Singularity Resolution in Isotropic Loop Quantum Cosmology: Recent
  Developments
IMSc/2007/03/2
Singularity Resolution in Isotropic Loop Quantum Cosmology:
Recent Developments
Ghanashyam Date1, ∗
1The Institute of Mathematical Sciences,
CIT Campus, Chennai-600 113, INDIA
Abstract
Since the past Iagrg meeting in December 2004, new developments in loop quantum cosmology
have taken place, especially with regards to the resolution of the Big Bang singularity in the
isotropic models. The singularity resolution issue has been discussed in terms of physical quantities
(expectation values of Dirac observables) and there is also an “improved” quantization of the
Hamiltonian constraint. These developments are briefly discussed.
This is an expanded version of the review talk given at the 24th IAGRG meeting in February
2007.
PACS numbers: 04.60.Pp,98.80.Jk,98.80.Bp
∗Electronic address: shyam@imsc.res.in
http://arxiv.org/abs/0704.0145v1
mailto:shyam@imsc.res.in
I. COSMOLOGY, QUANTUM COSMOLOGY, LOOP QUANTUM COSMOLOGY
Our current understanding of large scale properties of the universe is summarised by
the so called Λ−CDM Big Bang model – homogeneous and isotropic, spatially flat space-
time geometry with a positive cosmological constant and cold dark matter. Impressive as
it is, the model is based on an Einsteinian description of space-time geometry which has
the Big Bang singularity. The existence of cosmological singularities is in fact much more
general. There are homogeneous but anisotropic solution space-times which are singular
and even in the inhomogeneous context there is a general solution which is singular [1].
The singularity theorems give a very general argument for the existence of initial singularity
for an everywhere expanding universe with normal matter content, the singularity being
characterized as incompleteness of causal geodesic in the past. Secondly, in conjunction
with an inflationary scenario, one imagines the origin of the smaller scale structure to be
attributed to quantum mechanical fluctuations of matter and geometry. On account of both
the features, a role for the quantum nature of matter and geometry is indicated.
Quantum mechanical models for cosmological context were in fact constructed, albeit
formally. For the homogeneous and isotropic sector, the geometry is described by just the
scale factor and the extrinsic curvature of the homogeneous spatial slices. In the gravitational
sector, a quantum mechanical wave function is a function of the scale factor i.e. a function
on the (mini-) superspace of gravity. The scale factor being positive, the minisuperspace has
a boundary and wave functions need to satisfy a suitable boundary condition. Furthermore,
the singularity was not resolved in that the Wheeler-De Witt equation (or the Hamiltonian
constraint) which is a differential equation with respect to the scale factor, had singular
coefficient due to the matter density diverging near the boundary. Thus quantization per se
does not necessarily give a satisfactory replacement of the Big Bang singularity.
Meanwhile, over the past 20 years, a background independent, non-perturbative quan-
tum theory of gravity is being constructed starting from a (gauge-) connection formulation of
classical general relativity [2]. The background independence provided strong constraints on
the construction of the quantum theory already at the kinematical level (i.e. before imposi-
tion of the constraints) and in particular revealed a discrete and non-commutative nature of
quantum (three dimensional Riemannian) geometry. The full theory is still quite unwieldy.
Martin Bojowald took to step of restricting to homogeneous geometries and quantizing such
models in a loopy way. Being inherited from the connection formulation, the geometry is
described in terms of densitized triad which for the homogeneous and isotropic context is
described by p ∼ sgn(p)a2 which can also take negative values (encoding the orientation of
the triad). This means that the classical singularity (at p = 0) now lies in the interior of
the superspace. Classically, the singularity prevents any relation between the two regions
of positive and negative values of p. Quantum mechanically however, the wave functions
in these two regions, could be related. One question that becomes relevant in a quantum
theory is that if a wave function, specified for one orientation and satisfying the Hamilto-
nian constraint, can be unambiguously extended to the other orientation while continuing
to satisfy the Hamiltonian constraint. Second main implication of loop quantization is the
necessity of using holonomies – exponentials of connection variable c – as well defined op-
erators. This makes the Hamiltonian constraint a difference equation on the one hand and
also requires an indirect definition for inverse triad (and inverse volume) operators entering
in the definition of the matter Hamiltonian or densities and pressures. Quite interestingly,
the Hamiltonian constraint equation turns out to be non-singular (i.e. deterministic) and
in the effective classical approximation suggests interesting phenomenological implication
quite naturally. These two features in fact made LQC an attractive field.
We will briefly summarise the results prior to 2005 and then turn to more recent devel-
opments. An extensive review of LQC is available in [3]. For simplicity and definiteness, we
will focus on the spatially flat isotropic models.
II. SUMMARY OF PRE 2005 LQC
Classical model: Using coordinates adapted to the spatially homogeneous slicing of the
space-time, the metric and the extrinsic curvature are given by,
ds2 := −dt2 + a2(t)
(dr2 + r2dΩ2
. (1)
Starting from the usual Einstein-Hilbert action and scalar matter for definiteness, one can
get to the Hamiltonian as,
|detgµν |
φ̇2 − V (φ)
(−aȧ2) +
a3φ̇2 − V (φ)a3
pa = −
aȧ , pφ = V0a
3φ̇ , V0 :=
d3x ;
H(a, pa, φ, pφ) = Hgrav +Hmatter
+ a3V0V (φ)
Hmatter
Thus, H = 0 ↔ Friedmann Equation. For the spatially flat model, one has to choose a
fiducial cell whose fiducial volume is denoted by V0.
In the connection formulation, instead of the metric one uses the densitized triad i.e.
instead of the scale factor a one has p̃, |p̃| := a2/4 while the connection variable is related
to the extrinsic curvature as: c̃ := γȧ/2 (the spin connections is absent for the flat model).
Their Poisson bracket is given by {c̃, p̃} = (8πGγ)/(3V0). The arbitrary fiducial volume can
be absorbed away by defining c := V
0 c̃, p := V
0 p̃. Here, γ is the Barbero-Immirzi pa-
rameter which is dimensionless and is determined from the Black hole entropy computations
to be approximately 0.23 [4]. From now on we put 8πG := κ. The classical Hamiltonian is
then given by,
γ−2c2
|p|−3/2p2φ + |p|3/2V (φ)
. (5)
For future comparison, we now take the potential for the scalar field, V (φ) to be zero as
well.
One can obtain the Hamilton’s equations of motion and solve them easily. On the con-
strained surface (H = 0), eliminating c in favour of p and pφ, one has,
c = ± γ
, ṗ = ±
|pφ||p|−1/2 .
φ̇ = pφ|p|−3/2 , ṗφ = 0 , (6)
|p| ⇒ p(φ) = p∗e±
(φ−φ∗) (7)
Since φ is a monotonic function of the synchronous time t, it can be taken as a new “time”
variable. The solution is determined by p(φ) which is (i) independent of the constant pφ and
(ii) passes through p = 0 as φ → ±∞ (expanding/contracting solutions). It is immediate
that, along these curves, p(φ), the energy density and the extrinsic curvature diverge as
p → 0. Furthermore, the divergence of the density implies that φ(t) is incomplete i.e. t
ranges over a semi-infinite interval as φ ranges over the full real line 1. Thus a singularity is
signalled by a solution p(φ) passing through p = 0 in finite synchronous time (or equivalently
by the density diverging somewhere along the solution). A natural way to ensure that all
solutions are non-singular is to ensure that either of the two terms in the Hamiltonian
constraint are bounded. Question is: does a quantum theory replace the Big Bang singularity
by something non-singular?.
There are at least two ways to explore this question. One can imagine computing cor-
rections to the Hamiltonian constraint such that individual terms in the effective constraint
are bounded. Alternatively and more satisfactorily, one should be able to define suitable
observables whose expectation values will generate the analogue of p(φ) curves along which
physical quantities such as energy density, remain bounded. The former method was used
pre-2005 because it could be used for more general models (non-zero potential, anisotropy
etc). The latter has been elaborated in 2006, for the special case of vanishing potential.
Both methods imply that classical singularity is resolved in LQC but not in Wheeler-De
Witt quantum cosmology. We will first discuss the issue in terms of effective Hamiltonian
because it is easier and then discuss it in terms of the expectation values.
In the standard Schrodinger quantization, one can introduce wave functions of p, φ and
quantize the Hamiltonian operator by c→ i~κγ/3∂p , pφ → −i~∂φ, in equation (5). With a
choice of operator ordering, ĤΨ(p, φ) = 0 leads to the Wheeler-De Witt partial differential
equation which has singular coefficients.
The background independent quantization of Loop Quantum Gravity however suggest a
different quantization of the isotropic model. One should look for a Hilbert space on which
only exponentials of c (holonomies of the connection) are well defined operators and not ĉ.
Such a Hilbert space consists of almost periodic functions of c which implies that the triad
operator has every real number as a proper eigenvalue: p̂|µ〉 := 1
γℓ2Pµ|µ〉, ∀µ ∈ R , ℓ2P := κ~.
This has a major implication: inverses of positive powers of triad operators do not exist [5].
These have to be defined by using alternative classical expressions and promoting them to
quantum operators. This can be done with at least one parameter worth of freedom, eg.
|p|−1 =
8πGγl
{c, |p|l}
]1/(1−l)
, l ∈ (0, 1) . (8)
1 For the FRW metric, integral curves of ∂t are time-like geodesics and hence incompleteness with respect
to t is synonymous with geodesic incompleteness.
Only positive powers of |p| appear now. However, this still cannot be used for quantization
since there is no ĉ operator. One must use holonomies: hj(c) := e
iτi , where τi are
anti-hermitian generators of SU(2) in the jth representation satisfying Trj(τiτj) = −13j(j +
1)(2j + 1)δij, Λ
i is a unit vector specifying a direction in the Lie algebra of SU(2) and µ0 is
the coordinate length of the loop used in defining the holonomy. Using the holonomies,
|p|−1 = (8πGµ0γl)
j(j + 1)(2j + 1)
TrjΛ · τ hj{h−1j , |p|l}
, (9)
which can be promoted to an operator. Two parameters, µ0 ∈ R and j ∈ N/2, have crept
in and we have a three parameter family of inverse triad operators. The definitions are:
|̂p|−1
|µ〉 =
(Fl(q))
1−l |µ〉 , q := µ
Fl(q) :=
l + 2
(q + 1)l+2 − |q − 1|l+2
l + 1
(q + 1)l+1 − sgn(q − 1)|q − 1|l+1
Fl(q ≫ 1) ≈
Fl(q ≈ 0) ≈
l + 1
. (11)
All these operators obviously commute with p̂ and their eigenvalues are bounded above. This
implies that the matter densities (and also intrinsic curvatures for more general homogeneous
models), remain bounded at the classically singular region. Most of the phenomenological
novelties are consequences of this particular feature predominantly anchored in the matter
sector. We have thus two scales: p0 :=
P and 2jp0 :=
µ0(2j)ℓ
P. The regime |p| ≪ p0 is
termed the deep quantum regime, p≫ 2jp0 is termed the classical regime and p0 . |p| . 2jp0
is termed the semiclassical regime. The modifications due to the inverse triad defined above
are strong in the semiclassical and the deep quantum regimes. For j = 1/2 the semiclassical
regime is absent. Note that such scales are not available for the Schrodinger quantization.
The necessity of using holonomies also imparts a non-trivial structure for the gravitational
Hamiltonian. The expression obtained is:
Hgrav = −
8πGγ3µ30
ǫijkTr
hihjh
j hk{h−1k , V }
In the above, we have used j = 1/2 representation for the holonomies and V denotes the
volume function. In the limit µ0 → 0 one gets back the classical expression.
While promoting this expression to operators, there is a choice of factor ordering involved
and many are possible. We will present two choices of ordering: the non-symmetric one which
keeps the holonomies on the left as used in the existing choice for the full theory, and the
particular symmetric one used in [6].
Ĥnon−symgrav =
γ3µ30ℓ
sin2µ0c
V̂ cos
− cosµ0c
V̂ sin
Ĥsymgrav =
24i(sgn(p))
γ3µ30ℓ
sinµ0c
V̂ cos
− cos
V̂ sin
sinµ0c (14)
At the quantum level, µ0 cannot be taken to zero since ĉ operator does not exist. The action
of the Hamiltonian operators on |µ〉 is obtained as,
Ĥnon−symgrav |µ〉 =
(Vµ+µ0 − Vµ−µ0) (|µ+ 4µ0〉 − 2|µ〉+ |µ− 4µ0〉) (15)
Ĥsymgrav|µ〉 =
[|Vµ+3µ0 − Vµ+µ0 | |µ+ 4µ0〉+ |Vµ−µ0 − Vµ−3µ0 | |µ− 4µ0〉
− {|Vµ+3µ0 − Vµ+µ0 |+ |Vµ−µ0 − Vµ−3µ0 |} |µ〉] (16)
where Vµ := (
γℓ2P|µ|)3/2 denotes the eigenvalue of V̂ . Denoting quantum wave function
by Ψ(µ, φ) the Wheeler-De Witt equation now becomes a difference equation. For the non-
symmetric one we get,
A(µ+ 4µ0)ψ(µ+ 4µ0, φ)− 2A(µ)ψ(µ, φ) + A(µ− 4µ0)ψ(µ− 4µ0, φ)
3ℓ2PHmatter(µ)ψ(µ, φ) (17)
where, A(µ) := Vµ+µ0 − Vµ−µ0 and vanishes for µ = 0.
For the symmetric operator one gets,
f+(µ)ψ(µ+ 4µ0, φ) + f0(µ)ψ(µ, φ) + f−(µ)ψ(µ− 4µ0, φ)
3ℓ2PHmatter(µ)ψ(µ, φ) where, (18)
f+(µ) := |Vµ+3µ0 − Vµ+µ0 | , f−(µ) := f+(µ− 4µ0) , f0 := − f+(µ)− f−(µ) .
Notice that f+(−2µ0) = 0 = f−(2µ0), but f0(µ) is never zero. The absolute values have
entered due to the sgn(p) factor.
These are effectively second order difference equations and the Ψ(µ, φ) are determined
by specifying Ψ for two consecutive values of µ eg for µ = µ̂+ 4µ0, µ̂ for some µ̂. Since the
highest (lowest) order coefficients vanishes for some µ, then the corresponding component
Ψ(µ, φ) is undetermined by the equation. Potentially this could introduce an arbitrariness
in extending the Ψ specified by data in the classical regime (eg µ ≫ 2j) to the negative µ.
For the non-symmetric case, the highest (lowest) A coefficients vanish for their argument
equal to zero thus leaving the corresponding ψ component undetermined. However, this un-
determined component is decoupled from the others. Thus apart from admitting the trivial
solution ψ(µ, φ) := Φ(φ)δµ,0, ∀µ, all other non-trivial solutions are completely determined
by giving two consecutive components: ψ(µ̂, φ), ψ(µ̂+ 4µ0, φ).
For the symmetric case, due to these properties of the f±,0(µ), it looks as if the difference
equation is non-deterministic if µ = 2µ0 + 4µ0n, n ∈ Z. This is because for µ = −2µ0,
ψ(2µ0, φ) is undetermined by the lower order ψ’s and this coefficient enters in the deter-
mination of ψ(2µ0, φ). However, the symmetric operator also commutes with the parity
operator: (Πψ)(µ, φ) := ψ(−µ, φ). Consequently, ψ(2µ0, φ) is determined by ψ(−2µ0, φ).
Thus, we can restrict to µ = 2µ0 + 4kµ0, k ≥ 0 where the equation is deterministic.
In both cases then, the space of solutions of the constraint equation, is completely de-
termined by giving appropriate data for large |µ| i.e. in the classical regime. Such a de-
terministic nature of the constraint equation has been taken as a necessary condition for
non-singularity at the quantum level 2. As such this could be viewed as a criterion to limit
the choice of factor ordering.
By introducing an interpolating, slowly varying smooth function, Ψ(p(µ) := 1
γℓ2P), and
keeping only the first non-vanishing terms, one deduces the Wheeler-De Witt differential
equation (with a modified matter Hamiltonian) from the above difference equation. Making
a WKB approximation, one infers an effective Hamiltonian which matches with the classical
Hamiltonian for large volume (µ≫ µ0) and small extrinsic curvature (derivative of the WKB
phase is small). There are terms of o(~0) which contain arbitrary powers of the first derivative
of the phase which can all be summed up. The resulting effective Hamiltonian now contains
modifications of the classical gravitational Hamiltonian, apart from the modifications in the
matter Hamiltonian due to the inverse powers of the triad. The largest possible domain of
validity of effective Hamiltonian so deduced must have |p| & p0 [7, 8].
An effective Hamiltonian can alternatively obtained by computing expectation values of
2 For contrast, if one just symmetrizes the non-symmetric operator (without the sgn factor), one gets a
difference equation which is non-deterministic.
the Hamiltonian operator in semiclassical states peaked in classical regimes [9]. The leading
order effective Hamiltonian that one obtains is (spatially flat case):
non−sym
eff = −
B+(p)sin
2(µ0c) +
A(p)− 1
B+(p)
+Hmatter ;
B+(p) := A(p+ 4p0) + A(p− 4p0) , A(p) := (|p+ p0|3/2 − |p− p0|3/2) , (19)
γℓ2Pµ , p0 :=
γℓ2Pµ0 .
For the symmetric operator, the effective Hamiltonian is the same as above except that
B+(p) → f+(p) + f−(p) and 2A(p) → f+(p) + f−(p).
The second bracket in the square bracket, is the quantum geometry potential which is
negative and higher order in ℓP but is important in the small volume regime and plays a role
in the genericness of bounce deduced from the effective Hamiltonian [10]. This term is absent
in effective Hamiltonian deduced from the symmetric constraint. The matter Hamiltonian
will typically have the eigenvalues of powers of inverse triad operator which depend on the
ambiguity parameters j, l.
We already see that the quantum modifications are such that both the matter and the
gravitational parts in the effective Hamiltonian, are rendered bounded and effective dynamics
must be non-singular.
For large values of the triad, p ≫ p0, B+(p) ∼ 6p0
p − o(p−3/2) while A(p) ∼ 3p0
o(p−3/2). In this regime, the effective Hamiltonians deduced from both symmetric and non-
symmetric ordering are the same. The classical Hamiltonian is obtained for µ0 → 0. From
this, one can obtain the equations of motion and by computing the left hand side of the
Friedmann equation, infer the effective energy density. For p≫ p0 one obtains 3,
:= ρeff =
Hmatter
1− 8πGµ
Hmatter
, p := a2/4 . (20)
The effective density is quadratic in the classical density, ρcl := Hmatterp
−3/2. This modi-
fication is due to the quantum correction in the gravitational Hamiltonian (due to the sin2
feature). This is over and above the corrections hidden in the matter Hamiltonian (due to
the “inverse volume” modifications). As noted before, we have two scales: p0 controlled by
µ0 in the gravitational part and 2p0j in the matter part. For large j it is possible that we
3 For p in the semiclassical regime, one should include the contribution of the quantum geometry potential
present in the non-symmetric ordering, especially for examining the bounce possibility [8].
can have p0 ≪ p ≪ 2p0j in which case the above expressions will hold with j dependent
corrections in the matter Hamiltonian. In this semiclassical regime, the corrections from
sin2 term are smaller in comparison to those from inverse volume. If p ≫ 2p0j then the
matter Hamiltonian is also the classical expression. For j = 1/2, there is only the p ≫ p0
regime and ρcl is genuinely the classical density.
Let us quickly note a comparison of the two quantizations as reflected in the corresponding
effective Hamiltonians, particularly with regards to the extrema of p(t). For this, we will
assume same ambiguity parameters (j, l) in the matter Hamiltonian, (1/2)p2φF
jl and explore
the regime p & p0. The two effective Hamiltonians differ significantly in the semiclassical
regime due to the quantum geometry potential.
The equations of motion imply that pφ is a constant of motion, φ varies monotonically
with t and on the constraint surface, we can eliminate c in favour of p and pφ. Let us
focus on p(t) and in particular consider its possible extrema. It is immediate that ṗ = 0
implies sin(2µ0c) = 0. This leads to two possibilities: (A) sin(µ0c) = 0 or (B) cos(µ0c) =
0. A local minimum signifies a bounce while a local maximum signifies a re-collapse. The
value of p at an extremum, p∗, gets determined in terms of the constant pφ. The bounce/re-
collapse nature of an extremum depends upon whether p∗ is in the classical regime or in the
semiclassical regime and also on the case (A) or (B). Note that for the case (A) to hold,
it is necessary that the quantum geometry potential is present. Thus, for the symmetric
ordering, case (A) cannot be realised – it will imply pφ = 0.
An extremum determined by case (A): It is a bounce if p∗ is in in the semiclassical regime;
p∗ varies inversely with pφ while the corresponding density varies directly. p∗ being limited
to the semiclassical regime implies that pφ is also bounded both above and below, for such
an extremum to occur. It turns out that p∗ can be in the classical regime, provided pφ ∼ ℓ2P.
Thus, the non-symmetric constraint, at the effective level, can accommodate a bounce only
in the semiclassical regime and with large densities.
An extremum determined by case (B): It is bounce if p∗ is in the classical regime; p∗
varies directly with pφ and the corresponding density varies inversely. p∗ being limited to
the classical regime implies that pφ must be bounded below but can be arbitrarily large and
thus the density can be arbitrarily small. This is quite unreasonable and has been sited as
one of the reasons for considering the “improved” quantization (more on this later). If p∗ is
in the semiclassical regime, it has to be a re-collapse with pφ ∼ ℓ2P.
In the early works, one worked with the non-symmetric constraint operator and the sin2
corrections were not incorporated (i.e. µ0c≪ π/2 was assumed) and the phenomenological
implications were entirely due to the modified matter Hamiltonian. These already implied
genericness of inflation and genericness of bounce. These results were discussed at the
previous IAGRG meeting in Jaipur.
To summarize: LQC differs from the earlier quantum cosmology in three basic ways (a)
the basic variables are different and in particular the classical singularity is in the inte-
rior of the mini-superspace; (b) the quantization is very different, being motivated by the
background independent quantization employed in LQG; (c) there is a “parent” quantum
theory (LQG) which is pretty much well defined at the kinematical level, unlike the metric
variables based Wheeler-De Witt theory. The loop quantization has fundamentally distinct
implications: its discrete nature of quantum geometry leads to bounded energy densities
and bounded extrinsic and intrinsic curvatures (for the anisotropic models). These two
features are construed as “resolving the classical singularity”. Quite un-expectedly, the ef-
fective dynamics incorporating quantum corrections is also singularity-free (via a bounce),
accommodates an inflationary phase rather naturally and is well behaved with regards to
perturbations. Although there are many ambiguity parameters, these results are robust with
respect to their values.
III. POST 2004 ISOTROPIC LQC
Despite many attractive features of LQC, many points need to be addressed further:
• LQC being a constrained theory, it would be more appropriate if singularity resolution
is formulated and demonstrated in terms of physical expectation values of physical
(Dirac) operators i.e. in terms of “gauge invariant quantities”. This can be done at
present with self-adjoint constraint i.e. a symmetric ordering and for free, massless
scalar matter.
• There are at least three distinct ambiguity parameters: µ0 related to the fiducial
length of the loop used in writing the holonomies; j entering in the choice of SU(2)
representation which is chosen to be 1/2 in the gravitational sector and some large
value in the matter sector; l entering in writing the inverse powers in terms of Poisson
brackets. The first one was thought to be determined by the area gap from the full
theory. The j = 1/2 in the gravitational Hamiltonian seems needed to avoid high order
difference equation and larger j values are hinted to be problematic in the study of a
three dimensional model [11]. Given this, the choice of a high value of j in the matter
Hamiltonian seems unnatural4. For phenomenology however the higher values allowing
for a larger semiclassical regime are preferred. The l does not play as significant a role.
• The bounce scale and density at the bounce, implied by the effective Hamiltonian
(from symmetric ordering), is dependent on the parameters of the matter Hamiltonian
and can be arranged such that the bounce density is arbitrarily small. This is a
highly undesirable feature. Furthermore, the largest possible domain of validity of
WKB approximation is given by the turning points (eg the bounce scale). However,
the approximation could break down even before reaching the turning point. An
independent check on the domain of validity of effective Hamiltonian is thus desirable.
• A systematic derivation of LQC from LQG is expected to tighten the ambiguity pa-
rameters. However, such a derivation is not yet available.
A. Physical quantities and Singularity Resolution
When the Hamiltonian is a constraint, at the classical level itself, the notion of dynamics
in terms of the ‘time translations’ generated by the Hamiltonian is devoid of any physical
meaning. Furthermore, at the quantum level when one attempts to impose the constraint
as Ĥ|Ψ〉 = 0, typically one finds that there are no solutions in the Hilbert space on which Ĥ
is defined - the solutions are generically distributional. One then has to consider the space
of all distributional solutions, define a new physical inner product to turn it into a Hilbert
space (the physical Hilbert space), define operators on the space of solutions (which must
thus act invariantly) which are self-adjoint (physical operators) and compute expectation
values, uncertainties etc of these operators to make physical predictions. Clearly, the space
of solutions depends on the quantization of the constraint and there is an arbitrariness in
the choice of physical inner product. This is usually chosen so that a complete set of Dirac
4 For an alternative view on using large values of j, see reference [12].
observables (as deduced from the classical theory) are self-adjoint. This is greatly simplified
if the constraint has a separable form with respect to some degree of freedom 5. For LQC
(and also for the Wheeler-De Witt quantum cosmology), such a simplification is available
for a free, massless scalar matter: Hmatter(φ, pφ) :=
p2φ|p|−3/2. Let us sketch the steps
schematically, focusing on the spatially flat model for simplicity [6, 13].
1. Fundamental constraint equation:
The classical constraint equations is:
|p|+ 8πG p2φ |p|−3/2 = 0 = Cgrav + Cmatter ; (21)
The corresponding quantum equation for the wave function, Ψ(p, φ) is:
8πGp̂2φΨ(p, φ) = [B̃(p)]
−1ĈgravΨ(p, φ) , [B̃(p)] is eigenvalue of |̂p|−3/2 ; (22)
Putting p̂φ = −i~∂φ, p :=
µ and B̃(p) := (
)−3/2B(µ), the equation can be
written in a separated form as,
∂2Ψ(µ, φ)
= [B(µ)]−1
ℓ−1P Ĉgrav
Ψ(µ, φ) := − Θ̂(µ)Ψ(µ, φ). (23)
The Θ̂ operator for different quantizations is different. For Schrodinger quantization
(Wheeler-De Witt), with a particular factor ordering suggested by the continuum limit
of the difference equation, the operator Θ̂(µ) is given by,
Θ̂Sch(µ)Ψ(µ, φ) = −
|µ|3/2∂µ
µ ∂µΨ(µ, φ) (24)
while for LQC, with symmetric ordering, it is given by,
Θ̂LQC(µ)Ψ(µ, φ) = −[B(µ)]−1
C+(µ)Ψ(µ+ 4µ0, φ) + C
0(µ)Ψ(µ, φ)+
C−(µ)Ψ(µ− 4µ0, φ)
C+(µ) :=
∣∣ |µ+ 3µ0|3/2 − |µ+ µ0|3/2
∣∣ , (25)
C−(µ) := C+(µ− 4µ0) , C0(µ) := − C+(µ)− C−(µ) .
Note that in the Schrodinger quantization, the BSch(µ) = |µ|−3/2 diverges at µ = 0
while in LQC, BLQC(µ) vanishes for all allowed choices of ambiguity parameters. In
both cases, B(µ) ∼ |µ|−3/2 as |µ| → ∞.
5 A general abstract procedure using group averaging is also available.
2. Inner product and General solution:
The operator Θ̂ turns out to be a self-adjoint, positive definite operator on the space
of functions Ψ(µ, φ) for each fixed φ with an inner product scaled by B(µ). That
is, for the Schrodinger quantization, it is an operator on L2(R, BSch(µ)dµ) while for
LQC it is an operator on L2(RBohr, BBohr(µ)dµBohr). Because of this, the operator has
a complete set of eigenvectors: Θ̂ek(µ) = ω
2(k)ek(µ), k ∈ R, 〈ek|ek′〉 = δ(k, k′), and
the general solution of the fundamental constraint equation can be expressed as
Ψ(µ, φ) =
dk Ψ̃+(k)ek(µ)e
iωφ + Ψ̃−(k)ēk(µ)e
−iωφ . (26)
The orthonormality relations among the ek(µ) are in the corresponding Hilbert spaces.
Different quantizations differ in the form of the eigenfunctions, possibly the spectrum
itself and of course ω(k). In general, these solutions are not normalizable in L2(RBohr×
R, dµBohr × dµ), i.e. these are distributional.
3. Choice of Dirac observables:
Since the classical kinematical phase space is 4 dimensional and we have a single
first class constraint, the phase space of physical states (reduced phase space) is two
dimensional and we need two functions to coordinatize this space. We should thus look
for two (classical) Dirac observables: functions on the kinematical phase space whose
Poisson bracket with the Hamiltonian constraint vanishes on the constraint surface.
It is easy to see that pφ is a Dirac observable. For the second one, we choose a
one parameter family of functions µ(φ) satisfying {µ(φ), C(µ, c, φ, pφ)} ≈ 0. The
corresponding quantum definitions, with the operators acting on the solutions, are:
p̂φΨ(µ, φ) := −i~∂φΨ(µ, φ) , (27)
̂|µ|φ0Ψ(µ, φ) := ei
Θ̂(φ−φ0)|µ|Ψ+(µ, φ0) + e−i
Θ̂(φ−φ0)|µ|Ψ−(µ, φ0) (28)
On an initial datum, Ψ(µ, φ0), these operators act as,
̂|µ|φ0Ψ(µ, φ0) = |µ|Ψ(µ, φ0) , p̂φΨ(µ, φ0) = ~
Θ̂Ψ(µ, φ0) . (29)
4. Physical inner product:
It follows that the Dirac operators defined on the space of solutions are self-adjoint if
we define a physical inner product on the space of solutions as:
〈Ψ|Ψ′〉phys := “
dµB(µ)” Ψ̄(µ, φ)Ψ′(µ, φ) . (30)
Thus the eigenvalues of the inverse volume operator crucially enter the definition of
the physical inner product. For Schrodinger quantization, the integral is really an
integral while for LQC it is actually a sum over µ taking values in a lattice. The inner
product is independent of the choice of φ0.
A complete set of physical operators and physical inner product has now been specified
and physical questions can be phrased in terms of (physical) expectation values of
functions of these operators.
5. Semiclassical states:
To discuss semiclassical regime, typically one defines semiclassical states: physical
states such that a chosen set of self-adjoint operators have specified expectation values
with uncertainties bounded by specified tolerances. A natural choice of operators for
us are the two Dirac operators defined above. It is easy to construct semiclassical
states with respect to these operators. For example, a state peaked around, pφ = p
and |µ|φ0 = µ∗ is given by (in Schrodinger quantization for instance),
Ψsemi(µ, φ0) :=
(k−k∗)2
2σ2 ek(µ)e
iω(φ0−φ
∗) (31)
k∗ = −
3/2κ~−1p∗φ , φ
∗ = φ0 +−
3/2κℓn|µ∗| . (32)
For LQC, the ek(µ) functions are different and the physical expectation values are to
be evaluated using the physical inner product defined in the LQC context.
6. Evolution of physical quantities:
Since one knows the general solution of the constraint equation, Ψ(µ, φ), given
Ψ(µ, φ0), one can compute the physical expectation values in the semiclassical so-
lution, Ψsemi(µ, φ) and track the position of the peak as a function of φ as well as the
uncertainties as a function of φ.
7. Resolution of Big Bang Singularity:
A classical solution is obtained as a curve in (µ, φ) plane, different curves being labelled
by the points (µ∗, φ∗) in the plane. The curves are independent of the constant value
of p∗φ These curves are already given in (7).
Quantum mechanically, we first select a semiclassical solution, Ψsemi(p
∗ : φ) in
which the expectation values of the Dirac operators, at φ = φ0, are p
φ and µ
∗ re-
spectively. These values serve as labels for the semiclassical solution. The former one
continues to be p∗φ for all φ whereas 〈 ̂|µ|φ0〉(φ) =: |µ|p∗φ,µ∗(φ), determines a curve in the
(µ, φ) plane. In general one expects this curve to be different from the classical curve
in the region of small µ (small volume).
The result of the computations is that Schrodinger quantization, the curve |µ|p∗
,µ∗(φ),
does approach the µ = 0 axis asymptotically. However for LQC, the curve bounces
away from the µ = 0 axis. In this sense – and now inferred in terms of physical
quantities – the Big Bang singularity is resolved in LQC. It also turns out that for
large enough values of p∗φ, the quantum trajectories constructed by the above procedure
are well approximated by the trajectories by the effective Hamiltonian. All these
statements are for semiclassical solutions which are peaked at large µ∗ at late times.
Two further features are noteworthy as they corroborate the suggestions from the effective
Hamiltonian analysis.
First one is revealed by computing expectation value of the matter density operator,
ρmatter :=
̂(p∗φ)
2|p|−3, at the bounce value of |p|. It turns out that this value is sensitive to
the value of p∗φ and can be made arbitrarily small by choosing p
φ to be large. Physically this
is unsatisfactory as quantum effects are not expected to be significant for matter density very
small compared to the Planck density. This is traced to the quantization of the gravitational
Hamiltonian, in particular to the step which introduces the ambiguity parameter µ0. A novel
solution proposed in the “improved quantization”, removes this undesirable feature.
The second one refers to the role of quantum modifications in the gravitational Hamil-
tonian compared to those in the matter Hamiltonian (the inverse volume modification or
B(µ)). The former is much more significant than the latter. So much so, that even if one
uses the B(µ) from the Schrodinger quantization (i.e. switch-off the inverse volume mod-
ifications), one still obtains the bounce. So bounce is seen as the consequence of Θ̂ being
different and as far as qualitative singularity resolution is concerned, the inverse volume
modifications are un-important. As the effective picture (for symmetric constraint) showed,
the bounce occurs in the classical region (for j = 1/2) where the inverse volume corrections
can be neglected. For an exact model which seeks to understand as to why the bounces are
seen, please see [14].
B. Improved Quantization
The undesirable features of the bounce coming from the classical region, can be seen
readily using the effective Hamiltonian, as remarked earlier. To see the effects of modifi-
cations from the gravitational Hamiltonian, choose j = 1/2 and consider the Friedmann
equation derived from the effective Hamiltonian (20), with matter Hamiltonian given by
Hmatter =
p2φ|p|−3/2. The positivity of the effective density implies that p ≥ p∗ with p∗
determined by vanishing of the effective energy density: ρ∗ := ρcl(p∗) = (
8πGµ20γ
−1. This
leads to |p∗| =
4πGµ20γ
|pφ| and ρ∗ =
8πGµ20γ
)−3/2|pφ|−1. One sees that for large |pφ|,
the bounce scale |p∗| can be large and the maximum density – density at bounce – could be
small. Thus, within the model, there exist a possibility of seeing quantum effects (bounce)
even when neither the energy density nor the bounce scale are comparable to the corre-
sponding Planck quantities and this is an undesirable feature of the model. This feature is
independent of factor ordering as long as the bounce occurs in the classical regime.
One may notice that if we replace µ0 → µ̄(p) :=
∆/|p| where ∆ is a constant, then
the effective density vanishes when ρcl equals the critical value ρcrit := (
8πG∆γ2
)−1, which is
independent of matter Hamiltonian. The bounce scale p∗ is determined by ρ∗ = ρcrit which
gives |p∗| = (
2ρcrit
)1/3. Now although the bounce scale can again be large depending upon pφ,
the density at bounce is always the universal value determined by ∆. This is a rather nice
feature in that quantum geometry effects are revealed when matter density (which couples
to gravity) reaches a universal, critical value regardless of the dynamical variables describing
matter. For a suitable choice of ∆ one can ensure that a bounce always happen when the
energy density becomes comparable to the Planck density. In this manner, one can retain
the good feature (bounce) even for j = 1/2 thus “effectively fixing” an ambiguity parameter
and also trade another ambiguity parameter µ0 for ∆. This is precisely what is achieved by
the “improved quantization” of the gravitational Hamiltonian [15].
The place where the quantization procedure is modified is when one expresses the cur-
vature in terms of the holonomies along a loop around a “plaquette”. One shrinks the
plaquette in the limiting procedure. One now makes an important departure: the plaque-
tte should be shrunk only till the physical area (as distinct from a fiducial one) reaches
its minimum possible value which is given by the area gap in the known spectrum of area
operator in quantum geometry: ∆ = 2
3πγG~. Since the plaquette is a square of fiducial
length µ0, its physical area is µ
0|p| and this should set be to ∆. Since |p| is a dynamical
variable, µ0 cannot be a constant and is to be thought of a function on the phase space,
µ̄(p) :=
∆/|p|. It turns out that even with such a change which makes the curvature to
be a function of both connection and triad, the form of both the gravitational constraint and
inverse volume operator appearing in the matter Hamiltonian, remains the same with just
doing the replacement, µ0 → µ̄ defined above, in the holonomies. The expressions simplify
by using eigenfunctions of the volume operator V̂ := ˆ|p|
, instead of those of the triad.
The relevant expressions are:
v := Ksgn(µ)|µ|3/2 , K := 2
; (33)
V̂ |v〉 =
)3/2 ℓ3P
|v||v〉 , (34)
Ψ(v) := Ψ(v + k) , (35)
|̂p|−1/2
j=1/2,l=3/4
Ψ(v) =
)−1/2
K1/3|v|1/3
∣∣|v + 1|1/3 − |v − 1|1/3
∣∣Ψ(v) (36)
B(v) =
∣∣|v + 1|1/3 − |v − 1|1/3
∣∣3 (37)
Θ̂ImprovedΨ(v, φ) = −[B(v)]−1
C+(v)Ψ(v + 4, φ) + C0(v)Ψ(v, φ)+
C−(v)Ψ(v − 4, φ)
, (38)
C+(v) :=
|v + 2| | |v + 1| − |v + 3|| , (39)
C−(v) := C+(v − 4) , C0(v) := − C+(v)− C−(v) . (40)
Thus the main changes in the quantization of the Hamiltonian constraint are: (1) replace
µ0 → µ̄ :=
∆/|p| in the holonomies; (2) choose symmetric ordering for the gravitational
constraint; and (3) choose j = 1/2 in both gravitational Hamiltonian and the matter Hamil-
tonian (in the definition of inverse powers of triad operator). The “improvement” refers
to the first point. This model is singularity free at the level of the fundamental constraint
equation (even though the leading coefficients of the difference equation do vanish, because
the the parity symmetry again saves the day); the densities continue to be bounded above –
and now with a bound independent of matter parameters; the effective picture continues to
be singularity free and with undesirable features removed and the classical Big Bang being
replaced by a quantum bounce is established in terms of physical quantities.
C. Close Isotropic Model
While close model seems phenomenologically disfavoured, it provides further testing
ground for quantization of the Hamiltonian constraint. Because of the intrinsic (spatial)
curvature, the plaquettes used in expressing the Fij in terms of holonomies, are not bounded
by just four edges – a fifth one is necessary. This was attempted and was found to lead to
an “unstable” quantization. This difficulty was bypassed by using the holonomies of the
extrinsic curvature instead of the gauge connection which is permissible in the homogeneous
context. The corresponding, non-symmetric constraint and its difference equation was anal-
ysed for the massless scalar matter. Green and Unruh, found that solutions of the difference
equation was always diverging (at least for one orientation) for large volumes. Further, the
divergence seemed to set in just where one expected a re-collapse from the classical theory.
In the absence of physical inner product and physical interpretation of the solutions, it was
concluded that this version of LQC for close model is unlikely to accommodate classical
re-collapse even though it avoided the Big Bang/Big Crunch singularities.
Recently, this model has been revisited [16]. One went back to using the gauge connec-
tion and the fifth edge difficulty was circumvented by using both the left-invariant and the
right-invariant vector fields to define the plaquette. In addition, the symmetric ordering was
chosen and finally the µ0 → µ̄ improvement was also incorporated. Without the improve-
ment, there were still the problems of getting bounce for low energy density and also not
getting a reasonable re-collapse (either re-collapse is absent or the scale is marginally larger
than the bounce scale). With the improvement, the bounces and re-collapses are neatly
accommodated and one gets a cyclic evolution. In this case also, the scalar field serves as a
good clock variable as it continues to be monotonic with the synchronous time.
I have focussed on the singularity resolution issue in this talk. Other developments have
also taken place in the past couple of years. I will just list these giving references.
1. Effective models and their properties: The effective picture was shown to be non-
singular and since this is based on the usual framework of GR, it follows that energy
conditions must be violated (and indeed they are thanks to the inverse volume modifi-
cations). This raised questions regarding stability of matter and causal propagation of
perturbations. Golam Hossain showed that despite the energy conditions violations,
neither of the above pathologies result [17].
Minimally coupled scalar has been used in elaborating inflationary scenarios. However
non-minimally coupled scalars are also conceivable models. The singularity resolution
and inflationary scenarios continue to hold also in this case. Furthermore sufficient
e-foldings are also admissible [18].
In the improved quantization, one sets the ambiguity parameter j = 1/2 and shifts the
dominant effects to the the gravitational Hamiltonian. All the previous phenomeno-
logical implications however were driven by the inverse volume modifications in the
matter sector. Consequently, it is necessary to check if and how the phenomenology
works with the improved quantization. This has been explored in [19].
Using the effective dynamics for the homogeneous mode, density perturbations were
explored and power spectra were computed with the required small amplitude [20, 21].
As many of the phenomenology oriented questions have been explored using effective
Hamiltonian which incorporate quantum corrections from various sources (gravity,
matter etc). This motivates a some what systematic approach to constructing effective
approximations. This has been initiated in [22].
2. Anisotropic models: The anisotropic models provide further testing grounds for loop
quantization. At the difference equation level, the non-singularity has been checked
also for these models in the non-symmetric scheme. For the vacuum Bianchi I model,
there is no place for the inverse volume type corrections to appear at an effective
Hamiltonian level and the effective dynamics would continue to be singular. However,
once the gravitational corrections (sin2) are incorporated, the effective dynamics again
is non-singular and one can obtain the non-singular version of the (singular) Kasner
solution [23]. More recently, the Bianchi I model with a free, massless scalar is also
analysed in the improved quantization [24]. A perturbative treatment of anisotropies
has been explored in [25].
3. Inhomogeneities: Inhomogeneities are a fact of nature although these are small in the
early universe. This suggests a perturbative approach to incorporate inhomogeneities.
On the one hand one can study their evolution in the homogeneous, isotropic back-
ground (cosmological perturbation theory). One can also begin with a (simplified)
inhomogeneous model and try to see how a homogeneous approximation can become
viable. The work on the former has already begun. For the latter part, Bojowald has
discussed a simplified lattice model to draw some lessons for the homogeneous models.
In particular he has given an alternative argument for the µ0 → µ̄ modification which
does not appeal to the area operator [12].
IV. OPEN ISSUES AND OUT LOOK
In summary, over the past two years, we have seen how to phrase and understand the
fate of Big Bang singularity in a quantum framework.
Firstly, with the help of a minimally coupled, free, massless scalar which serves as a good
clock variable in the isotropic context, one can define physical inner product, a complete set
of Dirac observables and their physical matrix elements. At present this can be done only for
self-adjoint Hamiltonian constraint. Using these, one can construct trajectories in the (p, pφ)
plane which are followed by the peak of a semiclassical state as well as the uncertainties in
the Dirac observables. It so happens that these trajectories do not pass through the zero
volume – Big Bang is replaced by a Bounce. For close isotropic model, the Big Crunch is
also replaced by a bounce while retaining classically understood re-collapse. In conjunction
with the µ̄ improvement, the gravitational Hamiltonian can be given the the main role in
generating the bounce. A corresponding treatment in Schrodinger quantization (Wheeler-De
Witt theory), does not generate a bounce nor does it render the density, curvatures bounded.
Thus, quantum representation plays a significant role in the singularity resolution.
Secondly, the improved quantization motivated by the regulation of the Fij invoking the
area operator from the full theory (or by the argument from the inhomogeneous lattice
model), also leads the bounce to be “triggered” when the energy density reaches a critical
value (∼ 0.82ρPlanck) which is independent of the values of the dynamical variables. Close
model also gives the same critical value.
While the improvement is demonstrated to be viable in the isotropic context, the proce-
dure differs from that followed in the full theory. One may either view this as something
special to the mini-superspace model(s) or view it as providing hints for newer approaches
in the full theory.
A general criteria for “non-singularity” is not in sight yet and so also a systematic deriva-
tion of the mini-superspace model(s) from a larger, full theory.
Acknowledgements: I would like to thank Parampreet Singh for discussions regarding
the improved quantization as well as for running his codes for exploring bounce in the
semiclassical regime. Thanks are due to Martin Bojowald for comments on an earlier draft.
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	Cosmology, quantum cosmology, loop quantum cosmology
	Summary of pre 2005 LQC
	Post 2004 Isotropic LQC
	Physical quantities and Singularity Resolution
	Improved Quantization
	Close Isotropic Model
	Open Issues and Out look
	References
