IGPG–07/4–2
Lattice refining loop quantum cosmology,
anisotropic models and stability
Martin Bojowald∗
Institute for Gravitational Physics and Geometry, The Pennsylvania State University,
104 Davey Lab, University Park, PA 16802, USA
Daniel Cartin†
Naval Academy Preparatory School, 197 Elliot Street, Newport, RI 02841
Gaurav Khanna‡
Physics Department, University of Massachusetts at Dartmouth,
North Dartmouth, MA 02747
Abstract
A general class of loop quantizations for anisotropic models is introduced and
discussed, which enhances loop quantum cosmology by relevant features seen in in-
homogeneous situations. The main new effect is an underlying lattice which is being
refined during dynamical changes of the volume. In general, this leads to a new fea-
ture of dynamical difference equations which may not have constant step-size, posing
new mathematical problems. It is discussed how such models can be evaluated and
what lattice refinements imply for semiclassical behavior. Two detailed examples
illustrate that stability conditions can put strong constraints on suitable refinement
models, even in the absence of a fundamental Hamiltonian which defines changes of
the underlying lattice. Thus, a large class of consistency tests of loop quantum grav-
ity becomes available. In this context, it will also be seen that quantum corrections
due to inverse powers of metric components in a constraint are much larger than they
appeared recently in more special treatments of isotropic, free scalar models where
they were artificially suppressed.
1 Introduction
Loop quantum cosmology [1] was designed to test characteristic effects expected in the full
framework of loop quantum gravity [2, 3, 4]. Implementing symmetries at the kinematical
∗e-mail address: bojowald@gravity.psu.edu
†e-mail address: Cartin@naps.edu
‡e-mail address: gkhanna@UMassD.Edu
http://arxiv.org/abs/0704.1137v1
quantum level allows explicit treatments of the dynamical equations while preserving basic
features such as the discreteness of spatial geometry [5]. (See also [6, 7, 8, 9, 10, 11] for
recent work on symmetry reduction in quantum theories.) Indeed, several new, initially
surprising results were derived in different applications in cosmology and black hole physics.
By now many such models have been studied in detail.
As the relation of dynamics to that of a possible full framework without symmetries is
not fully worked out, detailed studies can be used to suggest improvements of the equa-
tions for physically viable behavior. Comparing results with full candidates for quantum
dynamics can then provide stringent self-consistency tests of the overall framework. It is to
be seen if, and how, such alterations of quantization procedures naturally result from a full
quantization. The first example of this type related to the stability behavior of solutions to
the difference equations of isotropic loop quantum cosmology, which was studied in [12, 13]
and was already restrictive for models with non-zero intrinsic curvature. Another limita-
tion, realized early on [14], occurs in the presence of a positive cosmological constant Λ.
In an exact isotropic model, the extrinsic curvature scale is given by k = ȧ =
8πGa2Λ/3
which, due to the factor of a2, can be large in a late universe although the local curva-
ture scale Λ might be small. Extrinsic curvature plays an important role since in a flat
isotropic model it appears in holonomies on which loop quantizations are based in such a
way that only eiαk with α ∈ R can be represented as operators, but not k itself [15]. Large
values of k would either require one to use extremely small α in the relevant operators, or
imply unexpected deviations from classical behavior. In fact, holonomies as basic objects
imply that the Hamiltonian constraint is quantized to a difference rather than differential
equation [16] since k in the Hamiltonian constraint (as in the Friedmann equation) is not
directly quantized but only exponentials eiαk. These are shift operators instead of differen-
tial operators. For a large, semiclassical universe a Wheeler–DeWitt wave function should
be a good approximation to the basic difference equation of loop quantum cosmology [17]
which, in a representation as a function of the momentum p = a2 conjugate to k, would be
oscillating on scales of the order (a
Λ)−1. This scale becomes shorter and shorter in an
expanding universe, eventually falling below the discreteness scale of the difference equa-
tion of loop quantum cosmology. At such a point, discreteness of spatial geometry would
become noticeable in the behavior of the wave function (independently of how physical
observables are computed from it) although the universe should be classical.
This does not pose a problem for the general formalism, because it only shows that the
specific quantization of the exact isotropic model used reaches its limits. Physically, this
can be understood as a consequence of a fixed spatial lattice being used throughout the
whole universe evolution. Exponentials eiαk in isotropic models derive from holonomies
he(A) = P exp(
Aiaτiė
adt) of the Ashtekar connection along spatial curves e. All the
freedom contained in choosing edges to capture independent degrees of freedom of the full
theory reduces, in isotropic models, to the single parameter α which suffices to separate
isotropic connections through all functions eiαk. The parameter α, from the full perspective,
is thus related to the edge length used in holonomies. Using a fixed and constant α is
analogous to using only edges of a given coordinate length, as they occur, for instance, in
a regular lattice. In the presence of a positive cosmological constant, for any α a value of k
will then be reached such that eiαk differs strongly from iαk. From the lattice perspective,
this means that the local curvature radius becomes comparable to or smaller than the
fixed lattice scale corresponding to α. Such a fixed lattice ceases to be able to support all
small-scale oscillations relevant for a semiclassical geometry.
This is not problematic if it occurs in a quantum regime where dynamics is indeed
expected to differ from the classical one, but it poses a problem in semiclassical regimes.
A better treatment has to refer to changing lattices, which is not easy to implement in
a straightforward quantization of purely homogeneous models. In a dynamical equation
closer to what is expected from the full framework, lattice refinements would take place
during the evolution since full Hamiltonian constraint operators generally create new ver-
tices of a lattice state in addition to changing their edge labels [18, 19]. While k increases
with increasing volume, the corresponding α decreases since the lattice is being refined all
the time. For a suitable lattice refinement, the increase in k can be balanced by the decrease
of α such that αk stays small and semiclassical behavior is realized for any macroscopic
volume even with Λ > 0. This provides an interesting relation between the fundamental
Hamiltonian, which is responsible for the lattice refinement, and semiclassical properties
of models. Testing whether an appropriate balance between increasing k and lattice re-
finements can be reached generically can thus provide stringent tests on the fundamental
dynamics even without using a precise full Hamiltonian constraint operator.
This feature of lattice refinements was not mimicked in the first formulations of loop
quantum cosmology [20, 21, 14, 22, 15] since the main focus was to understand small-volume
effects such as classical singularities [23, 24]. In this context, lattice refinements appear
irrelevant because only a few action steps of the Hamiltonian, rather than long evolution,
are sufficient to probe a singularity. By now, perturbative regimes around isotropic models
have been formulated in loop quantum cosmology which are inhomogeneous and thus must
take into account lattice states and, at least at an effective level, lattice refinements [9].
One special version, corresponding to lattices with a number of vertices growing linearly
with volume in a specific way referring to the area operator, has been studied in detail in
isotropic models with a free, massless scalar [25]. Although the complicated relation to a
full, graph-changing Hamiltonian constraint is still not fully formulated, such models allow
crucial tests of the local dynamics.
While isotropic models can easily be understood in terms of wave functions on a 1-
dimensional discrete minisuperspace in terms of oscillation lengths [26], anisotropic models
with higher-dimensional minisuperspaces can be more subtle. In such models, limitations
similar to that of a cosmological constant have been observed as possible instabilities
of solutions in classical regions or the lack of a sufficient number of semiclassical states
[27, 28, 29]. For the partial difference equations of anisotropic models in loop quantum
cosmology, stability issues can be much more severe than in isotropic models and thus
lead to further consistency tests which might help to restrict possible quantization freedom
(see, e.g., [30]). In this paper we therefore introduce the general setting of anisotropic
models taking into account lattice refinements of Hamiltonian constraint operators, fo-
cusing mainly on the anisotropic model which corresponds to the Schwarzschild interior.
As we will see, the type of difference equations in general changes since they can become
non-equidistant. This leads to new mathematical problems which we address here briefly,
leaving further analysis for future work. The examples presented here already show that
one can distinguish different refinement models by their stability properties. The refine-
ment model corresponding to [25] turns out to give unstable evolution of the Schwarzschild
interior, while a new version, whose vertex number also grows linearly with volume, is
stable. Compared to isotropic models which are sensitive only to how the vertex number
of a state changes with volume, anisotropic models allow one to test much more detailed
properties.
An appendix discusses subtleties in how homogeneous models faithfully represent inho-
mogeneous states, mainly regarding the magnitude of corrections arising from quantizations
of inverse metric components which often plays a large role in cosmological applications.
2 Difference equation for the Schwarzschild interior
with varying discreteness scale
Basic variables of a loop quantization are holonomies along lattice links and fluxes over
transversal surfaces. For the Schwarzschild interior [31], the connection used for holonomies
and the densitized triad used for fluxes take the form
Aiaτidx
a = c̃τ3dx+ (ãτ1 + b̃τ2)dϑ+ (−b̃τ1 + ãτ2) sinϑdϕ + τ3 cosϑdϕ (1)
Eai τ
= p̃cτ3 sin ϑ
+ (p̃aτ1 + p̃bτ2) sinϑ
+ (−p̃bτ1 + p̃aτ2)
. (2)
Coordinates (x, ϑ, ϕ) are adapted to the symmetry, with polar angles ϑ and ϕ along orbits
of the rotational symmetry subgroup, and τj = − i2σj in terms of Pauli matrices. Spatial
geometry is determined by the spatial line element, which in terms of the densitized triad
components is
ds2 =
p̃2a + p̃
|p̃c|
dx2 + |p̃c|dΩ2 (3)
obtained from qab = Eai E
i /| detEcj |. We will also use the co-triad eia, i.e. the inverse of
eai = E
| detEbj |,
eiaτidx
a = ecτ3dx+ (eaτ1 + ebτ2)dϑ+ (−ebτ1 + eaτ2) sinϑdϕ (4)
with components
sgnp̃c
p̃2a + p̃
|p̃c|
, eb =
|p̃c| p̃b
p̃2a + p̃
and ea =
|p̃c| p̃a
p̃2a + p̃
. (5)
The phase space is spanned by the spatial constants (ã, b̃, c̃, p̃a, p̃b, p̃c) ∈ R6 with non-
vanishing Poisson brackets
{ã, p̃a} = γG/L0 , {b̃, p̃b} = γG/L0 , {c̃, p̃c} = 2γG/L0
where G is the gravitational constant and γ the Barbero–Immirzi parameter [32, 33].
Moreover, L0 is the size of a coordinate box along x used in integrating out the fields in
d3xȦiaE
˙̃cp̃c +
bp̃b +
˙̃ap̃a
to derive the symplectic structure. The SU(2)-gauge transformations rotating a general
triad are partially fixed to U(1) by demanding the x-component of Eai to point in the
internal τ3-direction in (2). The U(1)-gauge freedom allows one to set ã = 0 = p̃a, still
leaving a discrete residual gauge freedom (b̃, p̃b) 7→ (−b̃,−p̃b). The remaining variables can
be rescaled by
(b, c) := (b̃, Loc̃) , (pb, pc) := (Lop̃b, p̃c) . (6)
to make the canonical structure L0-independent:
{b, pb} = γG , {c, pc} = 2γG . (7)
This rescaling is suggested naturally by holonomies, as written below, and fluxes which are
considered the basic objects in loop quantizations.
To express the elementary variables through holonomies, which unlike connection com-
ponents will be promoted to operators, it suffices to choose curves along the x-direction of
coordinate length τL0 and along ϑ of coordinate length µ since this captures all information
in the two connection components,
h(τ)x (A) = exp
∫ τLo
dxc̃τ3 = cos
+ 2τ3 sin
ϑ (A) = exp
dϑb̃τ2 = cos
+ 2τ2 sin
. (9)
The quantum Hilbert space is then based on cylindrical states depending on the connection
through countably many holonomies, which can always be written as almost periodic func-
tions f(b, c) =
µ,τ fµ,τ exp
(µb+ τc) of two variables. These form the set of functions
on the double product of the Bohr compactification of the real line, which is a compact
Abelian group. Its Haar measure defines the inner product of the (non-separable) Hilbert
space, in which states
〈b, c|µ, τ〉 = e i2 (µb+τc) µ, τ ∈ R . (10)
form an orthonormal basis. Holonomies simply act by multiplication on these states, while
densitized triad components become derivative operators
p̂b = −iγℓ2P
, p̂c = −2iγℓ2P
using the Planck length ℓP =
G~. They act as
p̂b|µ, τ〉 = 12 γℓ
P µ|µ, τ〉, p̂c|µ, τ〉 = γℓ2P τ |µ, τ〉 , (12)
immediately showing their eigenvalues.
To formulate the dynamical equation, one has to quantize the Hamiltonian constraint
d3x ǫijk(−F kab + γ2Ωkab)
EaiEbj
| detE|
where Ωkabτkdx
a ∧ dxb = − sinϑτ3dϑ ∧ dϕ is the intrinsic curvature of 2-spheres, while F kab
is the curvature computed from Aia ignoring the spin connection term sin ϑτ3dϕ. Following
standard procedures a Hamiltonian constraint operator can be expressed in the basic op-
erators. First, one replaces the inverse determinant of Eai by a Poisson bracket, following
[19],
ǫijkτ
ajEbk
| detE|
= − 1
K∈{x,ϑ,ϕ}
ǫabcωKc h
(δ)−1
K , V } (14)
with edge lengths ℓx0 = δL0 and ℓ
0 = δ, and left-invariant 1-forms ω
c on the symmetry
group manifold. For curvature components F kab one uses a holonomy around a closed loop
F iab(x)τi =
A(IJ)
IJ − 1) + O((b2 + c2)3/2
A) (15)
IJ = h
−1 (16)
and AIJ being the coordinate area of the loop, using the corresponding combinations of
ℓI0. In these expressions, a parameter δ has been chosen which specifies the length of edges
with respect to the background geometry provided by the symmetry group. Putting all
factors together and replacing Poisson brackets by commutators, one has
Ĥ(δ) = 2i(γ3δ3ℓ2P)
−1 tr
ǫIJK ĥ
(δ)−1
(δ)−1
K [ĥ
(δ)−1
K , V̂ ] + 2γ
2δ2τ3ĥ
x [ĥ
(δ)−1
x , V̂ ]
= 4i(γ3δ3ℓ2P)
8 sin
V̂ cos
− cos δb
V̂ sin
4 sin2
+ γ2δ2
V̂ cos
− cos δc
V̂ sin
which acts as
Ĥ(δ)|µ, τ〉 = (2γ3δ3ℓ2P)−1 [2(Vµ+δ,τ − Vµ−δ,τ ) (18)
×(|µ+ 2δ, τ + 2δ〉 − |µ+ 2δ, τ − 2δ〉 − |µ− 2δ, τ + 2δ〉+ |µ− 2δ, τ − 2δ〉)
+ (Vµ,τ+δ − Vµ,τ−δ)(|µ+ 4δ, τ〉 − 2(1 + 2γ2δ2)|µ, τ〉+ |µ− 4δ, τ〉)
on basis states. This operator can be ordered symmetrically, defining Ĥ
symm :=
(Ĥ(δ) +
Ĥ(δ)†), whose action is1
Ĥ(δ)symm|µ, τ〉 = (2γ3δ3ℓ2P)−1 [(Vµ+δ,τ − Vµ−δ,τ + Vµ+3δ,τ+2δ − Vµ+δ,τ+2δ)|µ+ 2δ, τ + 2δ〉
−(Vµ+δ,τ − Vµ−δ,τ + Vµ+3δ,τ−2δ − Vµ+δ,τ−2δ)|µ+ 2δ, τ − 2δ〉
−(Vµ+δ,τ − Vµ−δ,τ + Vµ−δ,τ+2δ − Vµ−3δ,τ+2δ)|µ− 2δ, τ + 2δ〉
+(Vµ+δ,τ − Vµ−δ,τ + Vµ−δ,τ−2δ − Vµ−3δ,τ−2δ)|µ− 2δ, τ − 2δ〉
(Vµ,τ+δ − Vµ,τ−δ + Vµ+4δ,τ+δ − Vµ+4δ,τ−δ)|µ+ 4δ, τ〉
−2(1 + 2γ2δ2)(Vµ,τ+δ − Vµ,τ−δ)|µ, τ〉
(Vµ,τ+δ − Vµ,τ−δ + Vµ−4δ,τ+δ − Vµ−4δ,τ−δ)|µ− 4δ, τ〉
. (19)
Transforming this operator to the triad representation obtained as coefficients of a wave
function |ψ〉 =
µ,τ ψµ,τ |µ, τ〉 in the triad eigenbasis and using the volume eigenvalues
Vµ,τ = 4π
|(p̂c)µ,τ |(p̂b)µ,τ = 2π(γℓ2P)3/2
|τ |µ ,
a difference equation
γ3/2δ3
(Ĥ(δ)symm|ψ〉)µ,τ = 2δ(
|τ + 2δ|+
|τ |) (ψµ+2δ,τ+2δ − ψµ−2δ,τ+2δ)
|τ + δ| −
|τ − δ|)
(µ+ 2δ)ψµ+4δ,τ − 2(1 + 2γ2δ2)µψµ,τ + (µ− 2δ)ψµ−4δ,τ
|τ − 2δ|+
|τ |) (ψµ−2δ,τ−2δ − ψµ+2δ,τ−2δ)
= 0 (20)
results for physical states. (For small µ the equation has to be specialized further due to
the remaining gauge freedom; see [31]. This is not relevant for our purposes.)
2.1 Relation to fixed lattices
Although there are no spatial lattices appearing in the exactly homogeneous context fol-
lowed here, the construction of the Hamiltonian constraint mimics that of the full theory.
States are then associated with spatial lattices, and holonomies refer to embedded edges
and loops. The parameter δ is the remnant of the loop size (in coordinates) used to act
with holonomies on a spatial lattice. As one can see, this parameter is important for the
resulting difference equation, determining its step-size. The above construction, using a
constant δ, can be seen as corresponding to a lattice chosen once and for all such that the
loop size is not being adjusted even while the total volume increases. As described in the
1Note that the first factor of 2 in the next-to-last line was missing in [31] and analogous places in
subsequent formulas. This turns out to be crucial for the stability analysis below. In particular, with the
corrected coefficient the quantization of the Schwarzschild interior in [31] is unstable for all values of γ.
Possible restrictions on γ, as suggested in [30] based on a difference equation with the wrong coefficient,
then do not follow easily but could be obtained from a more detailed analysis.
introduction, this ignores the possible creation of new lattice vertices and links, and can
be too rigid in certain semiclassical regimes.
To express this clearly, we now construct holonomies which are not simply along a single
edge of a certain length δ, but which are understood as holonomies along lattice links. We
keep our coordinate box of size L0 in the x-direction as well as the edge length ℓ0. If this
is a link in a uniform lattice, there are Nx = L0/ℓ0 lattice links in this direction, and a link
holonomy appears in the form
hx = exp(ℓ0c̃τ3) = exp(ℓ0cτ3/L0) = exp(cτ3/Nx) (21)
when computed along whole lattice edges. Thus, a constant coefficient 1/Nx in holonomies
corresponds to a fixed lattice whose number of vertices does not change when the volume
increases. Lattice refinements of an inhomogeneous lattice state, on the other hand, can be
mimicked by a parameterNx which depends on the phase space variables, most importantly
the triad components. If this is carried through, as we will see explicitly below, the step-size
of the resulting difference equation is not constant in the triad variables anymore.
2.2 Lattice refinements
Let us now assume that we have a lattice with N vertices in a form adapted to the
symmetry, i.e. there are Nx vertices along the x-direction (whose triad component pc gives
rise to the label τ) and N 2ϑ vertices in spherical orbits of the symmetry group (whose triad
component pb gives rise to the label µ). Thus, N = NxN 2ϑ .
Since holonomies in such a lattice setting are computed along single links, rather than
through all of space (or the whole cell of size L0), basic ones are hx = exp(ℓ
0 c̃τ3) and
hϑ = exp(ℓ
0 b̃τ2), denoting the edge lengths by ℓ
0 and keeping them independent of each
other in this anisotropic setting. Edge lengths are related to the number of vertices in
each direction by ℓx0 = L0/Nx and ℓϑ0 = 1/Nϑ. With the rescaled connection components
c = L0c̃ and b = b̃ we have basic holonomies
hx = exp(ℓ
0 cτ3) = exp(cτ3/Nx) , hϑ = exp(ℓϑ0bτ2) = exp(bτ2/Nϑ) . (22)
Using this in the Hamiltonian constraint operator then gives a difference equation whose
step-sizes are 1/NI .
So far, we only reinterpreted δ in terms of vertex numbers. We now turn our attention to
solutions to the Hamiltonian constraint which, in the full theory, usually changes the lattice
by adding new edges and vertices while triad eigenvalues increase. For larger µ and τ , the
Hamiltonian constraint thus acts on a finer lattice than for small values, and the parameter
N for holonomies appearing in the constraint operator is not constant on phase space but
triad dependent. Due to the irregular nature of lattices with newly created vertices such a
refinement procedure is difficult to construct explicitly. But it is already insightful to use
an effective implementation, using the derivation of the Hamiltonian constraint for a fixed
lattice, but assuming the vertex number N (µ, τ) to be phase space dependent. Moreover,
we include a parameter δ as before, which now takes a value 0 < δ < 1 and arises because
a graph changing Hamiltonian does not use whole lattice edges but only a fraction, given
by δ.2 Effectively assuming in this way that the lattice size is growing through the basic
action of the Hamiltonian constraint, we will obtain a difference equation whose step-size
δ/N is not constant in the original triad variables.
For the Schwarzschild interior, we have step sizes δ/Nϑ for µ and δ/Nx for τ . Go-
ing through the same procedure as before, we end up with an operator containing flux-
dependent holonomies instead of basic ones, e.g., Nx(µ, τ)hx = Nx(µ, τ) exp(cτ3/Nx(µ, τ))
which reduces to an Nx-independent connection component c in regimes where curvature
is small. Keeping track of all prefactors and holonomies in the commutator as well as the
closed loop, one obtains the difference equation
C+(µ, τ)
ψµ+2δNϑ(µ,τ)−1,τ+2δNx(µ,τ)−1 − ψµ−2δNϑ(µ,τ)−1,τ+2δNx(µ,τ)−1
+C0(µ, τ)
(µ+ 2δNϑ(µ, τ)−1)ψµ+4δNϑ(µ,τ)−1,τ − 2(1 + 2γ
2δ2Nϑ(µ, τ)−2)µψµ,τ
+ (µ− 2δNϑ(µ, τ)−1)ψµ−4δNϑ(µ,τ)−1,τ
+C−(µ, τ)
ψµ−2δNϑ(µ,τ)−1,τ−2δNx(µ,τ)−1 − ψµ+2δNϑ(µ,τ)−1,τ−2δNx(µ,τ)−1
= 0 . (23)
C±(µ, τ) = 2δNϑ(µ, τ)−1(
|τ ± 2δNx(µ, τ)−1|+
|τ |) (24)
C0(µ, τ) =
|τ + δNx(µ, τ)−1| −
|τ − δNx(µ, τ)−1| . (25)
(A total factor NxN 2ϑ for the number of vertices drops out because the right hand side is
zero in vacuum, but would multiply the left hand side in the presence of a matter term.)
3 Specific refinement models
For further analysis one has to make additional assumptions on how exactly the lattice
spacing is changing with changing scales µ and τ . To fix this in general, one would have
to use a full Hamiltonian constraint and determine how its action balances the creation
of new vertices with increasing volume. Instead of doing this, we will focus here on two
geometrically motivated cases. Technically simplest is a quantization where the number of
vertices in a given direction is proportional to the geometrical area of a transversal surface.
Moreover, the appearance of transversal surface areas is suggested by the action of the full
Hamiltonian constraint which, when acting with an edge holonomy, creates a new vertex
along this edge (changing NI for this direction) and changes the spin of the edge (changing
2A precise value can be determined only if a precise implementation of the symmetry for a fixed full
constraint operator is developed. Currently, both the symmetry reduction for composite operators and a
unique full constraint operator are lacking to complete this program and we have to work with δ as a free
parameter. This parameter is sometimes related to the lowest non-zero eigenvalue of the full area operator
[15, 25]. From the inhomogeneous perspective of lattice states used here, however, there is no indication
for such a relation.
the area of a transversal surface). It also agrees with [25, 34], although the motivation in
those papers, proposing to use geometrical areas rather than coordinate areas AIJ in (16),
is different.
Geometrically more intuitive is the case where the number of vertices in a given direction
is proportional to the geometrical extension of this direction.3 The resulting difference
equation will be more difficult to deal with due to its non-constant step-size, but naturally
gives rise to Misner-type variables. This case will also be seen to have improved stability
properties compared to the first one using areas. In both cases, N ∝ V is assumed, i.e.
the lattice size increases proportionally to volume. This is not necessary in general, and
we choose these two cases mainly for illustrative purposes. In fact, constant N as in [15]
and N ∝ V first used in [25] are two limiting cases from the full point of view, the first
one without creating new vertices and the second one without changing spin labels along
edges since local lattice volumes V/N remain constant. In general, both spin changes and
the creation of new vertices happen when acting with a Hamiltonian constraint operator.
Thus, one expects N ∝ V α with some 0 < α < 1 to be determined by a detailed analysis
of the full constraint and its reduction to a homogeneous model. Even assuming a certain
behavior of N (V ) without analyzing the relation to a full constraint leaves a large field to
be explored, which can give valuable consistency checks. We will not do this systematically
in this paper but rather discuss a mathematical issue that arises in any such case: initially,
one has to deal with difference equations of non-constant step-size which can be treated
either directly or by tranforming a non-equidistant difference equation to an equidistant
one. We first illustrate this for ordinary difference equations since partial ones, as they
arise in anisotropic models, can often be reduced to this case.
3.1 Ordinary difference equations of varying step-size
Let us assume that we have an ordinary difference equation for a function ψµ, which appears
in the equation with µ-dependent increments ψµ+δN1(µ)−1 . To transform this to a fixed step-
size, we introduce a new variable µ̃(µ) such that µ̃(µ+δ/N1(µ)) = µ̃(µ)+δµ̃′/N1(µ)+O(δ2)
has a constant linear term in δ. (For the isotropic equation, N1 is the vertex number only in
one direction. The total number of vertices in a 3-dimensional lattice is given by N = N 31 .)
This is obviously satisfied if we choose µ̃(µ) :=
∫ µN1(ν)dν. We then have
ψµ+δ/N1(µ) = ψ̃µ̃(µ+δ/N1(µ)) = ψ̃µ̃+δ+
δiN (i−1)1 /N i1
= ψ̃µ̃+δ +
ψ̃′ +O(δ3) (27)
3This behavior is introduced independently in [35] where “effective” equations, obtained by replacing
connection components in the classical constraint by sines and cosines of such components according to
how they occur in the quantized constraint, are analyzed for the Schwarzschild interior. The results are
complementary to and compatible with our stability analysis of the corresponding difference equations
below. We thank Kevin Vandersloot for discussions on this issue.
where N (i)1 denotes the i-th derivative of N1. Thus, up to terms of order at least δ2 the new
equation will be of constant step-size for the function ψ̃µ̃ := ψµ(µ̃). (The derivative ψ̃
µ̃ may not be defined for any solution to the difference equation. We write it in this form
since such terms will be discussed below in the context of a continuum or semiclassical
limit where derivatives would exist.)
It is easy to see that, for refining lattices, the additional terms containing derivatives of
the wave function are of higher order in ~ and thus correspond to quantum corrections. For
N1(µ) ∝ µq as a positive power of µ, which is the expected case from lattice refinements
related to the increase in volume, we have
µN1(µ)
4πγℓ2P
relating µ to an isotropic triad component p = 4πγℓ2Pµ/3 as it occurs in isotropic loop
quantum gravity [14]. Moreover,
ψ̃′ =
N1(µ)
= − i
N1(µ)
in terms of a curvature operator ĉ = 8πiγG~/3d/dp = 2id/dµ which exists in a continuum
limit [17]. Thus,
ψ̃′ ∝
)1+2q
With q positive (or just larger than −1/2) for a refining lattice, there is a positive power
of ~, showing that additional terms arising in the transformation are quantum corrections.
This has two important implications. First, it shows that the correct classical limit
is obtained if lattices are indeed refined, rather than coarsened, since q is restricted for
corrections to appear in positive powers of ~. In anisotropic models, as we will see, the
behavior is more complicated due to the presence of several independent variables. An
analysis of the semiclassical limit can then put strong restrictions on the behavior of lat-
tices. Secondly, we can implicitly define a factor ordering of the original constraint giving
rise to the non-equidistant difference equation by declaring that all quantum correction
terms arising in the transformation above should cancel out with factor ordering terms.
We then obtain a strictly equidistant equation in the new variable µ̃. For example, a
function N1(µ) =
|µ| gives µ̃ ∝ |µ|3/2 such that the transformed difference equation
will be equidistant in volume rather than the densitized triad component. For this special
case, factor orderings giving rise to a precisely equidistant difference equation have been
constructed explicitly in [25, 34].
3.2 Number of vertices proportional to transversal area
A simple difference equation results if the number of vertices is proportional to the transver-
sal area in any direction.4 In the x-direction we have transversal surfaces given by symmetry
4Since this refers to the area, it is the case which agrees with the motivation of [25, 34].
orbits of area pc, using the line element (3), and thus Nx ∝ τ . Transversal surfaces for
an angular direction are spanned by the x- and one angular direction whose area is pb,
giving Nϑ ∝ µ. Each minisuperspace direction has a step-size which is not constant but
independent of the other dimension. Moreover, due to the simple form one can transform
the equation to constant step-size by using independent variables τ 2 and µ2 instead of τ
and µ. Illustrating the general procedure given before, a function ψ̃τ2,µ2 acquires constant
shifts under the basic steps,
ψ̃(τ+nδ/τ)2 ,(µ+mδ/µ)2 = ψ̃τ2+2nδ+n2δ2/τ2,µ2+2mδ+m2δ2/µ2
= ψ̃τ2+2nδ,µ2+2mδ +O(τ
−2) +O(µ−2)
up to terms which can be ignored for large τ and µ. This is sufficient for a straightforward
analysis in asymptotic regimes. Moreover, higher order terms in the above equation come
with higher derivatives of the wave function in the form
γ2ℓ4P
ψ̃′ = −i(γℓ
since q = 1 compared to the discussion in Sec. 3.1. Due to the extra factors of ~ (or even
higher powers in further terms in the Taylor expansion) any additional term adding to the
constant shift of ψ̃τ2,µ2 can be attributed to quantum corrections in a semiclassical limit.
Accordingly, such terms can be avoided altogether by a judicious choice of the initial factor
ordering of operators.
3.3 Number of vertices proportional to extension
Geometrically more intuitive, and as we will see below dynamically more stable, is the
case in which the number of vertices in each direction is proportional to the extension of
that direction measured with the triad itself. This gives Nϑ ∝
|τ | and Nx ∝ µ/
|τ |,
using the classical co-triad (4). (One need not worry about the inverse τ since the effective
treatment of lattice refinements pursued here is not valid close to a classical singularity
where an already small lattice with a few vertices changes. Singularities in general can
only be discussed by a direct analysis of the resulting difference operators. Since only
a few recurrence steps are necessary to probe the scheme around a classical singularity,
equidistant difference operators are not essential in this regime. They are more useful in
semiclassical regimes where one aims to probe long evolution times as in the examples
below. Similar remarks apply to the horizon at µ = 0 which, although a classical region
for large mass parameters, presents a boundary to the homogeneous model used for the
Schwarzschild interior.) The behavior is thus more complicated than in the first case since
the step size of any of the two independent variables depends on the other variable, too.
First, it is easy to see, as before with quadratic variables, that the volume label ω = µ
changes (approximately) equidistantly with each iteration step which is not equidistant for
the basic variables µ and τ . But it is impossible to find a second, independent quantity
which does so, too. In fact, such a quantity f(µ, τ) would have to solve two partial
differential equations in order to ensure that
f(µ+nδNϑ(µ, τ)−1, τ+mδNx(µ, τ)−1) ∼ f(µ, τ)+nδNϑ(µ, τ)−1∂µf(µ, τ)+mδNx(µ, τ)−1∂τf(µ, τ)
changes only by a constant independent of τ and µ. This implies ∂µf(µ, τ) ∝
|τ | and
∂τf(µ, τ) ∝ µ/
|τ | whose only solution is f(µ, τ) ∝ µ
|τ | which is the volume ω.
We thus have to deal with non-equidistant partial difference equations in this case which
in general can be complicated. A possible procedure to avoid this is to split the iteration
in two steps since an ordinary difference equation can always be made equidistant as
above (cancelling quantum corrections by re-ordering). We first transform τ to the volume
variable ω which gives, up to quantum corrections, constant iteration steps for this variable.
With the second variable still present, a higher order difference equation
C0(µ, ω
2/µ2)(1 + 2δ/ω)µψµ(1+4δ/ω),ω+4δ + C+(µ, ω
2/µ2)ψµ(1+2δ/ω),ω+3δ
−C−(µ, ω2/µ2)ψµ(1+2δ/ω),ω+δ − 2C0(µ, ω2/µ2)(1 + 2γ2δ2µ2/ω2)µψµ,ω
−C+(µ, ω2/µ2)ψµ(1−2δ/ω),ω−δ + C−(µ, ω2/µ2)ψµ(1−2δ/ω),ω−3δ
+C0(µ, ω
2/µ2)(1− 2δ/ω)µψµ(1−4δ/ω),ω−4δ = 0 (28)
results with
C0(µ, ω
2/µ2) =
C±(µ, ω
2/µ2) = 2δ
1± 2δ
derived from the original coefficients (24). The structure of this difference equation is quite
different from the original one: not only is it of higher order, but now only one value of the
wave function appears at each level of ω, rather than combinations of values at different
values of µ. Note also that only the coefficient of the unshifted ψµ,ω depends on µ. This
form of the difference equation is, however, a consequence of the additional rotational
symmetry and is not realized in this form for fully anisotropic Bianchi models as we will
see below.
Proceeding with this specific case, we have to look at wave functions evaluated at
shifted positions µ(1 + mδ/ω) with integer m. At fixed ω = ω0, we are thus evaluating
the wave function at values of µ multiplied with a constant, instead of being shifted by a
constant as in an equidistant difference equation. This suggests to use the logarithm of
µ instead of µ itself as an independent variable, which is indeed the result of the general
procedure. After having transformed from τ to ω already, we have to use τ as a function
of µ and ω in the vertex number Nϑ, which is τ(µ, ω) = (ω/µ)2 after using ω = µ
Thus, Nϑ(µ, τ(µ, ω)) =
τ(µ, ω) = ω/µ now is not a positive power of the independent
variable µ and we will have to be more careful in the interpretation of correction terms
after performing the transformation. (The lattice is coarsened with increasing anisotropy
at constant volume.) Naively applying the results of Sec. 3.1 to q = −1 would suggest that
corrections come with inverse powers of ~ which would certainly be damaging for the correct
classical limit. However, the factors change due to the presence of the additional variable
ω0 even though it is treated as a constant. We have N ′ϑ/N 2ϑ = −1/ω0 = −(γℓ2P/2)3/2/V0
in terms of the dimensionful volume V , while it would just be a constant −1 without the
presence of ω. The additional factor of ~3/2 ensures that corrections come with positive
powers of ~ for the correct classical limit to be realized.
For any ω0, we thus transform ψ̃µ(1+mδ/ω0) to equidistant form by using
ψµ̃ = ψ̃µ(µ̃) with
µ̃(µ) = log µ. This transformation is possible since the second label ω0 is now treated as a
constant, rather than an independent variable of a partial difference equation. (Recall that
for the type of difference equation discussed here there is only one variable, the volume,
which is equidistant under all of the original discrete steps.) Despite of negative powers of
some variables in the vertex numbers, we have the correct classical limit in the presence
of ω. As before, the transformation is exact up to higher order terms which are quantum
and higher order curvature corrections. Defining the original constraint operator ordering
implicitly by the requirement that all those terms are cancelled allows us to work with an
equidistant difference equation.
3.4 Bianchi models
As mentioned before, the transformed difference equation does not become higher order
for fully anisotropic Bianchi models. In this case, we have three independent flux labels
µI , I = 1, 2, 3, and vertex numbers NI . Using vertex numbers proportional to the spatial
extensions for each direction gives N1 =
µ2µ3/µ1, N2 =
µ1µ3/µ2 and N3 =
µ1µ2/µ3.
As in the difference equation for the Schwarzschild interior, the difference equation for
Bianchi models [22] uses values of the wave function of the form ψµ1+2δ/N1,µ2+2δ/N2,µ3 . One
can again see easily that the volume ω =
|µ1µ2µ3| behaves equidistantly under the
increments,
ω(µ1 + 2δ/N1, µ2 + 2δ/N2, µ3) =
µ1 + 2δ
µ2 + 2δ
µ1µ2µ3 + 4δ
µ1µ2µ3 + 4δ2 = ω + 2δ +O(δ
The leading order term of the difference equation in ω results from a combination
C1ψµ1,µ2+2δ/N2,µ3+2δ/N3 + C2ψµ1+2δ/N1,µ2,µ3+2δ/N3 + C3ψµ1+2δ/N1,µ2+2δ/N2,µ3
≈ C1ψ̃µ1,µ2+2δ/N2,ω+2δ + C2ψ̃µ1+2δ/N1,µ2,ω+2δ + C3ψ̃µ1+2δ/N1,µ2+2δ/N2,ω+2δ
= C1ψ̃µ1,µ2(1+2δ/ω),ω+2δ + C2ψ̃µ1(1+2δ/ω),µ2 ,ω+2δ + C3ψ̃µ1(1+2δ/ω),µ2(1+2δ/ω),ω+2δ
=: Ĉ+ψ̃ω+2δ(µ1, µ2)
where we used 1/N1 =
µ1/µ2µ3 = µ1/ω and defined the operator Ĉ+ acting on the
dependence of ψ on µ1 and µ2. Thus, unlike for the Schwarzschild interior the difference
equation does not become higher order in ω, and the highest order term does have a
difference operator coefficient in the remaining independent variables.
The recurrence proceeds as follows: We have a partial difference equation of the form
Ĉ+ψ̃ω+2δ(µ1, µ2) + Ĉ0ψ̃ω(µ1, µ2) + Ĉ−ψ̃ω−2δ(µ1, µ2)
with difference operators Ĉ± and Ĉ0 acting on the dependence on µ1 and µ2. In terms of
initial data at two slices of ω we can solve recursively for Ĉ0ψ̃ω(µ1, µ2)+Ĉ−ψ̃ω−2δ(µ1, µ2) =:
φ(µ1, µ2) and then, in each ω-step, use boundary conditions to solve the ordinary difference
equation
Ĉ+ψ̃ω+2δ(µ1, µ2) = φ(µ1, µ2) .
Although the operator Ĉ+ itself is not equidistant, this remaining ordinary difference
equation can be transformed to an equidistant one by transforming µ1 and µ2 as in
Sec. 3.1 (using that ω is constant and fixed for this equation at any recursion step). With
µ3(µ1, µ2, ω) = ω
2/µ1µ2, we have lattice spacings N1(µ1, µ2, ω) = ω/µ1 and N2(µ1, µ2, ω) =
ω/µ2 in terms of ω which are already independent of each other. The two remaining vari-
ables µ1 and µ2 are thus transformed to equidistant ones by taking their logarithms as
encountered before.
Note the resemblance of the new variables, volume and two logarithms a metric compo-
nents at constant volume, to Misner variables [36]. This observation may be of interest in
comparisons with Wheeler–DeWitt quantizations where Misner variables have often been
used, making the Wheeler–DeWitt equation hyperbolic.
4 Application: Stability of the Schwarzschild interior
Now that we have several possibilities for the lattice spacings, we consider their effect on
the solutions of the Hamiltonian constraint. In particular, these solutions may have un-
desirable properties reminiscent of numerical instabilities, as it was indeed noticed for the
original quantization of the Schwarzschild interior in [28]. Also problems in the presence
of a positive cosmological constant, described in the introduction, are of this type. Re-
call that when one wishes to solve an ordinary differential equation, for example, there
are various discrete schemes that ensure errors do not propagate as the number of time
steps increases. Here we are in the opposite situation – instead of having the freedom to
pick the discrete version of a continuous equation, the discrete equation itself is what is
fundamental. Thus, like a badly chosen numerical recipe, some choices of the functions Nτ
and Nϑ in the constraint equation may quickly lead to solutions that are out of control,
and increase without bound. To test for this, we will use a von Neumann stability analy-
sis [28] on the possible recursion relations. The essential idea is to treat one of the relation
parameters as an evolution parameter, and decompose the rest in terms of orthogonal
functions, representing “spatial” modes of the solution. This will give rise to a matrix that
defines the evolution of the solution; if the matrix eigenvalues are greater than unity for
a particular mode, that mode is unstable. In particular, a relation
k=−M an+kψn+k = 0
is equivalent to a vector equation of the form ~vn = Q(n)~vn−1, where the column vector
~vn = (ψn+M , ψn+M−1, · · · , ψn−M+1)T . The evolution of an eigenvector ~w of the matrix Q(n)
is given by ~wn = λw ~wn−1. Thus, when the size of the corresponding eigenvalue |λw| > 1,
the values in the sequence associated to ~w will grow as well.
With this in mind, we consider the choices of Nx and Nϑ discussed previously, starting
with the case Nx = τ and Nϑ = µ. In the large µ, τ limit for this choice, the coefficients of
the Hamiltonian constraint become
C±(µ, τ) ∼
, C0(µ, τ) ∼
τ 3/2
In the asymptotic limit, the coefficients of the ψµ±2δ/µ,τ and ψµ,τ terms go to C0(µ, τ)µ.
As we saw in Section 3.2, we can choose a different set of variables in which the step sizes
are constant (up to ordering of the operators). Plugging these asymptotic values into the
Hamiltonian constraint, and changing variables to µ̃ = µ2/2 and τ̃ = τ 2/2 gives
4τ̃(ψµ̃+2δ,τ̃+2δ − ψµ̃−2δ,τ̃+2δ + ψµ̃−2δ,τ̃−2δ − ψµ̃+2δ,τ̃−2δ) + µ̃(ψµ̃+4δ,τ̃ − 2ψµ̃,τ̃ + ψµ̃−4δ,τ̃) = 0.
Because all the step sizes now are constants depending on δ, we define new parameters
m,n such that µ̃ = 2mδ and τ̃ = 2nδ. Using m as our evolution parameter and n as the
“spatial” direction, we decompose the sequence as ψ2mδ,2nδ = um exp(inω). With this new
function, the recursion relation is written as
2in(un+1 − un−1)− (m sin θ)un = 0.
This is equivalent to the vector equation
[ − im
sin θ 1
= Q(m,n)
. (31)
The eigenvalues of the matrix Q are
−im sin θ ±
16n2 −m2 sin2 θ
When the discriminant 16n2 − m2 sin2 θ ≥ 0, then |λ| = 1, and the solution is stable;
however, there are unstable modes when 16n2 − m2 sin2 θ < 0. The most unstable mode
corresponds to the choice sin θ = 1, giving instabilities in terms of the original variables
when µ > 2τ . In this regime, all solutions behave exponentially rather than oscillating.
This region includes parts of the classical solutions for the Schwarzschild interior even for
values of µ and τ for which one expects classical behavior to be valid. The presence of
instabilities implies, irrespective of the physical inner product, that quantum solutions
in those regions cannot be wave packets following the classical trajectory, and the correct
classical limit is not guaranteed for this quantization, which is analogous to that introduced
in [25, 34].
The situation is different when we consider the choices Nx =
|τ | and Nϑ = µ/
|τ |,
where we will find a lack of instability. There is no choice of variables that allows us to
asymptotically approach a constant spacing recursion relation, because of the mixing of
the µ and τ variables in the step size functions. Thus, we will make the assumption that
in the large µ, τ limit, the solution does not change much under step sizes δN−1x and δN
To see how this affects the resulting stability of the solutions, we will look at a simpler
example first. If we start with the Fibonacci relation Rτ ≡ ψτ+1 − ψτ − ψτ−1 = 0, then
the two independent solutions are of the form ψτ = κ
τ , where κ is either the golden ratio
φ = (1 +
5)/2 or else −φ−1. Only the latter solution meets the criterion for stability,
since |φ| > 1. When we change this relation to
R̃τ ≡ ψτ+1/τn − ψτ − ψτ−1/τn = 0, (32)
with n 6= 1, the situation changes – only one of the two solutions outlined above will solve
the relation asymptotically. In particular, when we examine the error R̃τ we get when we
plug κτ into the altered relation (32), i.e.
R̃τ = κ
τ (κ1/τ
n − 1− κ−1/τn),
the error is proportional to ψτ itself. As τ → ∞, therefore, the error for the κ = φ solution
grows without bound, while that of κ = −φ−1 goes to zero. Thus, we see in this situation
a relation between the stability and the asymptotic behavior of a solution.
Returning to the Schwarzschild relation, in the large µ, τ limit the coefficient functions
of the recursion relation are to leading order
C±(µ, τ) ∼ 4δ, C0(µ, τ) ∼
In turn, the relation itself becomes
4(ψµ+2δ/
τ ,τ+2δ
τ/µ − ψµ−2δ/√τ ,τ+2δ√τ/µ − ψµ+2δ/√τ ,τ−2δ√τ/µ + ψµ−2δ/√τ ,τ−2δ√τ/µ)
+(ψµ+4δ/
τ ,τ − 2ψµ,τ + ψµ−4δ/√τ ,τ ) = 0.
From this point on, we assume that we have a solution to this relation which does not vary
greatly when, for example, µ is changed by ±2δ/√µ, and similarly for τ . Both Nx and Nϑ
are constant to first order in shifts µ± 2δN−1x and similarly for τ , in the asymptotic limit.
Thus, we assume that α = 2δN−1x and β = 2δN
ϑ are constants, and use the scalings
µ = αm and τ = βn. When this is done, we get an equation similar to the case when
Nx = τ and Nϑ = µ, but with constant coefficients; this is the crucial difference that allows
stable solutions to the case here. Using the decomposition ψαm,βn = un exp(imθ), we arrive
at the matrix equation
[ − i
sin θ 1
. (33)
The matrix here has eigenvalues λ with |λ| = 1 for all m,n, so the solution is stable. Using
arguments as in the Fibonacci example, the non-equidistant equation of the second scheme
is shown to be stable.
5 Conclusions
Following [9], we explicitly introduced loop quantum cosmological models which take into
account the full lattice structure of inhomogeneous states. Such lattices are in general
refined by adding new vertices when acting with the Hamiltonian constraint. Thus, also
dynamical equations even in homogeneous models should respect this property. Several
interesting features arose: One obtains non-equidistant difference equations which, when
imposed for functions on the whole real line as in isotropic loop quantum cosmology,
are more restrictive than equidistant ones due to the absence of superselected sectors.
This leaves the singularity issue unchanged since for this one only needs to consider a
few steps in the equation. But a stability analysis of solutions and the verification of the
correct classical limit in all semiclassical regimes can be more challenging. We presented an
example for such an analysis, but also introduced a procedure by which one can transform
the resulting equations to equidistant ones up to quantum corrections, which is sufficient
for a semiclassical analysis. Interestingly, properties of the transformation itself provide
hints to the correct semiclassical behavior. As a side-result, we demonstrated that one
particular version of lattice refinements naturally gives rise to Misner-type variables.
It is our understanding that this general procedure of defining lattice refining models
mostly agrees with the intuition used specifically in isotropic models in [25], and adapted
to anisotropic ones in [34].5 However, there are some departures from what is assumed in
[25]. First, we do not see indications to refer to the area operator while the area spectrum
was not only used in [25] to fix the constant δ and the volume dependence of the step size
but in fact provided the main motivation. Secondly, due to this motivation [25] presents
a more narrow focus which from our viewpoint corresponds to only one single refinement
model. It has a vertex number proportional to volume, which is a limiting case not realized
by known full Hamiltonian constraints, and puts special emphasis on geometrical areas to
determine the vertex number. Finally, commutators for inverse volume operators are to be
treated differently from [25], taking into account a lattice refining model which would not
be possible in a purely homogeneous formulation. As shown in the appendix, this enlarges
expected quantum corrections to the classical functions.
We have discussed similar cases for illustration here, but keep a more general view-
point on the refinement as a function of volume. A preliminary stability analysis for
the Schwarzschild interior, consistent with [35] indeed suggests that a behavior different
from what is suggested in [25] is preferred, which indicates that models can provide tight
conditions for the general analysis of quantum dynamics. We emphasize that stability
arguments as used here are independent of physical inner product issues since they refer
to properties of general solutions. A general analysis as started here allows detailed tests
of the full dynamics in manageable settings, which can verify the self-consistency of the
framework of loop quantum gravity — or possibly point to limitations which need to be
better understood.
5We thank A. Ashtekar for discussions of this point.
Acknowledgements
We thank Kevin Vandersloot for discussions. This work was supported in part by NSF
grant PHY0554771. GK is grateful for research support from the University of Mas-
sachusetts and Glaser Trust of New York.
A Inverse volume terms in homogeneous models and
lattice refinement
We have seen in this paper, following [9], that Hamiltonian constraint operators with triad-
dependent parameters in holonomies allow one to model lattice refinements faithfully, with
interesting results and some improvements over the original non-refining models. However,
as always there are also some features of inhomogeneous states and operators which are
not present and difficult to mimic in homogeneous models. Thus, even models generalized
in this way by allowing for lattice refinement effects have to be interpreted with great care.
While qualitative effects can be investigated fruitfully to test the full framework, there is
no basis for drawing quantitative conclusions. The prime example is that of commutator
terms which appear ubiquitously in composite operators of loop quantum gravity, such as
the coefficients of difference equations or also matter Hamiltonians.
In the main construction of this paper we used holonomies associated with links of a lat-
tice, rather than edges of a fixed coordinate length. This allows us, effectively, to take into
account lattice refinements which change the number of vertices. It applies to holonomies
(16) along a closed loop used to quantize curvature components which determine the step
size of difference equations, but also to the link holonomies used in commutators to quan-
tize inverse triad components based on (14). What is not modeled in homogeneous models
is the fact that a lattice operator makes use of the local volume V̂v at a given vertex v
where the commutator is acting, rather than the total volume V̂ =
v V̂v of the whole
box in which the lattice is embedded. In a fully inhomogeneous setting the difference
does not matter since volume contributions from vertices not touched by the edge used in
a commutator drop out in the end, [he, V̂ ] =
v∈e[he, V̂v]. But in homogeneous models
there is a difference since volume contributions from different vertices, in an exactly ho-
mogeneous setting, are all identical. Thus, the total volume V = NVv is the number of
vertices multiplied with the local volume Vv. Then, [h, V̂v] rather than [h, V̂ ] is expected as
the contribution to constraint operators from the inhomogeneous perspective. In homoge-
neous models as in [25], on the other hand, [h, V̂ ] is more straightforward to use. We now
show that without corrections this would imply crucial deviations from the inhomogeneous
behavior.
It is easy to see that commutators differ depending on whether the local or total volume
is used. For simplicity of the argument, we proceed with an isotropic situation where
V = |p|3/2 in terms of the basic isotropic densitized triad component p. A local lattice flux,
for a surface S intersecting only a single link, would be ρ =
d2yp̃ = ℓ20p/L
0 = p/N 2/3
for links of coordinate length ℓ0, such that Vv = |ρ|3/2 = |p|3/2/N is the local volume.
(We again use a coordinate box of size L20 and introduce rescaled flux variables p = L
0p̃.)
Isotropic states are spanned by eiµc/2 where µ ∈ R is related to the triad eigenvalues by
pµ = 4πγℓ
Pµ/3. Using a link holonomy h ∼ eiℓ0c̃/2 = eic/2N
which as a multiplication
operator increases µ by 1/N 1/3, a commutator with the total volume will have eigenvalues
of the form
h−1[h, V̂ ] ∼ V (µ+1/N 1/3)−V (µ−1/N 1/3) = |p+4πγℓ2P/3N 1/3|3/2−|p−4πγℓ2P/3N 1/3|3/2 .
If the local volume is used, on the other hand, we have to refer to local edge labels µe =
3ρ/4πγℓ2P, rather than using the total p. Thus,
h−1e [he, V̂v] ∼ Vv(µe + 1)− Vv(µe − 1) = |ρ+ 4πγℓ2P/3|3/2 − |ρ− 4πγℓ2P/3|3/2
= N−1(|p+ 4πγℓ2PN 2/3/3|3/2 − |p− 4πγℓ2PN 2/3/3|3/2) . (35)
For large volume, p ≫ N , both expressions give the correct classical limit 3
N−1/3
expected from {eic/2N1/3 , |p|3/2}. However, quantum corrections, i.e. deviations from this
classical limit for finite p, are much larger for the second version using the local volume as
it would occur in an inhomogeneous quantization. The smooth classical function dV/dp in
a Poisson bracket appears in discretized form by the large step-size N2/3 in (35) rather than
the small one N−1/3 in (34). Perturbative corrections, derived by Taylor expanding the
difference terms and keeping higher order corrections to the classical expression, are thus
larger. (This can have cosmological implications [37, 38, 39].) Non-perturbative effects as
observed for the inverse scale factor operator in isotropic loop quantum cosmology which
has an upper bound at finite volume [40], start to arise for p ∼ N 2/3 when the local
volume is used but only at the much smaller p ∼ N−1/3 for the total volume. Since only
the local volume is relevant for inhomogeneous quantizations, quantum corrections from
inverse volume operators can be large.
Unfortunately, this effect is more difficult to mimic in exact homogeneous models unlike
the behavior of holonomies under lattice refinements and has therefore been overlooked in
[25]. The connection components appearing in holonomies can simply be divided by a
function N of triad components to implement shrinking edges due to subdivision. This is
not possible for the volume itself to use a local version in a homogeneous model since, if
we would divide the total volume by the appropriate function of triad components, only
a constant would remain for an N proportional to volume and the commutator would be
zero. The only way to have this effect faithfully implemented in a homogeneous model is
to use higher SU(2) representations for holonomies in commutators but not for holonomies
used in the loop to quantize curvature components. (This is not possible if one writes the
constraint as a single trace, tr(hαhe[h
e , V̂ ]) but can easily be done using the equivalent
form tr(τihα) tr(τihe[h
e , V̂ ]). We emphasize that higher representations for commutators
are advocated here only in exactly homogeneous models to mimic inhomogeneous effects.
Fully inhomogeneous operators usually need not refer to higher representations.) In a rep-
resentation of spin j, matrix elements of holonomies contain exponentials exp(imc/2N 1/3)
with −j ≤ m ≤ j, which increases the shifts in volume labels resulting from commutators.
Resulting expressions for commutators can be found in [41, 42]. If the representation label
j is of the order N , effects as they result from lattice refinements and using the local vol-
ume are correctly implemented. Accordingly, corrections from inverse triad components
quantized through commutators are much larger than they would otherwise be.6
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	Introduction
	Difference equation for the Schwarzschild interior with varying discreteness scale
	Relation to fixed lattices
	Lattice refinements
	Specific refinement models
	Ordinary difference equations of varying step-size
	Number of vertices proportional to transversal area
	Number of vertices proportional to extension
	Bianchi models
	Application: Stability of the Schwarzschild interior
	Conclusions
	Inverse volume terms in homogeneous models and lattice refinement

  A general class of loop quantizations for anisotropic models is introduced
and discussed, which enhances loop quantum cosmology by relevant features seen
in inhomogeneous situations. The main new effect is an underlying lattice which
is being refined during dynamical changes of the volume. In general, this leads
to a new feature of dynamical difference equations which may not have constant
step-size, posing new mathematical problems. It is discussed how such models
can be evaluated and what lattice refinements imply for semiclassical behavior.
Two detailed examples illustrate that stability conditions can put strong
constraints on suitable refinement models, even in the absence of a fundamental
Hamiltonian which defines changes of the underlying lattice. Thus, a large
class of consistency tests of loop quantum gravity becomes available. In this
context, it will also be seen that quantum corrections due to inverse powers of
metric components in a constraint are much larger than they appeared recently
in more special treatments of isotropic, free scalar models where they were
artificially suppressed.

Introduction
Loop quantum cosmology [1] was designed to test characteristic effects expected in the full
framework of loop quantum gravity [2, 3, 4]. Implementing symmetries at the kinematical
∗e-mail address: bojowald@gravity.psu.edu
†e-mail address: Cartin@naps.edu
‡e-mail address: gkhanna@UMassD.Edu
http://arxiv.org/abs/0704.1137v1
quantum level allows explicit treatments of the dynamical equations while preserving basic
features such as the discreteness of spatial geometry [5]. (See also [6, 7, 8, 9, 10, 11] for
recent work on symmetry reduction in quantum theories.) Indeed, several new, initially
surprising results were derived in different applications in cosmology and black hole physics.
By now many such models have been studied in detail.
As the relation of dynamics to that of a possible full framework without symmetries is
not fully worked out, detailed studies can be used to suggest improvements of the equa-
tions for physically viable behavior. Comparing results with full candidates for quantum
dynamics can then provide stringent self-consistency tests of the overall framework. It is to
be seen if, and how, such alterations of quantization procedures naturally result from a full
quantization. The first example of this type related to the stability behavior of solutions to
the difference equations of isotropic loop quantum cosmology, which was studied in [12, 13]
and was already restrictive for models with non-zero intrinsic curvature. Another limita-
tion, realized early on [14], occurs in the presence of a positive cosmological constant Λ.
In an exact isotropic model, the extrinsic curvature scale is given by k = ȧ =
8πGa2Λ/3
which, due to the factor of a2, can be large in a late universe although the local curva-
ture scale Λ might be small. Extrinsic curvature plays an important role since in a flat
isotropic model it appears in holonomies on which loop quantizations are based in such a
way that only eiαk with α ∈ R can be represented as operators, but not k itself [15]. Large
values of k would either require one to use extremely small α in the relevant operators, or
imply unexpected deviations from classical behavior. In fact, holonomies as basic objects
imply that the Hamiltonian constraint is quantized to a difference rather than differential
equation [16] since k in the Hamiltonian constraint (as in the Friedmann equation) is not
directly quantized but only exponentials eiαk. These are shift operators instead of differen-
tial operators. For a large, semiclassical universe a Wheeler–DeWitt wave function should
be a good approximation to the basic difference equation of loop quantum cosmology [17]
which, in a representation as a function of the momentum p = a2 conjugate to k, would be
oscillating on scales of the order (a
Λ)−1. This scale becomes shorter and shorter in an
expanding universe, eventually falling below the discreteness scale of the difference equa-
tion of loop quantum cosmology. At such a point, discreteness of spatial geometry would
become noticeable in the behavior of the wave function (independently of how physical
observables are computed from it) although the universe should be classical.
This does not pose a problem for the general formalism, because it only shows that the
specific quantization of the exact isotropic model used reaches its limits. Physically, this
can be understood as a consequence of a fixed spatial lattice being used throughout the
whole universe evolution. Exponentials eiαk in isotropic models derive from holonomies
he(A) = P exp(
Aiaτiė
adt) of the Ashtekar connection along spatial curves e. All the
freedom contained in choosing edges to capture independent degrees of freedom of the full
theory reduces, in isotropic models, to the single parameter α which suffices to separate
isotropic connections through all functions eiαk. The parameter α, from the full perspective,
is thus related to the edge length used in holonomies. Using a fixed and constant α is
analogous to using only edges of a given coordinate length, as they occur, for instance, in
a regular lattice. In the presence of a positive cosmological constant, for any α a value of k
will then be reached such that eiαk differs strongly from iαk. From the lattice perspective,
this means that the local curvature radius becomes comparable to or smaller than the
fixed lattice scale corresponding to α. Such a fixed lattice ceases to be able to support all
small-scale oscillations relevant for a semiclassical geometry.
This is not problematic if it occurs in a quantum regime where dynamics is indeed
expected to differ from the classical one, but it poses a problem in semiclassical regimes.
A better treatment has to refer to changing lattices, which is not easy to implement in
a straightforward quantization of purely homogeneous models. In a dynamical equation
closer to what is expected from the full framework, lattice refinements would take place
during the evolution since full Hamiltonian constraint operators generally create new ver-
tices of a lattice state in addition to changing their edge labels [18, 19]. While k increases
with increasing volume, the corresponding α decreases since the lattice is being refined all
the time. For a suitable lattice refinement, the increase in k can be balanced by the decrease
of α such that αk stays small and semiclassical behavior is realized for any macroscopic
volume even with Λ > 0. This provides an interesting relation between the fundamental
Hamiltonian, which is responsible for the lattice refinement, and semiclassical properties
of models. Testing whether an appropriate balance between increasing k and lattice re-
finements can be reached generically can thus provide stringent tests on the fundamental
dynamics even without using a precise full Hamiltonian constraint operator.
This feature of lattice refinements was not mimicked in the first formulations of loop
quantum cosmology [20, 21, 14, 22, 15] since the main focus was to understand small-volume
effects such as classical singularities [23, 24]. In this context, lattice refinements appear
irrelevant because only a few action steps of the Hamiltonian, rather than long evolution,
are sufficient to probe a singularity. By now, perturbative regimes around isotropic models
have been formulated in loop quantum cosmology which are inhomogeneous and thus must
take into account lattice states and, at least at an effective level, lattice refinements [9].
One special version, corresponding to lattices with a number of vertices growing linearly
with volume in a specific way referring to the area operator, has been studied in detail in
isotropic models with a free, massless scalar [25]. Although the complicated relation to a
full, graph-changing Hamiltonian constraint is still not fully formulated, such models allow
crucial tests of the local dynamics.
While isotropic models can easily be understood in terms of wave functions on a 1-
dimensional discrete minisuperspace in terms of oscillation lengths [26], anisotropic models
with higher-dimensional minisuperspaces can be more subtle. In such models, limitations
similar to that of a cosmological constant have been observed as possible instabilities
of solutions in classical regions or the lack of a sufficient number of semiclassical states
[27, 28, 29]. For the partial difference equations of anisotropic models in loop quantum
cosmology, stability issues can be much more severe than in isotropic models and thus
lead to further consistency tests which might help to restrict possible quantization freedom
(see, e.g., [30]). In this paper we therefore introduce the general setting of anisotropic
models taking into account lattice refinements of Hamiltonian constraint operators, fo-
cusing mainly on the anisotropic model which corresponds to the Schwarzschild interior.
As we will see, the type of difference equations in general changes since they can become
non-equidistant. This leads to new mathematical problems which we address here briefly,
leaving further analysis for future work. The examples presented here already show that
one can distinguish different refinement models by their stability properties. The refine-
ment model corresponding to [25] turns out to give unstable evolution of the Schwarzschild
interior, while a new version, whose vertex number also grows linearly with volume, is
stable. Compared to isotropic models which are sensitive only to how the vertex number
of a state changes with volume, anisotropic models allow one to test much more detailed
properties.
An appendix discusses subtleties in how homogeneous models faithfully represent inho-
mogeneous states, mainly regarding the magnitude of corrections arising from quantizations
of inverse metric components which often plays a large role in cosmological applications.
2 Difference equation for the Schwarzschild interior
with varying discreteness scale
Basic variables of a loop quantization are holonomies along lattice links and fluxes over
transversal surfaces. For the Schwarzschild interior [31], the connection used for holonomies
and the densitized triad used for fluxes take the form
Aiaτidx
a = c̃τ3dx+ (ãτ1 + b̃τ2)dϑ+ (−b̃τ1 + ãτ2) sinϑdϕ + τ3 cosϑdϕ (1)
Eai τ
= p̃cτ3 sin ϑ
+ (p̃aτ1 + p̃bτ2) sinϑ
+ (−p̃bτ1 + p̃aτ2)
. (2)
Coordinates (x, ϑ, ϕ) are adapted to the symmetry, with polar angles ϑ and ϕ along orbits
of the rotational symmetry subgroup, and τj = − i2σj in terms of Pauli matrices. Spatial
geometry is determined by the spatial line element, which in terms of the densitized triad
components is
ds2 =
p̃2a + p̃
|p̃c|
dx2 + |p̃c|dΩ2 (3)
obtained from qab = Eai E
i /| detEcj |. We will also use the co-triad eia, i.e. the inverse of
eai = E
| detEbj |,
eiaτidx
a = ecτ3dx+ (eaτ1 + ebτ2)dϑ+ (−ebτ1 + eaτ2) sinϑdϕ (4)
with components
sgnp̃c
p̃2a + p̃
|p̃c|
, eb =
|p̃c| p̃b
p̃2a + p̃
and ea =
|p̃c| p̃a
p̃2a + p̃
. (5)
The phase space is spanned by the spatial constants (ã, b̃, c̃, p̃a, p̃b, p̃c) ∈ R6 with non-
vanishing Poisson brackets
{ã, p̃a} = γG/L0 , {b̃, p̃b} = γG/L0 , {c̃, p̃c} = 2γG/L0
where G is the gravitational constant and γ the Barbero–Immirzi parameter [32, 33].
Moreover, L0 is the size of a coordinate box along x used in integrating out the fields in
d3xȦiaE
˙̃cp̃c +
bp̃b +
˙̃ap̃a
to derive the symplectic structure. The SU(2)-gauge transformations rotating a general
triad are partially fixed to U(1) by demanding the x-component of Eai to point in the
internal τ3-direction in (2). The U(1)-gauge freedom allows one to set ã = 0 = p̃a, still
leaving a discrete residual gauge freedom (b̃, p̃b) 7→ (−b̃,−p̃b). The remaining variables can
be rescaled by
(b, c) := (b̃, Loc̃) , (pb, pc) := (Lop̃b, p̃c) . (6)
to make the canonical structure L0-independent:
{b, pb} = γG , {c, pc} = 2γG . (7)
This rescaling is suggested naturally by holonomies, as written below, and fluxes which are
considered the basic objects in loop quantizations.
To express the elementary variables through holonomies, which unlike connection com-
ponents will be promoted to operators, it suffices to choose curves along the x-direction of
coordinate length τL0 and along ϑ of coordinate length µ since this captures all information
in the two connection components,
h(τ)x (A) = exp
∫ τLo
dxc̃τ3 = cos
+ 2τ3 sin
ϑ (A) = exp
dϑb̃τ2 = cos
+ 2τ2 sin
. (9)
The quantum Hilbert space is then based on cylindrical states depending on the connection
through countably many holonomies, which can always be written as almost periodic func-
tions f(b, c) =
µ,τ fµ,τ exp
(µb+ τc) of two variables. These form the set of functions
on the double product of the Bohr compactification of the real line, which is a compact
Abelian group. Its Haar measure defines the inner product of the (non-separable) Hilbert
space, in which states
〈b, c|µ, τ〉 = e i2 (µb+τc) µ, τ ∈ R . (10)
form an orthonormal basis. Holonomies simply act by multiplication on these states, while
densitized triad components become derivative operators
p̂b = −iγℓ2P
, p̂c = −2iγℓ2P
using the Planck length ℓP =
G~. They act as
p̂b|µ, τ〉 = 12 γℓ
P µ|µ, τ〉, p̂c|µ, τ〉 = γℓ2P τ |µ, τ〉 , (12)
immediately showing their eigenvalues.
To formulate the dynamical equation, one has to quantize the Hamiltonian constraint
d3x ǫijk(−F kab + γ2Ωkab)
EaiEbj
| detE|
where Ωkabτkdx
a ∧ dxb = − sinϑτ3dϑ ∧ dϕ is the intrinsic curvature of 2-spheres, while F kab
is the curvature computed from Aia ignoring the spin connection term sin ϑτ3dϕ. Following
standard procedures a Hamiltonian constraint operator can be expressed in the basic op-
erators. First, one replaces the inverse determinant of Eai by a Poisson bracket, following
[19],
ǫijkτ
ajEbk
| detE|
= − 1
K∈{x,ϑ,ϕ}
ǫabcωKc h
(δ)−1
K , V } (14)
with edge lengths ℓx0 = δL0 and ℓ
0 = δ, and left-invariant 1-forms ω
c on the symmetry
group manifold. For curvature components F kab one uses a holonomy around a closed loop
F iab(x)τi =
A(IJ)
IJ − 1) + O((b2 + c2)3/2
A) (15)
IJ = h
−1 (16)
and AIJ being the coordinate area of the loop, using the corresponding combinations of
ℓI0. In these expressions, a parameter δ has been chosen which specifies the length of edges
with respect to the background geometry provided by the symmetry group. Putting all
factors together and replacing Poisson brackets by commutators, one has
Ĥ(δ) = 2i(γ3δ3ℓ2P)
−1 tr
ǫIJK ĥ
(δ)−1
(δ)−1
K [ĥ
(δ)−1
K , V̂ ] + 2γ
2δ2τ3ĥ
x [ĥ
(δ)−1
x , V̂ ]
= 4i(γ3δ3ℓ2P)
8 sin
V̂ cos
− cos δb
V̂ sin
4 sin2
+ γ2δ2
V̂ cos
− cos δc
V̂ sin
which acts as
Ĥ(δ)|µ, τ〉 = (2γ3δ3ℓ2P)−1 [2(Vµ+δ,τ − Vµ−δ,τ ) (18)
×(|µ+ 2δ, τ + 2δ〉 − |µ+ 2δ, τ − 2δ〉 − |µ− 2δ, τ + 2δ〉+ |µ− 2δ, τ − 2δ〉)
+ (Vµ,τ+δ − Vµ,τ−δ)(|µ+ 4δ, τ〉 − 2(1 + 2γ2δ2)|µ, τ〉+ |µ− 4δ, τ〉)
on basis states. This operator can be ordered symmetrically, defining Ĥ
symm :=
(Ĥ(δ) +
Ĥ(δ)†), whose action is1
Ĥ(δ)symm|µ, τ〉 = (2γ3δ3ℓ2P)−1 [(Vµ+δ,τ − Vµ−δ,τ + Vµ+3δ,τ+2δ − Vµ+δ,τ+2δ)|µ+ 2δ, τ + 2δ〉
−(Vµ+δ,τ − Vµ−δ,τ + Vµ+3δ,τ−2δ − Vµ+δ,τ−2δ)|µ+ 2δ, τ − 2δ〉
−(Vµ+δ,τ − Vµ−δ,τ + Vµ−δ,τ+2δ − Vµ−3δ,τ+2δ)|µ− 2δ, τ + 2δ〉
+(Vµ+δ,τ − Vµ−δ,τ + Vµ−δ,τ−2δ − Vµ−3δ,τ−2δ)|µ− 2δ, τ − 2δ〉
(Vµ,τ+δ − Vµ,τ−δ + Vµ+4δ,τ+δ − Vµ+4δ,τ−δ)|µ+ 4δ, τ〉
−2(1 + 2γ2δ2)(Vµ,τ+δ − Vµ,τ−δ)|µ, τ〉
(Vµ,τ+δ − Vµ,τ−δ + Vµ−4δ,τ+δ − Vµ−4δ,τ−δ)|µ− 4δ, τ〉
. (19)
Transforming this operator to the triad representation obtained as coefficients of a wave
function |ψ〉 =
µ,τ ψµ,τ |µ, τ〉 in the triad eigenbasis and using the volume eigenvalues
Vµ,τ = 4π
|(p̂c)µ,τ |(p̂b)µ,τ = 2π(γℓ2P)3/2
|τ |µ ,
a difference equation
γ3/2δ3
(Ĥ(δ)symm|ψ〉)µ,τ = 2δ(
|τ + 2δ|+
|τ |) (ψµ+2δ,τ+2δ − ψµ−2δ,τ+2δ)
|τ + δ| −
|τ − δ|)
(µ+ 2δ)ψµ+4δ,τ − 2(1 + 2γ2δ2)µψµ,τ + (µ− 2δ)ψµ−4δ,τ
|τ − 2δ|+
|τ |) (ψµ−2δ,τ−2δ − ψµ+2δ,τ−2δ)
= 0 (20)
results for physical states. (For small µ the equation has to be specialized further due to
the remaining gauge freedom; see [31]. This is not relevant for our purposes.)
2.1 Relation to fixed lattices
Although there are no spatial lattices appearing in the exactly homogeneous context fol-
lowed here, the construction of the Hamiltonian constraint mimics that of the full theory.
States are then associated with spatial lattices, and holonomies refer to embedded edges
and loops. The parameter δ is the remnant of the loop size (in coordinates) used to act
with holonomies on a spatial lattice. As one can see, this parameter is important for the
resulting difference equation, determining its step-size. The above construction, using a
constant δ, can be seen as corresponding to a lattice chosen once and for all such that the
loop size is not being adjusted even while the total volume increases. As described in the
1Note that the first factor of 2 in the next-to-last line was missing in [31] and analogous places in
subsequent formulas. This turns out to be crucial for the stability analysis below. In particular, with the
corrected coefficient the quantization of the Schwarzschild interior in [31] is unstable for all values of γ.
Possible restrictions on γ, as suggested in [30] based on a difference equation with the wrong coefficient,
then do not follow easily but could be obtained from a more detailed analysis.
introduction, this ignores the possible creation of new lattice vertices and links, and can
be too rigid in certain semiclassical regimes.
To express this clearly, we now construct holonomies which are not simply along a single
edge of a certain length δ, but which are understood as holonomies along lattice links. We
keep our coordinate box of size L0 in the x-direction as well as the edge length ℓ0. If this
is a link in a uniform lattice, there are Nx = L0/ℓ0 lattice links in this direction, and a link
holonomy appears in the form
hx = exp(ℓ0c̃τ3) = exp(ℓ0cτ3/L0) = exp(cτ3/Nx) (21)
when computed along whole lattice edges. Thus, a constant coefficient 1/Nx in holonomies
corresponds to a fixed lattice whose number of vertices does not change when the volume
increases. Lattice refinements of an inhomogeneous lattice state, on the other hand, can be
mimicked by a parameterNx which depends on the phase space variables, most importantly
the triad components. If this is carried through, as we will see explicitly below, the step-size
of the resulting difference equation is not constant in the triad variables anymore.
2.2 Lattice refinements
Let us now assume that we have a lattice with N vertices in a form adapted to the
symmetry, i.e. there are Nx vertices along the x-direction (whose triad component pc gives
rise to the label τ) and N 2ϑ vertices in spherical orbits of the symmetry group (whose triad
component pb gives rise to the label µ). Thus, N = NxN 2ϑ .
Since holonomies in such a lattice setting are computed along single links, rather than
through all of space (or the whole cell of size L0), basic ones are hx = exp(ℓ
0 c̃τ3) and
hϑ = exp(ℓ
0 b̃τ2), denoting the edge lengths by ℓ
0 and keeping them independent of each
other in this anisotropic setting. Edge lengths are related to the number of vertices in
each direction by ℓx0 = L0/Nx and ℓϑ0 = 1/Nϑ. With the rescaled connection components
c = L0c̃ and b = b̃ we have basic holonomies
hx = exp(ℓ
0 cτ3) = exp(cτ3/Nx) , hϑ = exp(ℓϑ0bτ2) = exp(bτ2/Nϑ) . (22)
Using this in the Hamiltonian constraint operator then gives a difference equation whose
step-sizes are 1/NI .
So far, we only reinterpreted δ in terms of vertex numbers. We now turn our attention to
solutions to the Hamiltonian constraint which, in the full theory, usually changes the lattice
by adding new edges and vertices while triad eigenvalues increase. For larger µ and τ , the
Hamiltonian constraint thus acts on a finer lattice than for small values, and the parameter
N for holonomies appearing in the constraint operator is not constant on phase space but
triad dependent. Due to the irregular nature of lattices with newly created vertices such a
refinement procedure is difficult to construct explicitly. But it is already insightful to use
an effective implementation, using the derivation of the Hamiltonian constraint for a fixed
lattice, but assuming the vertex number N (µ, τ) to be phase space dependent. Moreover,
we include a parameter δ as before, which now takes a value 0 < δ < 1 and arises because
a graph changing Hamiltonian does not use whole lattice edges but only a fraction, given
by δ.2 Effectively assuming in this way that the lattice size is growing through the basic
action of the Hamiltonian constraint, we will obtain a difference equation whose step-size
δ/N is not constant in the original triad variables.
For the Schwarzschild interior, we have step sizes δ/Nϑ for µ and δ/Nx for τ . Go-
ing through the same procedure as before, we end up with an operator containing flux-
dependent holonomies instead of basic ones, e.g., Nx(µ, τ)hx = Nx(µ, τ) exp(cτ3/Nx(µ, τ))
which reduces to an Nx-independent connection component c in regimes where curvature
is small. Keeping track of all prefactors and holonomies in the commutator as well as the
closed loop, one obtains the difference equation
C+(µ, τ)
ψµ+2δNϑ(µ,τ)−1,τ+2δNx(µ,τ)−1 − ψµ−2δNϑ(µ,τ)−1,τ+2δNx(µ,τ)−1
+C0(µ, τ)
(µ+ 2δNϑ(µ, τ)−1)ψµ+4δNϑ(µ,τ)−1,τ − 2(1 + 2γ
2δ2Nϑ(µ, τ)−2)µψµ,τ
+ (µ− 2δNϑ(µ, τ)−1)ψµ−4δNϑ(µ,τ)−1,τ
+C−(µ, τ)
ψµ−2δNϑ(µ,τ)−1,τ−2δNx(µ,τ)−1 − ψµ+2δNϑ(µ,τ)−1,τ−2δNx(µ,τ)−1
= 0 . (23)
C±(µ, τ) = 2δNϑ(µ, τ)−1(
|τ ± 2δNx(µ, τ)−1|+
|τ |) (24)
C0(µ, τ) =
|τ + δNx(µ, τ)−1| −
|τ − δNx(µ, τ)−1| . (25)
(A total factor NxN 2ϑ for the number of vertices drops out because the right hand side is
zero in vacuum, but would multiply the left hand side in the presence of a matter term.)
3 Specific refinement models
For further analysis one has to make additional assumptions on how exactly the lattice
spacing is changing with changing scales µ and τ . To fix this in general, one would have
to use a full Hamiltonian constraint and determine how its action balances the creation
of new vertices with increasing volume. Instead of doing this, we will focus here on two
geometrically motivated cases. Technically simplest is a quantization where the number of
vertices in a given direction is proportional to the geometrical area of a transversal surface.
Moreover, the appearance of transversal surface areas is suggested by the action of the full
Hamiltonian constraint which, when acting with an edge holonomy, creates a new vertex
along this edge (changing NI for this direction) and changes the spin of the edge (changing
2A precise value can be determined only if a precise implementation of the symmetry for a fixed full
constraint operator is developed. Currently, both the symmetry reduction for composite operators and a
unique full constraint operator are lacking to complete this program and we have to work with δ as a free
parameter. This parameter is sometimes related to the lowest non-zero eigenvalue of the full area operator
[15, 25]. From the inhomogeneous perspective of lattice states used here, however, there is no indication
for such a relation.
the area of a transversal surface). It also agrees with [25, 34], although the motivation in
those papers, proposing to use geometrical areas rather than coordinate areas AIJ in (16),
is different.
Geometrically more intuitive is the case where the number of vertices in a given direction
is proportional to the geometrical extension of this direction.3 The resulting difference
equation will be more difficult to deal with due to its non-constant step-size, but naturally
gives rise to Misner-type variables. This case will also be seen to have improved stability
properties compared to the first one using areas. In both cases, N ∝ V is assumed, i.e.
the lattice size increases proportionally to volume. This is not necessary in general, and
we choose these two cases mainly for illustrative purposes. In fact, constant N as in [15]
and N ∝ V first used in [25] are two limiting cases from the full point of view, the first
one without creating new vertices and the second one without changing spin labels along
edges since local lattice volumes V/N remain constant. In general, both spin changes and
the creation of new vertices happen when acting with a Hamiltonian constraint operator.
Thus, one expects N ∝ V α with some 0 < α < 1 to be determined by a detailed analysis
of the full constraint and its reduction to a homogeneous model. Even assuming a certain
behavior of N (V ) without analyzing the relation to a full constraint leaves a large field to
be explored, which can give valuable consistency checks. We will not do this systematically
in this paper but rather discuss a mathematical issue that arises in any such case: initially,
one has to deal with difference equations of non-constant step-size which can be treated
either directly or by tranforming a non-equidistant difference equation to an equidistant
one. We first illustrate this for ordinary difference equations since partial ones, as they
arise in anisotropic models, can often be reduced to this case.
3.1 Ordinary difference equations of varying step-size
Let us assume that we have an ordinary difference equation for a function ψµ, which appears
in the equation with µ-dependent increments ψµ+δN1(µ)−1 . To transform this to a fixed step-
size, we introduce a new variable µ̃(µ) such that µ̃(µ+δ/N1(µ)) = µ̃(µ)+δµ̃′/N1(µ)+O(δ2)
has a constant linear term in δ. (For the isotropic equation, N1 is the vertex number only in
one direction. The total number of vertices in a 3-dimensional lattice is given by N = N 31 .)
This is obviously satisfied if we choose µ̃(µ) :=
∫ µN1(ν)dν. We then have
ψµ+δ/N1(µ) = ψ̃µ̃(µ+δ/N1(µ)) = ψ̃µ̃+δ+
δiN (i−1)1 /N i1
= ψ̃µ̃+δ +
ψ̃′ +O(δ3) (27)
3This behavior is introduced independently in [35] where “effective” equations, obtained by replacing
connection components in the classical constraint by sines and cosines of such components according to
how they occur in the quantized constraint, are analyzed for the Schwarzschild interior. The results are
complementary to and compatible with our stability analysis of the corresponding difference equations
below. We thank Kevin Vandersloot for discussions on this issue.
where N (i)1 denotes the i-th derivative of N1. Thus, up to terms of order at least δ2 the new
equation will be of constant step-size for the function ψ̃µ̃ := ψµ(µ̃). (The derivative ψ̃
µ̃ may not be defined for any solution to the difference equation. We write it in this form
since such terms will be discussed below in the context of a continuum or semiclassical
limit where derivatives would exist.)
It is easy to see that, for refining lattices, the additional terms containing derivatives of
the wave function are of higher order in ~ and thus correspond to quantum corrections. For
N1(µ) ∝ µq as a positive power of µ, which is the expected case from lattice refinements
related to the increase in volume, we have
µN1(µ)
4πγℓ2P
relating µ to an isotropic triad component p = 4πγℓ2Pµ/3 as it occurs in isotropic loop
quantum gravity [14]. Moreover,
ψ̃′ =
N1(µ)
= − i
N1(µ)
in terms of a curvature operator ĉ = 8πiγG~/3d/dp = 2id/dµ which exists in a continuum
limit [17]. Thus,
ψ̃′ ∝
)1+2q
With q positive (or just larger than −1/2) for a refining lattice, there is a positive power
of ~, showing that additional terms arising in the transformation are quantum corrections.
This has two important implications. First, it shows that the correct classical limit
is obtained if lattices are indeed refined, rather than coarsened, since q is restricted for
corrections to appear in positive powers of ~. In anisotropic models, as we will see, the
behavior is more complicated due to the presence of several independent variables. An
analysis of the semiclassical limit can then put strong restrictions on the behavior of lat-
tices. Secondly, we can implicitly define a factor ordering of the original constraint giving
rise to the non-equidistant difference equation by declaring that all quantum correction
terms arising in the transformation above should cancel out with factor ordering terms.
We then obtain a strictly equidistant equation in the new variable µ̃. For example, a
function N1(µ) =
|µ| gives µ̃ ∝ |µ|3/2 such that the transformed difference equation
will be equidistant in volume rather than the densitized triad component. For this special
case, factor orderings giving rise to a precisely equidistant difference equation have been
constructed explicitly in [25, 34].
3.2 Number of vertices proportional to transversal area
A simple difference equation results if the number of vertices is proportional to the transver-
sal area in any direction.4 In the x-direction we have transversal surfaces given by symmetry
4Since this refers to the area, it is the case which agrees with the motivation of [25, 34].
orbits of area pc, using the line element (3), and thus Nx ∝ τ . Transversal surfaces for
an angular direction are spanned by the x- and one angular direction whose area is pb,
giving Nϑ ∝ µ. Each minisuperspace direction has a step-size which is not constant but
independent of the other dimension. Moreover, due to the simple form one can transform
the equation to constant step-size by using independent variables τ 2 and µ2 instead of τ
and µ. Illustrating the general procedure given before, a function ψ̃τ2,µ2 acquires constant
shifts under the basic steps,
ψ̃(τ+nδ/τ)2 ,(µ+mδ/µ)2 = ψ̃τ2+2nδ+n2δ2/τ2,µ2+2mδ+m2δ2/µ2
= ψ̃τ2+2nδ,µ2+2mδ +O(τ
−2) +O(µ−2)
up to terms which can be ignored for large τ and µ. This is sufficient for a straightforward
analysis in asymptotic regimes. Moreover, higher order terms in the above equation come
with higher derivatives of the wave function in the form
γ2ℓ4P
ψ̃′ = −i(γℓ
since q = 1 compared to the discussion in Sec. 3.1. Due to the extra factors of ~ (or even
higher powers in further terms in the Taylor expansion) any additional term adding to the
constant shift of ψ̃τ2,µ2 can be attributed to quantum corrections in a semiclassical limit.
Accordingly, such terms can be avoided altogether by a judicious choice of the initial factor
ordering of operators.
3.3 Number of vertices proportional to extension
Geometrically more intuitive, and as we will see below dynamically more stable, is the
case in which the number of vertices in each direction is proportional to the extension of
that direction measured with the triad itself. This gives Nϑ ∝
|τ | and Nx ∝ µ/
|τ |,
using the classical co-triad (4). (One need not worry about the inverse τ since the effective
treatment of lattice refinements pursued here is not valid close to a classical singularity
where an already small lattice with a few vertices changes. Singularities in general can
only be discussed by a direct analysis of the resulting difference operators. Since only
a few recurrence steps are necessary to probe the scheme around a classical singularity,
equidistant difference operators are not essential in this regime. They are more useful in
semiclassical regimes where one aims to probe long evolution times as in the examples
below. Similar remarks apply to the horizon at µ = 0 which, although a classical region
for large mass parameters, presents a boundary to the homogeneous model used for the
Schwarzschild interior.) The behavior is thus more complicated than in the first case since
the step size of any of the two independent variables depends on the other variable, too.
First, it is easy to see, as before with quadratic variables, that the volume label ω = µ
changes (approximately) equidistantly with each iteration step which is not equidistant for
the basic variables µ and τ . But it is impossible to find a second, independent quantity
which does so, too. In fact, such a quantity f(µ, τ) would have to solve two partial
differential equations in order to ensure that
f(µ+nδNϑ(µ, τ)−1, τ+mδNx(µ, τ)−1) ∼ f(µ, τ)+nδNϑ(µ, τ)−1∂µf(µ, τ)+mδNx(µ, τ)−1∂τf(µ, τ)
changes only by a constant independent of τ and µ. This implies ∂µf(µ, τ) ∝
|τ | and
∂τf(µ, τ) ∝ µ/
|τ | whose only solution is f(µ, τ) ∝ µ
|τ | which is the volume ω.
We thus have to deal with non-equidistant partial difference equations in this case which
in general can be complicated. A possible procedure to avoid this is to split the iteration
in two steps since an ordinary difference equation can always be made equidistant as
above (cancelling quantum corrections by re-ordering). We first transform τ to the volume
variable ω which gives, up to quantum corrections, constant iteration steps for this variable.
With the second variable still present, a higher order difference equation
C0(µ, ω
2/µ2)(1 + 2δ/ω)µψµ(1+4δ/ω),ω+4δ + C+(µ, ω
2/µ2)ψµ(1+2δ/ω),ω+3δ
−C−(µ, ω2/µ2)ψµ(1+2δ/ω),ω+δ − 2C0(µ, ω2/µ2)(1 + 2γ2δ2µ2/ω2)µψµ,ω
−C+(µ, ω2/µ2)ψµ(1−2δ/ω),ω−δ + C−(µ, ω2/µ2)ψµ(1−2δ/ω),ω−3δ
+C0(µ, ω
2/µ2)(1− 2δ/ω)µψµ(1−4δ/ω),ω−4δ = 0 (28)
results with
C0(µ, ω
2/µ2) =
C±(µ, ω
2/µ2) = 2δ
1± 2δ
derived from the original coefficients (24). The structure of this difference equation is quite
different from the original one: not only is it of higher order, but now only one value of the
wave function appears at each level of ω, rather than combinations of values at different
values of µ. Note also that only the coefficient of the unshifted ψµ,ω depends on µ. This
form of the difference equation is, however, a consequence of the additional rotational
symmetry and is not realized in this form for fully anisotropic Bianchi models as we will
see below.
Proceeding with this specific case, we have to look at wave functions evaluated at
shifted positions µ(1 + mδ/ω) with integer m. At fixed ω = ω0, we are thus evaluating
the wave function at values of µ multiplied with a constant, instead of being shifted by a
constant as in an equidistant difference equation. This suggests to use the logarithm of
µ instead of µ itself as an independent variable, which is indeed the result of the general
procedure. After having transformed from τ to ω already, we have to use τ as a function
of µ and ω in the vertex number Nϑ, which is τ(µ, ω) = (ω/µ)2 after using ω = µ
Thus, Nϑ(µ, τ(µ, ω)) =
τ(µ, ω) = ω/µ now is not a positive power of the independent
variable µ and we will have to be more careful in the interpretation of correction terms
after performing the transformation. (The lattice is coarsened with increasing anisotropy
at constant volume.) Naively applying the results of Sec. 3.1 to q = −1 would suggest that
corrections come with inverse powers of ~ which would certainly be damaging for the correct
classical limit. However, the factors change due to the presence of the additional variable
ω0 even though it is treated as a constant. We have N ′ϑ/N 2ϑ = −1/ω0 = −(γℓ2P/2)3/2/V0
in terms of the dimensionful volume V , while it would just be a constant −1 without the
presence of ω. The additional factor of ~3/2 ensures that corrections come with positive
powers of ~ for the correct classical limit to be realized.
For any ω0, we thus transform ψ̃µ(1+mδ/ω0) to equidistant form by using
ψµ̃ = ψ̃µ(µ̃) with
µ̃(µ) = log µ. This transformation is possible since the second label ω0 is now treated as a
constant, rather than an independent variable of a partial difference equation. (Recall that
for the type of difference equation discussed here there is only one variable, the volume,
which is equidistant under all of the original discrete steps.) Despite of negative powers of
some variables in the vertex numbers, we have the correct classical limit in the presence
of ω. As before, the transformation is exact up to higher order terms which are quantum
and higher order curvature corrections. Defining the original constraint operator ordering
implicitly by the requirement that all those terms are cancelled allows us to work with an
equidistant difference equation.
3.4 Bianchi models
As mentioned before, the transformed difference equation does not become higher order
for fully anisotropic Bianchi models. In this case, we have three independent flux labels
µI , I = 1, 2, 3, and vertex numbers NI . Using vertex numbers proportional to the spatial
extensions for each direction gives N1 =
µ2µ3/µ1, N2 =
µ1µ3/µ2 and N3 =
µ1µ2/µ3.
As in the difference equation for the Schwarzschild interior, the difference equation for
Bianchi models [22] uses values of the wave function of the form ψµ1+2δ/N1,µ2+2δ/N2,µ3 . One
can again see easily that the volume ω =
|µ1µ2µ3| behaves equidistantly under the
increments,
ω(µ1 + 2δ/N1, µ2 + 2δ/N2, µ3) =
µ1 + 2δ
µ2 + 2δ
µ1µ2µ3 + 4δ
µ1µ2µ3 + 4δ2 = ω + 2δ +O(δ
The leading order term of the difference equation in ω results from a combination
C1ψµ1,µ2+2δ/N2,µ3+2δ/N3 + C2ψµ1+2δ/N1,µ2,µ3+2δ/N3 + C3ψµ1+2δ/N1,µ2+2δ/N2,µ3
≈ C1ψ̃µ1,µ2+2δ/N2,ω+2δ + C2ψ̃µ1+2δ/N1,µ2,ω+2δ + C3ψ̃µ1+2δ/N1,µ2+2δ/N2,ω+2δ
= C1ψ̃µ1,µ2(1+2δ/ω),ω+2δ + C2ψ̃µ1(1+2δ/ω),µ2 ,ω+2δ + C3ψ̃µ1(1+2δ/ω),µ2(1+2δ/ω),ω+2δ
=: Ĉ+ψ̃ω+2δ(µ1, µ2)
where we used 1/N1 =
µ1/µ2µ3 = µ1/ω and defined the operator Ĉ+ acting on the
dependence of ψ on µ1 and µ2. Thus, unlike for the Schwarzschild interior the difference
equation does not become higher order in ω, and the highest order term does have a
difference operator coefficient in the remaining independent variables.
The recurrence proceeds as follows: We have a partial difference equation of the form
Ĉ+ψ̃ω+2δ(µ1, µ2) + Ĉ0ψ̃ω(µ1, µ2) + Ĉ−ψ̃ω−2δ(µ1, µ2)
with difference operators Ĉ± and Ĉ0 acting on the dependence on µ1 and µ2. In terms of
initial data at two slices of ω we can solve recursively for Ĉ0ψ̃ω(µ1, µ2)+Ĉ−ψ̃ω−2δ(µ1, µ2) =:
φ(µ1, µ2) and then, in each ω-step, use boundary conditions to solve the ordinary difference
equation
Ĉ+ψ̃ω+2δ(µ1, µ2) = φ(µ1, µ2) .
Although the operator Ĉ+ itself is not equidistant, this remaining ordinary difference
equation can be transformed to an equidistant one by transforming µ1 and µ2 as in
Sec. 3.1 (using that ω is constant and fixed for this equation at any recursion step). With
µ3(µ1, µ2, ω) = ω
2/µ1µ2, we have lattice spacings N1(µ1, µ2, ω) = ω/µ1 and N2(µ1, µ2, ω) =
ω/µ2 in terms of ω which are already independent of each other. The two remaining vari-
ables µ1 and µ2 are thus transformed to equidistant ones by taking their logarithms as
encountered before.
Note the resemblance of the new variables, volume and two logarithms a metric compo-
nents at constant volume, to Misner variables [36]. This observation may be of interest in
comparisons with Wheeler–DeWitt quantizations where Misner variables have often been
used, making the Wheeler–DeWitt equation hyperbolic.
4 Application: Stability of the Schwarzschild interior
Now that we have several possibilities for the lattice spacings, we consider their effect on
the solutions of the Hamiltonian constraint. In particular, these solutions may have un-
desirable properties reminiscent of numerical instabilities, as it was indeed noticed for the
original quantization of the Schwarzschild interior in [28]. Also problems in the presence
of a positive cosmological constant, described in the introduction, are of this type. Re-
call that when one wishes to solve an ordinary differential equation, for example, there
are various discrete schemes that ensure errors do not propagate as the number of time
steps increases. Here we are in the opposite situation – instead of having the freedom to
pick the discrete version of a continuous equation, the discrete equation itself is what is
fundamental. Thus, like a badly chosen numerical recipe, some choices of the functions Nτ
and Nϑ in the constraint equation may quickly lead to solutions that are out of control,
and increase without bound. To test for this, we will use a von Neumann stability analy-
sis [28] on the possible recursion relations. The essential idea is to treat one of the relation
parameters as an evolution parameter, and decompose the rest in terms of orthogonal
functions, representing “spatial” modes of the solution. This will give rise to a matrix that
defines the evolution of the solution; if the matrix eigenvalues are greater than unity for
a particular mode, that mode is unstable. In particular, a relation
k=−M an+kψn+k = 0
is equivalent to a vector equation of the form ~vn = Q(n)~vn−1, where the column vector
~vn = (ψn+M , ψn+M−1, · · · , ψn−M+1)T . The evolution of an eigenvector ~w of the matrix Q(n)
is given by ~wn = λw ~wn−1. Thus, when the size of the corresponding eigenvalue |λw| > 1,
the values in the sequence associated to ~w will grow as well.
With this in mind, we consider the choices of Nx and Nϑ discussed previously, starting
with the case Nx = τ and Nϑ = µ. In the large µ, τ limit for this choice, the coefficients of
the Hamiltonian constraint become
C±(µ, τ) ∼
, C0(µ, τ) ∼
τ 3/2
In the asymptotic limit, the coefficients of the ψµ±2δ/µ,τ and ψµ,τ terms go to C0(µ, τ)µ.
As we saw in Section 3.2, we can choose a different set of variables in which the step sizes
are constant (up to ordering of the operators). Plugging these asymptotic values into the
Hamiltonian constraint, and changing variables to µ̃ = µ2/2 and τ̃ = τ 2/2 gives
4τ̃(ψµ̃+2δ,τ̃+2δ − ψµ̃−2δ,τ̃+2δ + ψµ̃−2δ,τ̃−2δ − ψµ̃+2δ,τ̃−2δ) + µ̃(ψµ̃+4δ,τ̃ − 2ψµ̃,τ̃ + ψµ̃−4δ,τ̃) = 0.
Because all the step sizes now are constants depending on δ, we define new parameters
m,n such that µ̃ = 2mδ and τ̃ = 2nδ. Using m as our evolution parameter and n as the
“spatial” direction, we decompose the sequence as ψ2mδ,2nδ = um exp(inω). With this new
function, the recursion relation is written as
2in(un+1 − un−1)− (m sin θ)un = 0.
This is equivalent to the vector equation
[ − im
sin θ 1
= Q(m,n)
. (31)
The eigenvalues of the matrix Q are
−im sin θ ±
16n2 −m2 sin2 θ
When the discriminant 16n2 − m2 sin2 θ ≥ 0, then |λ| = 1, and the solution is stable;
however, there are unstable modes when 16n2 − m2 sin2 θ < 0. The most unstable mode
corresponds to the choice sin θ = 1, giving instabilities in terms of the original variables
when µ > 2τ . In this regime, all solutions behave exponentially rather than oscillating.
This region includes parts of the classical solutions for the Schwarzschild interior even for
values of µ and τ for which one expects classical behavior to be valid. The presence of
instabilities implies, irrespective of the physical inner product, that quantum solutions
in those regions cannot be wave packets following the classical trajectory, and the correct
classical limit is not guaranteed for this quantization, which is analogous to that introduced
in [25, 34].
The situation is different when we consider the choices Nx =
|τ | and Nϑ = µ/
|τ |,
where we will find a lack of instability. There is no choice of variables that allows us to
asymptotically approach a constant spacing recursion relation, because of the mixing of
the µ and τ variables in the step size functions. Thus, we will make the assumption that
in the large µ, τ limit, the solution does not change much under step sizes δN−1x and δN
To see how this affects the resulting stability of the solutions, we will look at a simpler
example first. If we start with the Fibonacci relation Rτ ≡ ψτ+1 − ψτ − ψτ−1 = 0, then
the two independent solutions are of the form ψτ = κ
τ , where κ is either the golden ratio
φ = (1 +
5)/2 or else −φ−1. Only the latter solution meets the criterion for stability,
since |φ| > 1. When we change this relation to
R̃τ ≡ ψτ+1/τn − ψτ − ψτ−1/τn = 0, (32)
with n 6= 1, the situation changes – only one of the two solutions outlined above will solve
the relation asymptotically. In particular, when we examine the error R̃τ we get when we
plug κτ into the altered relation (32), i.e.
R̃τ = κ
τ (κ1/τ
n − 1− κ−1/τn),
the error is proportional to ψτ itself. As τ → ∞, therefore, the error for the κ = φ solution
grows without bound, while that of κ = −φ−1 goes to zero. Thus, we see in this situation
a relation between the stability and the asymptotic behavior of a solution.
Returning to the Schwarzschild relation, in the large µ, τ limit the coefficient functions
of the recursion relation are to leading order
C±(µ, τ) ∼ 4δ, C0(µ, τ) ∼
In turn, the relation itself becomes
4(ψµ+2δ/
τ ,τ+2δ
τ/µ − ψµ−2δ/√τ ,τ+2δ√τ/µ − ψµ+2δ/√τ ,τ−2δ√τ/µ + ψµ−2δ/√τ ,τ−2δ√τ/µ)
+(ψµ+4δ/
τ ,τ − 2ψµ,τ + ψµ−4δ/√τ ,τ ) = 0.
From this point on, we assume that we have a solution to this relation which does not vary
greatly when, for example, µ is changed by ±2δ/√µ, and similarly for τ . Both Nx and Nϑ
are constant to first order in shifts µ± 2δN−1x and similarly for τ , in the asymptotic limit.
Thus, we assume that α = 2δN−1x and β = 2δN
ϑ are constants, and use the scalings
µ = αm and τ = βn. When this is done, we get an equation similar to the case when
Nx = τ and Nϑ = µ, but with constant coefficients; this is the crucial difference that allows
stable solutions to the case here. Using the decomposition ψαm,βn = un exp(imθ), we arrive
at the matrix equation
[ − i
sin θ 1
. (33)
The matrix here has eigenvalues λ with |λ| = 1 for all m,n, so the solution is stable. Using
arguments as in the Fibonacci example, the non-equidistant equation of the second scheme
is shown to be stable.
5 Conclusions
Following [9], we explicitly introduced loop quantum cosmological models which take into
account the full lattice structure of inhomogeneous states. Such lattices are in general
refined by adding new vertices when acting with the Hamiltonian constraint. Thus, also
dynamical equations even in homogeneous models should respect this property. Several
interesting features arose: One obtains non-equidistant difference equations which, when
imposed for functions on the whole real line as in isotropic loop quantum cosmology,
are more restrictive than equidistant ones due to the absence of superselected sectors.
This leaves the singularity issue unchanged since for this one only needs to consider a
few steps in the equation. But a stability analysis of solutions and the verification of the
correct classical limit in all semiclassical regimes can be more challenging. We presented an
example for such an analysis, but also introduced a procedure by which one can transform
the resulting equations to equidistant ones up to quantum corrections, which is sufficient
for a semiclassical analysis. Interestingly, properties of the transformation itself provide
hints to the correct semiclassical behavior. As a side-result, we demonstrated that one
particular version of lattice refinements naturally gives rise to Misner-type variables.
It is our understanding that this general procedure of defining lattice refining models
mostly agrees with the intuition used specifically in isotropic models in [25], and adapted
to anisotropic ones in [34].5 However, there are some departures from what is assumed in
[25]. First, we do not see indications to refer to the area operator while the area spectrum
was not only used in [25] to fix the constant δ and the volume dependence of the step size
but in fact provided the main motivation. Secondly, due to this motivation [25] presents
a more narrow focus which from our viewpoint corresponds to only one single refinement
model. It has a vertex number proportional to volume, which is a limiting case not realized
by known full Hamiltonian constraints, and puts special emphasis on geometrical areas to
determine the vertex number. Finally, commutators for inverse volume operators are to be
treated differently from [25], taking into account a lattice refining model which would not
be possible in a purely homogeneous formulation. As shown in the appendix, this enlarges
expected quantum corrections to the classical functions.
We have discussed similar cases for illustration here, but keep a more general view-
point on the refinement as a function of volume. A preliminary stability analysis for
the Schwarzschild interior, consistent with [35] indeed suggests that a behavior different
from what is suggested in [25] is preferred, which indicates that models can provide tight
conditions for the general analysis of quantum dynamics. We emphasize that stability
arguments as used here are independent of physical inner product issues since they refer
to properties of general solutions. A general analysis as started here allows detailed tests
of the full dynamics in manageable settings, which can verify the self-consistency of the
framework of loop quantum gravity — or possibly point to limitations which need to be
better understood.
5We thank A. Ashtekar for discussions of this point.
Acknowledgements
We thank Kevin Vandersloot for discussions. This work was supported in part by NSF
grant PHY0554771. GK is grateful for research support from the University of Mas-
sachusetts and Glaser Trust of New York.
A Inverse volume terms in homogeneous models and
lattice refinement
We have seen in this paper, following [9], that Hamiltonian constraint operators with triad-
dependent parameters in holonomies allow one to model lattice refinements faithfully, with
interesting results and some improvements over the original non-refining models. However,
as always there are also some features of inhomogeneous states and operators which are
not present and difficult to mimic in homogeneous models. Thus, even models generalized
in this way by allowing for lattice refinement effects have to be interpreted with great care.
While qualitative effects can be investigated fruitfully to test the full framework, there is
no basis for drawing quantitative conclusions. The prime example is that of commutator
terms which appear ubiquitously in composite operators of loop quantum gravity, such as
the coefficients of difference equations or also matter Hamiltonians.
In the main construction of this paper we used holonomies associated with links of a lat-
tice, rather than edges of a fixed coordinate length. This allows us, effectively, to take into
account lattice refinements which change the number of vertices. It applies to holonomies
(16) along a closed loop used to quantize curvature components which determine the step
size of difference equations, but also to the link holonomies used in commutators to quan-
tize inverse triad components based on (14). What is not modeled in homogeneous models
is the fact that a lattice operator makes use of the local volume V̂v at a given vertex v
where the commutator is acting, rather than the total volume V̂ =
v V̂v of the whole
box in which the lattice is embedded. In a fully inhomogeneous setting the difference
does not matter since volume contributions from vertices not touched by the edge used in
a commutator drop out in the end, [he, V̂ ] =
v∈e[he, V̂v]. But in homogeneous models
there is a difference since volume contributions from different vertices, in an exactly ho-
mogeneous setting, are all identical. Thus, the total volume V = NVv is the number of
vertices multiplied with the local volume Vv. Then, [h, V̂v] rather than [h, V̂ ] is expected as
the contribution to constraint operators from the inhomogeneous perspective. In homoge-
neous models as in [25], on the other hand, [h, V̂ ] is more straightforward to use. We now
show that without corrections this would imply crucial deviations from the inhomogeneous
behavior.
It is easy to see that commutators differ depending on whether the local or total volume
is used. For simplicity of the argument, we proceed with an isotropic situation where
V = |p|3/2 in terms of the basic isotropic densitized triad component p. A local lattice flux,
for a surface S intersecting only a single link, would be ρ =
d2yp̃ = ℓ20p/L
0 = p/N 2/3
for links of coordinate length ℓ0, such that Vv = |ρ|3/2 = |p|3/2/N is the local volume.
(We again use a coordinate box of size L20 and introduce rescaled flux variables p = L
0p̃.)
Isotropic states are spanned by eiµc/2 where µ ∈ R is related to the triad eigenvalues by
pµ = 4πγℓ
Pµ/3. Using a link holonomy h ∼ eiℓ0c̃/2 = eic/2N
which as a multiplication
operator increases µ by 1/N 1/3, a commutator with the total volume will have eigenvalues
of the form
h−1[h, V̂ ] ∼ V (µ+1/N 1/3)−V (µ−1/N 1/3) = |p+4πγℓ2P/3N 1/3|3/2−|p−4πγℓ2P/3N 1/3|3/2 .
If the local volume is used, on the other hand, we have to refer to local edge labels µe =
3ρ/4πγℓ2P, rather than using the total p. Thus,
h−1e [he, V̂v] ∼ Vv(µe + 1)− Vv(µe − 1) = |ρ+ 4πγℓ2P/3|3/2 − |ρ− 4πγℓ2P/3|3/2
= N−1(|p+ 4πγℓ2PN 2/3/3|3/2 − |p− 4πγℓ2PN 2/3/3|3/2) . (35)
For large volume, p ≫ N , both expressions give the correct classical limit 3
N−1/3
expected from {eic/2N1/3 , |p|3/2}. However, quantum corrections, i.e. deviations from this
classical limit for finite p, are much larger for the second version using the local volume as
it would occur in an inhomogeneous quantization. The smooth classical function dV/dp in
a Poisson bracket appears in discretized form by the large step-size N2/3 in (35) rather than
the small one N−1/3 in (34). Perturbative corrections, derived by Taylor expanding the
difference terms and keeping higher order corrections to the classical expression, are thus
larger. (This can have cosmological implications [37, 38, 39].) Non-perturbative effects as
observed for the inverse scale factor operator in isotropic loop quantum cosmology which
has an upper bound at finite volume [40], start to arise for p ∼ N 2/3 when the local
volume is used but only at the much smaller p ∼ N−1/3 for the total volume. Since only
the local volume is relevant for inhomogeneous quantizations, quantum corrections from
inverse volume operators can be large.
Unfortunately, this effect is more difficult to mimic in exact homogeneous models unlike
the behavior of holonomies under lattice refinements and has therefore been overlooked in
[25]. The connection components appearing in holonomies can simply be divided by a
function N of triad components to implement shrinking edges due to subdivision. This is
not possible for the volume itself to use a local version in a homogeneous model since, if
we would divide the total volume by the appropriate function of triad components, only
a constant would remain for an N proportional to volume and the commutator would be
zero. The only way to have this effect faithfully implemented in a homogeneous model is
to use higher SU(2) representations for holonomies in commutators but not for holonomies
used in the loop to quantize curvature components. (This is not possible if one writes the
constraint as a single trace, tr(hαhe[h
e , V̂ ]) but can easily be done using the equivalent
form tr(τihα) tr(τihe[h
e , V̂ ]). We emphasize that higher representations for commutators
are advocated here only in exactly homogeneous models to mimic inhomogeneous effects.
Fully inhomogeneous operators usually need not refer to higher representations.) In a rep-
resentation of spin j, matrix elements of holonomies contain exponentials exp(imc/2N 1/3)
with −j ≤ m ≤ j, which increases the shifts in volume labels resulting from commutators.
Resulting expressions for commutators can be found in [41, 42]. If the representation label
j is of the order N , effects as they result from lattice refinements and using the local vol-
ume are correctly implemented. Accordingly, corrections from inverse triad components
quantized through commutators are much larger than they would otherwise be.6
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	Introduction
	Difference equation for the Schwarzschild interior with varying discreteness scale
	Relation to fixed lattices
	Lattice refinements
	Specific refinement models
	Ordinary difference equations of varying step-size
	Number of vertices proportional to transversal area
	Number of vertices proportional to extension
	Bianchi models
	Application: Stability of the Schwarzschild interior
	Conclusions
	Inverse volume terms in homogeneous models and lattice refinement
