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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