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Symmetry, Integrability and Geometry: Methods and ApplicationsSIGMA 8 (2012), 052, 31 pages
Discrete Gravity Models and Loop Quantum Gravity:
a Short Review?
Ma¨ ıt´ e DUPUIS†, James P. RYAN‡and Simone SPEZIALE§
†Institute for Theoretical Physics III, University of Erlangen-N¨ urnberg, Erlangen, Germany
E-mail: dupuis@theorie3.physik.uni-erlangen.de
‡MPI f¨ ur Gravitationsphysik, Am M¨ uhlenberg 1, D-14476 Potsdam, Germany
E-mail: james.ryan@aei.mpg.de
§Centre de Physique Th´ eorique, CNRS-UMR 7332, Luminy Case 907, 13288 Marseille, France
E-mail: simone.speziale@cpt.univ-mrs.fr
Received April 25, 2012, in final form August 06, 2012; Published online August 13, 2012
http://dx.doi.org/10.3842/SIGMA.2012.052
Abstract. We review the relation between Loop Quantum Gravity on a fixed graph and
discrete models of gravity. We compare Regge and twisted geometries, and discuss discrete
actions based on twisted geometries and on the discretization of the Plebanski action. We
discuss the role of discrete geometries in the spin foam formalism, with particular attention
to the definition of the simplicity constraints.
Key words: Loop Quantum Gravity; discrete gravity; Regge calculus; simplicity constraints;
twisted geometries
2010 Mathematics Subject Classification: 83C27; 83C45
1 Introduction
The success of lattice gauge theories suggests that a discrete formulation of general relativity can
play a major role in understanding the quantum theory. A discretized path integral is indeed
the starting point of approaches to quantum gravity such as quantum Regge calculus [80] and
(causal) dynamical triangulations [10]. In both cases, general relativity is discretized using Regge
calculus [102]. A useful alternative is to consider discrete actions based on connection variables.
This has been considered in the literature [46, 90], and it is one of the main rationales behind
the construction of spin foam models [100]. It requires a suitable discretization of the connection
variables, and in particular of the simplicity constraints needed to single out the metric degrees
of freedom. The action of general relativity based on connection variables allows a reformulation
of general relativity as a topological theory plus so-called “simplicity constraints”, which play an
essential role. Our first goal is to review the various discretizations of the simplicity constraints
which appeared in the literature.
One advantage of a discrete path integral based on connection variables is the possibility of
interpreting its boundary states as the spin network states of Loop Quantum Gravity (LQG).
This brings to the foreground the question of finding a discrete geometric interpretation for spin
networks, a program started long ago by Immirzi [85], and finally solved with the introduction
of twisted geometries [73], a suitable generalization of Regge geometries. Our second goal is to
review the relation between LQG and these two different discrete geometries.
Regge geometries can be recovered from twisted geometries imposing suitable shape matching
conditions, which guarantee the continuity of the piecewise-flat metric. Such conditions are not
?This paper is a contribution to the Special Issue “Loop Quantum Gravity and Cosmology”. The full collection
is available at http://www.emis.de/journals/SIGMA/LQGC.html
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2M. Dupuis, J.P. Ryan and S. Speziale
present in the canonical formulation of LQG, although evidence exists that they are imposed
dynamically, as we review below in Section 5. It has been further argued in [53] that these condi-
tions can be naturally introduced in the canonical framework as a discretization of the secondary
simplicity constraints. We will also review this proposal and the way these discretizations are
used to define connection-variable based path integrals.
In an effort to organize the review logically rather than historically, we will focus first on the
canonical theory, and leave the path integral for a later stage. We begin in Section 2 with a brief
overview of Regge calculus, where the fundamental variable is the metric, and its discretization
furnished by the edge lengths of a triangulation of spacetime. This will allow us to appreciate the
peculiarities of working with the connection as the fundamental variable. For instance, instead
of edge lengths, one typically ends up with discretizations involving other geometric quantities,
such as areas and angles.
Next, in Section 3 we review the relation between LQG on a fixed graph and twisted geomet-
ries. LQG is a continuous theory of quantum gravity, defined as a projective limit/direct sum
over graphs. Truncating the theory to a given graph captures only a finite number of degrees of
freedom, and these in turn may be used to describe a discretization of general relativity. Indeed,
from the viewpoint of LQG, there is a priori no need to interpret this set as discrete geometries.
The usual description of the truncated Hilbert space involves in fact continuous, albeit finite,
degrees of freedom. This is the traditional interpretation of distributional holonomies and fluxes
[12, 110, 117], and more recently an alternative but analogously continuous interpretation has
been proposed in [28, 69]. On the other hand, it has been shown that the same holonomies and
fluxes describe certain discrete geometries, more general than the one used in Regge calculus,
called twisted geometries [73]. They correspond to a collection of flat polyhedra, which define
in general discontinuous piecewise flat metrics and extrinsic curvature [29, 112]. In the special
case of a triangulation, if one further imposes suitable shape-matching conditions, continuity of
the metric is ensured and a Regge geometry is recovered. Imposing analogue shape-matching
conditions on an arbitrary graph extends a notion of Regge geometry to arbitrary cellular de-
compositions. However, while the first can be described in terms of edge lengths, the latter
must be described using areas and angles. The resulting picture of a relation between spin
networks and (the quantization of) discrete geometries has proved very useful to understand the
spin foam dynamics, and found applications in different contexts such as calculations of n-point
functions [31], cosmological [30] and black hole models [27].
In this initial part, there is no mentioning of simplicity constraints. Indeed, we are dealing
with ordinary SU(2) LQG, in which the simplicity constraints are already solved at the classical,
continuum level. The constraints enter the picture if we consider a covariant version of LQG,
in which the spin network states are based on the entire Lorentz group. In the rest of Section 3
we describe this formulation and how the simplicity constraints can be discretized. Their imple-
mentation leads to a notion of covariant twisted geometry, where the polyhedra have Lorentzian
curvature among them. This material paves the way for subsequent discussions concerning the
path integral action.
In Section 4 we review the construction of [53]: One starts with the Holst action, and dis-
cretizes it in terms of holonomies and fluxes. The variables are parametrized in way motivated
by Regge calculus. The procedure allows to study the shape-matching condition as part of
discretized secondary simplicity constraints, and perform a full reduction in which the shape-
matching conditions are imposed, obtaining a definition of Regge phase space, which has been
further developed in [51, 54]. The comparison of the approaches of Sections 3 and 4 offers
a deeper understanding of the relation between the space of discrete connections and Regge
calculus, as well as a different perspective on the simplicity constraints.
Finally in Section 5 we review the role of discretized actions in constructing spin foam models.
This is a rapidly evolving research area, and we content ourselves with reviewing some of the
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Discrete Gravity Models and Loop Quantum Gravity: a Short Review3
main ideas and the different discretization schemes proposed in the literature. Emphasis is put
on the role of the primary simplicity constraints, on the use of intuition from discrete geometries
in the way they are realized in the spin foam path integral, and on the emergence of the shape
matching conditions in the large spin limit.
2Regge calculus
A discrete version of general relativity was provided by Regge in [102]. Spacetime is triangulated
using a simplicial manifold ∆ and, as fundamental metric variables, one assigns the lengths of
all the edges, ?e:
M → ∆,gµν→ ?e.
This assignment induces a piecewise-linear flat metric on ∆: each tetrahedral 3-cell is flat
along with its boundary triangles and edges. The curvature is all concentrated into the notion
of a deficit angle ?t associated to each triangle t, and represents the failure of the sum of 4-
dimensional dihedral angles at t to equal 2π:1
?
where l denotes the set of edge lengths. It emerges that all aspects of this discrete geometry
can be reconstructed from the edge lengths. An n-dimensional dihedral angle θσ
angle, at an (n − 2)-dimensional hinge t, between two (n − 1)-simplices τ1 and τ2, within an
n-simplex σ:
?t(?) = 2π −
σ∈t
θσ
t(?),(2.1)
tconstitutes the
sinθσ
t(?) =
n
n − 1
V(n−2)
t
V(n−1)
τ1
(?)V(n)
(?)V(n−1)
τ2
σ
(?)
(?)
.
where we have written this formula in terms of the volumes of the various simplices involved.
In turn, these volumes may be specified in terms of their associated Cayley matrices:
(V(n))2=(−1)n+1
2n(n!)2detC(n),C(n)=
01
0
1
?2
0
1
?2
...
...
...
...
0
1
?2
1
2
n
?2
n+1
...
?2
2n−1
...
?2
(n2−n)/2
0
,
where {l1,...,l(n2−n)/2} is the subset of edges which constitute the n-simplex in question (and,
incidentally, the Cayley matrices are symmetric). If one further specifies Cartesian coordinates
on a 4-simplex σ, a flat metric can be explicitly written for σ as:
δµν=
1
V2
?
e
∂V2
∂?2
e
?µ
e?ν
e.
Coming to the dynamics, the action principle for Regge calculus is built through the direct
discretization of the Ricci scalar in terms of deficit angles and reads:
?
1A 4-dimensional dihedral angle θσ
t corresponds to the angle, at a triangle t, between two tetrahedra within
a 4-simplex σ.
SR(?) =
t∈∆
At(?)?t(?).(2.2)
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4M. Dupuis, J.P. Ryan and S. Speziale
If a boundary is present, then one needs the discrete equivalent of the Gibbons–Hawking bounda-
ry term. This boundary term has the same form as (2.2), except that the 2π in the definition (2.1)
replaced by π. Like the continuum Einstein–Hilbert action, (2.2) is unbounded. This stems from
the following pair of inequalities: 2π[1 − n4(t)/2] < ?t< 2π, where n4(t) is the number of 4-
simplices containing t. This implies that:
2πkAtot< S < 2πAtot,
with k = 1−max
arise for special configurations.
The equations of motion are:
?
Here the sum is over triangles t sharing the edge e and αt
within t, opposite to e. The action and the corresponding Regge equations provide an approxi-
mation to general relativity that is accurate to second-order [43]. To be more precise, one assigns
initial data to the simplicial boundary such that it possesses a unique solution in the interior.
Subsequently, one compares the obtained edge lengths ?ewith those given by the appropriate
geodesics of the continuum solution. Analytic and numerical results show that the difference be-
tween the two goes like the square of the typical length, thus smoothly to zero in the continuum
limit.
An important issue in the above argument concerns the symmetries of (2.2). These have been
studied in the literature [49, 123], and are the object of a dedicated research plan by Dittrich
and her group [14, 15, 50, 51] (see also [108]). We refer the reader to the recent review [50] and
mention here only some minimal facts. The natural invariance under (active) diffeomorphisms
of general relativity is destroyed by the discretization: generically, there are no displacements
of the lengths that preserve the metric and only trivial relabelings remain. In this sense, the
edge lengths are perfect gauge-invariant observables. However, a notion of gauge invariance can
re-emerge in the form of (possibly local) isometries of the discrete metric. The typical example
is the case in which the edge lengths describe a patch of flat spacetime. In this case, the action is
invariant under bounded vertex displacements preserving the flatness. This somewhat accidental
symmetry actually plays an important role in assuring that we recover diffeomorphism invariance
in the classical continuum limit: as one increases the number of simplices, while assuming that
fixed boundary data induce a unique classical solution with typical curvature scale, one hits
a point where the average curvature is approximately zero. Thus, vertex displacements are
always a symmetry in the continuum limit.
We conclude this quick overview of Regge calculus with some remarks, which will be useful
to keep in mind while moving on.
t∈∆{n4(t)}/2 and Atot=?
t∈∆
At. However, notice that one-sided boundedness can
t∈e
?t(?)cotαt
e(?) = 0.
eis the 2-dimensional dihedral angle
2.1 Area-angle Regge calculus
Motivated by LQG and spin foams, one can consider taking the areas of the triangles as funda-
mental variables, instead of the edge lengths. This was proposed in [97, 107] and some attempts
have been pursued in the literature [24, 98, 120]. Notice that a generic triangulation has more
triangles than edges and, even when the numbers match, the same area configuration can cor-
respond to different edge sets [24], thus constraints among the areas are needed to guarantee
that a unique set of edge lengths is reconstructed. The difficulty with this idea is that the re-
quired constraints are non-local with respect to the triangulation and no general form is known.
A solution to this problem has been found using a formulation in terms of areas and angles [55].
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Discrete Gravity Models and Loop Quantum Gravity: a Short Review5
Increasing the number of fundamental variables allows one to render the needed constraints in
a local manner. They can be written explicitly and the unique reconstruction of edge lengths
proved. The constraints are of two types: the closure constraints, say Cτ, local with respect to
the tetrahedra, and the gluing (a.k.a. shape-matching) constraints, say Cσ
to each pair of edges within a simplex σ. Using suitable Lagrange multipliers, the resulting
action reads [55]:
?
ee?, local with respect
S[At,φτ
e,λτ,µσ
ee?] =
t
At?t(φ) +
?
τ
λτCτ(A,φ) +
?
σ
?
ee?∈σ
µσ
ee?Cσ
ee?(φ).
2.2 On the choice of variables
Taking the lengths as fundamental variables is very natural and due to the automatic rigidity of
the simplices: specifying the edge lengths always specifies a unique n-simplex. Furthermore, the
formulae prescribing its geometry are quite simple as one can see from the above expressions.
There are however some drawbacks with this choice that become more evident when trying
to quantize the theory. The first one is that the space of edge length configurations is much
larger than the space of piecewise-linear flat metrics. To ensure that one is really recovering
a Riemannian (or pseudo-Riemannian) metric, triangle inequalities need to be imposed. These
guarantee the positivity of (space-like, for Lorentzian signature) simplicial volumes. While this
might be simple to deal with in the classical setting, such conditions need to be additionally
imposed in a path integral formulation, making it cumbersome to handle.
A second drawback is that the geometry is very rigidly Riemannian. There is, for instance,
no room for torsion. On the other hand, a number of approaches to quantum gravity, including
LQG, permit the presence of torsion, typically sourced by fermions. Modifications of Regge
calculus to include torsion have been considered in the literature [46, 57, 83, 114].
2.3 On the quantum theory
The Regge action is taken as a starting point for both quantum Regge calculus [80, 122] and,
when restricted to the sub case when all the edge lengths are the same (the relevant variables
then become just the numbers of simplices), for (causal) dynamical triangulations [10]. Both are
path integral approaches and have obtained quite interesting results, including evidence for the
existence of a continuum limit. On the other hand, the dynamical content of such a continuum
limit is still insufficiently known, which partly motivates the search for alternative discretization
schemes. Two specific difficulties of these approaches, related to the choice of variables, are the
following. The first is to construct a Hilbert space for the boundary states of the path integral.
The second concerns the unwieldy positive-metric conditions along with the ambiguities of the
path integral measure. See e.g. [56, 82] on the measure for quantum Regge calculus.
A possible answer to the above questions is provided by the discrete geometric interpretation
of LQG on a fixed triangulation. As we show below, the alternative description provided by
LQG based on connection variables can dispense with the triangle inequalities by implementing
them automatically, it allows for torsion, it has a well-defined Hilbert space and a prescription
for the path integral measure.
3 Canonical LQG and discrete geometries
Historically, Loop Quantum Gravity and discrete Regge calculus remained somewhat detached
from each other, despite several superficial similiarities. Ultimately, spin foam models facilitated
the development of a precise link between the two – since they both provide a dynamics for the
LQG on a fixed triangulation and approximate, in the large spin limit, exponentials of the Regge
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Discrete Gravity Models and Loop Quantum Gravity: a Short Review27
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