Covariant and Canonical Quantum Gravity

Corso di Laurea in Fisica

Tesi di Laurea Specialistica

Canonical and Covariant Quantum

Gravity

Candidato

Francesco Sardelli

Relatore

Relatore interno

Karim Noui

Pietro Menotti

From freedom to research 1

Organization of the text and notation 3

1 Just a change of variables 5

2 Quantisation program 7

3 The kinematical Hilbert space 9

4 The Gauss law 11

5 Diffeomorfism constraint 13

6 Spin foam approach 15

6.1 Classical BF theory and Plebanski action . . . 15 6.2 Path integral for BF theory . . . 16 6.3 A general definition of a Spin-Foam model . . . 17 7 The problem of the scalar constraint 19

8 The vertex 21

8.1 A General expression of the vertex . . . 21 8.2 Vertices of particular models . . . 25 9 The vertex and the physical scalar product 33 9.1 The topological model . . . 34 9.2 The Barrett-Crane model . . . 35 9.3 The Engle-Pereira-Rovelli model . . . 37

Non siamo fisici: siamo uomini. Stefano Bianco Let us suppose that the concept of individual does make sense. With this hypothesis, we are logically allowed to speak about each one of us and our existence.

Now, accepting that life has no pre-assigned aim (which is a more thought version of the statement: life has no meaning) entrusts a wonderful task to us and only us: choosing. The first choice that we have in front of us in each moment is killing ourselves or not. If we choose the latter, the fascinating existential phase space opens in front of us and even that very little part which we know of it excites us. At this point, some questions can rise: How do I wish to use my freedom? What do I want to do? What divertissement do I wish to choose?

At this level, if we think about the existential phase space as a manifold without no more structure on it, all choices are equivalent.

For example, we could choose our actions to be guided by aesthetical criteria, without caring about standard moralities. In this way, we can make our life itself an art work.

Or, we could do practical-theoretical research-action to understand why most of people in our society seem to become just a consuming-producing mechanism. To understand why they seem to be satisfied in such a condition. To understand if an authentic democracy is possible. To understand why it seems that dreaming and turning our dreams into projects is quite difficult today.

Or, we could do research to discover how Nature works. For example, we could wonder why the standard model or string theory or Loop Quantum Gravity seem not to predict the existence of themselves as cultural structures. Or, we can ask a much more simple question: what happens in Nature when energies are very high?

The text can be roughly divided into two parts. In the first one, from chapter 1 to chapter 6, we give a review of Loop Quantum Gravity theory and Spin Foam models. This review is an elaboration of the expositions contained in [1], [2] and [6]. In the second one, chapter 8 and 9, the original part of the thesis work is explained. Between the two parts, in chapter 7, the original work is summarized and motivations for it are given within the context of present research in quantum gravity.

In consideration of the importance of questions in research, I have not renounced to express ques-tions hither and thither in the text even without giving answers for them.

Let us say a few words about notation. Other explanations about it will be given when necessary in the text.

Indexes indicated with the letters I, J, K, L, ... are contracted with the metric tensor ηIJ ≡

diag(−1, +1, +1, +1). Indexes indicated with the letters µ, ν, ρ, σ, ... are contracted with the metric tensor gµν. This is not valid when we talk about tensors that belong to vector spaces of

group representations.

We indicate with the symbol ∗ the Hodge operation on a form.

We will use xµ_{, y}µ_{, ... to indicate the coordinates of the 4d spacetime manifold M and τ}a _{for the}

space-like submanifolds of M.

Sometimes, a graphical representation for tensorial calculus is used. Actually, this is not simply notation: a bit deeper concepts are behind it. More detailed explanations about it can be found, for example, in C.Rovelli’s book [1].

Just a change of variables

Let us take the action of general relativity with pure gravity: S[gµν] = 1 2k Z M R + ”boundary terms” (1.1) The notation is the usual one.

If we pass to ADM variables, the action becomes: 1 2kS[qab, π ab_{, N} a, N ] = Z dt Z Σt d3_x[πab_˙q ab− NaVa(qab, πab) − N S(qab, πab)] (1.2)

where Σtis a foliation of spacetime in terms of space-like three-dimension surfaces, qabis the metric

induced on those surfaces by gµν, πab are the momenta conjugate to qab, Na is the shift vector, N

is the lapse function, and Va _{and S are defined as follows:}

Va(qab, πab) = −2∇(3)b (q −1/2_πab_{) (1.3)} S(qab, πab) = −(q1/2[R(3)− q−1πcdπcd+ 1 2q −1_π2_{]) (1.4)}

where ∇(3)a is the covariant derivative compatible with qab, q is the determinant of qab, R(3) is the

scalar curvature of qaband π = πabqab.

Let us give the Poisson brackets between our canonical variables:

{πab(x), qab(y)} = 2kδa(cδd)b δ(x, y) (1.5)

{πab(x), πcd(y)} = 0 (1.6) {qab(x), qcd(y)} = 0 (1.7)

The variation of the action (1.2) with respect to Na and N gives the constraints:

Va(qab, πab) = 0 (1.8)

S(qab, πab) = 0 (1.9)

These constraints are non-polynomial with respect to the variables qab and πab. This fact makes

the procedure of quantization very difficult. To overcome this problem, one could think about reformulating classical GR with some new variables with which the constraints (or at least part of them) become polynomial. To reach such an aim, we will start from the Palatini first order formulation of GR: SP[e, ω] = 1 4k Z M ǫIJKLeI∧ eJ∧ FKL (1.10) where eI _{= e}I

µdxµ are four 1-forms such that gµν = eIµηIJeJν (where gµν is the metric variable of

the usual Einstein-Hilbert formulation of GR), ωJI = ωIJµdxµis a connection 1-form with values in

Now, it can be proved that we can add a term to the Palatini action without modifying the equations of motion, obtaining the so-called Holst action:

SH[e, ω] = SP[e, ω] − 1 2kγ Z M eI∧ eJ∧ FIJ (1.11)

where γ ∈ R \ {0} is fixed and is called the Barbero-Immirzi parameter.

Then, we would like to perform an hamiltonian analysis of Holst theory. Let us take a foliation Σt of the spacetime manifold M in terms of space-like three-dimension surfaces. At each point

P ∈ Σt, the tangent space TPΣt is a vector subspace of the tangent space TPM to M in the

same point P : let us call p the projector from TPM into TPΣt.Let us indicate an orthonormal

frame in TPΣt as ei. So, if we choose the basis ei in TPΣt and eI in TPM, the matrix elements

of the projection operator p will be indicated as pi

I. While, using the coordinate basis, the matrix

elements will be pa

µ. Let us call n the normalized vector field orthogonal to Σt.

Now, let us use the connection ω to define two connections on Σt:

Γia ≡ 1 2p µ apiIǫIJKLnJωµKL (1.12) Kai ≡ pµapiInJωIJµ (1.13) Γi _{is an so(3)-connection on Σ}

tand it is compatible with ei if ωIJ is compatible with eI. While,

Ki

a is the extrinsic curvature of Σtif the relation

de + ω ∧ e = 0 (1.14) holds. Finally, let us define another so(3)−connection 1-form:

Aia≡ Γia+ γKai (1.15)

We can formulate GR using the variable Ai

a as lagrangian variable. The canonical conjugate

variable to Ai a is: Eia≡ 1 2kγǫ abc_ǫ ijkejbekc (1.16)

And now, we are ready to perform a Legendre transform. We will skip the details of the calculation and give the final result:

S[Eia, Aia, Na, N, Nj] = Z dt Z Σt d3x[EjbA˙ j b−NbVb(Eia, Aia)−N S(Eia, Aia)−NjGj(Eia, Aia)] (1.17) where Gj(Eia, Aia) = DaEja (1.18) Vb(Eia, Aia) = EjaFab− (1 + γ2)KaiGi (1.19) S(Eia, Aia) = kγ2 2p| det(q)|E a iEjb[ǫ ij kF k ab− 2(1 + γ2)K[ai K j b]] + (γ 2_{+ 1)k∂} a Ea i p | det(q)| ! Gi (1.20) where Fab and Da are, respectively, the curvature and the covariant derivative of the connection

Ai

a, whereas det(q) is the determinant of the 3-metric qabon Σt.

The variation of the action (1.17) with respect to Na, N and Nj gives the constraints:

Gj(Eia, Aia) = 0 (1.21)

Vb(Eia, Aia) = 0 (1.22)

Quantisation program

We would like to obtain a quantum theory of pure gravity starting from the classical canonical formulation with the Ashtekar-Barbero variables. We would like to do it carrying out the following steps:

1) Defining an Hilbert space (the kinematical Hilbert space Hk) and promoting the canonical

variables to self-adjoint operators satisfying the usual commutation relations corresponding to the Poisson brackets.

2) Promoting the constraints to self-adjoint operators: in this way, we will obtain the quantum constraint equations.

3) Finding the solutions to the quantum constraint equations.

4) Defining the scalar product in the vector space of the solutions to the quantum constraint equa-tions. In this way we will obtain an Hilbert space which we call physical Hilbert space Hp. The

scalar product will give the transition amplitudes between states and the transition probabilities. 5) Finding gauge invariant observables and promoting them to self-adjoint operators.

The kinematical Hilbert space

Let us define the kinematical Hilbert space Hk. Let us consider the three dimensional surfaces Σtin

terms of which we foliated spacetime. If we disregard the metric on them, they are indistinguishable between each other as a mathematical object and we can drop the subscriptt. The idea to define

Hk is to consider the space of complex functionals of the connection fields Aia(x) on Σ. Then, one

can define a measure on these functionals and consider the space of square-integrable ones: the latter space will be Hk. We would like to take into account not only the smooth connection fields,

but also the ”distributional” ones. We need a notion of smearing: the holonomy of the connection along a path. With these ideas in mind, we will now construct Hk.

First, we call an oriented closed graph γ in Σ a union of oriented smooth paths in Σ such that each endpoint of them coincides with at least two other endpoints. Given an oriented closed graph, we call edge of γ each one of its paths and vertex each endpoint of the paths.

Let γ be an oriented closed graph in Σ and {ei}Ei=1the set of its edges. Let Uei(A) be the holonomy

of the connection Aiaalong the path ei. Given f a smooth function f : SU (2)E→ C, we can define

a functional Ψf[A] of the connection in the following way:

Ψf[A] ≡ f ((Uei(A))

E

i=1) (3.1)

Let us define Vγas the vector space of all such functionals Ψffor all possible smooth f : SU (2)E→

C_{. Then, let us define V as:}

V ≡M

γ∈Γ

Vγ (3.2)

where Γ is the set of all oriented closed graphs in Σ. The vector spaces Vγ satisfy the following

property: if γ ⊆ γ′_{, then V}

γ ⊆ Vγ′. In fact, let {e_i}E

i=1 be the edges of γ and {ei}Fi=E+1 the paths

of γ′_{\γ; if Φ ≡ Ψ}

f ∈ Vγ, with f : SU (2)E → C, we can consider ˜f : SU (2)F → C defined by

˜

f ((gi)Fi=1) = f ((gi)Ei=1) for any (gi)Fi=1. Then Φ = Ψ_f˜, that is Φ ∈ Vγ′.

Now, we want to give V the structure of an Hilbert space. So, first of all, we have to construct a scalar product in V . Given Ψ, Φ ∈ V , there exists an oriented closed graph β such that Ψ, Φ ∈ Vβ.

In fact, Ψ and Φ will be linear combinations of functionals constructed on some graphs. So, using the property of the Vγ’s just proved and other elementary facts, we can take as β simply the union

of those graphs. So, to define the scalar product it suffices to consider elements of a Vγ. Then,

given Ψ ≡ Ψf, Φ ≡ Ψf′ ∈ V_γ, using the usual Dirac notation we define:

hΦ|Ψi = Z _YE

i=1

dgif′((gi)Ei=1)f ((gi)Ei=1) (3.3)

where dg is the Haar measure of SU (2), the symbol ¯ indicates the complex conjugate and E is the number of edges of γ.

Now, we finally define Hk as the completion of V in the norm induced by the scalar product. Hk

We now promote our canonical variables to operators on Hk: Aia(x) → ˆAia(x) ≡ Aia(x) (multiplicative operator) (3.4) Eia(x) → ˆEia(x) ≡ i~ δ δAi a(x)

(functional derivative operator) (3.5) Our operators verify the desired commutation relations:

[ ˆAai(x), ˆEbj(y)] = i~δ a

bδijδ(x, y) (3.6)

(3.7) From now on, we will drop the symbol ˆ to distinguish the classical variables from the correspond-ing operators because this information will be clear from the context.

The Gauss law

Let us solve the first constraint:

DaEia(x)|Ψi = 0 (4.1)

which is known as Gauss law. Classically the Gauss law constraint is the generator of local SU (2) gauge transformations. We have a natural way to implement these transformations in the quantum theory. Let Ω : Σ → SU (2) be a local SU (2) gauge transformation: under it, the connection changes as follows:

A 7→ AΩ= ΩAΩ−1+ ΩdΩ−1 (4.2)

Then, for each Ω, we define the operator UΩ: Hk → Hk as:

UΩΨ[A] = Ψ[AΩ−1] (4.3)

It can be shown that UΩis a unitary operator in Hk and that UΩUΩ′ = U_Ω◦Ω′ for any Ω, Ω′ local

SU (2) gauge transformations. It can be also shown that the Gauss law quantum constraint (4.1) is simply the requirement that the functional Ψ is invariant under all the UΩ. Then, the idea to

solve the constraint (4.1) is the give a more rigorous construction of the following formal object: PG≡ ”

Z

DΩUΩ” (4.4)

that is the sum of the UΩoperators on all possible Ω. Here is a more rigorous definition of PG. Let

Ψ ∈ V . So, there is a γ and an f such that Ψ = Ψf ∈ Vγ. Let i ∈ {1, 2, ..., V } label the vertexes

of γ and L = {(i, j)|the vertex i and the vertex j are linked by an edge in γ}. Then: PGΨ ≡ Z _YV l=1 dΩif ((ΩiUe(i,j)(A)Ω −1 j )(i,j)∈L) (4.5)

It can be proved that PG really satisfies

UΩPGΨ = PGΨ (4.6)

for any Ψ and Ω. So, given a Ψ ∈ V , PGΨ is a solution of the constraint (4.1). Actually, it is

possible to prove that all the solutions of the constraint (4.1) in Hk are of the form PGΨ with

Ψ ∈ V . Moreover, PG is a projector, that is it verifies PG2 = PG and PG† = PG. Let us call HG the

image PG(Hk) of PG, which is an Hilbert subspace of Hk. Let us give a basis for HG.

Before doing this, we need to define the notion of intertwiner. Let us take n irreducible represen-tations Ri (i = 1, ..., n) of SU (2). Let Vi be their associated vector spaces. Let call R ≡Nni=1Ri

the tensor product representation. R can be decomposed into the direct sum of irreducible repre-sentations: R= m M j=1 Tj (4.7)

Let us call Wj the vector spaces of the representations Tj. Let us indicate all the trivial

define as intertwiners between the representations Ri all the vectors of the space W0≡Lkj=1Wj.

Let us put it in other words. The elements of the vector space Nni=1Vi =Lmj=1Wj are tensors

vα1...αn _{with n indexes: the intertwiners are just those tensor v}α1...αn _{that are invariant under the}

action of the group, that is, for all U ∈ SU (2):

1 Rα1 β1 (U ) . . . n Rαn βn (U )v β1...βn_{= v}α1...αn _(4.8)

where we have indicated with Riαβ (U ) the matrix that represents U in Ri.

Now, let us extend a bit the definition of intertwiner. Given a representation R, we indicate with R∗ the dual representation. Of course, V∗ is the vector space of R∗, if V is the vector space of R. Let us take two finite sets Ri (i = 1, ..., n) (with vector spaces Vi) and Si (i = 1, ..., m) (with

vector spaces Wi) of representations of SU (2). We call intertwiner between the representations Ri

and the dual representations S∗i, each tensor vα1...αnβ1...βm in the space

Nn

i=1Vi⊗Nmi=1Wi∗ that

is invariant under the action of the group (the definition of being invariant is the natural extension of (4.8) ).

Now, let us indicate withR the unitary irreducible representation (UIR) of SU (2) of spin quantumJ number J for all J ∈ {0,1

2, 1, 3

2, 2, . . .}. We call J

R∗the dual representations. Then, for each couple of finite sets of UIR, let us choose an orthonormal basis in the space of intertwiners between the representations of the first set and the dual ones corresponding to the second. So, we are ready to describe a basis for HG. The states of the basis are labeled by a so-called a colored graph, that is

the following triple:

1) an oriented closed graph γ;

2) an assignment of an irreducible representation

J(i,j)

R of SU (2) to each edge e(i,j) ((i, j) ∈ L) of γ;

3) to each vertex i, an assignation of an element ωi of the basis of the intertwiners between the

representations assigned to the edges that enter in i and the dual representations corresponding to the ones assigned to the edges that exit from i.

Now, given a colored graph, the wave functional of the corresponding state is: S[A] = ( O (i,j)∈L )R(i,j)(Ue(i,j)) · ( O i∈ ωi) (4.9)

where R(i,j)(Ue(i,j)) is the matrix that represents Ue(i,j) in

J(i,j)

R and the symbol · simply indi-cates the tensor index contraction. Such a state is called a spin network. Let us define VG ≡

Span({spin networks}): it will be useful later.

The next step that we will face is to solve the constraint of (1.22) and to do that we will start working directly with HG instead of Hk.

Diffeomorfism constraint

Let us solve the constraint

(EjaFab− (1 + γ2)KaiGi)|Ψi = 0 (5.1)

that is the so-called diffeomorfism constraint.

As in the case of the Gauss law, classically the constraint (5.1) is the generator of a certain group of transformations, which are combinations of certain SU (2) gauge transformations and the 3d diffeomorfisms. We have already obtained a space of states invariant under local SU(2) trasformations. So, now it will suffice to focus on 3d diffeomorfisms: let us implement them in the quantum theory. First of all, for technical reasons that we do not discuss here, we will extend the 3d diffeomorfisms group to the ”extended 3d diffeomorfisms” group, that is the set of any α : Σ → Σ such that α is continuous, invertible and α and α−1 _{are smooth everywhere except at}

zero or finitely many points. We will call D∗_{the set of the ”extended 3d diffeomorfisms”.}

Under an α ∈ D∗_{, the connection A transforms in the following way:}

A 7→ α⋆A (5.2)

where the problems that could arise in the finitely many isolated points where α can not be smooth are not important when we consider the functionals because of the smearing we do with the holonomy. Given α ∈ D∗, we define the corresponding operator Uα: Hk → Hk:

UαΨ[A] ≡ Ψ[(α−1)⋆A] (5.3)

Now, it can be proved that if a Ψ ∈ HGis invariant under the transformations Uαfor any α ∈ D∗,

it would be a solution of the constraint (5.1). But in HG there are not such Ψ. The problem

we have met here is analogous to the one that we meet if we want to find the eigenstates of the hamiltonian of the free particle in non-relativistic quantum mechanics: in L2_{(R) there are not any}

solutions and we have to use distributions. So, let us use an analogous procedure: let us make our language more powerful considering the space of distributions D ≡ {ϕ : VG → C| ϕ linear},

in which VG and HG are naturally immersed. There is a natural way to implement the D∗ in the

context of D using the Uα: let us define Υα: D → D:

(Υαϕ)(Ψ) ≡ ϕ(Uα−1Ψ) (5.4)

for any Ψ ∈ Hk.

Now, let us define the linear function PD : VG→ D in the following way:

(PDΨ)(Ψ′) ≡

X

α∈D∗

hUαΨ|Ψ′i (5.5)

for any Ψ, Ψ′_{∈ V}

G. Notice that the idea behind the definition (5.6) of PD is more or less the same

as the one intuitively expressed in the formula (4.4) for PG.

At a first sight, one could think that the sum in (5.6) involves really a lot of terms, infinitely many. But it is not like that: it has a finite number of terms! In fact, Ψ and Ψ′ are (finite) linear

combinations of spin networks. Then, if an α ∈ D∗ _{modifies the graph of a spin network state,}

we get a spin network orthogonal to the first one. If an α ∈ D∗ does not modify the graph of a spin network state, it could leave it unchanged or modifying the ordering and the orientation of the links: these operations can produce at most a finite multiplicity in the sum in (5.6).

Let us call KDthe image of PD. It can be easily proved that the states in KD are diff-invariant.

One could implement the SU (2) gauge transformations on D with a natural definition and show that the states in KD are invariant under those transformations, too. Actually, I have not checked

this last fact and I have not found it in the literature, but I am pretty sure that there would be no problems.

Finally, here is the definition of the scalar product on KD:

hPDΨ|PDΨ′i ≡ (PDΨ)(Ψ′) (5.6)

Let us briefly say that a basis for KD is labeled by equivalence classes of colored graphs,

according to which two of them are equivalent if their graphs differ by an extended diffeomorfism. So, only the topological information about the graph is important: one could expect that from a diff-invariant theory such as gravity. Moreover, even if HG is a huge space, KD is a separable

Spin foam approach

We would like now to solve the last remaining constraint (1.20): the scalar constraint. As one can see, its expression is complicated. Are there other variables in which also the expression of the last constraint (1.23) becomes somehow manageable, for example non-polynomial? Anyway, we could try to follow another route.

In non-relativistic and in special relativistic quantum theories one has at least two languages to speak about mechanics: the hamiltonian and the path integral formulation. So far, we have described a canonical approach to the problem of quantum gravity. One could try to develop a path integral approach. Let us see what the first problems that we meet along this way are. For example, let us formally set out the problem using the standard metric variables. We have a region R of the space-time manifold M bounded by the 3d surfaces Σ1and Σ2. Let us indicate the metric

on Σ1and Σ2 with qab and q′abrespectively and the corresponding quantum states with |qabi and

|q′

abi. Now, we could say that the transition amplitude hqab|qab′ i is given by:

hqab|qab′ i =

Z

g∈SR(q,q′)

Dg expiS[g] _(6.1)

where S[g] is the action of GR in the metric variables and SR(q, q′) is the set of all the metrics

on R that induce the fixed metrics qab and q_ab′ on Σ1 and Σ2 respectively. So, first of all, we

have to define the measure Dg and we do not know how to do give sense to such a functional integration. Moreover, we have to deal with gauge invariance, in the sense that we would like to perform the “sum over space-time histories” up to 4d diffeomorfisms and consider qab and qab′ up

to 3d diffeomorfisms.

A lot of other questions can be asked about the formula (6.2), but perhaps one of its most important problems is the fact that we do not have a definition of the quantum states |qabi and |qab′ i. But, we

succeed to define the states of pure gravity at least at the kinematical level in the context of LQG. So, for example, we could think about trying to develop a path integral formulation starting from a classical framework that is the covariant version of the Ashtekar variables: the Holst action. But we have not been able to define a path integral using the Holst action, so far.

Is there a classical theory which is is somehow close to gravity and for which we are able to define and handle a path integral formulation? Yes, there is: it is called BF theory. Are there others?

6.1

Classical BF theory and Plebanski action

In this section, we will explain how BF theory is related to GR.

Let us define classical BF theory. It can be defined in any finite dimension, but we will focus on 4 dimensional BF theory, in particular the one for the group SO(4). So, let us take a 4 dimensional differentiable manifold M. On M we will consider an SO(4) Lie-algebra valued 2-form BIJ _and

an SO(4) connection AIJ_{. The action of BF theory is the following:}

SBF[B, A] =

Z

M

where FIJ_{[A] is the curvature of the connection A}IJ_{. BF theory has is a topological theory, that}

is, it has no local degrees of freedom.

If we take BF theory and add some constraints on B, we will be very close to Euclidean GR. In fact, let us consider two Langrangian multiplier fields: a scalar λIJKLand a 4-form µ. We require

that λ satisfies λIJKL = −λJIKL = −λIJLK = λKLIJ. Then, we consider the SO(4) Plebanski

action:

SP l[B, A, λ, µ] =

Z

M

[BIJ∧ FIJ[A] + λIJKLBIJ∧ BKL+ µǫIJKLλIJKL] (6.3)

Variation with respect to µ gives ǫIJKL_λ

IJKL = 0. While, variation with respect to λ imposes

some constraints on the field B. Their solutions are:

B = ± ∗ (e ∧ e) and B = ±e ∧ e (6.4) where eI _{is a 1-form on M. If we substitute the first of these solutions in the SO(4) Plebanski}

action, we get the Palatini action of GR: SP a[e, A] =

Z

M

eI∧ eJ∧ ∗FIJ[A] (6.5)

6.2

Path integral for BF theory

Let us construct the path integral machinery for the BF theory.

First, we will discretize classical BF theory. To do this, we introduce a triangulation T of the manifold M. We will also consider the dual T⋆ _{of the triangulation T . Let us call B}

f ∈ so(4) the

integral of B on the face f , for any f ∈ T⋆_{. Then, let us call g}

e ∈ SO(4) the holonomy of the

connection A along the edge e, for any e ∈ T⋆_{. Now, we are ready to define the discretized version}

of the action of BF theory:

S⋄BF[Bf, ge] = X f ∈T⋆ BfIJtr[ Y e∈∂(f ) geτIJ] (6.6)

where we have called ∂(f ) the set of the edges the bound the face f and we have indicated the generators of SO(4) as τIJ. In this ”discretized” framework, we can define the path integral of

discretized BF theory as:

Z ≡Z Y f ∈T⋆ dBf Y e∈T⋆ dgeeiS⋄BF[Bf,ge] (6.7)

Integrating over Bf, we get:

Z =Z Y e∈T⋆ dge Y f ∈T⋆ δ( Y ε∈∂(f ) gε) (6.8)

Now, we can use the expansion of the delta function: δ(g) =X

(i,j)

d(i,j)tr[R(i,j)(g)] (6.9)

where the couple of half-integers (i, j) labels the unitary irreducible representations of SO(4), d(i,j)

is the dimension of the representation (i, j) and R(i,j)_{(g) is the matrix corresponding to g in the}

representation (i, j). So, using formula (6.9), we obtain:

Z = X ((i1,j1),...,(iF,jF)) Y f ∈T⋆ d(if,jf) Z Y e∈T⋆ dgetr[R(if,jf)( Y ε∈∂(f ) gε)] (6.10)

where F is the total number of faces in T⋆_{. The integration over the g}

evariables can be performed

Z =

ι1,...,ιE((i1,j1),...,(iF,jF))f ∈T⋆

d(if,jf)

v∈T⋆

{15j}v (6.12)

where v labels the vertexes in T⋆ _{and the symbol 15j stays for:}

{15j}v ≡ F((i1, j1), ..., (i10, j10), ι1, . . . , ι5) ≡ X α1...α10 ωα1α6α9α5 ι1 ω α2α7α10α1 ι2 ω α3α7α8α2 ι3 ω α4α9α7α3 ι4 ω α5α10α8α4 ι4 (6.13)

where (i1, j1), ..., (i10, j10) are the representations associated to the 10 faces bounded by the vertex

v, each one of the indexes αa (a = 1, . . . , 10) is in the representation (ia, ja) and ι1, . . . , ι5 is a

choice of basis elements for the 5 spaces of intertwiners, each one between four representations as one can read from the groups of indexes αa in each ωιb.

So, even if the calculations performed so far are quite formal and have a number of mathematical problems, we are somehow able to write a formal expression for the partition function of the BF theory path integral. Can we manage a similar calculation-reasoning with GR using the Plebanski action? We could try to do this by dealing with the constraints that make Plebanski theory different from BF theory. We could try to implement somehow these constraints directly at the quantum level. A number of Spin Foam models try to go through this way. By now, we do not know yet which is the right one. Moreover, one should clarify the meaning of the word right used in the previous sentence.

Now, after the example of BF theory, we can give a general definition of a Spin Foam model.

6.3

A general definition of a Spin-Foam model

A Spin-Foam model is basicly the assignment of a complex amplitude A(T ) to any triangulation T of a given four dimensional manifold M. The triangulation consists in the union ∪4

i=2Ti of the

set of its faces T2, the set of its tetrahedra T3 and the set of its 4-simplices T4. The amplitude A

is constructed from the representation theory of a given Lie group G that we assume compact for simplicity. To do so, one first colors each face f ∈ T2 with an unitary irreducible representation

(UIR) jf of G and each tetrahedron t ∈ T3 with intertwiners ιt between representations coloring

its four faces. Then, one associates an amplitude A2(jf) to each face f , an amplitude A3(ωt, jft)

to each tetrahedron t which depends on the intertwiner ωtand on the representations coloring its 4

faces ft, and an amplitude V (ωts, jfs) to each 4-simplex s which depends on the representations jfs

and ωts coloring its 10 faces fsand 5 tetrahedra ts. Finally, the spin-foam amplitude is formally

defined by the series

A(T ) ≡ X {jf},{ωt} Y f ∈T2 A2(jf) Y t∈T3 A3(ωt, jft) Y s∈T4 V (ωts, jfs) (6.14)

where the sum runs into a certain subset of UIR and intertwiners of G. The sum is a priori infinite and therefore the amplitude is only defined formally at this stage unless it is convergent. Notice that in all the models that have been studied in the literature, the amplitude A3 is assumed to

depends on the intertwiners ωt only. The function V is precisely the vertex amplitude of the

Spin-Foam model. To finish with this brief introduction of Spin-Foam models, let us mention that Spin-Foam models can also be defined for any dimensional manifold M.

The problem of the scalar

constraint

Let us go back to the question of solving the scalar constraint. This problem is that of finding a projector P from KD to the kernel of the scalar constraint and it is somehow equivalent to find

the physical scalar product in our Hilbert space. In fact, if we have P , the scalar product between two states |Ai, |Bi ∈ KD will be:

hB|Aiphysical= hB|P |Ai (7.1)

where on the right-hand side we have used the scalar product in KD. Actually, we do not know

precisely what kind of mathematical object P is. Perhaps, if we want a theory with a certain level of mathematical rigor, we have not to think P as an operator on an Hilbert space at all. Perhaps, we have to use a mathematical framework similar to the one used in the case of the diffeomorfism constraint. But we will skip this problem for the moment, pretending that it does not exist. In principle, a Spin Foam model should give precisely the value of the physical scalar product between spin network states: given two colored graphs σ1 and σ2, we should have:

hσ2|σ1iphysical=

X

T ∈S(σ∞,σ∈)

A(T ) (7.2)

where S(σ1, σ2) is the set of all triangulations T ∈ T (T is a subset of all triangulations specified

by the particular Spin Foam model) colored with UIR and intertwiners such that the bound of their dual ∂T⋆_{= σ}

1∪ σ2and A(T ) is of course the spin-foam model amplitude we are considering.

So, we can try to go through the following way to attach the scalar constraint problem: we can consider several Spin Foam model and look for an operator P in the LQG framework, such that the corresponding physical scalar product is equal to the transition amplitude computed with the Spin Foam model for any couple of spin network states:

hσ2|P |σ1i =

X

T ∈S(σ∞,σ∈)

A(T ) (7.3)

Then, one can see if the operator P is precisely the projector into the scalar constraint kernel. If for some Spin Foam model, the answer to this question is yes, in an optimistic perspective, one has solved two big problems at the same time: solving the last remaining constraint in LQG and finding the link between the canonical and the covariant approach to quantum gravity.

So, we will try to go through this way. But, we will start with a bit simpler problem. Instead of considering all the spin-foam amplitude, we will focus on the vertex amplitude only.

We will consider Euclidean Spin-Foam models associated to the group G = SU (2) × SU (2) (which is the double cover of SO(4)). They are characterized by their vertex amplitude V : the vertex amplitude is the weight associated to a 4-simplex; it is therefore a function V (Iij, ωi) of the

G-representations Iij coloring the 10 faces of the 4-simplex and of the G-intertwiners ωi associated

in the boundary of the 4-simplex. We want to interpret this vertex amplitude V as the physical scalar product between two spin-network states: the 1-tetrahedron state τ1 and the 4-tetrahedra

state τ4 associated to spin-networks respectively dual to one tetrahedron and to four tetrahedra

as illustrated in the figure (7.1). The free ends of these spin-networks coincinde and therefore τ1

and τ4are particular cylindrical functions of the same graph, denoted ˜Γ, as illustrated in the figure

(9.1). The graph ˜Γ is the union of the 4-simplex graph Γ with four free edges and it was introduced to take into account the free ends of the states τ1 and τ4.

ω1

ω2 ω3

ω4

ω5

Figure 7.1: Illustration of the 1-tetrahedron state τ1 on the left and the 4-tetrahedron state τ4 on the

right. Vertices, labelled by i∈ {1, 5}, are colored with intertwiners ωi and edges ℓij with representations

Iij. The 4 free ends are colored with representations I0i.

More precisely, we will construct on operator P acting on the space of cylindrical functions Cyl(˜Γ) such that its matrix elements are related to the vertex amplitude of Spin-Foam models as follows:

hτ4, P τ1i = N V (Iij, ωi) (7.4)

where N is an eventual normalization factor. In that sense, the matrix element hτ4, P τ1i would be

the physical scalar product between the kinematical states τ1and τ4. In fact, the bra-ket notation

for the physical scalar product might be misleading because mathematically P is a linear form on the space Cyl(˜Γ), i.e. P ∈ Cyl(˜Γ)∗_{, abusively called a “projector”, and the physical scalar}

product is hτ4, P τ1i = P (τ4τ1). In the context of Gelfand-Naimark-Segal theory (see the

Ashtekar-Lewandowski review [1] and references therein), P , if it satisfies some additional properties, would be a state and would allow to construct the whole physical Hilbert space in principle.

We will find a solution for the projector P for different Spin-Foam models: the topological SU (2) BF model whose vertex VBF is the 15j symbol of SU (2) (this system has no physical relevence);

the BC model whose vertex VBC is the so-called 10j symbol; the new model whose vertex VEP R

has been defined recently and also the FK model whose vertex construction is a direct extension of the EPR one. The projector PBF associated to the topological model is a multiplicative operator

which acts only on the edges of the spin-networks and imposes that the connection is flat. The projectors PBC and PEP Rrespectively associated to the BC and the EPR models act both on the

vertices (as derivative operators, in the sense that it involves left and right invariant derivatives) and on the edges of spin-networks. Note that we will construct one solution of P and we will not precisely address the question of the unicity.

To carry out this program, we have to analyze first the vertexes of the Spin Foam models that we are going to consider and to find some new formulas for them.

The vertex

We will consider exclusively the case where M is 4-dimensional and we study some properties of the vertex amplitude V only. Therefore, we will not mention the amplitudes A2 and A3when we

discuss the Spin-Foam models in the sequel; as a result, we will omit any discussion concerning the amplitude A and a fortiori the question of its convergence.

Furthermore, we will consider Euclidean Spin-Foam models only which are associated to the com-pact Lie groups G = SU (2) (for the topological model) or G = SU (2) × SU (2) (for the BC and EPR models). Letters I, J, · · · label unitary irreducible representations of the group G and the associated vector spaces are denoted UI, UJ, · · · . When G = SU (2), I is a half-interger whereas it

is a couple of half-integers when G = SU (2) × SU (2). Due to the compactness of G, each repre-sentation I is finite dimensional and associates to any g ∈ G a finite dimensional matrix which will be denoted RI_{(g) when G = SU (2) × SU (2) and D}I_{(g) in the other case. To a representation I}

is associated a contragredient (or a dual) representation I∗ _{such that R}I∗

(g) =t_RI_(g−1_{) and the}

same for the SU (2) representations DI∗

; it is common to identify U∗

I ≡ UI∗ to U_I. More precision

concerning the representation theory of the groups G will be given later.

The vertex V (Iij, ωi) is then a function of the 5 intertwiners ωicoloring the 5 tetrahedra (which

are ordered and labelled by i ∈ {1, 5}) of a 4-simplex and of the 10 representations (Iij)i<j of G

coloring the 10 faces at the intersections of the tetrahedra i and j; ωi: ⊗j>iUIij → ⊗j<iUIji is an

intertwiner between the representations Iij “meeting” at the tetrahedron i. In the next part, we

are going to show that the vertex amplitude of all the models we consider can be written as an integral over 10 copies of the 3-sphere S3 _{(which we identify with SU (2)) as follows:}

V (Iij, ωi) =

Z _Y

i<j

dxij
C(xij) V(Iij, ωi; xij) (8.1)

where C(xij) is a universal function, in the sense that it is model independent, which reads

C(xij) ≡

Z _Y5 i=1

dxi
δ(x−1ij xix−1j ) . (8.2)

V is a model dependent function of the variables xij. As we will see in the next Section, such a

formula will be crucial to link Spin-Foam models with Loop Quantum Gravity.

8.1

A General expression of the vertex

There exists many equivalent ways to define the vertex amplitude of a Spin-Foam model. For our purposes, it is convenient to view the vertex amplitude as a “Feynman graph” evaluation of a closed oriented graph which is dual to a 4-simplex. The dual of a 4-simplex Γ is in fact topologically equivalent to a 4-simplex and then consists in a set of 5 vertices linked by 10 edges: we endow the set of vertices with a linear ordering such that the vertices are labelled with an integer i ∈ {1, 5}; this ordering induces a natural orientation on the links, indeed the link ℓij between the edges i

following “Feynman rules”: each oriented link ℓij, with i < j, is associated to a UIR of G denoted

Iij (the opposite link ℓji is associated to the contragredient representation denoted for simplicity

Iji= Iij∗); each vertex i is associated to an intertwiner ωi: ⊗j>iUIij → ⊗j<iUIji. As a result, the

“Feynman evaluation” of such a graph is the scalar obtained by contracting the 10 propagators with the 5 intertwiners and gives the vertex amplitude which formally reads:

V (Iij, ωi) = h 5 O i=1 ωii ≡ X {eij} 5 Y i=1 hO j<i eji|ωi| O j>i eiji (8.3)

where eijruns over the finite set of a given orthonormal basis of UIij and we have used the standard

bra-ket notation to denote the vectors |eiji of UIij and the dual vectors heij|.

In order to have a more useful formula, it will be convenient to trivially identify ωi with an

element of Hom(⊗j6=iUIij, C) and then to notice that ωi is completely caracterized by a vector

vi∈ ⊗j6=iUI∗ij. These vectors can be written in the form vi=

P

(aij)α

(aij)

i ⊗j6=ivaij where (aij)j6=i

is a set whose elements label vectors vaij ∈ UIij, α

(aij)

i are complex numbers and the sum is finite.

The explicit relation between ωiand vi is the following:

ωi = hvi| Z dg O j6=i RIij_{(g) ∈ Hom(}O j6=i UIij, C) (8.4)

where we have used the SU (2) × SU (2) notations for the representations and R dg is the Haar measure of G. As a result, the vertex amplitude can be reformulated as a multi-integral over G according to the formula:

V (Iij, ωi) = X (aij) 5 Y i=1 α(aij) i Z ( 5 Y i=1 dgi) h⊗i<jvaij| O i<j RIij_(g ig−1j )| ⊗i>jvaiji (8.5)

which can be written in the following more compact well-known form V (Iij, ωi) = Z ( 5 Y i=1 dgi) (⊗5i=1vi) · ( O i<j RIij_(g ig−1j )) (8.6)

where the dot · denotes the appropriate contraction between the vectors vi and the matrices of

the representations. This vertex amplitude is in fact rather general and caracterizes partially a large class of Spin-Foam models. It is general because we have for the moment a total freedom in the choice of the representations and the intertwiners; it is nonetheless only partial because we do not consider the amplitudes associated to faces and tetrahedra. To go further in the study of this amplitude, we need to recall some basic results on the representation theory of SU (2) × SU (2). Representation theory of G: basic results

Let us start with the group SU (2): its representations are labelled by a half-integer, the spin I; they are finite dimensional of dimension dI = 2I + 1 and we denote by |I, ii with i ∈ [−I, I] the

vector of an orthonormal basis of UI. The group G = SU (2) × SU (2) is the double cover of SO(4).

Any of its elements g can be written as a couple (gL, gR) of two SU (2) group elements. Its Unitary

Irreducible Representations (UIR) are labelled by a couple of (integers or half-integers) spins (I, J): they are finite dimensional and the vector space UIJ = UI⊗ UJ of the representation (I, J) is the

tensor product of the two SU (2) representations vector spaces UI and UJ. Therefore, the family

of vectors (|I, ii ⊗ |J, ji)IJij form an orthonormal basis of UIJ. The action of g ∈ G in this basis

K=|I−J|

This decomposition provides indeed another orthonormal basis of UIJ, given by the family of

vectors (|K, ki)Kk where K ∈ [|I − J|, I + J] and k ∈ [−K, K] as usual. The changing basis

formulae are given in terms of the Clebsch-Gordan coefficients hKk|IiJji as follows: |Iii ⊗ |Jji =X

K,k

hKk|IiJii |Kki and |Kki =X

IJij

hKk|IiJii |Iii ⊗ |Jji . (8.9) To write the action of G on the basis elements |Kki, it is convenient to find the subgroup H ⊂ G which leaves the subspaces UK of the decomposition (8.8) invariant and then to identify G with

the space G ≃ H × (G/H). In fact, it is immediate to see that H ≃ SU (2), the coset G/H is isomorphic to the sphere S3_{and then we identify G with SU (2) × S}3_{. Notice that the identification}

we have just mentionned is not canonical because G admits many SU (2) subgroups; therefore, to make this identification well defined, one has to precise which SU (2) subgroup one is talking about. In our case, the SU (2) subgroup is the diagonal one, i.e. it is the group of the elements (gL, gR)

where gL= gR. As a result, the explicit mapping between G and SU (2) × S3is:

G −→ SU (2) × S3 (gL, gR) = (u, ux) 7−→ (u, x) = (gL, gL−1gR) . (8.10)

This mapping is of course invertible and its inverse is trivially given by:

SU (2) × S3−→ G (u, x) 7−→ (u, ux) . (8.11) The multiplication law (gL, gR)(g′L, gR′ ) = (gLgL′, gRg′R) induces the multiplication rule

(u, x)(u′, x′) = (uu′, u′−1xu′x′) (8.12) in the SU (2) × S3_{representation of G. In particular, the inverse of the element (u, x) is given by}

(u, x)−1= (u−1, ux−1u−1). The diagonal terms u ≡ (u, 1) and the pure spherical terms x ≡ (1, x) will be relevant in the following construction.

Let us now come back to the action of G on the the vectors |K, ki of the vector space UIJ;

this action is best written and simpler using the factorization SU (2) × S3 _{of G. Indeed, a simple}

calculation shows that

RIJKkLℓ(u) = RIJKkLℓ(u, u) = X m1,m2 hKk|Im1Jm2iDIm1,n1(gL)D J m2,n2(u)hIn1Jn2|Lℓi = δK,LD K kℓ(u) RIJKkLℓ(x) = RIJKkLℓ(1, x) = X ijj′

hKk|IiJji hIiJj′|Lℓi DJjj′(x) . (8.13)

where we have introduced the notation RIJ

KkLℓ(g) ≡ hKk|RIJ(g)|Lℓi for the SU (2) × SU (2) matrix

elements. As expected, we see that u ∈ SU (2) leaves any SU (2) representation spaces UK of the

decomposition (8.8) invariant whereas x moves the vectors from one SU (2) representation space to another. This closes the brief review on SU (2) × SU (2) representations theory.

The vertex amplitude as an integral over several copies ofS3

We make use of the basic properties on representations theory recalled above to write the general formula of the vertex amplitude (8.6) in the form (8.1). To do so, one splits the integrations over the group variables gi ∈ G in the formula (8.6) into integrations over the xi ∈ S3 variables and

integrations over the ui∈ SU (2) variables using the isomorphism (8.10) and one obtains:

V (Iij, ωi) = Z ( 5 Y i=1 dxi)( 5 Y i=1 dui) (⊗5i=1vi) · ( O i<j RIij_(u i)RIij(xixj−1)RIij(u−1j )) (8.14)

u1 u1 u1 u1 u1 u1 u1 u1 u−1₂ u−1₂ x1x−12 u2 u2 u2 u2 u2 u2 u−1_{3 u}−1 3 u−1₃ u−1₃ u3 u3 u3 u3 u−1₄ u−1₄ u−1₄ u−1₄ u−1₄ x1x−13 u−1₄ u4 u4 x1x−14 x1x−15 x2x−15 x2x−14 x2x−13 x3x−14 x3x−15 x4x−15 u−1₅ u−1₅ u−1₅ u−1₅ u−1₅ u−1₅ u−1₅ u−1₅ v1 v5 v2 v3 _v4

Figure 8.1: This picture is a graphical representation of the integrand in the formula (8.14) defining the vertex amplitude. Each line are doubled because it carries a representation of SU (2) × SU (2) and the single lines in the pair colored with (I, J) are colored by I and J separately. Furthermore, the single lines are endowed with bullets that represent the insertion of SU (2) group elements: the small ones are associated to diagonal elements ui∈ SU (2) whereas the big ones are associated to

spherical elements xix−1j ∈ S3. The vectors vi are represented by boxes and they are contracted

with the free ends of the graph.

where RI_{(u) ≡ R}I_{(u, 1) (resp. R}I_{(u, u)) and R}I_{(x) ≡ R}I_{(1, x) (resp. R}I_{(1, x)) are the matrices of}

SU (2) × SU (2) representations I in the SU (2) × S3 _{(resp. SU (2) × SU (2)) formulations. To have}

a “geometrical” intuition of this formula, we give a graphical representation of the integrand in the figure (8.1) below. In the models we are going to consider explicitly in the sequel, we can perform the integrations over the ui variables; therefore we formally perform the integration over the ui’s

in the general formula (8.14) and we obtain a formula for the vertex amplitude as an integral over 5 copies of S3 _only: V (Iij, ωi) = Z ( 5 Y i=1 dxi) (⊗5i=1νi) · ( O i<j RIij_(x ix−1j )) . (8.15)

The integrations over the five SU (2) variables ui have been hidden in the following definition of

the vectors νi∈ ⊗j6=iUI∗ij:

νi ≡ X (aij) α(aij) i Z du (⊗j>ihvaij|R Iij_(u i)) ⊗ (⊗j<iRIji(u)−1|vaiji) (8.16)

u−1₂ u−1₂ u−1₂ u−1₃ u−1₃ u−1₄ u−1₅ u−1₅

Figure 8.2: Structure of the node i = 1. Four pairs of edges are attached at each node of the graph: each edge are colored with a SU (2) representation. The bullets illustrate the inclusions of SU (2) variables ui or S3 variables xix−1j . Notice that, in the SU (2) × SU (2) formulation, each pair of

lines is associated to the element (gL, gR), gL corresponding to the left line and gR to the right

one.

Before considering specific examples, let us add one more important remark. The vertex am-plitude can be trivially reformulated as an integral over 10 copies of G as follows:

V (Iij, ωi) = Z (Y i<j dxij) C(xij) (⊗5i=1νi) · ( O i<j RIij_(x ij)) (8.17)

where the contraint C(xij) is a distribution which imposes, rougthly speaking, xijto be a

“cobound-ary”, i.e. of the form xix−1j . An explicit formula for C(xij) is simply given by the integral:

C(xij) = Z ( 5 Y i=1 dxi) Y i6=j δ(x−1ij xix−1j ) (8.18)

where δ is the SU (2) delta distribution. It is possible to perform the above integration whose result is simply given by the product of five delta distributions:

C(xij) = δ(x123) δ(x234) δ(x345) δ(x451) δ(x512) (8.19)

where xijk= xijxjkxkiand, by convention, xij= x−1ji . The interpretation of the constraint C(xij)

will become clear in the last chapter where we make the link with the canonical quantization. To conclude, we underline that we have finally found the desired formula (8.1) for the vertex ampli-tude with the announced expression of the distribution C(xij) and the model dependent function

V(Iij, ωi; xij) = (⊗5i=1νi) · (Ni<jRIij(xij)) is a particular contraction of five SU (2) matrices.

8.2

Vertices of particular models

This part is devoted to study some aspects of the vertex amplitude (8.15) for the topological model, the BC model and the EPR model. In fact, these models differ only by the choice of the intertwiners ωi or equivalently the vectors viwhich are their building blocks. Thus, to understand

the construction of these models and their differences, one has to understand the definition of their associated intertwiners. For that purpose, let us start by recalling basic properties of intertwiners. First of all, in Spin-Foam models, we are interested in 4-valent intertwiners only. The 4-valent intertwiners between four given representations form a (normed) vector space of finite dimension. In the case where G = SU (2), one can exhibit three canonical (natural) orthogonal basis (labelled by an index ǫ ∈ {+, −, 0} that indicates the “coupling channel”) presented in the figure (8.3). Whatever the basis we choose, any of its element is completely characterized by the representation appearing in the intermediate channel in the tensor product decomposition. Therefore, one often identifies the element of each basis with a representation. We will use the notations ιǫ(α) to denote

these results to construct basis of SU (2) × SU (2) 4-valent intertwiners. In particular, one can naturally exhibit nine “tensor product” basis labelled by a couple (ǫ, ǫ′). However, we will consider in the sequel only the three basis of the type (ǫ, ǫ) which will be labelled by a single ǫ for simplicity: elements of the basis ǫ are denoted ιǫ(α) as in the SU (2) case but with the difference that α is now

a couple of SU (2) representations.

ι+(α) ι−(α) ι0(α)

Figure 8.3: The three canonical basis of the space of 4-valent intertwiners. The intermediate channel is endowed with the representation α.

Now, we are ready to define the intertwiner ωifor the model we are interested in. Afterwards,

we are going to make the general abstract formula of the vertex amplitude more concrete and more useful for studying its properties.

The topological model

We start with the simplest, certainly the more mathematically precise but non-physical model. The topological model is the path integral for discretized BF theory with gauge group SU (2). As we have seen before in the case of SO(4), given a triangulation T of a 4-dimensional manifold M, one can discretize the BF action to be well-defined on this triangulation and the path integral ZBF(T ) of the discretized action can be formulated as a state sum or equivalently a Spin-Foam

model: ZBF(T ) = X {jf},{ωt} Y f ∈T2 dim(jf) Y t∈T3 dim(ωt)−1 Y s∈T4 VBF(ωts, jfs) (8.20)

where we have used notations of (6.14); we have identified the intertwiners ωtwith the associated

representation and VBF is the vertex amplitude completely defined by the graph (8.4). This

ω1 ω2 ω3 ω4 ω5 I12 I13 I14 I15 I23 _I 24 I25 I34 I35 I45

 I35 I24 I34 ω3 ω4 K   ω1 I14 I15 I25 ω5 K   I45 ω5 I34 I14 ω4 K  . (8.21) The 6j symbols are the totally symmetrized 6j symbols defined, for example, in the chapter 6 of the book [20]. Note that the sum is finite and then the vertex amplitude is well-defined. However, the state sum is generally divergent; it can be made convergent by gauge fixing or by turning classical groups into quantum groups.

We have voluntarily not given neither the interwiners ωBF

i nor the vectors videfining the model

according to the previous Section. Indeed, such a formulation is not very useful for the topological model and the description of the previous Section is naturally adpated for SU (2) × SU (2) Spin-Foam models and not really for SU (2) Spin-Spin-Foam models.

The Barrett-Crane model

The Barrett-Crane model has been constructed as a step towards the covariant quantization of four dimensional pure Euclidean or Lorentzian gravity `a la Plebanski. Here, we consider exclusively the Euclidean case. The BC model is then a state sum associated to a triangulation T of a 4-manifold M which is supposed to reproduce the path integral ZP l(T ) of a discretized version of the Plebanski

action. However, the link between the BC model and gravity is somehow misleading. Indeed, the BC state sum has been constructed heuristically as a modification of the SU (2)×SU (2) topological state sum according to the following rules: representations coloring the faces of the 4-simplex are supposed to be simple, i.e. of the form (Iij, Iij); the intertwiners ωiBC associated to the tetrahedra

are also called simple or BC intertwiners we will recall the definition in the sequel; the vertex amplitude VBC associated to the 4-simplices are the so-called 10j symbols whose definition will

also be recalled later. The BC model does not say anything concerning the amplitudes A2and A3

associated to the faces and the tetrahedra of the triangulation. However, many arguments lead to certain expressions of A2 and A3 and the corresponding state sums have been numerically tested

[23]. Anyway, we will not consider these amplitudes.

Let us concentrate on the construction of the vertex amplitude VBC whose basic ingredient is

the simple intertwiner. A simple n-valent intertwiner is such that any of its decompositions into 3-valent intertwiners introduce only simple representations in the intermediate channel. The simple intertwiner has been studied intensively in the literature; in particular it was shown to be unique up to a global normalization [21]. This property makes clear that the vertex amplitude of the BC model is a function V (Iij, ωiBC) of only 10 representations and it is called a 10j symbol. To precisely

define the simple intertwiner ωBC

i , it is more convenient to start with the formula (8.4) which shows

that ωBC

i is completely determined by the choice of a “simple” vector vBCi ∈ ⊗j6=iUI∗ijJij where

(Iij, Jij) is a SU (2) × SU (2) UIR . If (Iij, Jij) is a simple representation, i.e. Iij = Jij, then the

associated vector space admits an unique normalized (diagonal) SU (2) invariant vector w (or |wi) which we identify with its dual hw| ∈ V∗

Iij. In that case, indeed, the decomposition (8.8) of UIijJij

into SU (2) representations contains the space U0 which is the one dimensional space of diagonal

SU (2) invariant states. The simple vector is in fact the tensor product of these invariant vectors: ωBC

i = w⊗4. As a result, the expression of the simple intertwiner in the tensor product basis reads:

ωBC i = 1 Q j6=i p dIij X α dαιǫ(α) (8.22)

where the sum runs over simple representations α ≡ (α, α) only and is finite. An important property is that the previous sum is independent on the choice of the basis ǫ. Using this formula of the simple intertwiner, one finds immediately the vertex amplitude of the BC model

VBC(Iij, ωiBC) = 1 Q i6=jdIij X α dαVBF(Iij, α)2 (8.23)

as a sum of BF amplitudes VBF which are SU (2) 15j symbols. The sum runs over simple

repre-sentations only and is independent on the choice of the intertwiners defining the 15j symbol. Such a formula is too cumbersome to be useful and one prefers to use the integral formulation (8.17) of the amplitude to study its physical properties. This integral formula simplifies indeed drastically because the SU (2) integral defining νi (8.16) becomes trivial due to the SU (2) invariance of the

vectors vi, and reads

VBC(Iij, ωi) = Z ( 5 Y i=1 dxi) hw⊗10| O i<j RIij_{(1, x} ix−1j ) |w ⊗10_{i. (8.24)}

Using the second equations in (8.13), one obtains the following integral formula for the 10j symbol: VBC(Iij, ωiBC) = Z Y i6=j dxij χIij(xij) dIij C(xij) = Z _Y5 i=1 dxi Y i<j χIij(xix −1 j ) dIij (8.25) where χI(x) is the SU (2) character of x in the representation I. Up to some normalization factors,

the previous formula coincides with the Euclidean 10j symbols. This integral formulation was very useful to study the classical behavior of the Euclidean BC model. Let us finish this brief presentation of the BC model with two important remarks.

Remark 1. The previous calculation can be done in a completely graphical way. Indeed, the “black” boxes representing the vectors vBC

i in (8.1) reduce to the following form

vBCi =

Iij IijIikIik IilIilIimIim Iij Iij IikIik Iil Iil IimIim

0 0 0 0 (8.26)

where the dashed lines represent spin 0 representation. We see explicitely that vBC

i project into

diagonal SU (2) invariant vectors. Furthermore, the 3j vectors involving a spin 0 representation are proportional to the “identity” according to the following pictorial rule

= pdIij.

Iij Iij

0 Iij Iij

(8.27) As a result, one immediately obtains the pictorial representation of the BC vertex amplitude which is given by the product of the normalization factorQ_i<jd−1Iij and the graph in Figure (8.5).

The graph consists in 10 disconnected loops colored by representations Iijwhich makes obvious that

the vertex amplitude integrand is, up to a normalization, the product of 10 characters χIij(xij).

Remark 2. There is another equivalent expression for the vertex amplitude which was very useful to study the classical behavior of the vertex amplitude found by Freidel and Louapre [22]. We will not use this formula, but it is still interesting to mention it at least to ask the question whether a similar formula exists for the EPR model. This formula is based on the simple fact that the character χI(x) depends only on the conjugacy class θ ∈ [0, π] of x = Λh(θ)Λ−1: Λ ∈ SU (2)/U (1)

and

h(θ) = diag(eiθ_{, e}−iθ₎

. This fact leads after some calculations to an expression of the vertex amplitude as an integral over the conjugacy classes:

VBC(Iij, ωiBC) = Z (Y i6=j dθij sin(dIijθij) dIij ) eC(θij) . (8.28)

The notation eC holds for the “Fourier transform” of the distribution C; it is a distribution as well given by:

x1x−13 x1x4 x2x−14 x2x−13 x3x−14 x3x−15 x4x−15

Figure 8.5: Pictorial representation of the BC vertex integrant up to the normalization factor Q

i<jd −1

Iij. The graph is made of 10 disconnected unknots colored with representations Iij. In each

loop is inserted a S3_{element of the form x} ix−1j .

The Engle-Pereira-Rovelli model

The BC model has been considered as the most promising Spin-Foam model for a long time: its definition is simple, it has a quite appealing physical interpretation and admits the good classical limit [22, 23, 24] in the sense that the associated vertex amplitude tends to the Regge action in the classical limit, apart from a term due to degenerate contributions, and it was also successful in reproducing the correct asymptotic behavior of the diagonal components of the graviton propagator [11, 25]. Nevertheless, it has been recently realized that the model does not satisfy the required properties to reproduce at the semi-classical limit the non-diagonal components of the propagator [10]. The reasons of this failure have been deeply investigated and a quest for a new model have been started. Recent researches have led to the so-called EPR model which has been argued to be a serious candidate. This Section is devoted to recall the basis of this model in the Euclidean sector.

As in the BC framework, Engle, Pereira and Rovelli have proposed a formula for the vertex amplitude VEP R only. To construct VEP R, one starts by coloring the faces of the 4-simplex by

simple representations and the tetrahedra i by specific intertwiners denoted ωEP R

i . We propose to

define ωEP R

i throught its associated vector viEP Raccording to the formula (8.4). To do so, to each

simple representation (Iij, Iij), we associate the projector I2Iij : UIijIij → U2Iij from the SU (2) ×

SU (2) representation’s vector space UIijIij into the vector space of the SO(3) representation of

spin 2Iij. In the standard bra-ket notation, the projector reads I2Iij =

P

m|m 2Iijihm IijIij|; it is

clear that it can be trivially identified to its dual I∗

2Iij = I2Iji. Then, the vector vi is constructed

from this projector as follows:

viEP R ≡ ιǫ(αi) (

O

j6=i

I_2I

ij) (8.30)

where ιǫ(αi) is a SO(3) intertwiner, viewed as an element of the tensor product ⊗j6=iV2I∗ij,

carac-terized by ǫ ∈ {0, +, −} and the SO(3) representation αi as illustrated in the figure (8.3). As the

vector vEP Ri is totally determined by a SO(3) representation αi and a choice of basis ǫ, we will

i i+ i− 2j1 2j2 2j3 2j4 j1 j2 j2 j1 j3 j4 j4 j3

Figure 8.6: EPR fusion coefficients. The edges are colored with SU (2) representations and the vertices with symmetric SU (2) 3j symbols. The picture illustrates the coefficient f (ωi, Iij, ιǫ(α))

for Iij= {j1, j2, j3, j4}, ωi is caracterized by i (and some ǫ) and α = (i+, i−).

the following: vEP Ri = Iij IijIikIik IilIilIimIim Iij Iij IikIik Iil Iil IimIim 2Iim 2Iil 2Iij 2Iik αi (8.31)

Note that we made a particular choice for ǫ to draw the picture; another choice would lead to a different contraction of the four edges colored by the representations 2Iij. Contrary to the BC

model, the EPR intertwiner between four given representations Iij is not unique for it depends on

αi and ε, both belonging to a finite set.

Now, it is possible to decompose the EPR intertwiner in any tensor product basis of the space of 4-valent SU (2) × SU (2) intertwiners. We are interested in its decomposition in the basis of the type (ǫ, ǫ) whose elements are denoted ιǫ(α) After some simple calculation, we recover the following

expression of the EPR intertwiner given in the literature: ωEP R

i =

X

α

f (ωi, Iij, ιǫ(α)) ιǫ(α) (8.32)

where the coefficient f is graphically “represented” in the Figure (8.6) and the sum is finite and runs over SU (2) × SU (2) representations α with a fixed chosen basis ǫ. In the notation of Engle-Pereira-Rovelli, α is denoted (i+, i−) and the representation defining ωi is denoted i. Note that

the sum (8.32) is not restricted to simple representations.

Now, we have all the ingredients to compute the vertex amplitude VEP R(Iij, ωEP Ri ) for the

EP R model. From the expression (8.32), we show immediately that: VEP R(Iij, ωiEP R) =

X

α=(i+,i−)

f (ωi, Iij, ιǫ(i+, i−)) VBF(Iij, i+) VBF(Iij, i−) (8.33)

where VBF(Iij, i±) are the SU (2) 15j symbols which depends on the representations Iij and α

but also on the choice of the basis ǫ which has not been explicitely written. The sums runs over SU (2) × SU (2) representations α with a fixed ǫ. Such a formula is rather complicated and one might prefer working instead with an integral formula of the form (8.3). To obtain such a formula, one has to separate in the integral (8.6) the variables ui from the variables xi as in (8.14) and

x1x−13 x1x4 x2x−14 x2x−13 x3x−14 x3x−15 x4x−15

Figure 8.7: Picturial representation of the EPR argument in the integral formula: vertices are labelled by i = 1, · · · , 5 where i = 1 is the top vertex and the others are enumerated according to the anti-clockwise orientation; edges are then oriented and are labelled by (ij) with i < j. The doubled lines are colored with simple representations (Iij, Iij). The lines (ij) in the same pair are

linked to a line colored with the representation 2Iij. At each vertex, the four single lines are linked

with a line of representation ωi.

Afterwards, the vertex amplitude reduces to the formula: VEP R(Iij, ωi) = N

Z Y

i6=j

dxijC(xij) V(Iij, ωi; xij) (8.34)

where the amplitude V is a function of the 10 variables xij and is graphically represented in the

Figure (8.7). This formula is the EPR counterpart of the formula (8.25) for the BC model. It will appear very useful in the next chapter to make a contact with Loop Quantum Gravity. It might also be useful to study the classical and semi-classical properties of the EPR model as it is the case for the BC model.

A direct generalization: the Freidel-Krasnov models

This Section is devoted to present a very direct generalization of the EPR model. This gener-alization leads to a large class of Spin-Foam models to which belong both the EPR and the BC models.

To motivate the construction of FK models, let us recall that the vector vEP Ri , necessary to

define the EPR intertwiner ωEP R

i , has been constructed making use of a projector I2Iij from the

vector space of the SU (2) × SU (2) simple representation (Iij, Iij) into the SO(3) vector space

representation U2Iij. A direct generalization would be to define a vector v

gen

i using instead, at

each vertex i, projectors IKi

j from VIijIij into the SO(3) representation UKij for any representation

Ki

j∈ [0, 2Iij]. The formal expression of the general vector is then the following:

vigen ≡ ιǫ(αi) (

O

j6=i

Ki

j) . (8.35)

Ki

j. It is represented by the following diagram

v_igen = JijJijJikJikJilJilJimJim JijJij JikJik JilJil JimJim Ki m Ki l Ki j Kki αi (8.36)

This leads to a vertex amplitude very similar to the EPR one. In particular, its integral formula takes the same form of (8.34) where the normalization factor is changed into N = (dI1

2dI54dω1dω1)

−1

and the function V is represented by the same graph drawn in the Figure (8.7) with different spin labels.

As a consequence, we get a large class of Spin-Foam models vertex amplitudes Vigen which

depends not only on the 10 representations Iij coloring the faces of the 4-simplex but also depends

on 5 other representations per tetrahedron i which have been denoted αi, Kji. Up to now, only

special cases of such models have been studied: the BC model where Ki

j= αi= 0, the EPR model

where Ki j = K

j

i = 2Iijand αiis a free parameter. Thus, either we choose to project into the trivial

representation either into the hightest representation. The FK model consists in another choice of the representations Ki

j and αi.

Many arguments lead to the fact that the EPR intertwiners define the good physical model, namely the one which should reproduce the discretized path integral of the Euclidean Plebanski theory.

The vertex and the physical scalar

product

In this chapter we will propose a link between (covariant) Spin-Foam models and (canonical) Loop Quantum Gravity.

More precisely, we consider the Spin-Foam associated to the 4-simplex graph denoted Γ. Its amplitude is given, up to some eventual irrelevant normalization factors, by the vertex amplitude V . From the general boundary (covariant) formulation point of view, Γ is viewed as a graph inter-polating between two kinematical boundary states which are τ1 and τ4 as schematically depicted

in the figure (9.1). In fact, as shown in the figure (9.1), τ1 and τ4 belong to the space Cyl(eΓ)

where eΓ is the union of Γ with four free ends. These free ends have been added for technical purposes only. Notice that Γ can be equivalently interpreted as the graph interpolating between two different graphs that would be denoted τ2(with two vertices) and τ3(with three vertices). For

that, one would need to introduce also some free ends at the graph Γ. From the canonical point

z2 z1 z3 z4 x21 x31 x41 x51 x24 x35 x34 x23 x45 x25 y5 y2 y3 y4

Figure 9.1: Representation of the graph eΓ. The subgraphs associated to τ1 and τ4 have been

underlined and the group variables associated to each edge have been emphasized.

of view, the states τ1 and τ4 are considered schematically as cylindrical functions on the graph

e

Γ. Therefore, one naturally asks the question whether it exists a “physical projector” P acting on the space Cyl(eΓ) such that its matrix element hτ4, P τ1i constructed from the kinematical scalar

product gives the vertex amplitude. The notation hτ4, P τ1i can be misleading because P has in

fact to be viewed as a state in the sense of Gelfand-Naimark-Segal (GNS), i.e. P is a linear form on Cyl(eΓ) and the physical scalar product reads hτ4, P τ1i = P (τ4τ1). We abusively use the same

notation for the projector viewed as a “ matricial operator” or a linear form. To be interpreted as a GNS state, P has to satisfy additional properties, like the positivity, that we will not discuss here. We show that it is possible to construct explicitely such an operator P for the topological, the BC and the EPR models. The “projector” for the FK model can also be obtained immediately