Papers in Physics, vol. 4, art. 040004 (2012)

Received: 17 October 2011, Accepted: 18 March 2012
Edited by: J. Pullin
Reviewed by: L. Freidel, Perimeter Institute for Theoretical Physics,

Waterloo, Canada
Licence: Creative Commons Attribution 3.0
DOI: 10.4279/PIP.040004

www.papersinphysics.org

ISSN 1852-4249

Invited review: The new spin foam models and quantum gravity

Alejandro Perez1∗

In this article, we give a systematic definition of the recently introduced spin foam models
for four-dimensional quantum gravity, reviewing the main results on their semiclassical
limit on fixed discretizations.

I. Introduction

The quantization of the gravitational interaction
is a major open challenge in theoretical physics.
This review presents the status of the spin foam ap-
proach to the problem. Spin foam models are defini-
tions of the path integral formulation of quantum
general relativity and are expected to be the co-
variant counterpart of the background independent
canonical quantization of general relativity known
as loop quantum gravity [1–3].

This article focuses on the definition of the
recently introduced Engle-Pereira-Rovelli-Livine
(EPRL) model [4,5] and the closely related Freidel-
Krasnov (FK) model [6]. An important original
feature of the present paper is the explicit deriva-
tion of both the Riemannian and the Lorentzian
models, in terms of a notation that exhibits the
close relationship between the two, at the algebraic
level, that might signal a possible deeper relation-
ship at the level of transition amplitudes.

We will take Plebanski’s perspective in which
general relativity is formulated as a constrained BF
theory (for a review introducing the new models
from a bottom-up perspective see Ref. [7]; for an

extended version of the present review including a
wide collection of related work see Ref. [8]). For
that reason, it will be convenient to start this re-
view by introducing the exact spin foam quantiza-
tion of BF in the following section. In Section III,
we present the EPRL model in both its Riemannian
and Lorentzian versions. A unified treatment of the
representation theory of the relevant gauge groups
is presented in that section. In Section IV, we in-
troduce the FK model and discuss its relationship
with the EPRL model. In Section V, we describe
the structure of the boundary states of these models
and emphasize the relationship with the kinemati-
cal Hilbert space of loop quantum gravity. In Sec-
tion VI, we give a compendium of important issues
(and associated references) that have been left out
but which are important for future development.
Finally, in Section VII, we present the recent en-
couraging results of the nature of the semiclassical
limit of the new models.

∗E-mail: perez@cpt.univ-mrs.fr

1 Centre de Physique Théorique, Campus de Luminy, 13288 Marseille, France. Unité Mixte de Recherche (UMR 6207)
du CNRS et des Universités Aix-Marseille I, Aix-Marseille II, et du Sud Toulon-Var; laboratoire afilié à la FRUMAM
(FR 2291).

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Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

II. Spin foam quantization of BF
theory

We will start by briefly reviewing the spin foam
quantization of BF theory. This section will be
the basic building block for the construction of the
models of quantum gravity that are dealt with in
this article. The key idea is that the quantum tran-
sition amplitudes (computed in the path integral
representation) of gravity can be obtained by suit-
ably restricting the histories that are summed over
in the spin foam representation of exactly solvable
BF theory. We describe the nature of these con-
straints at the end of this section.

Here, one follows the perspective of Ref. [9]. Let
G be a compact group whose Lie algebra g has an
invariant inner product, here denoted 〈〉, and M
a d-dimensional manifold. Classical BF theory is
defined by the action

S[B, ω] =

∫

M

〈B ∧ F(ω)〉, (1)

where B is a g valued (d−2)-form, ω is a connection
on a G principal bundle over M. The theory has no
local excitations: All the solutions of the equations
of motion are locally related by gauge transforma-
tions. More precisely, the gauge symmetries of the
action are the local G gauge transformations

δB = [B, α] , δω = dωα, (2)

where α is a g-valued 0-form, and the ‘topological’
gauge transformation

δB = dωη, δω = 0, (3)

where dω denotes the covariant exterior derivative
and η is a g-valued 0-form. The first invariance is
manifest in the form of the action, while the sec-
ond one is a consequence of the Bianchi identity,
dωF (ω) = 0. The gauge symmetries are so vast
that all the solutions to the equations of motion
are locally pure gauge. The theory has only global
or topological degrees of freedom.

For the time being, we assume M to be a com-
pact and orientable manifold. The partition func-
tion, Z, is formally given by

Z =

∫

D[B]D[ω] exp(i

∫

M

〈B ∧ F (ω)〉). (4)

Formally integrating over the B field in (4), we
obtain

Z =

∫

D[ω] δ (F (ω)) . (5)

The partition function Z corresponds to the ‘vol-
ume’ of the space of flat connections on M.

In order to give a meaning to the formal expres-
sions above, we replace the d-dimensional mani-
fold M with an arbitrary cellular decomposition ∆.
We also need the notion of the associated dual 2-
complex of ∆ denoted by ∆⋆. The dual 2-complex
∆⋆ is a combinatorial object defined by a set of ver-
tices v ∈ ∆⋆ (dual to d-cells in ∆) edges e ∈ ∆⋆

(dual to (d−1)-cells in ∆) and faces f ∈ ∆⋆ (dual
to (d−2)-cells in ∆). In the case where ∆ is a sim-
plicial decomposition of M, the structure of both
∆ and ∆⋆ is illustrated in Figs. 1, 2 and 3 in two,
three, and four dimensions, respectively.

Figure 1: On the left: A triangulation and its
dual in two dimensions. On the right: The dual
two complex; faces (shaded polygon) are dual to
0-simplices in 2d.

Figure 2: On the left: A triangulation and its
dual in three dimensions. On the right: The dual
two complex; faces (shaded wedge) are dual to 1-
simplices in 3d.

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Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

Figure 3: On the left: a triangulation and its dual
in four dimensions. On the right: the dual two com-
plex; faces (shaded wedge) are dual to triangles in
4d. The shaded triangle dual to the shaded face is
exhibited.

For simplicity, we concentrate on the case when
∆ is a triangulation. The field B is associated with
Lie algebra elements Bf assigned to faces f ∈ ∆

⋆.
We can think of it as the integral of the (d−2)-form
B on the (d−2)-cell dual to the face f ∈ ∆⋆, namely

Bf =

∫

(d−2)−cell

B. (6)

In other words, Bf can be interpreted as the
‘smearing’ of the continuous (d−2)-form B on the
(d−2)-cells in ∆. We use the one-to-one correspon-
dence between faces f ∈ ∆⋆ and (d−2)-cells in ∆ to
label the discretization of the B field Bf . The con-
nection ω is discretized by the assignment of group
elements ge ∈ G to edges e ∈ ∆

⋆. One can think of
the group elements ge as the holonomy of ω along
e ∈ ∆⋆, namely

ge = P exp(−

∫

e

ω), (7)

where the symbol “P exp ” denotes the path-order-
exponential that reminds us of the relationship of
the holonomy with the connection along the path
e ∈ ∆⋆.

With this, the discretized version of the path in-
tegral (4) is

Z(∆) =

∫

∏

e∈∆⋆

dge
∏

f∈∆⋆

dBf e
iBf Uf

=

∫

∏

e∈∆⋆

dge
∏

f∈∆⋆

δ(ge1 · · · gen ), (8)

where Uf = ge1 · · · gen denotes the holonomy
around faces, and the second equation is the result
of the B integration: It can be, thus, regarded as
the analog of (5). The integration measure dBf is
the standard Lebesgue measure, while the integra-
tion in the group variables is done in terms of the
invariant measure in G (which is the unique Haar
measure when G is compact). For given h ∈ G and
test function F (g), the invariance property reads as
follows

∫

dgF (g) =

∫

dgF (g−1) =

∫

dgF (gh)

=

∫

dgF (hg) (9)

The Peter-Weyl’s theorem provides a useful for-
mula or the Dirac delta distribution appearing in
(8), namely

δ(g) =
∑

ρ

dρTr[ρ(g)], (10)

where ρ are irreducible unitary representations of
G. From the previous expression, one obtains

Z(∆) =
∑

C:{ρ}→{f}
∫

∏

e∈∆⋆

dge
∏

f∈∆⋆

dρf Tr
[

ρf (g
1
e . . . g

N

e )
]

. (11)

Integration over the connection can be performed
as follows. In a triangulation ∆, the edges e ∈ ∆⋆

bound precisely d different faces. Therefore, the
ge’s in (11) appear in d different traces. The rele-
vant formula is

P einv(ρ1, · · · , ρd)

:=

∫

dge ρ1(ge) ⊗ ρ2(ge) ⊗ · · · ⊗ ρd(ge). (12)

For compact G, using the invariance (and normal-
ization) of the the integration measure (9), it is easy
to prove that P einv = (P

e
inv )

2 is the projector onto
Inv[ρ1 ⊗ ρ2 ⊗ · · · ⊗ ρd]. In this way, the spin foam
amplitudes of SO(4) BF theory reduce to

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Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

ZBF (∆) =
∑

Cf :{f}→ρf
∏

f∈∆⋆

dρf
∏

e∈∆⋆

P einv (ρ1, · · · , ρd). (13)

In other words, the BF amplitude associated to a
two-complex ∆⋆ is simply given by the sum over
of all possible assignments of irreducible represen-
tations of G to faces of the number obtained by
the natural contraction of the network of projectors
P einv, according to the pattern provided defined by
the two-complex ∆⋆.

There is a nice graphical representation of the
partition function of BF theory that will be very
useful for some calculations. On the one hand, us-
ing this graphical notation one can easily prove the
discretization independence of the BF amplitudes.
On the other hand, this graphical notation will sim-
plify the presentation of the new spin foam models
of quantum gravity that will be considered in the
following sections. This useful notation was intro-
duced by Oeckl [10,11] and used in Ref. [12] to give
a general proof of the discretization independence
of the BF partition function and the Turaev-Viro
invariants for their definition on general cellular de-
compositions.

We will present this notation in detail: The idea
is to represent each representation matrix appear-
ing in (11) by a line (called a wire) labeled by an
irreducible representation, and integrations on the
group by a box (called a cable). The traces in Eq.
(11) imply that there is a wire, labeled by the rep-
resentation ρf , winding around each face f ∈ ∆

⋆.
In addition, there is a cable (integration on the
group) associated with each edge e ∈ ∆⋆. As in
(13), there is a projector P einv, which is the pro-
jector in Inv[ρ1 ⊗ ρ2 ⊗ · · · ⊗ ρd] associated to each
edge. This will be represented by a cable with d
wires, as shown in (14). Such graphical represen-
tation allows for a simple diagrammatic expression
of the BF quantum amplitudes.

P einv(ρ1, ρ2, ρ3, · · · , ρd) ≡

ρ1ρ2ρ3
· · ·

ρd

(14)

The case of physical interest is d = 4. In such
case, edges are shared by four faces; each cable has

now four wires. The cable wire diagram giving the
BF amplitude is dictated by the combinatorics of
the dual two complex ∆⋆. From Fig. 3, one gets

ZBF (∆) =
∑

Cf :{f}→ρf

∏

f∈∆⋆

dρ

ρ1
ρ2

ρ3
ρ4

ρ5
ρ6

ρ7

ρ8

ρ9
ρ10

. (15)

The 10 wires corresponding to the 10 faces f ∈ ∆⋆,
sharing a vertex v ∈ ∆⋆, are connected to the neigh-
boring vertices through the 5 cables (representing
the projectors in (13) and Fig. 14) associated to
the 5 edges e ∈ ∆⋆, sharing the vertex v ∈ ∆⋆.

a. SU (2) × SU (2) BF theory: a starting point for
4d Riemannian gravity.

We now present the BF quantum amplitudes in the
case G = SU (2) × SU (2). This special case is of
fundamental importance in the construction of the
gravity models presented in the following sections.
The product form of the structure group implies
the simple relationship ZBF (SU (2) × SU (2)) =
ZBF (SU (2))

2. Nevertheless, it is important for us
to present this example in an explicit way as it will
provide the graphical notation that is needed to
introduce the gravity models in a simple manner.
The spin foam representation of the BF partition
function follows from expressing the projectors in
(15) in the orthonormal basis of intertwiners, i.e.,
invariant vectors in Inv[ρ1 ⊗ · · · ⊗ ρ4]. From the
product form of the structure group, one has

ρ1 ρ2ρ3 ρ4

=

j−1 j
−
2 j

−
3 j

−
4 j

+
1 j

+
2 j

+
3 j

+
4

=
∑

ι−ι+

j−1j
−
2 j

−
3 j

−
4

ι−

j+1j
+
2 j

+
3 j

+
4

ι+

,

(16)

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Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

where ρf = j
−
f ⊗ j

+
f , j

±
f and ι

± are half integers la-
beling left and right representations of SU (2) that
defined the irreducible unitary representations of
G = SU (2) × SU (2). We have used the expression
of the right and left SU (2) projectors in a basis of
intertwiners, namely

j1 j2 j3 j4

=
∑

ι

j1 j2 j3 j4

ι , (17)

where the four-leg objects on the right hand side
denote the invariant vectors spanning a basis of
Inv[j1 ⊗ · · · ⊗ j4], and ι is a half integer, labeling
those elements. Accordingly, when replacing the
previous expression in (15), one gets

ZBF (∆) =
∑

Cf :{f}→ρf

∏

f∈∆⋆

dj−
f

dj+
f

, (18)

and equivalently,

ZBF (∆) =
∑

Cf :{f}→ρf

∏

f∈∆⋆

dj−
f

dj+
f

∑

Ce:{e}→ιe

(19)

from which we finally obtain the spin foam repre-
sentation of the SU (2) × SU (2) partition function

as a product of two SU (2) amplitudes, namely

ZBF (∆) =
∑

Cf :{f}→ρf

∏

f∈∆⋆

dj−
f

dj+
f

∑

Ce:{e}→ιe

∏

v∈∆⋆

ι−1

ι−2

ι−3

ι−4

ι−5

j−1

j−2

j−3

j−4

j−5

j−6

j−7

j−8 j−9

j−10

ι+1

ι+2

ι+3

ι+4
ι+5

j+1

j+2

j+3

j+4

j+5

j+6

j+7

j+8
j+9

j+10

(20)

Extra remarks on four-dimensional BF theory

The state sum (11) is generically divergent (due to
the gauge freedom analogous to (3)). A regular-
ized version defined in terms of SUq(2) × SUq(2)
was introduced by Crane and Yetter [13, 14]. As in
three dimensions, if an appropriate regularization
of bubble divergences is provided, (11) is topolog-
ically invariant and the spin foam path integral is
discretization independent.

As in the three-dimensional case, BF theory can
be coupled to topological defects [15] in any di-
mension. In the four-dimensional case, defects are
string-like [16] and can carry extra degrees of free-
dom, such as topological Yang-Mills fields [17]. The
possibility that quantum gravity could be defined
directly from these simple kinds of topological the-
ories has also been considered outside spin foams
[18] (for which the UV problem described in the in-
troduction is absent). This is attractive and should,
in my view, be considered further.

It is also possible to introduce one-dimensional
particles in four-dimensional BF theory and grav-
ity, as shown in Ref. [19].

Two-dimensional BF theory has been used as the
basic theory in an attempt to define a manifold in-
dependent model of QFT in Ref. [20]. It is also
related to gravity in two dimensions in two ways:
On the one hand, it is equivalent to the so-called
Jackiw-Teitelboim model [21,22], on the other hand
it is related to usual 2d gravity via constraints in a
way similar to the one exploited in four dimensions
(see next section). The first relationship has been
used in the canonical quantization of the Jackiw-
Teitelboim model in Ref. [23]. The second rela-
tionship has been explored in Ref. [24].

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Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

Three-dimensional BF theory and the spin foam
quantization presented above are intimately related
to classical and quantum gravity in three dimen-
sions (for a classic reference see Ref. [25]). The
state sum, as presented above, matches the quan-
tum amplitudes first proposed by Ponzano and
Regge in the 60’s, based on their discovery of the
asymptotic expressions of the 6j symbols [26], of-
ten referred to as the Ponzano-Regge model. Di-
vergences in the above formal expression require
regularization. Natural regularizations are avail-
able so that the model is well-defined [27–29]. For
a detailed study of the divergence structure of
the model, see Refs. [30–32]. The quantum de-
formed version of the above amplitudes lead to
the so-called Turaev-Viro model [33], which is ex-
pected to correspond to the quantization of three-
dimensional Riemannian gravity in the presence of
a non-vanishing positive cosmological constant. For
the definition of observables in the latter context,
as well as in the analogue four-dimensional analog,
see Ref. [34].

The topological character of BF theory can be
preserved by the coupling of the theory with topo-
logical defects that play the role of point particles.
In the spin foam literature, this has been consid-
ered from the canonical perspective in Refs. [35,36]
and from the covariant perspective extensively by
Freidel and Louapre [37]. These theories have been
proved by Freidel and Livine to be dual, in a suit-
able sense, to certain non-commutative fields theo-
ries in three dimensions [38, 39].

Concerning coupling BF theory with non-
topological matter, see Refs. [40, 41] for the case of
fermionic matter, and Ref. [42] for gauge fields. A
more radical perspective for the definition of mat-
ter in 3d gravity is taken in Ref. [43]. For three-
dimensional supersymmetric BF theory models, see
Refs. [44, 45]

Recursion relations for the 6j vertex amplitudes
have been investigated in Refs. [46, 47]. They pro-
vide a tool for studying dynamics in spin foams of
3d gravity and might be useful in higher dimensions
[48].

i. The coherent states representation

In this section, we introduce the coherent state rep-
resentation of the SU (2) and Spin(4) path integral
of BF theory. This will be particularly important

for the definition of the models defined by Freidel
and Krasnov in Ref. [6] that we will address in Sec-
tion IV as well as in the semiclassical analysis of the
new models reported in Section VII. The relevance
of such representation for spin foams was first em-
phasized by Livine and Speziale in Ref. [49].

a. Coherent states

Coherent states associated with the representation
theory of a compact group have been studied by
Thiemann and collaborators [50,51,51–59], see also
Ref. [60]. Their importance for the new spin foam
models was put forward by Livine and Speziale in
Ref. [49], where the emphasis was put on coher-
ent states of intertwiners or the so-called quantum
tetrahedron (see also [61]). Here we follow the pre-
sentation of [6].

In order to build coherent states for Spin(4), we
start by introducing them in the case of SU (2).
Starting from the representation space Hj of di-
mension dj ≡ 2j + 1, one can write the resolution
of the identity in terms of the canonical orthonor-
mal basis |j, m〉 as

1j =
∑

m

|j, m〉〈j, m|, (21)

where −j ≤ m ≤ j. There exists an over complete
basis |j, g〉 ∈ Hj , labeled by g ∈ SU (2), such that

1j = dj

∫

SU(2)

dg |j, g〉〈j, g|, (22)

The states |j, g〉 ∈ Hj are SU (2) coherent states
defined by the action of the group on maximum
weight states |j, j〉 (themselves coherent), namely

|j, g〉 ≡ g|j, j〉 =
∑

m

|j, m〉D
j
mj (g), (23)

where D
j
mj (g) are the matrix elements of the uni-

tary representations in the |j, m〉 (Wigner matri-
ces). Equation (22) follows from the orthonor-
mality of unitary representation matrix elements,
namely

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Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

dj

∫

SU(2)

dg |j, g〉〈j, g|,

= dj
∑

mm′

|j, m〉〈j, m′|

∫

SU(2)

dg D
j
mj (g)D

j
m′j (g)

=
∑

m

|j, m〉〈j, m|, (24)

where in the last equality we have used the or-
thonormality of the matrix elements. The decom-
position of the identity (22) can be expressed as
an integral on the two-sphere of directions S2 =
SU (2)/U (1) by noticing that D

j
mj (g) and D

j
mj (gh)

differ only by a phase for any group element h from
a suitable U (1) ⊂ SU (2). Thus, one has

1j = dj

∫

S2
dn |j, n〉〈j, n|, (25)

where n ∈ S2 is integrated with the invariant mea-
sure of the sphere. The states |j, n〉 form (an over-
complete) basis in Hj . SU (2) coherent states have
the usual semiclassical properties. Indeed, if one
considers the generators J i of su(2), one has

〈j, n|Ĵ i|j, n〉 = j ni, (26)

where ni is the corresponding three-dimensional
unit vector for n ∈ S2. The fluctuations of Ĵ 2

are also minimal with ∆J 2 = ~2j, where we have
restored ~ for clarity. The fluctuations go to zero
in the limit ~ → 0 and j → ∞, while ~j is kept
constant. This kind of limit will be used often as
a notion of semiclassical limit in spin foams. The
state |j, n〉 is a semiclassical state describing a vec-
tor in R3 of length j and of direction n. It will
be convenient to introduce the following graphical
notation for Eq. (25)

j
= dj

∫

S2

dn

j

n (27)

Finally, an important property of SU (2) coherent
states stemming from the fact that

|j, j〉 = | 1
2
, 1

2
〉| 1

2
, 1

2
〉 · · · | 1

2
, 1

2
〉 ≡ | 1

2
, 1

2
〉⊗2j

is that

|j, n〉 = | 1
2
, n〉⊗2j . (28)

The above property will be of key importance in
constructing effective discrete actions for spin foam
models. In particular, it will play a central role in
the study of the semiclassical limit of the EPRL
and FK models studied in Sections III, and IV. In
the following subsection, we provide an example for
Spin(4) BF theory.

b. Spin(4) BF theory: Amplitudes in the coherent
state basis

Here we study the coherent states representation of
the path integral for Spin(4) BF theory. The con-
struction presented here can be extended to more
general cases. The present case is, however, of par-
ticular importance for the study of gravity models
presented in Sections III, and IV. With the intro-
duction of coherent states, one achieves the most
difficult part of the work. In order to express the
Spin(4) BF amplitude in the coherent state rep-
resentation, one simply inserts a resolution of the
identity in the form (25) on each and every wire
connecting neighboring vertices in the expression
(18) for the BF amplitudes. The result is

ZBF (∆) =
∑

Cf :{f}→ρf

∏

f∈∆⋆
dj−

f
dj+

f

∫
∏

e∈∈∆⋆ dj−
ef

dj+
ef

dn−ef dn
+
ef

n
−
1

n
+
1

n
−
2

n
+
2

n
−
3

n
+
3

n
−
4

n
+
4

,(29)

where we have explicitly written the n± ∈ S
2 in-

tegration variables only on a single cable. One ob-
serves that there is one n± ∈ S

2 per each wire
coming out at an edge e ∈ ∆⋆. As wires are in
one-to-one correspondence with faces f ∈ ∆⋆, the
integration variables n±ef ∈ S

2 are labeled by an

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Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

edge and face subindex. In order to get an expres-
sion of the BF path integral in terms of an affective
action, we restore, at this stage, the explicit group
integrations represented by the boxes in the previ-
ous equation. One gets

ZBF (∆) =
∑

Cf :{f}→ρf

∏

f∈∆⋆
dj−

f
dj+

f

∫

∏

e∈∆⋆ dj−
ef

dj+
ef

dn−ef dn
+
ef

∏

v∈∆⋆
∏

e,e′∈v

dg−ef dg
+
ef (〈n

−
ef |(g

−)−1ef g
−
e′f |n

−
e′f 〉)

2j
−

f

(〈n+ef |(g
+)−1ef g

+
e′f |n

+
e′f 〉)

2j+
f , (30)

where we have used the coherent states property
(28), and |n±〉 is a simplified notation for | 1

2
, n±〉.

The previous equation can be finally written as

ZBF (∆) =
∑

Cf :{f}→ρf

∏

f∈∆⋆
dj−

f
dj+

f

∫

∏

e∈∆⋆ dj−
ef

dj+
ef

dn−ef dn
+
ef dg

−
ef dg

+
ef

exp (Sd
j±,n±

[g±]), (31)

where the discrete action

Sdj±,n± [g
±] =

∑

v∈∆⋆

Svjv ,nv [g
±] (32)

with

Svj,n[g] =

5
∑

a<b=1

2jab ln 〈nab|g
−1
a gb| nba〉, (33)

and the indices a, b label the five edges of a given
vertex. The previous expression is equal to the form
(11) of the BF amplitude. In the case of the gravity
models studied in what follows, the coherent state
path integral representation will be the basic tool
for the study of the semiclassical limit of the models
and the relationship with Regge discrete formula-
tion of general relativity.

ii. The relationship between gravity and
BF theory

The field theory described in the present section has
no local degrees of freedom. It represents the sim-
plest example of a topological field theory in four
dimensions. The interest of this theory for gravity

models stems from the fact that an action for the
gravitational degrees of freedom (basically equiva-
lent to general relativity in the first order formu-
lation) can be obtained by supplementing a 4d BF
theory action with internal gauge group SL(2, C)
(Lorentzian) or Spin(4) (Riemannian) with the fol-
lowing set of quadratic constraints on the B-field

ǫIJKLB
IJ
µν B

KL
ρσ − e ǫµνρσ ≈ 0, (34)

where e ≡ σ2(1/4!)ǫIJKLB
IJ
µν B

KL
ρσ ǫ

µνρσ where
σ2 = ±1 according to whether one is in the Rie-
mannian or Lorentzian case. More generally, a one-
parameter family of gravity actions can be obtained
from the imposition of the previous constraints on
the following modified BF action

Sγ (B, ω) =

∫

M

〈(⋆B +
1

γ
B) ∧ F (ω)〉, (35)

where γ is the Immirzi parameter. The strategy
behind the definition of the new spin foam models
for quantum gravity consists of imposing these con-
straints on the path integral of BF theory on the
momenta J = ⋆B + 1

γ
B conjugated to ω. In order

to impose the Plebanski constraints above, it will
be convenient to express the B field in terms of the
momenta J, namely

B =
γ

1 − σ2γ2
(J − γ⋆J). (36)

The imposition of the constraints (34) on the BF
path integral on a fixed discretization can be done
in two different ways: by directly restricting the
spin foam configurations (this is the EPRL ap-
proach described in the following section), or by
restricting the semiclassical values of the B field in
the coherent state representation of the BF path in-
tegral (this is the FK strategy described in Section
IV).

III. The Engle-Pereira-Rovelli-
Livine (EPRL) model

In this section, we introduce the Engle-Pereira-
Rovelli-Livine (EPRL) model [4, 5]. The section is
organized as follows: The relevant representation
theory is introduced in Subsection i. In Subsection
ii, we present and discuss the linear simplicity con-
straints —classically equivalent to the Plebanski
constraints—and discuss their implementation in

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Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

the quantum theory. In Subsection iii, we introduce
the EPRL model of Riemannian gravity. In Subsec-
tion iv, we prove the validity of the quadratic Ple-
banski constraints—reducing BF theory to general
relativity—directly in the spin foam representation.
In Subsection v, we present the coherent state rep-
resentation of the Riemannian EPRL model. In
Subsection vi, we describe the Lorentzian model.
The material of this section will also allow us to
describe the construction of the closely related (al-
though derived from a different logic) Riemannian
FK constructed in Ref. [6]. The idea that linear
simplicity constraints are more convenient for deal-
ing with the constraints that reduce BF theory to
gravity was pointed out by Freidel and Krasnov in
this last reference.

i. Representation theory of Spin(4) and
SL(2, C) and the canonical basis

In this section, we present the representation theory
of the groups Spin(4) and SL(2, C) that is neces-
sary for the definition of the new spin foam mod-
els for Riemannian and Lorentzian gravity, respec-
tively. To emphasize the highly symmetric struc-
ture of the two, we present them in a unified no-
tation where a parameter σ = 1 for the Rieman-
nian sector and σ = i for the Lorentzian one. The
simple relationship between the two might be a
hint to a possible relationship between model am-
plitudes in a spirit similar to the interesting link
between Euclidean and Lorentzian QFT provided
by Wick rotations 1. Unitary irreducible repre-
sentations Hp,k of Spin(4) and SL(2, C) are la-
beled by two parameters, p and k. In the case of
Spin(4) = SU (2) × SU (2), the unitary irreducible
representations are finite-dimensional and the la-
bels p and k can be expressed in terms of the half
integers labeling the right and left SU (2) unitary
representations j±, as follows

p = j+ + j− + 1 k = |j+ − j−|. (37)

In the SL(2, C) case, the unitary irreducible repre-
sentations are infinite-dimensional and one has

p ∈ R+ k ∈ N/2. (38)

The two Casimirs are C1 =
1
2
JIJ J

IJ = L2 + σ2K2

and C2 =
1
2

⋆JIJ J
IJ = K ·L where Li are the gener-

ators of an arbitrary rotation subgroup and Ki are
the generators of the corresponding boosts. The
Casimirs act on |p, k〉 ∈ Hp,k, as follows

C1|p, k〉 =
1

2
(k2 + σ2p2 − 1) |p, k〉

C2|p, k〉 = pk |p, k〉. (39)

For details on the representation theory of
SL(2, C), see Refs. [63–65]. The definition of the
EPRL model requires the introduction of an (ar-
bitrary) subgroup SU (2) ⊂ Spin(4) or SU (2) ⊂
SL(2, C), according to whether one is working in
the Riemannian or in the Lorentzian sector. This
subgroup corresponds to the internal gauge group
of the gravitational phase space in connection vari-
ables in the time gauge (see Ref. [8] for details).
Hence, in the quantum theory, the representation
theory of this SU (2) subgroup will be important.
This importance will soon emerge as apparent from
the imposition of the constraints that define the
EPRL model. The link between the unitary repre-
sentations of SL(2, C) and those of SU (2) is given
by the decomposition

Hp,k =

p−1
⊕

j=k

Hj =

j++j−
⊕

j=|j+−j−|

Hj , (40)

for the Riemannian sector, and

Hp,k =

∞
⊕

j=k

Hj , (41)

for the Lorentzian sector. As the unitary irre-
ducible representations of the subgroup SU (2) ∈
Spin(4) and SU (2) ∈ SL(2, C) are essential for un-
derstanding the link of the EPRL model and the
operator canonical formulation of LQG, it will be
convenient to express the action of the generators
of the Lie algebra of the corresponding group in a
basis adapted to the above equation. In order to do
this, we first notice that the Lie algebra spin(4) and
sl(2, C) can be characterized in terms of the gener-
ators of a rotation subgroup Li and the remaining

1Such explicit relationship between gravity amplitudes in the Euclidean and Lorentzian sectors can be established by
analytic continuation in 3d [62].

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Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

boost generators Ki, as follows

[L3, L±] = ± L± [L+, L−] = 2 L3

[L+, K+] = [L−, K−] = [L3, K3] = 0

[K3, L±] = ± K± [L±, K∓] = ±2 K3

[L3, K±] = ± K±

[K3, K±] = ±σ
2L±

[K+, K−] = 2σ
2L3, (42)

where K± = K
1 ± iK2 and L± = L

1±iL2, respec-
tively. The action of the previous generators in the
basis |p, k; j, m〉 can be shown to be

L3|p, k; j, m〉 = m|p, k; j, m〉,

L+|p, k; j, m〉 =
√

(j + m + 1)(j − m)

|p, k; j, m + 1〉,

L−|p, k; j, m〉 =
√

(j + m)(j − m + 1)

|p, k; j, m − 1〉,

K3|p, k; j, m〉 = αj
√

j2 − m2|p, k; j − 1, m〉

+γj m|p, k; j, m〉

−αj+1
√

(j + 1)2 − m2

|p, k; j + 1, m〉,

K+|p, k; j, m〉 = αj
√

(j − m)(j − m − 1)

|p, k; j − 1, m + 1〉

+γj
√

(j − m)(j + m + 1)

|p, k; j, m + 1〉

+αj+1
√

(j + m + 1)(j + m + 2)

|p, k; j + 1, m + 1〉,

K−|p, k; j, m〉 = −αj
√

(j + m)(j + m − 1)

|p, k; j − 1, m − 1〉

+γj
√

(j + m)(j − m + 1)

|p, k; j, m − 1〉

−αj+1
√

(j − m + 1)(j − m + 2)

|p, k; j + 1, m − 1〉, (43)

where

γj =
kp

j(j + 1)
, αj = σ

√

(j2 − k2)(j2 + p2)

j2(4j2 − 1)
(44)

The previous equations will be important in what
follows: they will allow for the characterization of

the solutions of the quantum simplicity constraints,
in both the Riemannian and Lorentzian models, in
a direct manner. This concludes the review of the
representation theory that is necessary for the def-
inition of the EPRL model.

ii. The linear simplicity constraints

As first shown in Ref. [6], the quadratic Plebanski
simplicity constraints—and more precisely in their
dual version presented below (34)—are equivalent
in the discrete setting to the linear constraint on
each face of a given tetrahedron

Dif = L
i
f −

1

γ
Kif ≈ 0, (45)

where the label f makes reference to a face f ∈
∆⋆, and where (very importantly) the subgroup
SU (2) ⊂ Spin(4) (or SL(2, C)) that is necessary
for the definition of the above constraints is chosen
arbitrarily at each tetrahedron, equivalent on each
edge e ∈ ∆⋆. Such choice of the rotation subgroup
is the precise analog of the time gauge in the canon-
ical analysis of general relativity. The EPRL model
is defined by imposing the previous constraints as
operator equations on the Hilbert spaces defined by
the unitary irreducible representations of the inter-
nal gauge group that take part in the state-sum of
BF theory. We will show in Subsection iv that the
models constructed on the requirement of a suitable
imposition of the linear constraints (45) satisfy the
usual quadratic Plebanski constraints—that reduce
BF theory to general relativity—in the path inte-
gral formulation (up to quantum corrections which
are neglected in the usual semiclassical limit).

From the commutation relations (42), from pre-
vious section, we can easily compute the commu-
tator of the previous tetrahedron constraints and
conclude that, in fact, it does not close, namely

[Dif , D
j
f ′ ] = δf f ′ ǫ

ij
k

[

(1 + σ
2

γ2
)Lkf −

2
γ
Kke

]

= 2δee′ ǫ
ij

kD
k + δee′

σ2−γ2

γ2
ǫ
ij

kL
k
f . (46)

The previous commutation relations imply that the
constraint algebra is not closed and cannot there-
fore be imposed as operator equations on the states
summed over in the BF partition function in gen-
eral. There are two interesting exceptions to the
previous statement:

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Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

1. The first one is to take γ = ±σ. This corre-
sponds to the description of the model in terms
of self-dual or anti-self-dual variables. Unfor-
tunately, the construction of the new models is
not well defined in this case for the Lorentzian
theory and leads to a trivial result in the Rie-
mannian sector: SU (2) BF theory.

2. The second possibility is to work in the sector
where Lif = 0. This choice leads to the Barret-
Crane model [66], where the degrees of freedom
of BF theory seem over constrained: Boundary
states satisfying the BC constraints are a very
small subset of the allowed boundary states
in LQG. This is believed to be problematic if
gravity is to be recovered at low energies.

The EPRL model is obtained by restricting the rep-
resentations appearing in the expression of the BF
partition function so that at each tetrahedron the
linear constraint (45) is the strongest possible way
that is compatible with the uncertainties relations
stemming from (46). In addition, one would add
the requirement that the state-space of the tetrahe-
dra is compatible with the state-space of the anal-
ogous excitation in the canonical context of LQG,
so that arbitrary states in the kinematical state of
LQG have non-trivial amplitudes in the model.

Due to the fact that the constraints Dif do not
form a closed (first class) algebra in the generic
case, one needs to devise a weaker sense in which
they are to be imposed. One possibility is to con-
sider the Gupta-Bleuler criterion consisting of se-
lecting a suitable class of states for which the ma-
trix elements on Dif vanish. One notices from (43)
that if we chose the subspace Hj ⊂ Hp,k, we would
have

〈p, k, j, q|D3f |p, k, j, m〉 = δq,mm(1 −
γj
γ

)

〈p, k, j, q|D±f |p, k, j, m〉 = δq±1,m

×
√

(j ± m + 1)(j ∓ m)(1 −
γj
γ

).

The matrix elements of the linear constraints van-
ish in this subclass if one chooses

γj =
pk

j(j + 1)
= γ (47)

There are two cases:

1. Case γ < 1: Following Ref. [67], in this case
one restricts the representations to

Riemannian: p = j + 1, k = γj.

Lorentzian: p = γ(j + 1), k = j. (48)

which amounts to choosing the maximum
weight component j = p − 1 in the expansion
(41). In the Riemannian case, the above choice
translates into j± = (1 ± γ)j/2 for the SU(2)
right and left representations. Notice that the
solutions to the simplicity constraints in the
Riemannian and Lorentzian sectors look very
different for γ < 1. Simple algebra shows that
condition (47) is met. There are indeed other
solutions [68] to the Gupta-Bleuler criterion in
this case.

2. Case γ > 1: In this case, according to Ref.
[69], one restricts the representations to

Riemannian: p = γ(j + 1), k = j.

Lorentzian: p = γ(j + 1), k = j. (49)

which amounts to choosing the minimum
weight component j = k in the expansion (41).
For the Riemannian case, we can write the so-
lutions in terms of j± = (γ ± 1) j

2
+ γ−1

2
. No-

tice that for γ > 1 there is complete symme-
try between the solutions of the Riemannian
and Lorentzian sectors. In my opinion, this
symmetry deserves further investigation as it
might be an indication of a deeper connection
between the Riemannian and Lorentzian mod-
els (again, such relationship is a fact in 3d
gravity [62].

Another criterion for weak imposition can be devel-
oped by studying the spectrum of the Master con-
straint Mf = Df ·Df . Strong imposition of the con-
straints Dif would amount to looking for the kernel
of the master constraint Mf . However, generically,
the positive operator associated with the master
constraint does not contain the zero eigenvalue in
the spectrum due to the open nature of the con-
straint algebra (46).

It is convenient, as in Ref. [70], to express the
master constraint in a manifestly invariant way. In
order to get a gauge invariant constraint one starts

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Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

from the master constraint and uses the Dif = 0
classically to write it in terms of Casimirs, namely

Mf = (1 + σ
2γ2)C2 − 2C1γ,

where C1 and C2 are the Casimirs given in Eq. (39).
The minimum eigenvalue condition is

Riemannian: p = j, k = γj.

Lorentzian: p = γj, k = j. (50)

The minimum eigenvalue is mmin = ~
2γj(γ2 − 1)

for the Riemannian case and mmin = γ for the
Lorentzian case. The master constraint criterion
works better in the Lorentzian case, as pointed out
in Ref. [70]. More recently, it has been shown that
the constraint solutions p = γj and k = j also
follow naturally from a spinor formulation of the
simplicity constraints [71–73]. The above criterion
is used in the definition of the EPRL model.

It is important to point out that the Rieman-
nian case imposes strong restrictions on the allowed
values of the Immirzi parameter if one wants the
spin j ∈ N/2 to be arbitrary (in order to have all
possible boundary states allowed in LQG). In this
case, the only possibilities are γ = N or γ = 1.
This restriction is not natural from the viewpoint
of LQG. Its relevance, if any, remains mysterious
at this stage.

Summarizing, in the Lorentzian (Riemannian)
EPRL model one restricts the SL(2, C) (Spin(4))
representations of BF theory to those satisfying

p = γj k = j (51)

for j ∈ N/2. From now on, we denote the subset of
admissible representation

Kγ ⊂ Irrep(SL(2, C))(Irrep(Spin(4))) (52)

The admissible quantum states Ψ are elements of
the subspace Hj ⊂ Hγj,j (i.e., minimum weight
states) which satisfy the constraints (45) in the fol-
lowing semiclassical sense:

(Kif − γL
i
f )Ψ = Osc, (53)

where the symbol Osc (order semiclassical) denotes
a quantity that vanishes in limit ~ → 0, j → ∞

with ~j =constant. In the Riemannian case, the
previous equation can be written as

[(1 − γ)J i+ − (1 + γ)J
i
−]Ψ = Osc, (54)

which in turn has a simple graphical representa-
tion in terms of spin-network grasping operators,
namely

−(1 + γ) +(1 − γ)

k

j− j+

= Osc

k

j− j+

(55)

The previous equation will be of great importance
in the graphical calculus that will allow us to show
that the linear constraint imposed here, at the level
of states, implies the vanishing of the quadratic Ple-
banski constraints (34) and their fluctuations, com-
puted in the path integral sense, in the appropriate
large spin semiclassical limit.

iii. Presentation of the Riemannian EPRL
amplitude

Here we complete the definition of the EPRL mod-
els by imposing the linear constraints on the BF
amplitudes constructed in Section II. We will also
show that the path-integral expectation value of the
Plebanski constraints (34), as well as their fluctua-
tions, vanish in a suitable semiclassical sense. This
shows that the EPRL model can be considered as a
lattice definition of the a quantum gravity theory.

We start with the Riemannian model for which
a straightforward graphical notation is available.
The first step is the translation of Eq. (40)—
for p and k satisfying the simplicity constraints—
in terms of the graphical notation introduced in
Section II. Concretely, for γ < 1, one has j± =
(1 ± γ)j/2 ∈ Kγ and (40) becomes

(1 − γ)
j
2

(1 + γ) j
2=

j
⊕

α=γj

α

(1 − γ)
j
2

(1 − γ)
j
2

(1 + γ) j
2

(1 + γ) j
2

(56)

For γ > 1 we have

(γ − 1) j
2

(1 + γ) j
2=

γj
⊕

α=j

α

(γ − 1) j
2

(γ − 1) j
2

(1 + γ) j
2

(1 + γ) j
2

(57)

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Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

The implementation of the linear constraints of
Subsection ii consists of restricting the representa-
tions ρf of Spin(4) (appearing in the state sum am-
plitudes of BF theory, as written in Eq. (18)) to the
subclass ρf ∈ Kγ ⊂ Irrep(Spin(4)), defined above,
while projecting to the highest weight term in (56)
for γ < 1. For γ > 1, one must take the minimum
weight term in (57) . The action of this projection
will be denoted Yj : H(1+γ)j/2,|(1−γ)|j/2 → Hj ,
graphically

Yj





|γ − 1| j
2

(1 + γ) j
2



 =
j

. (58)

Explicitly, one takes the expression of the BF par-
tition function (13) and modifies it by replacing the
projector P einv(ρ1, · · · , ρ4) with ρ1, · · · ρ4 ∈ Kγ by
a new object

P eeprl(j1, · · · , j4) ≡ P
e
inv(ρ1 · · · ρ4)

×(Yj1 ⊗ · · · ⊗ Yj4 )P
e
inv (ρ1 · · · ρ4) (59)

with j1, · · · j4 ∈ N/2, implementing the linear con-
straints described in the previous section. Graphi-
cally, the modification of BF theory that produces
the EPRL model corresponds to the replacement

P einv(ρ1 · · · ρ4) =

P eeprl(j1 · · · j4) = (60)

on the expression (18), where we have dropped
the representation labels from the figure for sim-
plicity. We have done the operation (58) on each
an every of the four pairs of representations. The
Spin(4) integrations represented by the two boxes
at the top and bottom of the previous graphical ex-
pression restore the full Spin(4) invariance as the
projection (58) breaks this latter symmetry for be-
ing based on the selection of a special subgroup
SU (2) ⊂ Spin(4) in its definition (see Subsection c

for an important implication). One should simply
keep in mind that green wires in the previous two
equations and in the ones that follow are labeled
by arbitrary spins j (which are being summed over
in the expression of the amplitude (61)), while red
and blue wires are labeled by j+ = (1 + γ)j/2 and
j− = |1 − γ|j/2, respectively. With this, (18) is
modified to

ZEeprl(∆) =
∑

ρf ∈K

∏

f∈∆⋆

d|1−γ| j
2
d(1+γ) j

2

×
∏

e

P eeprl(j1, · · · , j4) =

=
∑

ρf ∈K

∏

f∈∆⋆

d|1−γ| j
2
d(1+γ) j

2

×
w

,(61)

The previous expression defines the EPRL model
amplitude.

a. The spin foam representation of the EPRL am-
plitude

Now we will work out the spin foam representation
of the EPRL amplitude which, at this stage, will
take no more effort than the derivation of the spin
foam representation for Spin(4) BF theory, as we
went from Eq. (18) to Eq. (20) in Section II. The
first step is given in the following equation

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Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

=

=
∑

ι
ι ῑ (62)

which follows, basically, from the invariance of the
Haar measure (9) (in the last line, we have used
(17)). More precisely, the integration of the sub-
group SU (2) ∈ Spin(4), represented by the green
box on the right, can be absorbed by suitable re-
definition of the integration on the right and left
copies of SU (2), represented by the red and blue
boxes, respectively. With this, we can already write
the spin foam representation of the EPRL model,
namely

ZEeprl(∆) =
∑

jf

∑

ιe

∏

f∈∆⋆
d|1−γ| j

2
d(1+γ) j

2

×
∏

v∈∆⋆

ι1

ι2

ι3

ι4

ι5

, (63)

where the vertex amplitude (graphically repre-
sented) depends on the 10 spins j associated to the
face-wires and the 5 intertwiners associated to the
five edges (tetrahedra). As in previous equations,
we have left the spin labels of wires implicit for no-
tational simplicity. We can write the previous spin
foam amplitude in another form by integrating out
all the projectors (boxes) explicitly. Using (17), we
get

=
∑

ι+ι−ι

y

z

x
(64)

thus replacing this in (61), we get

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Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

ZEeprl(∆) =
∑

jf

∏

f∈∆⋆
d|γ−1| j

2
d(γ+1) j

2

∑

ιe

∏

v∈∆⋆

∑

ι
−

1 ···ι
−

5

∑

ι
+
1 ···ι

+
5

5
∏

a=1
f ιa

ι
−
a ,ι

+
a

(65)

ι
−

1

ι
−

2

ι
−

3

ι
−

4

ι
−

5

|1−γ|
j1
2

|1−γ|
j2
2

|1−γ|
j3
2

|1−γ|
j4
2

|1−γ|
j5
2

|1−γ|
j6
2

|1−γ|
j7
2

|1−γ|
j8
2
|1−γ|

j9
2

|1−γ|
j10
2

ι
−

1

ι
−

2

ι
−

3

ι
−

4

ι
−

5

|1+γ|
j1
2

|1+γ|
j2
2

|1+γ|
j3
2

|1+γ|
j4
2

|1+γ|
j5
2

|1+γ|
j6
2

|1+γ|
j7
2

|1+γ|
j8
2

|1+γ|
j9
2

|1+γ|
j10
2

where the coefficients f ι
ι+ι−

are the so-called fusion
coefficients which already appear in their graphical
form in (64), more explicitly

f ιι+ι− (j1, · · · , j4) =

ι+

ι−

ι

|1−γ|
j1
2

|1−γ|
j2
2

|1−γ|
j3
2

|1−γ|
j4
2

|1+γ|
j1
2

|1+γ|
j2
2

|1+γ|
j3
2

|1+γ|
j4
2

j1

j2

j3

j4

(66)

The previous Eq. (66) is the form of the EPRL
model as derived in Ref. [5].

iv. Proof of validity of the Plebanski con-
straints

In this section, we prove that the quadratic con-
straints are satisfied in the sense that their path
integral expectation value and fluctuation vanish
in the appropriate semiclassical limit.

a. The quadratic Plebanski constraints

The quadratic Plebanski constraints are

ǫIJKLB
IJ
µν B

KL
ρσ − e ǫµνρσ ≈ 0. (67)

The constraints in this form are more suitable for
the translation into the discrete formulation. More

precisely, according to (6), the smooth fields BIJµν
are now associated with the discrete quantities
BIJ

triangles
, or equivalently BIJf as faces f ∈ ∆

⋆ are
in one-to-one correspondence to triangles in four
dimensions. The constraints (67) are local con-
straints valid at every spacetime point. In the dis-
crete setting, spacetime points are represented by
four-simplexes or (more addapted to our discus-
sion) vertices v ∈ ∆⋆. With this, the constraints
(67) are discretized as follows:

Triangle (or diagonal) constraints:

ǫIJKLB
IJ
f B

KL
f = 0, (68)

for all f ∈ v, i.e., for each and every face of the 10
possible faces touching the vertex v.

Tetrahedron constraints:

ǫIJKLB
IJ
f B

KL
f ′ = 0, (69)

for all f, f ′ ∈ v, so that they are dual to triangles
sharing a one-simplex, i.e., belonging to the same
tetrahedron out of the five possible ones.

4-simplex constraints:

ǫIJKLB
IJ
f B

KL
f̄

= ev, (70)

for any pair of faces f, f̄ ∈ v that are dual to trian-
gles sharing a single point. The last constraint will
require a more detailed discussion. At this point,
let us point out that the constraint (70) is inter-
preted as a definition of the four volume ev of the
four-simplex. The constraint requires such defini-
tion to be consistent, i.e., the true condition is

ǫIJKLB
IJ
f B

KL
f̄

= ǫIJKLB
IJ
f ′ B

KL
f̄ ′

= ǫIJKLB
IJ
f ′′ B

KL
f̄ ′′

= · · · = ev (71)

for all five different possible pairs of f and f̄ in a
four simplex, and where we assume the pairs f -f̄
are ordered in agreement with the orientation of
the complex ∆⋆.

b. The path integral expectation value of the Ple-
banski constraints

Here we prove that the Plebanski constraints are
satisfied by the EPRL amplitudes in the path inte-
gral expectation value sense.

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Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

The triangle constraints:

We start from the simplest case: The triangle (or
diagonal) constraints (68). We choose a face f ∈ v
(dual to a triangle) in the cable-wire-diagram of
Eq. (61). This amounts to choosing a pair of
wires (right and left representations) connecting
two nodes in the vertex cable wire diagram. The
two nodes are dual to the two tetrahedra—in the
four simplex dual to the vertex—sharing the chosen
triangle. Equation (36) shows that

ǫIJKLB
IJ
f B

KL
f

∝ (1 + γ)2 J−f · J
−
f − (1 − γ)

2J +f · J
+
f , (72)

where J±f denotes the self-dual and anti-self-dual

parts of ΠIJf . The path integral expectation value
of the triangle constraint is then

〈(1 + γ)2J−f · J
−
f − (1 − γ)

2J +f · J
+
f 〉 ∝ (73)

(1 + γ)2

w

−(1 − γ)2

w

= Osc,

where the double graspings on the anti-self-dual
(blue) wire and the self-dual (red) wire represent
the action of the Casimirs J−f · J

−
f and J

+
f · J

+
f , on

the cable-wire diagram of the corresponding vertex.
Direct evaluation shows that the previous diagram
is proportional to ~2jf which vanishes in the semi-
classical limit ~ → 0, j → ∞ with ~j =constant.
We use the notation already adopted in (54) and
call such quantity Osc. This proves that the trian-
gle Plebanski constraints are satisfied in the semi-
classical sense.

The tetrahedra constraints:

The proof of the validity of the tetrahedra con-
straints (69). In this case we also have

(1 + γ)2

w

(74)

−(1 − γ)2

w

= Osc.

where we have chosen an arbitrary pair of faces. In
order to prove this, let us develop the term on the
right. The result follows from

=

=
(1 + γ)

|1 − γ|
+ Osc

=
(1 + γ)2

(1 − γ)2
+ Osc

=
(1 + γ)2

(1 − γ)2
+ Osc, (75)

where in the first line we have used the fact that
the double grasping can be shifted through the
group integration (due to gauge invariance (9)).

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Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

In the first and second terms on the second line,
we have used Eq. (55) to move the graspings on
self-dual wires to the corresponding anti-self-dual
wires. Equation (75) immediately follows the pre-
vious one; the argument works in the same way
for any other pair of faces. Notice that the first
equality in Eq. (75) implies that we can view the
Plebanski constraint as applied in the frame of the
tetrahedron as well as in a Lorentz invariant frame-
work (the double grasping defines an intertwiner
operator commuting with the projection P einv rep-
resented by the box). An analogous statement also
holds for the triangle constraints (73).

The 4-simplex constraints

Now we show the validity of the four simplex con-
straints in their form (71). As we will show below,
this last set of constraints follow from the Spin(4)
gauge invariance of the EPRL node (i.e., the va-
lidity of the Gauss law) plus the validity of the
tetrahedra constraints (69). Gauge invariance of
the node takes the following form in graphical no-
tation:

+

+ + = 0, (76)

where the above equation represents the gauge in-
variance under infinitesimal left SU (2) rotations.
An analogous equation with insertions on the right
is also valid. The validity of the previous equation
can, again, be related to the invariance of the Haar
measure used in the integration on the gauge group
that defines the boxes (9).

Now we choose an arbitrary pair f and f̄ (where
f̄ is one of the three possible faces whose dual tri-
angle only shares a point with the one correspond-
ing to f ) and will show how the four volumen ev
defined by it equals the one defined by any other
admissible pair. The first step is to show that we
get the same result using the pairs f -f̄ and f - ¯̄f ,
where ¯̄f is another of the three admissible faces op-
posite to f . The full result follows from applying
the same procedure iteratively to reach any admis-
sible pair. It will be obvious from the treatment
given below, that this is possible. Thus, for a given
pair of admissible faces, we have

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Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

ev = (1 + γ)
2

w

− (1 − γ)2

w

= −(1 + γ)2





















w

+

w

+

w





















+(1 − γ)2





















w

+

w

+

w





















= −(1 + γ)2

w

+ (1 − γ)2

w

+ Osc, (77)

where going from the first line to the second and
third lines we have simply used (76) on the bot-
tom graspings on the right and left wires. The last
line results from the validity of (69). Notice that
the second terms in the second and third lines add
up to Osc, as well as the third terms in the sec-
ond and third line. There is an overall minus sign
which amounts for an orientation factor. It should
be clear that we can apply the same procedure to
arrive at any admissible pair.

c. Peprl is not a projector

We will study in detail the object P eeprl(j1, · · · , j4).
We see that it is made of two ingredients. The
first one is the projection to the maximum weight
subspace Hj for γ > 1 in the decomposition of
Hj+,j− for j

± = (1 ± γ)j/2 (j± = (γ ± 1)j/2 for

γ > 1) in terms of irreducible representations of
an arbitrarily chosen SU (2) subgroup of Spin(4).
The second ingredient is to eliminate the depen-
dence on the choice of subgroup by group averaging
with respect to the full gauge group Spin(4). This
is diagrammatically represented in (60). However,
P eeprl(j1, · · · , j4) is not a projector, namely

P eeprl(j1, · · · , j4)
2 6= P eeprl(j1, · · · , j4). (78)

Technically, this follows from (59) and the fact that

[P einv (ρ1 · · · ρ4), (Yj1 ⊗ · · · ⊗ Yj4 )] 6= 0 (79)

i.e., the projection imposing the linear constraints
(defined on the frame of a tetrahedron or edge)

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Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

and the Spin(4) (or Lorentz) group averaging—
rendering the result gauge invariant—do not com-
mute. The fact that the P eeprl(j1, · · · , j4) is not a
projection operator has important consequences in
the mathematical structure of the model:

1. From (61) one can immediately obtain the fol-
lowing expression for the EPRL amplitude

Zeprl(∆) =
∑

ρf ∈K

∏

f∈∆⋆

d|1−γ| j
2
d(1+γ) j

2

×
∏

e

P eeprl(j1, · · · , j4). (80)

This expression has the formal structure of ex-
pression (13) for BF theory. The formal sim-
ilarity, however, is broken by the fact that
P eeprl(j1, · · · , j4) is not a projection operator.
From the formal perspective, there is the possi-
bility for the amplitudes to be defined in terms
of a network of projectors (as in BF theory).
This might provide an interesting structure
that might be of relevance in the definition of
a discretization independent model. On the
contrary, the failure of P eeprl(j1, · · · , j4) to be
a projector may lead, in my opinion, to dif-
ficulties in the limit where the complex ∆ is
refined: The increasing of the number of edges
might produce either trivial or divergent am-
plitudes 2.

2. Another difficulty associated with
P eeprl(j1, · · · , j4)

2 6= P eeprl(j1, · · · , j4) is the
failure of the amplitudes of the EPRL model,
as defined here, to be consistent with the ab-
stract notion of spin foams as defined in [74].
This is a point of crucial importance under cur-
rent discussion in the community. The point
is that the cellular decomposition ∆ has no
physical meaning and is to be interpreted as a
subsidiary regulating structure to be removed
when computing physical quantities. Spin
foams configurations can fit in different ways
on a given ∆, yet any of these different em-
beddings represent the same physical process
(like the same gravitational field in different
coordinates). Consistency requires the spin

foam amplitudes to be independent of the em-
bedding, i.e., well-defined on the equivalence
classes of spin foams as defined by Baez in
Ref. [74] (the importance of these consistency
requirements was emphasized in Ref. [75]).
The amplitude (80) fails this requirement due
to P eeprl(j1, · · · , j4)

2 6= P eeprl(j1, · · · , j4).

d. The Warsaw proposal

If one sees the above as difficulties, then there is a
simple solution, at least in the Riemannian case. As
proposed in Ref. [76, 77], one can obtain a consis-
tent modification of the EPRL model by replacing
P eeprl in (80) by a genuine projector P

e
w, graphically

P ew(j1 · · · j4) =
∑

αβ

Inv







α β







α
β , (81)

It is easy to check that by construction

(P ew(j1 · · · j4))
2 = P ew(j1 · · · j4). (82)

The variant of the EPRL model proposed in Refs.
[76, 77] takes then the form

Zeprl(∆) =
∑

jf

∏

f∈∆⋆

d|1−γ| j
2
d(1+γ) j

2

×
∏

e

P ew(j1, · · · , j4) (83)

=
∑

jf

∑

ιev

∏

f∈∆⋆

d|1−γ| j
2
d(1+γ) j

2

×
∏

e∈∆⋆

geιevs ι
e
vt

∏

v∈∆⋆

ι1v

ι2v

ι3v

ι4v

ι5v

.

2This is obviously not clear from the form of (80). We are extrapolating the properties of (P e
eprl

)N for large N to those

of the amplitude (80) in the large number of edges limit implied by the continuum limit.

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Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

Thus, in the modified EPRL model, edges e ∈ ∆⋆

are assigned pairs of intertwiner quantum numbers
ιevs and ι

e
vt

and an edge amplitude given by the
matrix elements geιevs ,ι

e
vt

(where vs and vt stand for

the source and target vertices of the given oriented
edge). The fact that edges are not assigned a sin-
gle quantum number is not really significative; one
could go to a basis of normalized eigenstates of P ew
and rewrite the modified model above as a spin
foam model where edges are assigned a single (ba-
sis element) quantum number. As the nature of
such basis and the quantum geometric interpreta-
tion of its elements are not clear at this stage, it
seems simpler to represent the amplitudes of the
modified model in the above form.

The advantages of the modified model are im-
portant. However, a generalization of the above
modification of the EPRL model in the Lorentzian
case is still lacking. Notice that this modification
does not interfere with the results on the semiclas-
sical limit (to leading order), as reviewed in Section
VII. The reason for this is that the matrix elements
geαβ → δαβ in that limit [78].

v. The coherent states representation

We have written the amplitude defining the EPRL
model by constraining the state sum of BF theory.
For semiclassical studies that we will review in Sec-
tion VII, it is convenient to express the EPRL am-
plitude in terms of the coherent states basis. The
importance of coherent states in spin foam models
was put forward in Ref. [49] and explicitly used
to re-derive the EPRL model in Ref. [79]. The
coherent state technology was used by Freidel and
Krasnov in [6] to introduce a new kind of spin foam
models for gravity: The FK models. In some cases,
the FK model is equivalent to the EPRL model.
We will review this in detail in Section IV.

The coherent state representation of the EPRL
model is obtained by replacing (27) in each of the
intermediate SU (2) (green) wires in the expression
(61) of the EPRL amplitudes, namely

(84)

=

∫

[S2]4

4
∏

I=1

djI dnI
n1n1n2n2n3n3n4n4

The case γ < 1

In this case, the coherent state property (28) im-
plies

n1n1n2n2n3n3n4n4

=

n1n1

n1 n1

n2n2

n2 n2

n3n3

n3 n3

n4n4

n4 n4

, (85)

where we have used, in the last line, the fact
that for γ < 1 the representations j of the sub-
group SU (2) ∈ Spin(4) are maximum weight, i.e.,
j = j+ + j−. Doing this at each edge, we get

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Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

ZEeprl(∆) =
∑

jf

∏

f∈∆⋆

dj−
f

dj+
f

∫

∏

e∈∈∆⋆

djef dnef

n1

n1

n2

n2

n3

n3

n4

n4

, (86)

where we have explicitly written the n ∈ S2 inte-
gration variables on a single cable. The expression
above is very similar to the coherent states repre-
sentation of Spin(4) BF theory given in Eq. (29).
In fact, one gets the above expression if one starts
from the expression (29) and sets n+ef = n

−
ef = nef

while dropping, for example, all the sphere inte-
grations corresponding to the n+ef (or equivalently

n−ef ). Moreover, by construction, the coherent
states participating in the previous amplitude sat-
isfy the linear constraints (45) in expectation val-
ues, namely

〈j, nef |D
i
f |j, nef 〉

= 〈j, nef |(1 − γ)J
+i
f + (1 + γ)J

−i
f |j, nef 〉

= 0. (87)

Thus, the coherent states participating in the above
representation of the EPRL amplitudes solve the
linear simplicity constraints in the usual semiclas-
sical sense. The same manipulations leading to (89)
in Section II lead to a discrete effective action for
the EPRL model, namely

Z γ<1eprl =
∑

jf

∏

f∈∆⋆

d
(1−γ)

jf
2

d
(1+γ)

jf
2

(88)

×

∫

∏

e∈∆⋆

djef dnef dg
−
ef dg

+
ef exp (S

γ<1

j±,n
[g±]),

where the discrete action

Sγ<1
j±,n

[g±] (89)

=
∑

v∈∆⋆

(Sv
(1−γ)

jf
2

,n
[g−] + Sv

(1+γ)
jf
2

,n
[g+])

with

Svj,n[g] (90)

=

5
∑

a<b=1

2jab ln 〈nab|g
−1
a gb| nba〉,

and the indices a, b label the five edges of a given
vertex. The previous expression is exactly equal
to the form (11) of the BF amplitude. In the case
of the gravity models presented here, the coher-
ent state path integral representation (analogous
to (31)) will be the basic tool for the study of the
semiclassical limit of the models and the relation-
ship with Regge discrete formulation of general rel-
ativity.

The case γ > 1

The case γ > 1 is more complicated [80]. The
reason for this is that the step (85), directly lead-
ing to the discrete action in the previous case, is
no longer valid, as the representations of the sub-
group SU (2) ∈ Spin(4) are now minimum instead
of maximum weight. However, the representations
j+ = j− + j are maximum weight. We can, there-
fore, insert coherent states resolution of the identity
on the right representations and get:

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Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

n1

n2

n3

n4

(91)

=

∫

[S3]4

4
∏

I=1

d
(1+γ)

jI
2

dmI m1
m2

m3

m4

n1

n2

n3

n4

=

∫

[S3]4

4
∏

I=1

d
(1+γ)

jI
2

dmI m1

m1

m1

m2

m2

m2

m3

m3

m3

m4

m4

m4

n1

n2

n3

n4

,

where we are representing the relevant part of the
diagram appearing in Eq. (85). In the last line,
we have used j+ = j + j− (i.e., maximum weight),
and the graphical notation m n ≡ 〈m|n〉 as it
follows from our previous conventions. With this,
one gets

Z γ>1eprl = (92)
∑

jf

∏

f∈∆⋆

d
(1−γ)

jf
2

d
(1+γ)

jf
2

×

∫

∏

e∈∆⋆

djef d(1+γ)
jef
2

dnef dmef dg
−
ef dg

+
ef

× exp (Sγ>1
j±,n,m

[g±]),

where the discrete action

Sγ>1
j±,n,m

[g±] =
∑

v∈∆⋆

Svj±,n,m[g
±] (93)

with

Svj±,n,m[g
±] (94)

=
∑

1≤a<b≤5

[

jab(1 + γ) log(〈mab|g
+
ab|mba〉)

+jab(γ − 1) log(〈mab|g
−
ab|mba〉)

+2jab (log(〈nab|mab〉) + log(〈mba|nba〉))] .

a. Some additional remarks

It is important to point out that the commuta-
tion relations of basic fields—reflecting the sim-
ple algebraic structure of spin(4)—used here are
the ones induced by the canonical analysis of BF
theory presented previously. The presence of con-
straints generally modifies canonical commutation
relations, in particular in the presence of second
class constraints. For some investigation of the is-
sue in the context of the EPRL and FK models,
see Ref. [69]. In Ref. [81], it is pointed out that
the presence of secondary constraints in the canon-
ical analysis of Plebanski action should be trans-
lated into additional constraints in the holonomies
of the spin foam models here considered (see also
Ref. [82]). A possible view is that the simplicity
constraints are here imposed for all times and thus
secondary constraints should be imposed automat-
ically.

There are alternative derivations of the models
presented in the previous sections. In particular,
one can derive them from a strict Lagrangean ap-
proach of Plebanski’s action. Such viewpoint is
taken in Refs. [83–85]. The path integral formula-
tion of Plebansky theory using commuting B-fields
was studied in Ref. [86], where it is shown that
only in the appropriate semiclassical limit the am-
plitudes coincide with the ones presented in the pre-
vious sections (this is just another indication that
the construction of the models has a certain semi-
classical input; see below). The spin foam quanti-
zation of the Holst formulation of gravity via cubu-
lations was investigated in Ref. [87]. The simplicity
constraints can also be studied from the perspective
of the U (N ) formulation of quantum geometry [88].
Such U (N ) treatment is related to previous work
[89, 90] which has been extended to a completely
new perspective on quantum geometry with possi-
ble advantageous features [91, 92]. For additional
discussion on the simplicity constraints, see Ref.

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Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

[93].

vi. Presentation of the EPRL Lorentzian
model

As it was briefly discussed in Section III, unitary
irreducible representations of SL(2, C) are infinite
dimensional and labeled by a positive real number
p ∈ R+ and a half-integer k ∈ N/2. These represen-
tations are the ones that intervene in the harmonic
analysis of square integrable functions of SL(2, C)
[64]. Consequently, one has an explicit expression
of the delta function distribution (defined on such
test function), namely

δ(g) =
∑

k

∫

R+

dp (p2 + k2)
∑

j,m

D
p,k
jmjm(g) (95)

where D
p,k
jmj′m′ (g) with j ≥ k and j ≥ m ≥ −j

(similarly for the primed indices) are the matrix
elements of the unitary representations p − k in
the so-called canonical basis [63]. One can use the
previous expression, the Lorentzian version of Eq.
(11), in order to introduce a formal definition of
the BF amplitudes, which now would involve inte-
gration of the continuous labels pf , in addition to
sums over discrete quantum numbers such as k, j
and m. The Lorentzian version of the EPRL model
can be obtained from the imposition of the linear
simplicity constraints to this formal expression. As
the continuum labels pf are restricted to pf = γjf ,
the Lorentzian EPRL model becomes a state-sum
model as its Riemannian relative. Using the follow-
ing graphical notation

D
p,k
jmj′m′ (g) =

p

k
j′,m′j,m (96)

the amplitude is

ZLeprl(∆)

=
∑

jf

∏

f∈∆⋆

(1 + γ2)j2f ,

where the boxes now represent SL(2, C) integra-
tions with the invariant measure. The previous am-
plitude is equivalent to its spin foam representation

ZLeprl(∆) =
∑

jf

∑

ιe

∏

f∈∆⋆

(1 + γ2)j2f

∏

v∈∆⋆

ι1

ι2

ι3

ι4

ι5

,

The vertex amplitude is divergent due to the pres-
ence of a redundant integration over SL(2, C). It
becomes finite by dropping an arbitrary integra-
tion, i.e., removing any of the 5 boxes in the vertex
expression [94].

a. The coherent state representation

It is immediate to obtain the coherent states repre-
sentation of the Lorentzian models. As in the Rie-
mannian case, one simply inserts resolution of the
identities (22) on the intermediate SU (2) (green)
wires in (97), from where it results

ZLeprl(∆) =
∑

jf

∏

f∈∆⋆

(1 + γ2)j2 (97)

∫

∏

e∈∈∆⋆

djef dnef
n1n1n2 n2n3n3n4 n4 ,

IV. The Freidel-Krasnov (FK)
model

Shortly after the appearance of the paper in Ref.
[4], Freidel and Krasnov [6] introduced a set of new
spin foam models for four-dimensional gravity us-
ing the coherent state basis of the quantum tetra-
hedron of Livine and Speziale [49]. The idea is to

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Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

impose the linearized simplicity constraints (45) di-
rectly as a semiclassical condition on the coherent
state basis. As we have seen above, coherent states
are quantum states of the right and left tetrahe-
dra in BF theory which have a clear-cut semiclassi-
cal interpretation through their property (26). We
have also seen that the imposition of the linear con-
straints (45) a la EPRL is in essence semiclassi-
cal as they are strictly valid only in the large spin
limit. In the FK approach one simply accepts from
the starting point that, due to their property of a
non-defining set that is closed under commutation
relations, the Plebansky constraints are to be im-
posed semiclassically. One defines new models by
restricting the set of coherent states entering in the
coherent state representation of Spin(4) BF theory
(29) to those that satisfy condition (45) in expec-
tation values. They also emphasize how the model
[4] corresponds, indeed, to the sector γ = ∞ which
has been shown to be topological [95].

The case γ < 1

For γ < 1, the vertex amplitude is identical to the
EPRL model. This is apparent in the coherent
state expression of the EPRL model (88). Thus,
we have

Z γ<1f k (∆) =
∑

jf

∏

f∈∆⋆

d|1−γ| j
2
d(1+γ) j

2
(98)

∏

e∈∆⋆

∫

d(1+γ) j
2
d

(γ−1)
jef
2

dnef

n1n1

n1 n1

n2n2

n2 n2

n3n3

n3 n3

n4n4

n4 n4

.

From the previous expression, we conclude that the
vertex amplitudes of the FK and EPRL model co-
incide for γ < 1

Aγ<1v f k = A
γ<1

v eprl. (99)

Notice, however, that different weights are assigned
to edges in the FK model. This is due to the

fact that one is restricting the Spin(4) resolution
of identity in the coherent basis in the previous ex-
pression, while in the EPRL model the coherent
state resolution of the identity is used for SU (2)
representations. This difference is important and
has to do with the still unsettled discussion con-
cerning the measure in the path integral represen-
tation.

The case γ > 1

For the case γ > 1, the FK amplitude is given by

Z γ>1f k (∆) =
∑

jf

∏

f∈∆⋆

d|1−γ| j
2
d(1+γ) j

2

∏

e∈∆⋆

∫

d(1+γ) j
2
d

(γ−1)
jef
2

dnef (100)

n1n1

−n1 −n1

n2n2

−n2 −n2

n3n3

−n3 −n3

n4n4

−n4 −n4

.

The study of the coherent state representation of
the FK model for γ > 1, in comparison with Eq.
(92) for the EPRL model, clearly shows the differ-
ence between the two models in this regime.

Z γf k =
∑

jf

∏

f∈∆⋆

d
(1−γ)

jf
2

d
(1+γ)

jf
2

∫

∏

e∈∆⋆

d
|1−γ|

jef
2

d
(1+γ)

jef
2

dnef dg
−
ef dg

+
ef

exp (Sf k γ
j±,n

[g±]), (101)

where the discrete action

Sf k γ
j±,n

[g±] =
∑

v∈∆⋆

[

Sv
(1−γ)

jf
2

,n
[g−]

+Sv
(1+γ)

jf
2

,s(γ)n
[g+]

]

, (102)

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Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

where s(γ) = sign(1 − γ) and

Svj,n[g] =

5
∑

a<b=1

2jab ln 〈nab|g
−1
a gb| nba〉, (103)

with the indices a, b labeling the five edges of a
given vertex.

V. Boundary data for the new mod-
els and relationship with the
canonical theory

So far, we have considered cellular complexes with
no boundaries. Transition amplitudes are expected
to be related to the definition of the physical scalar
product. In order to define them, one needs to
consider complexes with boundaries. Boundary
states are defined on the boundary of the dual
two-complex ∆⋆ that we denote ∂∆⋆. The object
∂∆⋆ is a one-complex (a graph). According to the
construction of the model (Section III), boundary
states are in one-to-one correspondence with SU (2)
spin networks. This comes simply from the fact
that links (one-cells) ℓ ∈ ∂∆⋆ inherit the spins la-
bels (unitary irreducible representations of the sub-
group SU (2)) from the boundary faces while nodes
(zero-cells) n ∈ ∂∆⋆ inherit the intertwiner levels
from boundary edges.

At this stage, one can associate the boundary
data with elements of a Hilbert space. Being in
one-to-one correspondence with SU (2) spin net-
works, a natural possibility is to associate to them
an element of the kinematical Hilbert space of
LQG. More precisely, with a given colored bound-
ary graph γ with links labeled by spins jℓ and nodes
labeled by intertwiners ιn, we associate a cylindri-
cal function Ψγ,{jℓ},{ιn} ∈ L

2(SU (2)Nℓ ), where
here Nℓ denotes number of links in the graph γ.
In this way, the boundary Hilbert space associated
with ∂∆⋆ is isomorphic (if one used the natural AL
measure) with the Hilbert space of LQG truncated
to that fixed graph. Moreover, geometric opera-
tors, such as volume and area defined in the covari-
ant context, are shown to coincide with the corre-
sponding operators defined in the canonical formu-
lation [67, 96]. Now, if cellular complexes are dual
to triangulations, then the boundary spin networks
can have at most four-valent nodes. This limita-
tion can be easily overcome: As in BF theory, the

EPRL amplitudes can be generalized to arbitrary
complexes with boundaries given by graphs with
nodes of arbitrary valence. The extension of the
model to arbitrary complexes has been first stud-
ied in Refs. [97, 98]. It has also been revisited in
Refs. [68].

Alternatively, one can associate the boundary
states with elements of L 2(Spin(4)Nℓ ) (in the
Riemannian models)—or carefully define the ana-
log of spin network states as distributions in the
Lorentzian case (see Refs. [99] for some insights on
the problem of defining a gauge invariant Hilbert
space of graphs for non-compact gauge groups). In
this case, one gets special kinds of spin network
states. These are a subclass of the so-called pro-
jected spin networks introduced in Refs. [100, 101]
in order to define a heuristic quantization of the
(non-commutative and very complicated) Dirac al-
gebra of a Lorentz connection formulation of the
phase space of gravity [100,102–107]. The fact that
this special subclass of projected spin networks ap-
pears naturally as a boundary state of the new spin
foams is shown in Ref. [108].

Due to their similarity to γ < 1, the same re-
lationship between boundary data and elements
of the kinematical Hilbert space holds for the FK
model. However, such simple relationship does not
hold for the model in the case γ > 1.

It is important to mention that the knotting
properties of boundary spin networks do not seem
to play a role in present definitions of transition
amplitudes [109].

VI. Further developments and re-
lated models

The spin foam amplitudes discussed in the previ-
ous sections have been introduced by constraining
the BF histories through the simplicity constraints.
However, in the path integral formulation, the pres-
ence of constraints has the additional effect of mod-
ifying the weights with which those histories are to
be summed: Second class constraints modify the
path integral measure (in the spin foam context
this issue was raised in Ref. [75]). As pointed out
before, this question has not been completely set-
tled in the spin foam community yet. The explicit
modification of the formal measure in terms of con-
tinuous variables for the Plebansky action was pre-

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Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

sented in Ref. [110]. A systematic investigation
of the measure in the spin foam context was at-
tempted in Ref. [111] and [112]. As pointed out
in Ref. [75], there are restrictions in the mani-
fold of possibilities coming from the requirement
of background independence. The simple BF mea-
sure chosen in the presentation of the amplitudes
in the previous sections satisfies these requirements.
There are other consistent possibilities; see, for in-
stance, Ref. [113] for a modified measure which re-
mains extremely simple and is suggested from the
structure of LQG.

An important question is the relationship be-
tween the spin foam amplitudes and the canoni-
cal operator formulation. The question of whether
one can reconstruct the Hamiltonian constraints
out of spin foam amplitudes has been analysed in
detail in three dimensions. For the study of quan-
tum three-dimensional gravity from the BF per-
spective, see Ref. [114]. We will, in fact, present
this perspective in detail in the three dimensional
part of this article. For the relationship with the
canonical theory using variables that are natural
from the Regge gravity perspective, see [115, 116].
There are generalizations of Regge variables more
adapted to the interpretation of spin foams [117].
In four dimensions,the question has been investi-
gated in Ref. [118] in the context of the new spin
foam models. In the context of group field theo-
ries, this issue is explored in Ref. [119]. Finally,
spin foams can, in principle, be obtained directly
from the implementation of the Dirac program us-
ing path integral methods. This has been explored
in Refs. [120, 121], from which a discrete path in-
tegral formulation followed [122]. The question of
the relationship between covariant and canonical
formulations in the discrete setting has been ana-
lyzed also in Ref. [123].

By construction, all tetrahedra in the FK and
EPRL models are embedded in a spacelike hyper-
surface and hence have only spacelike triangles. It
seems natural to ask the question of whether a more
general construction allowing for timelike faces is
possible. The models described in previous sections
have been generalized in order to include timelike
faces in the work of F. Conrady [124–126]. An ear-
lier attempt to define such models in the context
of the Barrett-Crane model can be found in Refs.
[127].

The issue of the coupling of the new spin foam

models to matter remains to a large extend un-
explored territory. Nevertheless, some results can
be found in the literature. The coupling of the
Barrett-Crane model (the γ → ∞ limit of the
EPRL model) to Yang-Mills fields was studied in
Ref. [128]. More recently, the coupling of the
EPRL model to fermions has been investigated in
Refs. [129, 130]. A novel possibility of unification
of the gravitational and gauge fields was recently
proposed in Ref. [131].

The introduction of a cosmological constant in
the construction of four-dimensional spin foam
models has a long history. Barrett and Crane in-
troduced a vertex amplitude [132], in terms of the
Crane and Yetter model [13], for BF theory with
cosmological constant. The Lorentzian quantum
deformed version of the previous model was stud-
ied in Ref. [133]. For the new models, the coupling
with a cosmological constant is explored in terms
of the quantum deformation of the internal gauge
symmetry in Refs. [134, 135], as well as (indepen-
dently) in Ref. [136]. The asymptotics of the vertex
amplitude are shown to be consistent with a cos-
mological constant term in the semiclassical limit
in Ref. [137].

The spin foam approach applied to quantum cos-
mology has been explored in Refs. [138–143]. The
spin foam formulation can also be obtained from
the canonical picture provided by loop quantum
cosmology (see Ref. [144] and references therein).
This has been explored systematically in Refs.
[145–148].

As we have discussed in the introduction of the
new models, Heisenberg uncertainty principle pre-
cludes the strong imposition of the Plebanski con-
straints that reduce BF theory to general relativ-
ity. The results of the semiclassical limit of these
models seem to indicate that metric gravity should
be recovered in the low energy limit. However, it
seems likely that the semiclassical limit could be re-
lated to certain modifications of Plebanski’s formu-
lation of gravity [149–153]. A simple interpretation
of the new models in the context of the bi-gravity
paradigm proposed in Ref. [154] could be of inter-
est.

As it was already pointed out in Ref. [74], spin
foams can be interpreted in close analogy to Feyn-
man diagrams. Standard Feynman graphs are gen-
eralized to 2-complexes and the labeling of propa-
gators by momenta to the assignment of spins to

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Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

faces. Finally, momentum conservation at vertices
in standard feynmanology is now represented by
spin-conservation at edges, ensured by the assign-
ment of the corresponding intertwiners. In spin
foam models, the non-trivial content of amplitudes
is contained in the vertex amplitude which, in the
language of Feynman diagrams, can be interpreted
as an interaction. This analogy is indeed realized
in the formulation of spin foam models in terms of
a group field theory (GFT) [155, 156].

The GFT formulation resolves, by definition, the
two fundamental conceptual problems of the spin
foam approach: Diffeomorphism gauge symmetry
and discretization dependence. The difficulties are
shifted to the question of the physical role of λ and
the convergence of the corresponding perturbative
series.

This idea has been studied in more detail in three
dimensions. In Ref. [157], the scaling properties of
the modification of the Boulatov group field the-
ory introduced in Ref. [158] were studied in de-
tail. In a further modification of the previous model
(known as colored tensor models [159], new tech-
niques based on a suitable 1/N expansion imply
that amplitudes are dominated by spherical topol-
ogy [160]. Moreover, it seems possible that the con-
tinuum limit might be critical as in certain matrix
models [161–165]. However, it is not yet clear if
there is a sense in which these models correspond
to a physical theory. The naive interpretation of
the models is that they correspond to a formula-
tion of 3d quantum gravity including a dynamical
topology.

VII. Results on the semiclassical
limit of EPRL-FK models

Having introduced the relevant spin foam models in
the previous sections, we now present the results of
the large spin asymptotics of the spin foam ampli-
tudes suggesting that on a fixed discretization the
semiclassical limit of the EPRL-FK models is given
by Regge’s discrete formulation of general relativity
[80, 166].

The semiclassical limit of spin foams is based on
the study of the the large spin limit asymptotic be-
havior of coherent state spin foam amplitudes. The
notion of large spin can be defined by the rescaling
of quantum numbers and Planck constant accord-

ing to j → λj and ~ → ~/λ and taking λ >> 1. In
this limit, the quantum geometry approximates the
classical one when tested with suitable states (e.g.,
coherent states). However, the geometry remains
discrete during this limiting process as the limit is
taken on a fixed regulating cellular structure. That
is why one usually makes a clear distinction be-
tween semiclassical limit and the continuum limit.
In the semiclassical analysis presented here, one can
only hope to make contact with discrete formula-
tions of classical gravity. Hence, the importance of
Regge calculus in the discussion of this section.

The key technical ingredient in this analysis is
the representation of spin foam amplitudes in terms
of the coherent state basis introduced in Subsec-
tion i. Here we follow Refs. [80, 166–169]. The
idea of using coherent states and discrete effective
actions for the study of the large spin asymptotics
of spin foam amplitudes was put forward in Refs.
[170, 171]. The study of the large spin asymptotics
has a long tradition in the context of quantum grav-
ity, dating back to the study of Ponzano-Regge [26].
More directly related to our discussion, here are the
early works [172,173]. The key idea is to use asymp-
totic stationary phase methods for the amplitudes
written in terms of the discrete actions presented
in the previous section.

In this section, we review the results of the analy-
sis of the large spin asymptotics of the EPRL vertex
amplitude for both the Riemannian and Lorentz-
tian models. We follow the notation and terminol-
ogy of Ref. [80] and related papers.

b. SU(2) 15j-symbol asymptotics

As SU (2) BF theory is quite relevant for the con-
struction of the EPRL-FK models, the study of the
large spin asymptotics of the SU (2) vertex ampli-
tude is a key ingredient in the analysis of [80]. The
coherent state vertex amplitude is

15j(j, n) (104)

=

∫ 5
∏

a=1

dga
∏

1≤a≤b≤5

〈nab|g
−1
a gb|nba〉

2jab ,

which depends on 10 spins jab and 20 normals
nab 6= nba. The previous amplitude can be ex-
pressed as

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Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

15j(j, n) =

∫ 5
∏

a=1

dga
∏

1≤a≤b≤5

exp Sj,n[g], (105)

Sj,n[g] =

5
∑

a<b=1

2jab ln 〈nab|g
−1
a gb| nba〉, (106)

and the indices a, b label the five edges of a given
vertex. The previous expression is equal to the
form (11) of the BF amplitude. In the case of the
EPRL model studied in Section III, the coherent
state representation—see Eqs. 88, 92, and 97—
is the basic tool for the study of the semiclassical
limit of the models and the relationship with Regge
discrete formulation of general relativity.

In order to study the asymptotics of (105), one
needs to use extended stationary phase methods
due to the fact that the action (106) is complex
(see Refs. [170, 171]). The basic idea is that, in
addition to stationarity, one requires real part of
the action to be maximal. Points satisfying these
two conditions are called critical points. As the real
part of the action is negative definite, the action at
critical points is purely imaginary.

Notice that the action (106) depends parametri-
cally on 10 spins j and 20 normals n. These pa-
rameters define the so-called boundary data for the
four simplex v ∈ ∆⋆. Thus, there is an action prin-
ciple for every given boundary data. The number of
critical points and their properties depend on these
boundary data, hence the asymptotics of the vertex
amplitude is a function of the boundary data. Dif-
ferent cases are studied in detail in Ref. [80]. Here
we present their results in the special case where the
boundary data describe a non-degenerate Regge ge-
ometry for the boundary of a four simplex. These
data are referred to as Regge-like, and satisfy the
gluing constraints. For such boundary data, the
action (106) has exactly two critical points, leading
to the asymptotic formula

15j(λj, n) ∼
1

λ12

[

N+ exp(i
∑

a<b

λjabΘ
E
ab)

+N− exp(−i
∑

a<b

λjabΘ
E
ab)

]

, (107)

where Θab are the appropriate dihedral angles de-
fined by the four-simplex geometry. Finally, the
N± are constants that do not scale with λ.

c. The Riemannian EPRL vertex asymptotics

The previous result, together with the fact that the
EPRL amplitude for γ < 1 is a product of SU (2)
amplitudes with the same n in the coherent state
representation (88), implies the asymptotic formula
for the vertex amplitude to be given by the unbal-
anced square of the above formula [167], namely

Aeprlv ∼
1

λ12

[

N+e
i
(1−γ)

2

P

a<b

λjabΘ
E
ab

+N− e
−i

(1−γ)
2

P

a<b

λjabΘ
E
ab

]

×

[

N+ e
i
(1+γ)

2

P

a<b

λjabΘ
E
ab

+N− e
−i

(1+γ)
2

P

a<b

λjab Θ
E
ab

]

.

One can write the previous expression as

Aeprlv ∼
1

λ12
[

2N+N− cos
(

SE
Regge

)

+N 2+ e
i 1

γ
SERegge + N 2− e

−i 1
γ

SERegge
]

. (108)

where

SE
Regge

=
∑

a<b

λγjabΘ
E
ab (109)

is the Regge-like action for λγjab = Aab, the ten tri-
angle areas (according to the LQG area spectrum
[1, 2]). Remarkably, the above asymptotic formula
is also valid for the case γ > 1 [80]. The first term
in the vertex asymptotics is in essence the expected
one: It is the analog of the 6j symbol asymptotics
in three-dimensional spin foams. Due to their ex-
plicit dependence on the Immirzi parameter, the
last two terms are somewhat strange from the the-
oretical point of view of the continuum field. How-
ever, this seems to be a peculiarity of the Rieman-
nian theory alone, as shown by the results of Ref.
[166] for the Lorentzian models. Non-geometric
configurations are exponentially suppressed

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Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

d. Lorentzian EPRL model

To each solution, one can associate a second so-
lution corresponding to a parity related 4-simplex
and, consequently, the asymptotic formula has two
terms. It is given, up to a global sign, by the ex-
pression

Aeprlv ∼
1

λ12

[

N+ exp

(

iλγ
∑

a<b

jabΘ
L
ab

)

+N− exp

(

−iλγ
∑

a<b

jabΘ
L
ab

)]

, (110)

where N± are constants that do not scale. Non-
geometric configurations are exponentially sup-
pressed

In Ref. [171], Freidel and Conrady gave a de-
tailed description of the coherent state representa-
tion of the various spin foam models described so
far. In particular, they provided the definition of
the effective discrete actions associated to each case
which we presented in (101). This provides the ba-
sic elements for setting up the asymptotic analysis
presented in Ref. [170] (the first results of the semi-
classical limit of the new spin foam models). This is
similar to the studies of the asymptotic of the ver-
tex amplitude reviewed above, but more general
in the sense that the semiclassical limit of a full
spin foam configuration (involving many vertices)
is studied. The result is technically more complex
as one studies now critical points of the action as-
sociated to a colored complex which, in addition
of depending on group variables g, depends on the
coherent state parameters n. The authors of Ref.
[170] write Eq. (101) in the following way:

Z γf k =
∑

jf

∏

f∈∆⋆

d
(1−γ)

jf
2

d
(1+γ)

jf
2

W
γ
∆⋆ (jf ), (111)

where

W
γ
∆⋆ (jf ) (112)

=

∫

∏

e∈∆⋆

d
|1−γ|

jef
2

d
(1+γ)

jef
2

dnef dg
−
ef dg

+
ef

× exp (Sf k γ
j±,n

[g±]).

They show that those solutions of the equations of
motion of the effective discrete action that are non-
geometric (i.e., the contrary of Regge-like) are not
critical and hence exponentially suppressed in the
scaling jf → λjf with λ >> 1. If configurations are
geometric (i.e., Regge-like), one has two kinds of
contributions to the amplitude asymptotics: Those
coming from degenerate and non-degenerate con-
figurations. If one (by hand) restricts to the non-
degenerate configurations, then one has

W
γ
∆⋆ (jf ) ∼

c

λ(33ne−6nv−4nf )

× exp(iλSE
Regge

(∆⋆, jf )), (113)

where ne, nv, and nf denote the number of edges,
vertices, and faces in the two complex ∆⋆, respec-
tively. There are recent works by M. Han in which
asymptotics of general simplicial geometry ampli-
tudes are studied in the context of the EPRL model
[174, 175].

The problem of computing the two-point func-
tion and higher correlation functions in the con-
text of spin foam has received a lot of attention
recently. The framework for the definition of the
correlation functions in the background indepen-
dent setting has been generally discussed by Rov-
elli in Ref. [176], and correspods to a special ap-
plication of a more general proposal investigated
by Oeckl [177–184]. It was then applied to the
Barrett-Crane model in Refs. [185–187], where it
was discovered that certain components of the two-
point function could not yield the expected result
compatible with Regge gravity in the semiclassi-
cal limit. This was used as the main motivation
for the weakening of the imposition of the Pleban-
ski constraints, leading to the new models. Soon
thereafter, it was argued that the difficulties of the
Barrett-Crane model where indeed absent in the
EPRL model [188]. The two-point function for the
EPRL model was calculated in Ref. [189] and it
was shown to produce a result in agreement with
that of Regge calculus [190,191], in the limit γ → 0.

The fact that, for the new model, the double scal-
ing limit γ → 0 and j → ∞ with γj= constant de-
fines the appropriate regime where the fluctuation
behave as in Regge gravity (in the leading order)
has been further clarified in Ref. [192]. This in-
dicates that the quantum fluctuations in the new
models are more general than simply metric fluc-

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Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

tuations. The fact that the new models are not
metric at all scales should not be surprising as we
know that the Plebanski constraints that produce
metric general relativity out of BF theory have been
implemented only semiclassically (in the large spin
limit). At the deep Planckian regime, fluctuations
are more general than metric. However, it is not
clear at this stage why this is controlled by the Im-
mirzi parameter.

All the previous calculations involve a complex
with a single four-simplex. The first computation
involving more than one simplex was performed in
Refs. [187, 193], for the case of the Barrett-Crane
model. Certain peculiar properties were found and
it is not clear at this stage whether these issues re-
main in the EPRL model. Higher order correlation
functions have been computed in Ref. [194], the
results are in agreement with Regge gravity in the
γ → 0 limit.

VIII. Acknowledgements

I would like to thank the help from many people in
the field that have helped me in various ways. I am
grateful to Eugenio Bianchi, Carlo Rovelli and Si-
mone Speziale for the many discussions on aspects
and details of the recent literature. Many detailed
calculations that contributed to the presentation of
the new models in this review were done in collab-
oration with Mercedes Velázquez to whom I would
like to express my gratitude. I would also like to
thank You Ding, Florian Conrady, Laurent Freidel,
Muxin Han, Merced Montesinos for their help and
valuable interaction.

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versity Press, Cambridge (UK) (2004), Pag.
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[2] T Thiemann, Modern canonical quantum
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[3] A Ashtekar, J Lewandowski, Background in-
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[4] J Engle, R Pereira, C Rovelli, The loop-
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