phase of e
s
jkj
is exactly con-
stant, whence contributions of
e
jkj
and e
jkj
are also
exactly equal.
(iii) On each lattice L
j"j
the incoming and outgoing
components are rotated with respect to each other
by an angle
e
s
jkj
j
L
j"j
!
!1
z
e
i
e
jkj
e
i
e
jkj
; (5.9)
where z
are some complex constants satisfying
jz
jjz
j, while the phases
are functions of "
and !. In general, for " 0 or " 2
o
,
need
not equal
.
4. Evaluation of the integral in (4.15)
Now that we have the symmetric basis functions e
s
jkj
at
our disposal, we can obtain the desired physical states by
directly evaluating the integral in (4.15).
We wish to construct physical states which are sharply
peaked at a phase space point on a classical trajectory of
the expanding universe at a late time (e.g., ‘‘now’’). The
form of the integrand in (4.15), the expression (4.16) of the
physical inner product, the functional form of !k and
standard facts about coherent and squeezed states in quan-
tum mechanics provide a natural strategy to select an
appropriate
~
k. If we set
~
ke
kk
?
2
=2
2
e
i!
?
(5.10)
with suitably small , the final state will be sharply peaked
at p
p
?
p
?
16G@
2
3
1=2
k
?
(5.11)
and the parameter
?
will determine the value
?
of the
Dirac observable
d
jj
o
at which the state will be peaked at
time
o
. As mentioned in Sec. III B, to obtain a state which
is semiclassical at a late time, we need a large value of p
:
p
?
@ in the classical units, c G 1. Therefore, we
need k
?
1, whence the functions
~
k of interest will
be negligibly small for k>0. Therefore, without loss of
physical content, we can set them to zero on the positive k
axis. This is why the explicit form of eigenfunctions e
s
jkj
is
not required in our analysis.
Thus, the integral we wish to evaluate is
;
Z
1
0
dke
kk
?
2
=2
2
e
s
k
e
i!k
?
;
(5.12)
where is the spread of the Gaussian. The details of the
numerical evaluation can be summarized as follows.
(i) For a generic ", the e
s
jkj
were found numerically
following the procedure specified in Secs. VA 2
and VA 3. For the exceptional cases, " 0, 2
o
,
in order to avoid loss of precision in the region
where e
s
jkj
is very small, we provided ‘‘initial
values’’ of e
s
k
j
at " and " 4
o
using (5.8). On the k axis we chose a set fk
j
g of
points which are uniformly distributed across the
interval k
?
10; k
?
10. In numerical simu-
lations, the number l of points in the set fk
j
g ranged
between 2
11
and 2
13
.
(ii) Next, for each k
j
, we calculated e
s
jk
j
j
i
for 2
f" 4n
o
:n 2fN; ...;Ngg where N is a large
constant 50p
?
.
(iii) Finally, we evaluated (5.12) using fast Fourier
transform. The result was a set of profiles
i
;
j
where
j
3=4G
p
j l=2=k
l
k
1
. This is a positive frequency solution to the
LQC Eq. (4.4).
Our next task is to analyze properties of these solutions.
Given any one
j
;
i
, we chose instants of time and
calculated the norm and the expectation values of our Dirac
observable j ^j
and
^
p
using
kk
2
X
i
2L
"
B
i
j;
i
j
2
; (5.13a)
h
d
jj
i
1
kk
2
X
i
2L
"
B
i
j
i
jj;
i
j
2
; (5.13b)
h
^
p
i
1
kk
2
X
i
2L
"
B
i
;
i
@
i
@
;
i
:
(5.13c)
Finally, the dispersions were evaluated using their defini-
tions:
h
d
jj
i
2
jh
d
j
2
j
ih
d
jj
i
2
j;
h
^
p
i
2
jh
^
p
2
ih
^
p
i
2
j:
(5.14)
These calculations were performed for 16 different
choices of " and for 10 values of p
up to a maximum
of p
10
3
, and for 5 different choices of the dispersion
QUANTUM NATURE OF THE BIG BANG: AN ... PHYSICAL REVIEW D 73, 124038 (2006)
124038-19
parameter . Figure 7 provides an example of a state
constructed via this method. The expectation values of
h
d
jj
i are shown in Fig. 8.
Results are discussed in Sec. V C below.
B. Evolution in
We also can regard the quantum constraint (4.4) as an
initial value problem in time and solve it by carrying out
the evolution. Conceptually, this approach is simpler
since it does not depend on the properties of the eigen-
functions of . However, compared to the direct evaluation
of the integral (4.15), this method is technically
more difficult because it entails solving a large number
of coupled differential equations. Nonetheless, to demon-
strate the robustness of results, we carried out the
evolution as well. This subsection summarizes the
procedure.
1. method of integration
At large jj the difference equation is well approxi-
mated by the WDW equation which is a hyperbolic partial
differential equation. However, since couples in dis-
crete steps, for a given " sector (4.4) is just a system of a
countable number of coupled ordinary differential equa-
tions of the second order.
8
Technical limitations require restriction of the domain of
integration to a set j "j4N
o
, where 1 N 2 Z.
This restriction makes the number of equations finite.
However one now needs to introduce appropriate boundary
conditions. The fundamental equation on the boundary is
i@
s
p
, where s 1 ( 1) for the forward
(backward) evolution in . It is difficult to calculate
p
at each time step. Therefore, just on the boundary
itself, this equation was simplified to
@
; s
G=3
p
jj2
o
;
4 sgn
o
;: (5.15)
This is the discrete approximation of the continuum opera-
tor @
s
16G=3
p
@
which itself is an excellent
approximation to the fundamental equation when the
boundary is far. The boundary condition requires the solu-
tion to leave the domain of integration. (For, to make the
evolution deterministic in the domain of interest, it is
important to avoid waves entering the integration domain
from the boundary.) The boundary was chosen to lie suffi-
ciently far from the location (in ) of the peak of the initial
wave packet. Its position was determined by requiring that
the value of the wave function at the boundary be less than
10
n
times than at its value of the peak, and n ranged
between 9 and 24 in different numerical simulations.
Three different methods were used to specify the initial
data and @
at
o
. These are described in
Sec. V B 2. The data were then evolved using the fourth
order adaptive Runge-Kutta method. To estimate the nu-
merical error due to discretization of time evolution, two
sup-norms were used:
j
1
2
j
I
sup
j
i
"jN
o
j
1
2
j
sup
j
i
"jN
o
j
2
j
; (5.16)
and
j
1
2
j
II
sup
j
i
"jN
o
jj
1
jj
2
jj
sup
j
i
"jN
o
j
2
j
: (5.17)
0.5
1
1.5
2
0
2.0*10
3
4.0*10
3
6.0*10
3
8.0*10
3
1.0*10
4
1.2*10
4
20
40
60
φ
µ/µ
| |
Ψ
ο
FIG. 7 (color online). Plot of the wave function ;
obtained by directly evaluating the right side of (4.15).
Parameters are p
500, p
=p
0:05, and "
o
.
0.5
1
1.5
2
0 2000 4000 6000 8000 10000 12000
LQC
classical
φ
µ/µ
ο
FIG. 8 (color online). Expectation values and dispersion of
d
jj
for the wave function presented in Fig. 7 are compared
with classical trajectories.
8
Unfortunately for the " 0 sector the equation is singular at
0, so the analysis of this subsection will not go through.
This sector was handled by the direct evaluation of the integral
representation of the solution, presented in the last subsection.
ASHTEKAR, PAWLOWSKI, AND SINGH PHYSICAL REVIEW D 73, 124038 (2006)
124038-20
Figure 9 shows an example of the results of convergence
tests for the solution corresponding to p
10
3
with
initial spread in p
of 3% (used in Figs. 12–15). One
can see that the phase of is more sensitive to numerical
errors than its absolute value. Therefore, although the
accuracy is high for the solution ; itself, it is even
higher for the mean values and dispersions of
d
jj
o
and
^
p
.
2. Initial data
Although we are interested in positive frequency solu-
tions, to avoid having to take square roots of at each
‘‘time step,’’ the second order evolution Eq. (4.4) was used.
Thus, the initial data consist of the pair and its time
derivative @
, specified at some time
o
. The positive
frequency condition was incorporated by specifying, in the
initial data, the time derivative of in terms of . Since
, we will restrict ourselves to the positive
axis. The idea is to choose semiclassical initial data
peaked at a point p
?
;
?
on a classical trajectory at
time
o
, with p
?
G@
2
p
;
?
1, and evolve
them. To avoid philosophical prejudices on what the state
should do at or near the big bang, we specify the data on the
expanding classical trajectory at late time (i.e., now) and
ask it to be semiclassical like the observed universe.
The idea that the data be semiclassical was incorporated
in three related but distinct ways.
(1) Method I. —This procedure mimics standard quan-
tum mechanics. Since c; are canonically conju-
gate on the phase space, we chose to be a
Gaussian (with respect to the measure defined by the
WDW inner product) and peaked at large
?
and the
value c
?
of c at the point on the classical trajectory
determined by
?
;p
?
;
o
):
j
o
:
Njj
3=4
e
2
=2~
2
e
ic
?
=2
;
(5.18)
where N is a normalization constant. The initial
value of @
j
o
was calculated using the classical
Hamilton’s equations of motion
@
j
o
sgnk
?
16G
3
s
?
?
~
2
i!
?
?
j
o
; (5.19)
where !
?
p
?
=@.
(2) Method II.—This procedure takes advantage of the
fact that (3.19) provides a WDW physical state
which is semiclassical at late times. The idea is to
calculate and @
at
o
and use their
restrictions to lattices L
"
as the initial data for
LQC, setting !
16G=3
p
jkj as in the discussion
following (3.19). Thus, the initial data used in the
simulations were of the form
j
o
?
1=4
e
2
=2ln
2
jj=j
?
j
e
ik
?
lnjj=j
?
j
;
(5.20)
@
j
o
sgnk
?
16
3
s
2
ln
jj
j
?
j
i!k
?
j
o
: (5.21)
This choice is best suited to comparing the LQC
results with those of the WDW theory. The spreads
~ of method I and of method II are related to the
initial spread j
o
as follows:
~
?
2
p
j
o
: (5.22)
(3) Method III.—To facilitate comparison with the di-
rect evaluation of the integral solution described in
Sec. VA, a variation was made on method II.
Specifically, in the expression (3.19) of , the
WDW basis eigenfunctions
e
k
were rotated by
multiplying them with a k- and "-dependent phase
factor defined in Eq. (5.9)
e
jkj
哫 e
i
e
jkj
: (5.23)
These phases were first found numerically using the
method specified in Appendix B and then functions
10
-5
10
-4
10
-3
10
-2
10
-1
10
5
10
6
N
M’
FIG. 9 (color online). Error functions j
M
0
M
j
I
(upper
curve) and j
M
0
M
j
II
(lower curve) are plotted as a
function of time steps. Here
M
0
refers to the final profile of
wave function for simulation with M
0
time steps.
M
refers to
the final profile for the finest evolution (approximately 2:88
10
6
time steps). In both cases, the evolution started at 0 and
the final profile refers to 1:8.
QUANTUM NATURE OF THE BIG BANG: AN ... PHYSICAL REVIEW D 73, 124038 (2006)
124038-21
of the form
A lnBk Ck D (5.24)
(where A, B, C, D are real constants) were fitted to
the results. After subtraction of the zeroth and the
first order terms in the expansion in k around k
?
(which, respectively, correspond to a constant phase
and a shift of origin of ) the resulting function was
used to rotate the basis
e
k
appearing in (3.19) via
(5.23). The expression on the right side of Eq. (3.19)
and its derivative were then integrated numeri-
cally at
o
.
In the simulations, 15 different values of p
?
were used
ranging between 10
2
and 10
5
.
?
was always greater than
2:5p
. The dispersion was allowed to have five different
values. These simulations involved four different, ran-
domly chosen values of ".
C. Results and comparisons
We now summarize the results obtained by using the two
constructions specified in Secs. VA and V B. The qualita-
tive results are robust and differences lie in the finer
structure. In particular, evolutions of initial data con-
structed from the three methods described in Sec. V B 2
yield physical states ; which are virtually identical
except for small differences in the behavior of relative
dispersions of the Dirac observables. The numerical evalu-
ation of the integral (4.15) in Sec. VA 4 yielded results very
similar to those obtained in Sec. V B 2 using initial data of
method III. An example of results is presented in Figs. 10
and 11.
Highlights of the results can be summarized as follows:
(1) The state remains sharply peaked throughout the
evolution. However, as shown in Fig. 16, while the
product p
is nearly constant for large ,
there is a substantial increase near 0.
(2) The expectation values of ^
and
^
p
are in good
agreement with the classical trajectories, until the
increasing matter density approaches a critical
value. Then, the state bounces from the expanding
branch to a contracting branch with the same value
of h
^
p
i. (See Fig. 11). This phenomena occurs
universally, i.e., in every " sector, for all three
methods of choosing the initial data and for any
choice of p
G@
2
p
. In this sense the classical
big bang is replaced by a quantum bounce. Note that
this is in striking contrast with the situation with the
WDW theory we encountered in Sec. III B, even
when the initial data is chosen using method II
which is tailored to the WDW theory. As indicated
in Appendix A, the existence of the bounce can be
heuristically understood from an ‘‘effective theory.’’
The detailed numerical work supports that descrip-
tion, thereby providing a justification for the ap-
proximation involved.
(3) If the state is peaked on the expanding branch
?
exp
16G=3
p
o
in the distant
future, due to the bounce it is peaked on a contract-
ing branch in the distant past, given by
Dp
?
?
exp
16G=3
p
o
, where
Dp
2
o
p
2
=12G@
2
. Thus, for large jj the
solution ; exhibits reflection symmetry
(about
o
1
2
lnDp
?
). However, it is not
exactly reflection symmetric (compare [60]).
(4) As a consistency check, we verified that the norm
and the expectation value h
^
p
i are preserved during
the entire evolution. Furthermore, the dispersion
also remains small throughout the evolution,
although the precise behavior depends on the
method of specification of the initial data. Dif-
ferences arise primarily near the bounce point and
manifest themselves through the behavior of the
0
0.2
0.4
0.6
0.8
1
-1.8
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
1.0*10
4
2.0*10
4
3.0*10
4
4.0*10
4
5.0*10
4
6.0*10
4
7.0*10
4
8.0*10
4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
µ/µ
o
φ
Ψ
| |
FIG. 10 (color online). The absolute value of the wave func-
tion obtained by evolving an initial data of method II. For clarity
of visualization, only the values of jj greater than 10
4
are
shown. Being a physical state, is symmetric under !.
In this simulation, the parameters were " 2
o
, p
?
10
4
, and
p
=p
?
7:5 10
3
.
-1.8
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0 2*10
4
4*10
4
6*10
4
8*10
4
1*10
5
LQC
classical
φ
µ/µ
o
FIG. 11 (color online). The expectation values (and disper-
sions) of
d
jj
are plotted for the wave function in Fig. 10 and
compared with expanding and contracting classical trajectories.
ASHTEKAR, PAWLOWSKI, AND SINGH PHYSICAL REVIEW D 73, 124038 (2006)
124038-22
relative dispersion = as a function of . These
are illustrated (for all three methods as well as for
the direct evaluation of Sec. VA) in Fig. 12. Finally,
as argued in Appendix A, since the state is sharply
peaked, the value of = can be related to via
Eq. (A13). One finds that the product p
is
essentially constant in the region away from the
bounce but grows significantly near the bounce.
The differences can be summarized as follows:
(i) In method I the initial state is a minimum
uncertainty state in ; c but does not mini-
mize the uncertainty in ; p
at any value
of . The relative spread in remains ap-
proximately constant in the regions where
h ^
i is large and increases near the bounce
point monotonically. The wave function in-
terpolates ‘‘smoothly’’ between expanding
and contracting branches (see Fig. 13). On
the other hand the product p
of un-
certainties has a value much higher than 1=2,
grows quickly near the bounce and settles
down to constant value after it. See Fig. 16.
(ii) The state obtained by evolving the initial
data constructed from method II has mini-
mal uncertainty in ; p
. = is ap-
proximately constant for large hb
i and it
decreases quickly near the bounce point,
reaching its minimal value shortly before
the bounce point. After the bounce, it grows
and stabilizes at the value of the relative
spread found for data constructed using
method I (for the same values of p
?
and
initial =). Behavior of the wave func-
tion is also different from that in the previous
case. Near the bounce point its value grows
to form a bulge (see Fig. 14). The product
-1.2
-1.1
-1
-0.9
-0.8
-0.7
-0.6
200
400
600
800
1000
1200
0.1
0.2
0.3
0.4
0.5
0.6
φ
µ/µ
o
Ψ
| |
FIG. 13 (color online). A zoom on the absolute value of the
wave function near the bounce point. Initial data was specified
using method I.
0
0.02
0.04
0.06
0.08
0.1
-1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0
Method I
Method II
Method III
Direct
φ
∆µ/µ
FIG. 12 (color online). Comparisons between the relative dis-
persions = as functions of for all three methods specify-
ing initial data and the result of direct construction.
-1.2
-1.1
-1
-0.9
-0.8
-0.7
-0.6
200
400
600
800
1000
1200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
φ
Ψ
| |
µ/µ
o
FIG. 14 (color online). A zoom on the absolute value of the
wave function near the bounce point. Initial data was specified
using method II.
-1.2
-1.1
-1
-0.9
-0.8
-0.7
-0.6
200
400
600
800
1000
1200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
φ
µ/µ
o
Ψ
| |
FIG. 15 (color online). A zoom on the absolute value of the
wave function near the bounce point. Initial data was specified
using method III.
QUANTUM NATURE OF THE BIG BANG: AN ... PHYSICAL REVIEW D 73, 124038 (2006)
124038-23
p
remains almost constant for as long
as the results of the numerical measurement
are reliable (see Fig. 16) approaching a con-
stant after the bounce, with a somewhat
higher value. The heuristic estimate on
p
(see Appendix A) agrees with these
results. Finally, we also found that the in-
crease of the relative spread depends on p
?
and initial =.
(iii) The state obtained by evolving the initial
data constructed from method III does not
have minimum uncertainty in ; p
. The
behavior of = is similar to the previous
case except that it becomes equal to
p
=p
at the bounce point and its asymp-
totic value in the contracting branch is the
same as its starting value. Thus, the spread is
symmetric with respect to reflection in
around the bounce point. The difference be-
tween the value of = for large hb
i and
the one corresponding to the minimum un-
certainty state (for the same p
?
and p
)is
a function of p
and p
. The wave func-
tion forms a symmetric bulge near the
bounce point and the value p
remains
constant within the regime of validity of its
estimation (see Fig. 16).
(iv) The direct construction of Sec. VA yields
results similar to those obtained by using the
third method to choose the initial data for the
evolution.
(v) The differences between the relative disper-
sion = resulting from different methods
of choosing initial data also can be estimated
using the solutions to the effective dynami-
cal equations (for details of the method see
Appendix A).
We will conclude with two remarks.
(i) Let us return to the comparison between the results
of LQC and the WDW theory in light of our nu-
merical results. As we remarked in Sec. II B, on
functions which we used to construct the
semiclassical initial data, the leading term in the
difference
^
C
grav
^
C
wdw
grav
between the actions of
the two constraint operators goes as O
2
o
0000
.
Now, in LQC
o
is fixed (
o
3
3
p
=2) and on
semiclassical states
0000
k
?2
=
4
. Hence the
difference is negligible only in the regime
k
?4
=
4
1. In our simulations, k
?
10
4
whence
the differences are guaranteed to be negligible only
for 10
4
, i.e., well away from the bounce. But
let us probe the situation in greater detail. Let us
regard
o
as a mathematical parameter which can
be varied and shrink it. In the limit
o
! 0,
^
C
grav
C
wdw
grav
should tend to zero. Since there
is no bounce in the WDW theory, we are led to ask:
Would the LQC bounce continue to exist all the way
to
o
0 or is there a critical value at which the
bounce stops? The answer is that for any finite value
of
o
there is a bounce. However, if we keep the
physical initial data the same, we find that as we
decrease
o
the solution follows the classical tra-
jectory into the past more and more and bounce is
pushed further and further into the past. In the limit
as
o
goes to zero, the wave function follows the
classical trajectory into infinite past, i.e., the bounce
never occurs. This is the sense in which the WDW
result is recovered in the limit
o
! 0.
(ii) Numerical simulations show that the matter density
at the bounce points is inversely proportional to the
expectation value h
^
p
ip
?
of the Dirac observ-
able
^
p
: Given two semiclassical states with
h
^
p
ip
?
and
p
?
, we have
crit
=
crit
p
?
=p
?
.
Therefore, this density can be made small by choos-
ing sufficiently large p
?
. Physically, this is unrea-
sonable because one would not expect departures
from the classical theory until matter density be-
comes comparable to the Planck density. This is a
serious weakness of our framework. Essentially
every investigation within LQC we are aware of
has this— or a similar—drawback but it did not
become manifest before because the physics of the
singularity resolution had not been analyzed sys-
tematically. The origin of this weakness can be
traced back to details of the construction of the
Hamiltonian constraint operator, specifically the
precise manner in which the operator corresponding
to the classical field strength F
i
ab
was introduced.
The physical ideas that in LQC the operator corre-
sponding to the field strength F
i
ab
should be defined
through holonomies, and that quantum geometry
does not allow us to shrink the loop to zero size,
seem compelling. However, the precise manner in
which the value of
o
was determined using the
0
1
2
3
4
5
6
7
8
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Method I
Method II
Method III
∆φ∆
p
φ
φ
FIG. 16 (color online). The uncertainty product p
for
three methods of specifying the initial data.
ASHTEKAR, PAWLOWSKI, AND SINGH PHYSICAL REVIEW D 73, 124038 (2006)
124038-24
area gap is not as systematic and represents only a
‘‘first stab’’ at the problem. In [36] we will discuss
an alternate and more natural way of implementing
this idea. The resulting Hamiltonian constraint has a
similar form but also important differences.
Because of similarities, the qualitative conclusions
of this analysis —including the occurrence of the
quantum bounce —are retained but the differences
are sufficiently important to replace the expression
of the critical density by
0
crit
3=8G
2
where, as before, 2
3
p
‘
2
Pl
is the area gap.
Since
0
crit
is of Planck scale and is independent of
parameters associated with the semiclassical state,
such as p
?
, the departures from classical theory now
appear only in the Planck regime. This issue is
discussed in detail in [36].
VI. DISCUSSION
We will first present a brief summary and then compare
our analysis with similar constructions and results which
have appeared in the literature. However, since this
literature is vast, to keep the discussion to a manageable
size, these comparisons will be illustrative rather than
exhaustive.
In general relativity, gravity is encoded in space-time
geometry. A basic premise of LQG is that geometry is
fundamentally quantum mechanical and its quantum as-
pects are central to the understanding of the physics of the
Planck regime. In the last three sections, we saw that LQC
provides a concrete realization of this paradigm. In our
model every classical solution is singular and the singu-
larity persists in the WDW theory. The situation is quite
different in LQC. As in full LQG, the kinematical frame-
work of LQC forces us to define curvature in terms of
holonomies around closed loops. The underlying quantum
geometry of the full theory suggests that it is physically
incorrect to shrink the loops to zero size because of the
‘‘area gap’’ . This leads to a replacement of the WDW
differential equation by a difference equation whose step
size is dictated by . Careful numerical simulations dem-
onstrated in a robust fashion that the classical big bang is
now replaced by a quantum bounce. Thus, with hindsight,
one can say that although the WDW theory is quite similar
to LQC in its structure, the singularity persists in the WDW
theory because it ignores the quantum nature of geometry.
9
A. Emergent time
In this paper we isolated the scalar field as the
emergent time and used it to motivate and simplify various
constructions. However, we would like to emphasize that
this—or indeed any other—choice of emergent time is not
essential to the final results. In the classical theory (for any
given value of the constant of motion p
), we can draw
dynamical trajectories in the plane without singling
out an internal clock. A complete set of Dirac observables
can be taken to be p
, and either jj
o
or j
o
.
10
What
these observables measure is correlations and their speci-
fication singles out a point in the reduced phase space.
However, we do not have to single out a time variable to
define them. The same is true in quantum theory. To have
complete control of physics, we need to construct the
physical Hilbert space H
phy
and introduce on it a com-
plete set of Dirac observables. Again, both of these steps
can be carried out without singling out as emergent time.
For example, the scalar product can be constructed using
group averaging which requires only the knowledge of the
full quantum constraint and its properties, and not its
decomposition into a ‘‘time evolution part’’ @
2
and an
operator on the ‘‘true degrees of freedom.’’ Once the
scalar product is constructed, we can introduce a complete
set of Dirac observables consisting of
^
p
, and
d
jj
o
or
d
j
o
. Again, what matters is the correlations. This and
related issues have been discussed exhaustively in the
quantum gravity literature in relativity circles. In particu-
lar, a major part of a conference proceedings [62] was
devoted to it in the late eighties and several exhaustive
reviews also appeared in the nineties (see in particular
[63,64]).
However, thanks to our knowledge of how quantum
theory works in static space-times, singling out as the
emergent time turned out to be extremely useful in practice
because it provided guidance at several intermediate steps.
In particular, it directly motivated our choice of L
2
R
Bohr
;
Bd
Bohr
L
2
R; d as our auxiliary Hilbert space;
streamlined the detailed definition of operators represent-
ing the Dirac observables; and facilitated the subsequent
selection of the inner product by demanding that these be
self-adjoint. More importantly, by enabling us to regard the
constraint as an evolution equation, it transformed the
‘‘frozen formalism’’ to a familiar language of evolution
and enabled us to picture and interpret the bounce and
associated physics more easily. Indeed, following the
lead of early LQC papers, initially we tried to use as
time and ran in to several difficulties: specification of
physically interesting data became nonintuitive and cum-
9
Differences arise in two places. The first occurs in the matter
part of the constraint and stems from the fact that the functions
B representing the eigenvalues of the operator
d
1=jpj
3=2
in
LQG is different from the corresponding
B of the WDW
theory. The second comes from the role of quantum geometry in
the gravitational part of the Hamiltonian constraint, emphasized
above. In our model, qualitatively new features of the LQG
quantum dynamics can be traced back to the second. In particu-
lar, the bounce would have persisted even if we had used
B in
place of B in the analysis presented in this paper.
10
In the closed models, care is needed to specify the latter
because they can not be defined globally. But this issue is well
understood in the literature, especially through Rovelli’s contri-
butions [61]. See also [33].
QUANTUM NATURE OF THE BIG BANG: AN ... PHYSICAL REVIEW D 73, 124038 (2006)
124038-25
bersome; one could not immediately recognize the occur-
rence of the bounce; and the physics of the singularity
resolution remained obscure.
Our specific identification and use of emergent time
differs in some respects from that introduced earlier in
the literature. For example, in the context of the WDW
theory, there is extensive work on isolating time in the
WKB approximation (see e.g., [51,65]). By contrast, a key
feature of our emergent time is that it is not restricted to
semiclassical regimes: We first isolated the scalar field
as a variable which serves as a good internal clock away
from the singularity in the classical theory, but then showed
that the form of the quantum Hamiltonian constraint is
such that can be regarded as emergent time in the full
quantum theory, without a restriction that the states be
semiclassical or stay away from the singularity. The idea
of identifying an emergent time and exploiting the result-
ing ‘‘deparametrization’’ to select an inner product on the
space of solutions to the Hamiltonian constraint is not new
[33,52,53,62–64]. However, several of the concrete pro-
posals turn out to have serious deficiencies (for a further
discussion see, e.g., [66,67]). The idea of using a matter
field to define emergent time is rather old. In the frame-
work of geometrodynamics, it was carried out in detail for
dust in [68]. A proposal to use a massless scalar field as
time was also made in the framework of LQG [69] but its
implementation remained somewhat formal. In particular,
it is unlikely that the required gauge conditions can be
imposed globally in the phase space, and the modifications
in the construction of the physical scalar product that are
necessary to accommodate more local constructions were
not spelled out. More recently, a massless scalar field was
used as the internal clock in quantum cosmology in the
connection dynamics framework [70]. However, the focus
of discussion there is on the Kodama state in inflationary
models. Because of the inflationary potential, the quantum
constraint has explicit time dependence and the construc-
tion of the physical inner product is technically much more
subtle. In particular, a viable inner product cannot depend
on any auxiliary structure such as the choice of an instant
of time. These issues do not appear to have been fully
addressed in [70].
B. Resolutions of the big bang singularity
The issue of obtaining singularity free cosmological
models has drawn much attention over the years. The
discovery of singularity theorems sharpened this discus-
sion and there is a large body of literature on how one may
violate one or more assumptions of these theorems, thereby
escaping the big bang. Proposals include the use of matter
which violate the standard energy conditions, addition of
higher derivative corrections to the Einstein-Hilbert action,
and introduction of higher-dimensional scenarios inspired
by string theory. To facilitate comparison with the model
discussed in this paper, we will restrict ourselves to spa-
tially noncompact situations.
Already in the seventies, Bekenstein investigated a
model where the matter source consisted of incoherent
radiation and dust, interacting with a conformal massless
scalar field (which can have a negative energy density).
He showed that Einstein’s equations admit solutions
which are free of singularities [71]. In the eighties,
Narlikar and Padmanabhan found a singularity free solu-
tion to Einstein’s equation with radiation and a negative
energy massless scalar field (called the ‘‘creation field’’) as
source, and argued that the resulting model was consistent
with the then available observations [72]. Such investiga-
tions were carried out entirely in the paradigm of classical
relativity and the key difference from the standard
Friedmann-Robertson-Walker models arose from the use
of ‘‘nonstandard’’ matter sources. Our analysis, by con-
trast, uses a standard massless scalar field and every solu-
tion is singular in the classical theory. The singularity is
resolved because of quantum effects.
Another class of investigations starts with actions con-
taining higher derivative terms which are motivated by
suitable fundamental considerations. For example, to guide
the search for an effective theory of gravity which is viable
close to the Planck scale, Mukhanov and Brandenberger
proposed an action with higher order curvature terms for
which all isotropic cosmological solutions are nonsingular,
even when coupled to matter [73]. The modifications to
Einstein’s equations are thought of as representing quan-
tum corrections. However one continues to work with dif-
ferential equations formulated in the continuum. By con-
trast, our investigation is carried out in the framework of a
genuine quantum theory with a physical Hilbert space,
Dirac observables, and detailed calculations of expectation
values and fluctuations. Departures from classical general
relativity arise directly from the quantum nature of geome-
try. The final results are also different: While solutions in
[73] asymptotically approach de Sitter space, in our analy-
sis the classical big bang is replaced by a quantum bounce.
Perhaps the most well-known discussions of bounces
come from the pre-big bang cosmology and ekpyrotic/
cyclic models. The pre-big bang model uses the string
dilaton action and exploits the scale factor duality to
postulate the existence of a superinflating pre-big bang
branch of the Universe, joined to the radiation dominated
post-big bang branch [4]. However, the work was carried
out in the framework of perturbative string theory and the
transition from the pre-big bang to post-big branch was
postulated. The initial hope was that nonperturbative
stringy effects would enforce such a transition. However,
as of now, such mechanisms have not been found [7].
Although subsequent investigations have shown that a
bounce can occur in simplified models [74] or by using
certain effective equations (see, e.g., [75]), it is not yet
clear that this is a consequence of the fundamental theory.
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124038-26
The ekpyrotic and the more recent cyclic models [5,6] are
motivated by certain compactifications in string theory and
feature a five-dimensional bulk space-time with four-
dimensional branes as boundaries. In the ekpyrotic model,
the collision between a bulk brane with a boundary brane is
envisioned as a big bang. A key difficulty is the singularity
associated with this collision [8] (which can be avoided but
at the cost of violating the null energy condition [6]). In the
cyclic model, collision occurs between the boundary
branes [6]; however, it has been shown that the singularity
problem persists [76]. Thus, a common limitation of these
models is that the branch on ‘‘our side’’ of the big bang is
not joined deterministically to the branch on the other side.
In LQC by contrast, the quantum evolution is fully deter-
ministic. This is possible because the approach is non-
perturbative and does not require a space-time continuum
in the background.
Finally, the idea of a bounce has been pursued also in the
context of braneworld models. In the original Randall-
Sundrum scenario, the Friedmann equation is modified
by addition of a term on the right side which is quadratic
in density:
_
a
2
=a
2
8G=31 =2 where is
brane tension. However, since >0, the sign of the qua-
dratic term in is positive whence
_
a cannot vanish and
there is no bounce. To obtain a bounce, the correction
should be negative, i.e., make a ‘‘repulsive’’ contribution.
One way to reverse the sign is to introduce a second time-
like dimension in the bulk [77]. However, this strategy does
not appear to descend from fundamental considerations
and the physical meaning of the second timelike direction
is also unclear. Another avenue is to consider a bulk with a
charged black hole. A nonvanishing charge leads to terms
in the modified Friedmann equation which are negative.
_
a
can now vanish and a bounce can occur [78]. However, it
was shown that transition from contraction to expansion
for the brane trajectory occurs in the Cauchy horizon of the
bulk which is unstable to small excitations, thus the brane
encounters singularity before bouncing [79]. In LQC, by
contrast, while the Friedmann equation is effectively modi-
fied, the corrections come from quantum geometry and
they are automatically negative.
C. Extensions
A major limitation of our analysis—shared by all other
current investigations in quantum cosmology—is that the
theory is not developed by a systematic truncation of full
quantum gravity. This is inevitable because we do not have
a satisfactory quantum gravity theory which can serve as
an unambiguous starting point. The viewpoint is rather that
one should use lessons learned from mini- and midi-
superspace analysis to work one’s way up to more general
situations, especially to reduce the large number of ambi-
guities that exist in the dynamics of the full theory.
Even within quantum cosmology, our detailed analysis
was restricted to a specific mini-superspace model. In more
complicated models, differences are bound to arise. For
example, the full solution is not likely to remain so sharply
peaked on the classical trajectory till the bounce point and
even the existence of a bounce is not a priori guaranteed,
especially when inhomogeneities are added. However, the
methods developed in the paper can be applied to more
general situations. First, one could consider anisotropies.
Now, the main structural difference is that the operator
will no longer be positive definite. However, a detailed
analysis shows that what matters is just the operator jj,
obtained by projecting the action of to the positive
eigenspace in its spectral decomposition. Therefore, our
analytical considerations should go through without a
major modification. The numerical simulations will be
more complicated because we have to solve a higher-
dimensional difference equation (involving four variables
in place of two). Another extension will involve the in-
clusion of nontrivial potentials for the scalar field. Now,
generically will no longer be a monotonic function on
the classical trajectories and one would not be able to use it
as internal time globally. In the quantum theory, the op-
erator becomes ‘‘time dependent’’ (i.e. depends on ),
and the mathematical analogy between the quantum con-
straint and the Klein-Gordon equation in a static space-
time is no longer valid. Nonetheless, one can still use the
group averaging procedure [33,52] to construct the physi-
cal Hilbert space. For a general potential, a useful notion of
time will naturally emerge only in the semiclassical re-
gimes. For specific potentials (such as the quadratic one
used in chaotic inflation) one should be able to use methods
that have been successfully employed in the quantization
of model systems [54,55] (in particular, a pair of harmonic
oscillators constrained to have a fixed total energy).
Incorporation of spherical inhomogeneities seems to be
within reach since a significant amount of technical
groundwork already has been laid [80]. Incorporation of
general inhomogeneities, on the other hand, will be sub-
stantially more difficult. Background dependent treatments
have suggested that results obtained in the mini-superspace
approximation may be qualitatively altered once field
theoretical complications are unleashed (see, e.g., [81]).
However, already in the anisotropic case, there is a quali-
tative difference between perturbative and nonperturbative
treatments. Specifically, if anisotropies are treated as per-
turbations of a background isotropic model, the big bang
singularity is not resolved while if one treats the whole
problem nonperturbatively, it is [82]. Therefore, definitive
conclusions cannot be reached until detailed calculations
have been performed in inhomogeneous models. However,
if a quantum bounce does generically replace the big bang
singularity, it would be possible to explore the relation
between the effective descriptions of LQG and the
Hartle-Hawking ‘‘no boundary’’ proposal [83]. For, in the
effective description, the extrinsic curvature would vanish
at the bounce. Therefore, generically it may be possible to
QUANTUM NATURE OF THE BIG BANG: AN ... PHYSICAL REVIEW D 73, 124038 (2006)
124038-27
attach to the Lorentzian, post-bounce effective solution
representing the Universe at late times, a Riemannian
prebounce solution without boundary. If so, it would be
very interesting to analyze the sense in which this
Riemannian solution captures the physics of the prebounce
branch of the full quantum evolution.
Finally, it is instructive to recall the situation with sin-
gularities in classical general relativity. There, singularities
first appeared in highly symmetric situations. For a number
of years, arguments were advanced that this is an artifact of
symmetry reduction and generic, nonsymmetric solutions
will be qualitatively different. However, singularity theo-
rems by Penrose, Hawking, Geroch, and others showed
that this is not correct. An astute use of the differential
geometric Raychaudhuri equation revealed that singular-
ities first discovered in the simple, symmetric solutions are
in fact a generic feature of classical general relativity. A
fascinating question is whether the singularity resolution
due to quantum geometry is also generic in an appropriate
sense [84]. Is there a general equation in quantum geometry
which implies that gravity effectively becomes repulsive
near generic spacelike singularities, thereby halting the
classical collapse? If so, one could construct robust argu-
ments, now establishing general ‘‘singularity resolution
theorems’’ for broad classes of situations in quantum grav-
ity, without having to analyze models, one at a time.
ACKNOWLEDGMENTS
We would like to thank Martin Bojowald, Jim Hartle,
and Pablo Laguna for discussions. This work was sup-
ported in part by the NSF grant Nos. PHY-0354932 and
PHY-0456913, the Alexander von Humboldt Foundation,
and the Eberly research funds of Penn State.
APPENDIX A: HEURISTICS
Quantum corrections to the classical equations can be
calculated using ideas from a geometric formulation of
quantum mechanics where the Hilbert space is regarded as
(an infinite-dimensional) phase space, the symplectic
structure being given by the imaginary part of the
Hermitian inner product (see, e.g., [85]). This ‘‘quantum
phase space’’ has the structure of a bundle with the classi-
cal phase space as the base space, and all states with the
same expectation values for the canonically conjugate
operators
^
q
i
;
^
p
i
as an (infinite-dimensional) fiber. Thus,
any horizontal section provides an embedding of the clas-
sical phase space into the quantum phase space. In the case
of a harmonic oscillator (or free quantum fields) coherent
states constitute horizontal sections which are furthermore
preserved by the full quantum dynamics. In the semiclas-
sical sector defined by these coherent states, the effective
Hamiltonian coincides with the classical Hamiltonian and
there are no quantum corrections to classical dynamics. For
more general systems, using suitable semiclassical states
one may be able to find horizontal sections which are
preserved by the quantum Hamiltonian flow to a desired
accuracy (e.g. in a @ expansion). The effective Hamiltonian
governing this flow—the expectation value of the quantum
Hamiltonian operator in the chosen states, calculated to the
desired accuracy—is generally different from the classical
Hamiltonian. In this case, dynamics generated by the ef-
fective Hamiltonian provides systematic quantum correc-
tions to the classical dynamics [23] (see also [24,25,32]).
This procedure has been explicitly carried out in LQC
for various matter sources [23,86]. For a massless scalar
field, the leading order quantum corrections are captured in
the following effective Hamiltonian constraint [86]:
C
eff
6
2
2
o
jpj
1=2
sin
2
o
c8GBpp
2
; (A1)
where Bp is the eigenvalue of
d
1=jpj
3=2
operator given by
(4.3).
11
For jj
o
, Bp can be approximated as
Bp
6
8‘
2
Pl
3=2
jj
3=2
1
5
96
2
o
2
O
4
o
4
:
(A2)
The leading order term is 1=p
3=2
, thus Bp quickly ap-
proaches its classical value for jj
o
, corrections
being significant only in the genuinely quantum region in
the vicinity of 0. From now on, we will ignore the
quantum corrections to Bp.
To obtain the equations of motion, we need the effective
Hamiltonian H
eff
. As usual it is obtained simply by a
rescaling of C
eff
which gives H
eff
the dimensions of
energy and ensures that the matter contribution to it is
the standard matter Hamiltonian:
H
eff
C
eff
16G
3
8G
2
2
o
jpj
1=2
sin
2
o
c
p
2
2p
3=2
:
(A3)
Then, the Hamilton’s equation for
_
p become
_
p fp; H
eff
g
8G
3
@H
eff
@c
2jpj
1=2
o
sin
o
ccos
o
c: (A4)
Further, since the Hamiltonian constraint implies that H
eff
of (A3) vanishes, we have
sin
2
o
c
8
2
2
o
G
6jpj
2
p
2
; (A5)
which, on using Eq. (A4), provides the modified Friedmann
11
In the literature, eigenvalues Bp often contain a half-integer
j, a parameter representing a quantization ambiguity. In view of
the general consistency arguments advanced in [87], we have set
its value to its minimum, i.e. j 1=2.
ASHTEKAR, PAWLOWSKI, AND SINGH PHYSICAL REVIEW D 73, 124038 (2006)
124038-28
equation for the Hubble parameter H:
H
2
_
p
2
4p
2
8G
3
1
crit
;
where
crit
3
8G
2
2
o
3=2
2
p
p
:
(A6)
To obtain the dynamical trajectory, we also need the
Hamilton’s equation for ,
_
f; H
eff
g
p
p
3=2
: (A7)
By combining Eq. (A4) and (A7) we obtain the effective
equation of motion in the - plane:
d
d
16G
3
s
1
crit
1=2
: (A8)
The classical Friedmann dynamics results if we set
crit
1. Equations (A6) and (A8) suggest that the LQC effects
significantly modify the Friedmann dynamics once the
matter density reaches a critical value,
crit
. In the classical
dynamics, the Hubble parameter H cannot vanish (except
in the trivial case with p
0). In the modified dynamics,
on the other hand, H vanishes at
crit
. At this point,
the Universe bounces. Thus, the bounce predicted by
Eq. (A6) has its origin in quantum geometry. (The critical
value at which this bounce occurs is given by
crit
6=8G@
2
p
p
o
.) As pointed out in the main text, a
physical limitation of the present framework is that if p
is chosen to be sufficiently large, the critical density
crit
can be small.
To obtain the effective equations, several approxima-
tions were made [23,86] which are violated in the deep
Planck regime. Nonetheless, the resulting picture of the
bounce is consistent with the detailed numerical analysis.
In fact, within numerical errors the trajectory
1
2
exp
16G
3
s
o
Dp
exp
16G
3
s
o
(A9)
obtained by integrating Eq. (A8) approximates the expec-
tation values of
d
jj
quite well. (As in the main text,
Dp
2
o
p
2
=12G@
2
). An illustrative plot of this ge-
neric behavior is shown in Fig. 17. Therefore, in retrospect,
this analysis can be taken as a justification for the validity
of the approximation throughout the evolutionary history
of semiclassical states used in this paper. However, by its
very nature, the effective description cannot reproduce the
interesting features exhibited by quantum states captured
in Figs. 13–16.
However, the effective description can be used to pro-
vide an intuitive understanding of the behavior of various
uncertainties discovered through numerical analysis. Let
us first note that the position of the bounce point depends
linearly on the value of p
. Next, consider two nearby
solutions with slightly different p
which asymptote to the
same WDW solution for the expanding branch in the
distant future. We wish to know the way in which
=
:
1
2
=
2
changes in the back-
ward evolution as the two wave functions asymptote to
WDW solutions in the distant past. This relative difference
can be found using Eq. (A9) and is given by
2
p
p
p
p
2
; (A10)
where p
is the difference between values of p
of the
two classical trajectories. A heuristic estimate on the rela-
tive difference in can be compared with the relative
dispersion = obtained from the method II in the
evolution of Sec. V B. It turns out that the estimate in
Eq. (A10) provides a reasonably good upper bound to the
relative dispersion found numerically (see Fig. 18). A
similar comparison can be made for method III of
Sec. V B. In this case the corresponding solution to
Eq. (A8) is
1
2
Dp
1=2
exp
16G
3
s
o
Dp
1=2
exp
16G
3
s
o
(A11)
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1000 1500 2000 2500 3000 3500
µ
φ
µ/µ
φ
o
FIG. 17 (color online). Expectation values (and dispersions) of
d
jj
are plotted near the bounce point, together with classical
and effective trajectories (fainter and darker dots, respectively).
While the classical description fails in this region, the effective
description provides an excellent approximation to the exact
quantum evolution. In this plot, p
3000 and " 2
o
.
QUANTUM NATURE OF THE BIG BANG: AN ... PHYSICAL REVIEW D 73, 124038 (2006)
124038-29
with relative difference
p
p
: (A12)
A comparison with the relative dispersions in numerical
analysis is shown in Fig. 19. As in the case of construction
of coherent states via method II, the above estimate serves
as an upper bound for relative dispersions computed by
numerical analysis.
Finally, in the numerical analysis, an important issue is
concerned with the behavior of dispersions of our Dirac
observables ^
and
^
p
and the product p
. Intuitive
understanding of our numerical results of Fig. 16 can be
gained by casting Eq. (A8) in the form
3
16G
s
1
crit
1=2
: (A13)
Since = can be determined numerically, we can then
estimate throughout the evolution. The factor 1
crit
1=2
is approximately equal to unity for
crit
or
equivalently for
crit
. However, near the bounce
point, 1
crit
1=2
1. In methods II and III of con-
structing initial states described in Sec. V B 2, this change
compensates the corresponding decrease in = and
leads to a nearly constant value of . However, for
method I, since = increases monotonically, the fluc-
tuation increases significantly near the bounce point.
APPENDIX B: ISSUES IN NUMERICAL ANALYSIS
Here we will spell out the way in which the WDW limit
of the eigenfunctions of were found.
Consider a general eigenfunction
e
!
of (for !
2
G=3). It is always a linear combination of basis functions
e
jkj
, e
jkj
[where k
2
3=16G!
2
1=16] defined in
Eq. (3.5). For later convenience, let us express the linear
combination as
e
!
r
e
i
e
jkj
r
e
i
e
jkj
; (B1)
where r
, , are real numbers. Since each e
jkj
is a
product of an ‘‘amplitude’’ jj
1=4
=4 and a ‘‘phase’’
e
ijkjlnjj
, it is natural to rescale e
!
:
~
e
!
:
4jj
1=4
e
!
: (B2)
In terms of coefficients defined in (B1), we have
~
e
!
e
i
r
r
cos jkjlnjj
ir
r
sin jkjlnjj: (B3)
The values of
~
e
!
trace out an ellipse on the complex
plane, parameterized by lnjj. The length of semimajor
and semiminor axis of this ellipse is equal to, respectively,
r
r
sup
j
~
e
!
j; jr
r
jinf
j
~
e
!
j; (B4)
whereas the phase is related to positions of maxima of
~
e
!
as follows:
je
!
jr
r
, jkjlnjjn; n 2 Z:
(B5)
The remaining phase is just the phase of
~
e
!
at maximum.
The sign of r
r
is, on the other hand, determined by
the direction of the rotation of the curve as increases.
The method specified above allows us to calculate the
decomposition in
e
k
basis of a function e
!
specified in the
form of numerical data (i.e. array of values at a sufficiently
large domain). The same algorithm can be applied to
identify the WDW limit of any eigenfunction of the LQC
operator . Indeed given an eigenfunction e
!
sup-
ported on the lattice L
j"j
(or L
j"j
) one can again define
~
e
!
analogously to
~
e
!
and find its (local) extrema for large
. (For definiteness, we restrict our consideration to find-
1000
1200
1400
1600
1800
2000
2200
2400
2600
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0
0.05
0.1
0.15
0.2
0.25
p
φ
∆
p
φ
/
p
φ
FIG. 18 (color online). The darker lattice shows the difference
between final and initial relative spreads = for states
obtained by evolving initial data of method II. The upper, fainter
lattice shows the heuristic bound = given by Eq. (A10).
800
1000
1200
1400
1600
1800
2000
2200
2400
2600
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
p
φ
∆
p
φ
/
p
φ
FIG. 19 (color online). The darker lattice shows the difference
between final and initial relative spreads = for states
obtained by evolving initial data of method III. The upper,
fainter lattice shows the heuristic bound = given by
Eq. (A12).
ASHTEKAR, PAWLOWSKI, AND SINGH PHYSICAL REVIEW D 73, 124038 (2006)
124038-30
ing the limit on the positive side; however, this method
can be used of course also for the negative domain.)
Next, the positions of extremas and values of
~
e
!
at them
can be used to calculate coefficients r
r
, , at each
extremum independently. If
~
e
!
!
~
e
!
then these coeffi-
cients form sequences fr
r
g
i
; fr
r
g
i
; fg
i
; fg
i
which converge to the analogous coefficients correspond-
ing to
~
e
!
as !1. Finding the WDW limit of
~
e
!
reduces
then to finding the limit of fr
r
g
i
; fr
r
g
i
; fg
i
; fg
i
.
In actual numerical work the following method was
used:
(i) After given eigenfunction e
!
was calculated using
(4.6) the positions of extrema fg
i
were found.
(ii) Around the extrema the function e
!
supported on
L
j"j
(or L
j"j
) was extended to neighborhoods of
fg
i
via polynomial interpolation. Then the posi-
tions and values of extrema were recalculated with
use of this extension. This allowed us to construct
sequences converging to the WDW limit much
more quickly than the ones constructed in the first
step. The motivation for this construction is the
expectation that for sufficiently large the values
of e
!
at L
j"j
should be good estimates of its WDW
limit (being a regular function defined on entire
R
).
(iii) Extrema found in the previous step were next used
to calculate sequences fr
r
g
i
; fr
r
g
i
;
fg
i
; fg
i
in a way analogous to that specified by
Eqs. (B4) and (B5) and in the description below
them.
(iv) Finally, the limits of coefficients at 1= ! 0 were
calculated by polynomial extrapolation.
APPENDIX C: AN ALTERNATE PHYSICAL
HILBERT SPACE
In this appendix we will construct a physical Hilbert
space H
0
phy
in LQC which is qualitatively different from
the spaces H
"
phy
constructed in Sec. IV B. In its features, it
interpolates between these and the Hilbert space H
wdw
phy
of
Sec. III B. For completeness, we will first explain why a
new representation of the algebra of Dirac observables can
arise, then summarize the results and finally compare and
contrast them with those obtained in Secs. III B and IV B.
The first part is somewhat technical but we have organized
the presentation such that the readers can go directly to the
summary without loss of continuity.
Let us begin by recalling the situation for a general
system with a single constraint C. In the refined version
of Dirac quantization [34], one introduces an auxiliary
Hilbert space H
aux
and represents the constraint by a
self-adjoint operator
^
C on it. The technically difficult
task is to choose a dense subspace of H
aux
such that
for all f, g 2 ,
f
j
:
Z
1
1
dhe
i
^
C
fj (C1)
is a well-defined element of
?
, such that the action
f
jgi of
f
j2
?
on jgi2 yields a Hermitian
scalar product on the space of solutions
f
j to the quan-
tum constraint (see, e.g. [33,34,88]). Results in Sec. IV B
were obtained using L
2
R
Bohr
;Bd
Bohr
L
2
R; d
for H
aux
, and the space of rapidly decreasing functions
f; in this H
aux
for . In the LQC literature, is
sometimes called Cyl and
?
is taken to be its algebraic
dual, denoted by Cyl
?
.
The construction given above is rather general. For the
model under consideration, we can extend this construction
by using an entirely different subspace of
?
for the
auxiliary Hilbert space. This is possible because by duality
the action of
^
C can be extended to all of
?
. Let us set
H
0
aux
:
L
2
R
2
;Bdd. This is a subspace of
?
because each 2 H
0
aux
defines a linear map from to C:
jfi
:
X
Z
1
1
dB
;f;8 2H
0
aux
;
(C2)
where the sum over converges because f 2 has
support only on a countable number of points on the
axis and a rapid falloff. The dual action of
^
C on H
0
aux
can
now be calculated: Since
j
^
Cfi
X
Z
1
1
dB
@
2
f
@
2
B
1
C
f 4
o
;
C
o
f; C
f 4
o
(C3)
it follows from the definitions of C
that
^
C;
@
2
@
2
B
1
C
4
o
;
C
o
; C
4
o
@
2
f
@
2
; (C4)
It is straightforward to verify that
^
and
^
C are self-adjoint
on H
0
aux
. Therefore, we can carry out group averaging on
H
0
aux
and obtain a new physical Hilbert space. As in the
WDW theory of Sec. III B, there are two superselected
sectors. We will work with the positive frequency sector
and denote it by H
0
phy
. The Dirac observables
^
p
and
d
jj
o
on act by duality on H
0
aux
and descend naturally
to H
0
phy
.
The final results can be summarized as follows: The new
physical Hilbert space H
0
phy
is the space of functions
; satisfying the positive frequency equation:
QUANTUM NATURE OF THE BIG BANG: AN ... PHYSICAL REVIEW D 73, 124038 (2006)
124038-31
i@
p
, with finite norm:
kk
02
phy
Z
o
dBj; j
2
(C5)
and the action of the Dirac observables is the standard one:
d
jj
o
; e
i
p
o
;
o
; and
^
p
; i@
@;
@
:
(C6)
Note that the final physical theory is different from both
the WDW theory of Sec. III B and the ‘‘standard’’ LQC
theory of Sec. IV B. Since the inner product (C5) involves
an integral rather than a sum, states ; now have
support on continuous intervals of the axis as in the
WDW theory, rather than on a countable number of points
as in LQC. However, the states satisfy the LQC type
positive frequency equation i@
p
, where the
operator on the right is the square root of a positive, self-
adjoint difference operator rather than of a differential
operator, and the measure determining the inner product
also involves B from LQC rather than
B from the
WDW theory. Thus, the dynamical operator is the same as
in LQC. In particular, as in Sec. V, the quantum states
exhibit a big bounce. However, since typical states are
continuous on the axis, the spectrum of the Dirac ob-
servable
d
jj
o
is now continuous. In essence, the states
; have support on two-dimensional continuous re-
gions of the plane as in the WDW theory but their
dynamics is dictated by a difference operator as in LQC.
When the cosmological constant is nonzero, the analog of
this physical Hilbert space appears to be a natural home to
analyze the role of the Kodama state in quantum cosmol-
ogy [70].
In the literature on polymer representations, nonrelativ-
istic quantum mechanics of point particles and the quan-
tum theory of a Maxwell theory have been discussed in
some detail [46,89]. In the first case, the standard
Schro
¨
dinger Hilbert space L
2
R; dx, and in the second
case, the standard Fock space turned out to be subspaces
of Cyl
?
which were especially helpful for semiclassical
analysis. The present auxiliary Hilbert space H
0
aux
?
is completely analogous to these. Therefore, the resulting
H
0
phy
may be more useful for semiclassical considera-
tions. Indeed, since it does not refer to any ", no coarse
graining is required to carry out the semiclassical analysis.
Therefore, the analog of this construction may well be
useful in more general contexts in full LQG.
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