Download Isotropic restriction in Group Field Theory condensates

Master 2 Sciences de la matière
École Normale Supérieure de Lyon
Université Claude Bernard Lyon I
Internship april-july 2015
Mathieu Martin
Isotropic restriction in
Group Field Theory condensates
Abstract:
In this report, we motivate and introduce the group field theory (GFT) formalism from the point
of view of quantum gravity: GFT is an attempt to define a path integral for quantum gravity and a
proposal for the microscopic quantum dynamics of spacetime. We present a path to quantum cosmology offered by GFT. The effective dynamics of some suitable quantum states can be understood
as quantum cosmology equations. GFT condensates are simple states that can be interpreted as
macroscopic homogeneous spatial geometries. The main part of the report is dedicated to the clarification of the isotropic limit of the GFT condensate states. In analogy with classical cosmology,
we expect that, in the isotropic limit, one single geometric variable captures the relevant dynamics
of the system. In a simple model for three-dimensional quantum gravity, we present a procedure
which allows to extract the dynamics of the condensates with respect to the isotropic variable. The
procedure is general and can be extended to more complicated models in any dimension.
Key words: group field theory, quantum gravity, quantum cosmology
Supervisors:
Dr. Lorenzo Sindoni &
Dr. Daniele Oriti
Max Planck Institute for Gravitational Physics
Science Park Potsdam-Golm
Am Mühlenberg 1
D-14476 Potsdam-Golm
http://www.aei.mpg.de/14014/AEI_Potsdam-Golm
Acknowledgements - Remerciements
With a little hindsight, I realise that how much I have learnt during this internship.
First of all, I offer my warmest thanks to Lorenzo Sindoni for his constant guidance, his numerous
explanations, his infinite patience, and his kindness. I would like to thank Daniele Oriti for welcoming
me in his group and ensuring that all went well for me. I warmly thank the members of the group
"Microscopic Quantum Structure and Dynamics of Spacetime" for numerous enlightening and enriching
discussions on many different topics. I also thank the various people of the Quantum Gravity division
of the Albert Einstein Institute with whom I had the opportunity to exchange.
It has been a real pleasure to work in such favorable conditions.
Je remercie toutes les personnes du département de gravité quantique de l’Institut Albert Einstein
qui ont facilité mon intégration. Je tiens en particulier à remercier Daniele Oriti pour l’accueil au
sein de son groupe et pour avoir veillé à ce que tout se passe bien pour moi. Un immense merci à
Lorenzo Sindoni pour sa gentillesse, sa patience et ses nombreuses explications. Je remercie également
chalheureusement tous les membres du groupe "Structure quantique microscopique et dynamique de
l’espace-temps" pour de nombreuses discussions éclairantes et très enrichissantes. Travailler à l’Institut
Max Planck fut une expérience très positive.
Je remercie enfin l’équipe pédagogique de l’ENS Lyon pour m’avoir aidé à trouver ce stage, et pour
m’avoir donné une formation qui me permette d’intégrer un projet de recherche si passionnant.
i
Contents
1 Group field theory: from quantum gravity to cosmology
1.1 General introduction . . . . . . . . . . . . . . . . . . . . . .
1.1.1 The search for quantum gravity . . . . . . . . . . . .
1.1.2 A group field theory for quantum gravity . . . . . .
1.2 From general relativity to group field theory. . . . . . . . . .
1.2.1 Canonical loop quantum gravity . . . . . . . . . . .
1.2.2 The covariant approach to quantum gravity . . . . .
1.2.3 Group field theory as a general framework . . . . . .
1.3 The group field theory formalism . . . . . . . . . . . . . . .
1.3.1 The Boulatov model . . . . . . . . . . . . . . . . . .
1.3.2 Interpretation . . . . . . . . . . . . . . . . . . . . . .
1.4 Homogeneous cosmology as group field theory condensates .
1.4.1 Classical and quantum cosmology . . . . . . . . . . .
1.4.2 Group field cosmology . . . . . . . . . . . . . . . . .
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1
1
1
1
2
2
4
7
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10
11
2 Isotropic sector of Group Field Theory condensates
2.1 Classical dynamics of the GFT condensates . . . . . . .
2.2 A toy model: 2d GFT in reduced configuration space . .
2.3 Isotropic restriction for 3d GFT condensates . . . . . . .
2.4 Isotropic restriction for 4d GFT condensates: an outlook
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13
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14
15
19
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3 Conclusion
20
A About SU(2)
A.1 Definition and representations
A.2 Recoupling theory . . . . . .
A.3 Haar measure . . . . . . . . .
A.4 Harmonic anaylsis . . . . . .
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21
21
22
23
24
B Complements in LQG and simplicial geometry
B.1 A graphical representation for spin networks and spin foams . . . . . . . . . . . . . . .
B.2 Simplicial geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
25
25
C Some intermediate calculations
C.1 Simplification of the action in the reduced configuration space of a two-dimensional
group field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.2 Three-dimensional group field theory in reduced configuration space . . . . . . . . . .
27
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27
28
Chapter 1
Group field theory: from quantum gravity
to cosmology
1.1
1.1.1
General introduction
The search for quantum gravity
In the beginning of the twentieth century, quantum mechanics and general relativity (GR) have
deeply modified our conceptual representation of some key notions such as space, time, causality
and matter. The two theories have been applied with solid experimental success and are now well
established. However we do not know a foundation capable of supporting both theories and predicting
what happens in the physical regime in which they are both relevant, the Planck-scale regime. The
problem of quantum gravity is the problem of making the synthesis of two fundamental theories in a
single language. In this synthesis the notions of space and time need to be deeply reshaped in order
to take into account what we have learned from both quantum mechanics and GR.
Background independence. GR is much more than the field theory of one particular force with
a peculiar gauge invariance. The key lesson we have learned from GR is that spacetime and the
gravitational field are the same entity. There is no background metric spacetime over which physics
happens. We think that a full theory of quantum gravity should be background independent: the
theory should explain the properties of spacetime itself, of its geometry and, maybe, its topology. The
quanta of the relevant field would not live in spacetime: they should build spacetime themselves.
The tensions between GR and quantum mechanics. Quantum field theory (QFT) relies on
a fixed spacetime background. However, this non-dynamical background is incompatible with GR.
In turn, GR is formulated in terms of Riemannian geometry, assuming that the metric is smooth
and deterministic, whereas quantum mechanics requires that any dynamical field should manifest
discreteness at small scales, and be governed by probabilistic laws. Therefore there should be quanta
of space and time. What does it mean? What is quantum spacetime?
1.1.2
A group field theory for quantum gravity
A background independent QFT for quantum gravity? It is easy to see why we may want
to use a QFT formalism for quantum gravity: QFT is the best formalism we have to describe physics
at both microscopic and mesoscopic scales, as it was shown in the context of high energy physics and
condensed matter. We only know how to define QFTs on given background manifolds. Though, as
highlighted in the previous section, the underlying background structure should not be spacetime. So
we can try to identify some background structures that are already present in GR. Such structures
have been identified and lie at the heart of Loop Quantum Gravity (LQG).
Group field theory. Group field theories (GFTs) are quantum field theories of spacetime (as opposed to QFTs on spacetime) in which the base manifold is taken to be a Lie group (or, equivalently,
its Lie algebra). GFTs are a proposal to describe the dynamics of both the topology and geometry
of spacetime in combinatorial, algebraic and simplicial terms. A GFT is not a direct quantization of
classical GR, but GFTs encompass ideas and insights of other background independent approaches to
non-perturbative quantum gravity, namely LQG and simplicial quantum gravity. GFT provides new
tools in an unifying framework that may allow to go beyond the limitations of each approach.
1
CHAPTER 1. GROUP FIELD THEORY: FROM QUANTUM GRAVITY TO COSMOLOGY
Spacetime and cosmology in group field theory GFT aims at providing a better understanding
of the fundamental nature of space and time. Extracting physical predictions from GFT, and recovering
spacetime physics from its microscopic description provided by GFT are very challenging tasks. The
cosmological sector is very appealing, since the reduction in the number of degrees of freedom may
allow to perform calculations leading to physical predictions, and bridge the gap between GFT and
spacetime physics.
The usual approach to quantum cosmology is to quantize a symmetry-reduced version of GR [1].
In GFT, a different road is being explored. It consists of deriving the effective quantum dynamics of
a class of states which admit a macroscopic interpretation as smooth homogeneous spatial geometries.
GFT condensates are a proposal for such states [2].
Research project. In spite of their simple nature, the effective dynamics of GFT condensates is
still intricate. The isotropic sector of the GFT condensates is just beginning to be explored [3]. The
work presented in this report is a contribution to this topic of research.
1.2
From general relativity to group field theory.
We motivate group field theory from the point of view of quantum gravity. In section 1.2.1, we
shortly review the program of canonical LQG [4, 5, 6]. We provide in section 1.2.2 some insights of the
covariant approach to quantum gravity and divide it in two parts: on the one hand, simplicial quantum
gravity [7] and, on the other hand, the spinfoam theory [8]. We finally explain in section 1.2.3 how
GFTs encompass ideas from all these approaches, using the variables introduced in LQG and giving a
simplicial description of spacetime.
1.2.1
Canonical loop quantum gravity
General relativity as a hamiltonian theory of connections. GR is often described as a dynamical theory of metrics, in a formalism based on Riemannian geometry. The dynamics of the
gravitational field gµν is ruled by the Einstein-Hilbert action. Another formulation can be given in
terms of Riemann-Cartan geometry; GR becomes a dynamical theory of connections. The basic variables in this formalism are taken to be the co-frame field eIµ and the so(1, 3)-valued connection ωµIJ .
Their dynamics is classically ruled by the Holst action.
Loop quantum gravity (LQG) is based on the quantization of hamiltonian GR. In the hamiltonian
formalism, one foliates the 4d spacetime manifold M into a family of 3d surfaces Σt . Under this
foliation, variables on M are split into 3+1 decomposition. When performing the Legendre transform
on the Holst action, two fundamental conjugate variables arise: the Ashtekar-Barbero connection Aia
and the densitized inverse triad E˜ia . The connection A encodes parallel transport on Σt . The densitized
triad E enables to define geometric quantities such as areas and volumes embedded in Σt .
In Dirac’s terminology, the dynamical content of GR is captured by three first class constraints: the
Gauss constraint, the diffeomorphism constraint and the scalar constraint. The constraints repectively
generate local SU(2) rotations, 3d diffeormorphisms and time translations. The constraints vanish in
the classical theory, because these transformations are gauge invariances of the theory. The role of the
scalar constraint is to restrict the space of the kinematic states of the theory to that of physical states,
the ones that follow gauge orbits: there is no global time evolution.
Quantization is carried out promoting canonical coordinates to operators and turning the constraints into operators.
New discrete variables. The classical variables which are to be quantized are the holonomies hγ
of the connection and the fluxes ES of the densitized triad Ẽ on closed 2-surfaces S. A holonomy
hγ is a SU(2) group element interpreted as the parallel transport of the connection A along a given
path γ. The holonomy transforms nicely under SU(2) gauge transformations. With the indices f and
i labeling the endpoints of γ, its transformation law is simply given by:
hγ → h0γ = gf hγ gi −1
2
(1.1)
Isotropic restriction in Group Field Theory condensates
Mathieu Martin
where g is a SU(2) group element. The same remark holds for the transformation of the holonomy
under diffeomorphisms. In the quantum theory, holonomies and fluxes are promoted as operators on
the Hilbert space, whose construction is summarized in figure 1.1 and presented below.
Hkin
cyl
HG
SU(2) spin net
Diff
Hinv
Hphys
s-knot Dynamics
Figure 1.1: Construction of the physical Hilbert space of LQG. The kinematical Hilbert space Hkin is
built using functions of holonomies on graphs embedded in Σ. We implement the three constraints one
after another to obtain the physical Hilbert space Hphys .
Kinematical Hilbert Space. The basic elements of the kinematical Hilbert space Hkin are functions
of holonomies defined on oriented graphs embedded in a 3d manifold Σ. They are called cylindrical
functions. We can define an inner product between any cylindrical functions, which has the essential
property to be invariant under SU(2) gauge transformations and 3d diffeomorphisms.
HG is the SU(2) gauge-invariant subspace of Hkin . A basis of Hkin is found using the Peter-Weyl
decomposition of integrable functions on SU(2) (see appendix A.4). Within this decomposition, it is
easy to identify the gauge-invariant states which form a basis of HG , the spin networks. Spin networks
are graphs with decorated links and nodes (cf. figure B.1 in appendix). A link l is attached with a
half-integer jl labeling a representation of SU(2); a node n is attached with in , a SU(2)-intertwiner
compatible with the different representations meeting at n.
The Hilbert diffeomorphism invariant subspace of HG is noted Hinv . Spin networks are embedded
in the 3d manifold Σ. Roughly speaking, we identify two graphs as gauge equivalent if they can be
deformed to each other. A s-knot is an equivalence class of gauge equivalent spin networks; it does not
live in space, it is a quantum state of space. The remaining degrees of freedom are contained in the
combinatorial structure of the graph and the algebraic data attached to its edges and nodes.
Quantum geometry. It is possible to construct operators on the Hilbert space that have a geometrical interpretation: they can measure areas or volumes. Spin networks are eigenstates of such
operators, which turn out to have discrete spectra: the geometry is quantized.
The volume operator has contributions only from the nodes of a spin network, while the area
operator has contributions only from its links. Therefore, each node represents a quantum of volume
and each link represents a quantum of area. A spin network with N nodes and L links is interpreted as
an ensemble of N quanta of volume located in the manifold around the nodes, and separated from one
another by the adjacent surfaces of L quanta of area. The volume of each chunk of space is determined
by the quantum number in , and the area of an adjacent surface is determined by the quantum number
jl . A spin network state thus defines a quantized 3d metric.
Once the diffeomorphism invariance is implemented, the geometric interpretation becomes even
more appealing. S-knots represent different quantized 3d geometries formed by abstract chunks of
space. They do not live on a 3d manifold: the chunks are only localized with respect to one other.
Their spatial relation is determined by the connectivity of the s-knot, whereas areas and volumes are
determined by the coloring of the s-knot.
Quantum dynamics. Hinv is the Hilbert space of states that are invariant under space diffeomorphisms and SU(2) gauge transformations. Their quantum dynamics (invariant under spacetime
diffeomorphisms) is supposed to be revealed by the implementation of the scalar constraint. The dynamical states belong to the physical Hilbert space Hphys . Though, the scalar constraint is intricate
and we do not fully control the quantum dynamics of spacetime. At this stage, we only precise that the
scalar constraint only acts on the nodes of a s-knot. The theory predicts correlations between physical
observables; it does not predict their evolution with respect to an external time variable.
Remark. From now on, as in most of the GFT literature, we will call "spin networks" graphs that
are in fact not embedded in a manifold and that are actually abstract spin networks.
3
CHAPTER 1. GROUP FIELD THEORY: FROM QUANTUM GRAVITY TO COSMOLOGY
1.2.2
The covariant approach to quantum gravity
Conventional QFTs admit two formulations: the canonical and the path-integral (or covariant)
formalisms. Feynman’s formulation of quantum mechanics suggests to define in the gravitational
context transition amplitudes between 3d boundary states of the gravitational field by integrating over
histories (i.e. spacetime geometries interpolating between a fixed boundary ∂M), with a weight given
by an action for GR. We would like to make sense of the formal transition amplitude
Z
Z
p
i
1
[Dg] exp
(1.2)
SGR [g]
,
SGR [g] =
dD x |g|(−R + Λ)
~
16πG
∂M fixed
However we lack a well-defined measure integration for the space of continuous geometries. One way
to circumvent this problem is to discretize the geometry. A discretization is an approximation in which
we truncate the d.o.f. On the other side, various arguments suggest that discrete structures could be
at the origin of a more fundamental theory, GR emerging only in a continuum approximation.
a) Simplicial quantum gravity
Simplicial geometry. Regge introduced an elegant discretization of GR which is called Regge calculus and is based on simplicial geometry (see appendix B.2). In dimension D ≥ n, a n-simplex is
the convex hull of its n + 1 vertices which are points in a manifold connected by n(n + 1)/2 segments.
One can glue D-simplices along matching boundary (D − 1)-simplices to obtain a triangulation of a
D-dimensional manifold. The metric is supposed to be flat (that means that the curvature vanishes)
everywhere except on the (D − 2)-simplices called the hinges. Curvature can be thought as generated
on the hinges thanks to the notion of deficit angle.
For example, the building blocks of a 2d-triangulated manifold are triangles which are glued along
common sides, the curvature being generated at the vertices of the triangles.
Regge proposed an action SRegge (T ) for classical GR on triangulation T . The Regge action turns out
to converge to the Einstein-Hilbert action when the triangulation converges to a Lorentzian manifold.
In order to define discretized path integrals, two alternative paths have been mainly explored, both
relying on simplicial geometry. To understand the differences between the two approaches, called
quantum Regge calculus and dynamical triangulations, one should bear in mind that a triangulation
T is the specification of a lattice L and all the edge lengths {le }. See [7] for a comprehensive review.
Quantum Regge calculus. In quantum Regge calculus, the theory is defined by a discrete gravity
path integral on a fixed lattice L. So, in this spirit, once the lattice is fixed, we sum over the admissible
edge lengths (i.e. all the admissible metrics 1 ) with a given (arbitrary) prescription for the measure.
Y Z
i
ZQRC (L) =
[Dle ]e h̄ SRegge (le ,L)
(1.3)
edges e
The definition of the path integral for quantum gravity is to be understood in the limit of an infinitely
refined lattice: Z = lim ZQRC (L). However, due to the diffeomorphism invariance of GR, it is not
L refined
sure that there is a proper way to define such a limit.
Dynamical triangulations. In this approach, we consider only equilateral triangulations of the
manifold: all the lengths are fixed and equal. That defines an ultra-violet cut-off for the theory. The
summation over all random equilateral triangulations is the discrete analogue to the integral over all
possible geometries.
X Z
X
1 i SRegge (T )
i
[Dg] exp
SGR [g] →
e~
(1.4)
~
CT
∂M fixed
equil. triang. T
∂T fixed
topologies
∂M fixed
The sum is weighted by a symmetry factor CT . A sum over topologies of spacetime in naturally
encoded in the sum over triangulations.
1
The edge lengths must obey some conditions so that the signature of the reconstructed metric corresponds to a
Lorentzian metric.
4
Isotropic restriction in Group Field Theory condensates
Mathieu Martin
Matrix and tensor models. Matrix [9] and tensor [10] models provide a way to generate sums over
equilateral triangulations through their perturbative Feynman graph expansions.
In matrix models, basic objects are taken to be matrix elements Mij of N ×N matrices. Graphically,
matrix elements label the two endpoints of a line. Compared to the action of a point particle, the
underlying idea is to have Feynman graphs that correspond to 2d structures. The price to pay is to
drop the usual assumption of locality of the interaction. Let’s consider the following action S and its
associated formal partition function Z perturbatively expanded in Feynman graphs ZΓ :
Z
X
1
g
S = Mij Mji − √ Mij Mjk Mkl
,
Z = [DM ]e−S(M ) =
ZΓ
(1.5)
2
N
Γ
Since matrices carry two indices, Feynman graphs have edges made of two lines called strands. Those
graphs, called ribbons graphs, are made of vertices and edges, but also of faces (called 2-cells) formed
by closed strands. We can give a simplicial representation to each diagram by introducing the notion
of dual triangulation. The dual triangulation ∆ of a graph Γ is obtained by respectively replacing each
face, edge and vertex of the graph by the corresponding vertex, edge, and face. See figure 1.2. Due to
the cubic nature of the interaction, the faces of the dual graph are triangles, taken to be equilateral.
Figure 1.2: A piece of Feynman diagram for a matrix model in
both its direct representation (full line) and its dual simplicial
representation (dashed line). The two parallel lines of propagation correspond to the two indices of the matrix. Adapted
from [7].
Due to the very simple nature of 2d gravity, the Feynman amplitudes of connected graphs ZΓc ,
the graphs generated by F = ln(Z), can be related to discrete 2d Euclidean gravity path integrals:
ZΓc ∼ e−SRegge (g∆ ) , the two coupling constants G and Λ of gravity being encoded in the coupling
constant of the matrix model g and the size parameter N . We see that the free energy F from the
matrix model point of view is actually the partition function from the 2d gravity point of view. In
other words, matrix models achieve to define of a measure on continuous geometries thanks to discrete
methods.
Interestingly, the free energy F admits a topological expansion. All the triangulations sharing the
same topology, characterized by the genus h in 2d, have the same contribution to the partition function.
X
X
X
1 −SRegge (g∆ )
F =
N 2−2h Fh
,
Fh =
e
(1.6)
C∆
h
connected Ω
∆
∂∆=Ω, h∆ =h
where C∆ is a symmetry factor. Matrix models were extensively studied as toy models for simplicial
quantum gravity. The matrix model formulation allows to reach a continuum phase when sending N
to infinity, which corresponds to a weak coupling regime of 2d gravity.
Tensor models generalize the idea of matrix models to higher dimensions. In the dual (simplicial)
representation, the tensor represents a (D − 1)-simplex, and the combinatorial structure of the interaction allows the identification of a vertex to the gluing of (D + 1) (D − 1)-simplices. Each edge of
a Feynman diagram is made of D strands. In dimension D = 3, the interaction is represented by the
gluing of four triangles along their sides to form a tetrahedron.
Unfortunately, when looking at the link with simplicial gravity, tensor models do not reproduce
some nice features present in matrix models. This is mainly due to a lack of geometric information
encoded in the amplitudes of the tensor models. Gravity in 3d is topological: there is no local degree
of freedom (d.o.f.). Still, it is much more richer than in 2d and we need a richer theory to reproduce
all the d.o.f. of the metric. GFTs rely on the combinatorics of tensor models but, as field theories,
they interestingly possess further geometrical degrees of freedom.
5
CHAPTER 1. GROUP FIELD THEORY: FROM QUANTUM GRAVITY TO COSMOLOGY
b)
Spin foams
Regge’s approach is based on metric variables. The spinfoam (SF) formalism [11] aims at defining
a path integral formulation of quantum gravity using connection variables. The idea is to define the
dynamics of LQG with transitions amplitudes between (abstract) spin network states |si.
X
Aσ
(1.7)
hs1 |s2 i =
σ:s1 →s2
The transition amplitude is represented as a formal sum over all possible histories of spin networks. To
each history σ, also called spin foam, is assigned a spinfoam amplitude Aσ . Boundary states represent
quantum spatial geometries, while spin foams are interpretated as quantum spacetimes. Spacetime is
thus obtained as a spin network history. This history can be decomposed in elementary steps. Taking
seriously the fact that the scalar constraint of canonical LQG only acts at the nodes of LQG states,
the individual steps in the history of a spin network can be seen as the split or the joining of the edges,
which locally changes the number of nodes. We call vertices the points where the edges branch. A
spin foam σ = (Γ, jf , ie ) is a 2-complex Γ (collection of faces bounded by edges joining at vertices)
with representations jf and intertwiners ie of a Lie group attached to its faces and edges, in such a
way that any slice or any boundary of it, corresponding to a spatial hypersurface, will be given by a
spin network. Edges and faces of a spin foam respectively correspond to the worldlines of a node and
worldsurfaces of a link of a spin network. See figure B.2 in appendix. One can assume as a kind of
locality principle that the amplitudes Aσ can be factorized over elementary contributions Av called
vertex amplitudes and which only depends on the group variables attached to the vertex v:
Y
X
XY
Aσ = w(Γ)
Av (jf , ie )
, so that:
hs1 |s2 i =
w(Γ)
Av (jf , ie )
(1.8)
v
Γ: ∂Γ=s1 ∪s2
jf ,ie v
|
{z
A(Γ)
}
where w(Γ) is a weight only depending of the 2-complex Γ. We summed over a set of 2-complexes Γ,
and all the representations j together with all compatible intertwiners i. In most of the current models
the combinatorial structure of the spin foam is restricted to be topologically dual to an abstract simplicial complex of appropriate dimension, so that to each spin foam corresponds a simplicial quantum
spacetime, the representations attached to the 2-complex providing quantum geometric information to
the simplicial complex. At this stage, two main steps need to be accomplished.
Step 1: we need to find a way to compute the vertex amplitude Av (jf , ie ). This step defines a
SF model. Here we sketch the general strategy. First, we choose an action for gravity to be quantized
(3d or 4d gravity, with or without cosmological constant). In 3d, gravity is topological and can be recast
as a BF theory. In 4d, the starting point is the Plebanski action, which recasts GR as a BF theory
with additional simplicity constraints. Second, we define an action in terms of discretized connection
variables on a triangulated manifold, such that, in the same way as for Regge calculus, we classically
recover GR on a smooth manifold when the triangulation is refined. Actually, the refinement is not
needed for topological theories since, in that case, the action is invariant under all transformations of a
triangulation that conserve the topology. Third, we quantize the discrete theory through path-integrals
methods. Fourth, and only in 4d, we implement the constraints at the quantum level.
Step 2: we need to define a consistent way to sum the spinfoam amplitudes in formula 1.8, by
defining a class of 2-complexes to sum over and fixing the relative weights w(Γ). We can try to see if
a prescription can emerge from the canonical theory, but this has not been fully achieved yet. As we
will discuss in the next section, GFTs give such a prescription.
One then has an implementation of a sum-over-histories for gravity in a combinatorial and algebraic
context. Spin foams also arise in the hamiltonian representation of LQG, but the link between the
covariant and Hamiltonian approach is not fully established yet and remains an active research topic.
6
Isotropic restriction in Group Field Theory condensates
1.2.3
Mathieu Martin
Group field theory as a general framework
It is now time to bring together all the ingredients previously introduced. The aim of GFT was
historically to extend the nice properties that matrix models share with 2d gravity to higher dimensions
and that tensor models fail to reproduce. A GFT is a non-local quantum field theory whose basis
manifold is not spacetime; it is taken to be a Lie group. This ensures the background independence of
the theory. In GFT for D-dimensional gravity, the field is defined on D copies on a Lie group G.
φ : (g1 , ..., gD ) ∈ GD → φ(g1 , ..., gD ) ∈ C
(1.9)
The field can be depicted graphically as a D-valent vertex dual to a (D − 1)-simplex. The D group
elements are dually associated with the D (D − 2)-boundaries of the D − 1 simplex represented by the
field. The closure of these (D − 2)-boundaries to form a (D − 1)-simplex is a consequence of a gauge
invariance imposed on the field. In the following section, we will make explicit this geometric picture
in the case D = 3 and write down the gauge invariance. This object will be the building block of
the quantum space in GFT. At the first quantized level, the field becomes the wave-function dictating
the geometry of a (D − 1)-simplex. A simplicial space built out of N such (D − 1)-simplices is then
described by the tensor product of N such wave-functions, with suitable constraints implementing their
gluing, i.e. the fact that some of their (D − 2)-boundaries are identified.
The GFT action shares the same combinatorics of tensor models. Spacetime is seen as emerging
from the interaction of fundamental building blocks of space. The interaction term describes the
interaction of D + 1 (D − 1)-simplices to form a D-simplex by gluing along their (D − 2)-boundaries
(arguments of the fields), that are pairwise linked by the interaction vertex. A D-dimensional simplicial
spacetime is constructed by gluing these elementary D-simplices. As a field theory, a GFT contains
more d.o.f. which are in turn given the interpretation of geometric data.
Here GFTs follows the path traced by LQG, in that they seek to describe quantum geometry in terms
of algebraic data, i.e. group and representation variables, stemming from the formulation of GR in
terms of connections. The group variables represent parallel transport of a connection along elementary
paths dual to the (D − 2)-boundaries. The Peter-Weyl decomposition of the GFT field into irreducible
representations gives a representation of the GFT field in terms of the quantum numbers associated
to the quantized geometry of the simplex it represents. Actually, we see that in this "momentum
space", the GFT states are labelled by spin networks of the group G. The gauge invariance in GFT
previously evoked actually corresponds to the Gauss constraint imposed on cylindrical functions in
LQG. In the second-quantized picture, GFT particles are D-valent open spin networks vertices (or
equivalently (D − 1)-simplices) that can be created or annihilated by the field operators, and a spin
network (simplicial space) is built enforcing gluing conditions which encode the connectivity of a graph.
The quantum dynamics of GFTs is perturbatively defined by the expansion of the formal partition
function in Feynman graphs. The GFT partition function contains a sum over geometric data and
triangulations, embracing the ideas from quantum Regge calculus and dynamical triangulations, and
completing the picture provided by tensor models. The sum naturally encodes a sum over topologies.
The Feynman diagrams Γ are, by construction, spin foams dual to simplicial complexes:
Z
X λV (Γ)
ZGFT = Dφe−SGFT (φ) =
A(Γ)
(1.10)
sym(Γ)
Γ
where V (Γ) is the number of vertices in Γ, sym(Γ) is a symmetry factor, and A(Γ) is the Feynman
amplitude of Γ, given in momentum space by a spinfoam model (cf. equation 1.8). It was shown
that to each spinfoam model corresponds a GFT action, through the specification of the kinetic and
interaction kernels of a GFT. The GFT approach subsumes spin foams in their perturbative expansion
since they offer a prescription to sum spin foams amplitudes associated to a given boundary through
their partition function (cf. Step 2 of the previous section). However, there is much more in a QFT
than in its perturbative expansion. Our point of view is not to consider GFT just as an auxiliary field
theory for spin foams, but rather as a fundamental formulation of quantum gravity.
7
CHAPTER 1. GROUP FIELD THEORY: FROM QUANTUM GRAVITY TO COSMOLOGY
1.3
The group field theory formalism
We are now ready to introduce the GFT formalism with a simple model for 3d Riemannian quantum
gravity: the Boulatov model [12]. We present the model in section 1.3.1 and provide an interpretation
of the formalism in section 1.3.2. General introductions of GFTs are given in [13, 14, 15, 16, 17].
1.3.1
The Boulatov model
Boulatov proposed the first GFT model in 1992. There is a very interesting link between the
Boulatov model and 3d Riemannian quantum gravity. The Feynman amplitudes of the Boulatov
model match the partition functions of the Ponzano-Regge spinfoam model for 3d Riemannian quantum
gravity. Once regularized, the model may be regarded as a non-perturbative definition of 3d simplicial
quantum gravity. We encourage the reader which is not familiar with GFT to have a look at appendix
A before reading this section. Some notations and useful formulas are introduced therein.
The field. We consider a real field φ defined on three copies of SU(2) and invariant under simultaneous right shifts of all variables by h ∈ SU(2).
∀h ∈ SU(2),
φ(g1 , g2 , g3 ) = φ(g1 h, g2 h, g3 h)
This gauge invariance can be imposed by projection P :
Z
φ(g1 , g2 , g3 ) = P φ(g1 , g2 , g3 ) ,
(1.11)
dhφ(g1 h, g2 h, g3 h)
(1.12)
SU(2)
The invariant field decomposes in SU(2) representations (see appendix A.4) as:
Z
X
j1 j2 j3
j1
j2
j3
φ(g1 , g2 , g3 ) =
φm1 m2 m3 ;k1 k2 k3 Dm1 n1 (g1 )Dm2 n2 (g2 )Dm3 n3 (g3 ) dhDnj11 k1 (h)Dnj22 k2 (h)Dnj33 k3 (h)
{j},{m},{n},{k}
=
X
j1
j2
j3
Ajm1 1j2mj32 m3 Dm
(g1 )Dm
(g2 )Dm
(g3 )
1 n1
2 n2
3 n3
{j},{m},{n}
j1 j2 j3
n1 n2 n3
(1.13)
where Dj stands for the Wigner matrix in the spin-j representation. Weused A.31 to get
the 3j-symbol.
P
j
j
j
1
2
3
We defined new Fourier coefficients: Ajm1 1j2mj32 m3 = {k} φjm1 1j2mj32 m3 ;k1 k2 k3
. The reality of
k1 k2 k3
1 j2 j3
.
the field implies: Ajm1 1j2mj32 m3 = (−1)j1 +j2 +j3 +m1 +m2 +m3 Aj−m
1 −m2 −m3
Dynamics. The action of our model is:
Z
Z
λ
1
3
[dg] φ(g1 , g2 , g3 )φ(g3 , g2 , g1 ) −
[dg]6 φ(g1 , g2 , g3 )φ(g3 , g4 , g5 )φ(g5 , g2 , g6 )φ(g6 , g4 , g1 )
S3d =
2
4!
(1.14)
=
1X
2
X
λX
j1 j2 j3
j1 j2 j3
2
j1 +j4
|Am1 m2 m3 | −
(−1)
×
j4 j5 j6
4!
{j} {−jk ≤mk ≤jk }
P
k mk
X
(−1)
{j}
3 j4 j5
5 j2 j6
6 j4 j1
Ajm1 1j2mj32 m3 Aj−m
Aj−m
Aj−m
3 m4 m5
5 −m2 m6
6 −m4 −m1
(1.15)
{−jk ≤mk ≤jk }
where we used the decomposition of the real field in Fourier modes, and the definition of the 6j-symbol
given in appendix A.2. The equation of motion reads:
Z
λ
dg4 dg5 dg6 φ(g3 , g4 , g5 )φ(g5 , g2 , g6 )φ(g6 , g4 , g1 ) = 0
(1.16)
φ(g3 , g2 , g1 ) −
3!
8
Isotropic restriction in Group Field Theory condensates
Mathieu Martin
The action 1.14 is used to write down a formal path integral, whose Feynman expansion generates
all possible oriented 3d simplicial complexes Γ weighted with corresponding Ponzano-Regge partition
functions for 3d Riemannian gravity:
Z3d =
X
λ
V (Γ)
APR (Γ)
APR (Γ) =
XY
j
Γ
Y jT
1
(2jL + 1)
jT4
T ∈Γ
L∈Γ
jT2
jT5
jT3
jT6
(1.17)
P
Q
where j Q
is the sum over all possible colorings of the links of Γ, L∈Γ is the product over all the links
in Γ, and T ∈Γ is the product over all tetrahedra in Γ. In the spinfoam theory, the Ponzano-Regge
model involves some insatisfactory divergences; the issue was cured in the Turaev-Viro model. The
issue can also be cured in GFT, by considering the quantum group SUq (2) instead of SU(2) [12]. Some
complications arise when one seeks to build a relevant GFT for 4d quantum gravity. In section 2.4, we
will explain where the differences lie between 3d GFT and 4d GFT.
1.3.2
Interpretation
Geometrical interpretation. In 3d GFT, the field φ possesses three arguments, so each edge of a
Feynman graph possesses three strands running parallel to it. Each of the three strands running along
the edges carry a group element and can be understood to be dual to each of the three sides of a triangle.
The closure of the triangle is captured by the 3j-symbol in equation 1.13, and its geometry is captured
in the Fourier coefficients A. The Boulatov model can naturally be called a second quantization of a flat
triangle: the GFT field φ is the wave-function of a quantized flat triangle, and the path integral provides
an interacting theory for such quantum geometric d.o.f. It is remarkable that the dynamical content
of 3d quantum gravity is fully encoded in the 6j-symbol attached to the tetrahedron representing an
interaction process. The propagator, which is trivial in the Boulatov model, gives a prescription for
the gluing of two triangles. Four edges meet at each vertex and the interaction forces the strands
recombine. This is identified with the gluing of four triangles to form a tetrahedron. See figure 1.3.
g1 g2 g3
g1
B
g2
g6
A
φ
g1
g4
g6
g3
g1
g2
g3
g1
g2
g3
g3
g4
g5
g1
g2
g4
g3
g5
g2 g2 g5
Figure 1.3: Diagramatic representation of (A) the field φ and its propagation and (B) the interaction
vertex. Black lines: direct space. Grey lines: dual space.
Each strand of a Feynman graph forms a closed loop which can be interpreted as the boundary of
a 2d disk. These data are enough to reconstruct a topological 2-complex Γ: the vertices and edges
of this complex correspond to vertices and edges or the Feynman graph, the 2-boundary of the faces
of Γ correspond to the strands of the Feynman graph. Taking the dual of Γ, we can reconstruct a
triangulated a 3d (pseudo-)manifold. Thus, our Feynman graphs are clearly dual to 3d triangulations.
Note that we consider here abstract complexes, i.e. complexes that are not embedded in a manifold.
Lie algebra variables. GFT was first formulated in terms of a field defined on a group manifold,
and we will stick to this representation in this report until section 2.4. We simply mention here that
the GFT field can equivalently be defined on a Lie algebra manifold. The Lie algebra variables are
interpreted as metric variables, leading to Feynman graphs being directly interpreted as simplicial
complexes. The non-commutative group Fourier transform relates the two formalisms. This new
formulation was introduced in [18].
9
CHAPTER 1. GROUP FIELD THEORY: FROM QUANTUM GRAVITY TO COSMOLOGY
Quantum spacetime in GFT. In our opinion, GFT should not necessarily be understood as the
result of the direct quantization of GR. Instead, we believe that GFTs are candidates for the microscopic
dynamics of fundamental building blocks of spacetime, while GR emerges as an effective macroscopic
dynamics for large collection of them. The perturbative Feynman expansion of GFTs, which is where
insights from LQG and simplicial gravity appear, is very useful for a physical interpretation of the
GFT system, of its quanta and field theoretic data. However, technically speaking, the perturbative
expansion is only useful for describing the GFT system in its few-particle regime. If we are interested in
describing the many-particle behaviour of the same system, we should move away from the no-particle
Fock vacuum. In our approach, continuous spacetime would emerge from a collective behaviour of
many GFT quanta, thus far from the Fock vacuum.
It turns out that, in GFT, field-theoretic techniques are available to deal with the many-particles
regime. Then, the conceptual strategy is to treat quantum spacetime as a sort of condensed matter
system. We expect this system to be in a certain stable GFT phase, and close to equilibrium (because
otherwise we would have probably already noticed a failure of the continuum description of spacetime).
From this perspective, the breakdown of GR in cosmology or in black hole physics would result
from a phase transition occurring in the GFT system. This description of continuous spacetime is
very similar to the description of a fluid, close to equilibrium, governed by hydrodynamical equations.
Thus, we seek to describe quantum spacetime as a quantum fluid of GFT particles. This interpretation
is largely detailed in [19].
Issues linked to renormalization are clearly beyond the scope of this work. We simply mention that
renormalization techniques may be very helpful to address the problem of continuum physics in GFT.
We have just provided a rather speculative vision of quantum spacetime in a GFT language. More
specifically, the problem is to identify suitable quantum states that admit an interpretation in terms
of continuum spacetimes and geometries, and to extract their effective dynamics.
It is natural to first look for GFT quantum states which will be interpreted as homogeneous geometries. These are most likely the simplest case to consider, and hand they form the basis for studying
effective cosmology from the GFT formalism. This path is developed in the next section.
1.4
Homogeneous cosmology as group field theory condensates
GFT offers a completely new approach to cosmology. Before sketching the general strategy followed
in GFT to extract cosmological data, we summarize the main relevant ideas in classical and quantum
cosmology.
1.4.1
Classical and quantum cosmology
Classical cosmology. Cosmology is the study of the dynamics of the universe at very large scale. It
appears that, at very large scale, the universe is approximately homogeneous, isotropic and spatially
flat. And, at large scales, gravity is the dominant interaction. A cosmological model is a symmetryreduced version of GR. In the hamiltonian formalism, 4d spacetime can be represented as a foliation of
3d hypersurfaces. The configuration space, the space spanned by the canonical variables, is generally
the infinite-dimensional space of all 3-geometries; it is called superspace. A finite-dimensional configuration space is called minisuperspace. Models of homogeneous, isotropic and spatially flat empty
universes are described by one single degree of freedom: the scale factor a(t), or its canonically conjugate variable, the Hubble rate H(t) = aȧ . Matter fields can also be considered, and inhomogeneities
are usually added perturbatively. The metric solving the equations of GR and describing the evolution
of a spatially flat homogeneous and isotropic expanding universe is given by:
ds2 = −N 2 (t)dt2 + a2 (t)d~x2
(1.18)
There is a freedom in the choice of the lapse function N (t) which enforces the Hamiltonian constraint.
Therefore, one single dynamical variable gives information about the relative expansion of the universe:
the scale factor a(t). Its dynamics is ruled by the Friedmann equations.
10
Isotropic restriction in Group Field Theory condensates
Mathieu Martin
Quantum cosmology. The most pressing issue in classical cosmology is the presence of a singularity
at the beginning of the Big Bang, where the curvature blows up to infinity. Quantum cosmology is
an attempt to cure this problem [1]. The usual path, introduced by Wheeler and De-Witt, is to
quantize a symmetry-reduced version of classical hamiltonian GR formulated in metric variables. The
quantization has never been carried out in the full superspace, but ill-defined quantities become welldefined in minisuperspace. One introduces a wave-function ψ, the "wave-function of the universe" or
"quantum cosmology wave-function", which is a functional on the minisuperspace. Its dynamics is
governed by the Wheeler-de-Witt equation, Ĥψ = 0, the equivalent of the scalar constraint in LQG.
The Wheeler-de-Witt equation for homogeneous and isotropic vacuum universes reads:
~ ∂
∂
Λ3
a
−a+
ψ(a) = 0
(1.19)
a ∂a
∂a
3
In the connection formulation of GR, the quantization in minisuperspace led to the canonical loop
quantum cosmology program.
Loop quantum cosmology (LQC). There are two ways for applying LQG to cosmology. One is
based on canonical LQG [20, 21], the other one is based on the spinfoam formalism [8]. As in classical
cosmology, both try to identify few large-scale relevant degrees of freedom.
In canonical LQC, we quantize the classically symmetry-reduced phase space of GR, following ideas
and techniques from the full theory of LQG. Owing to the classical symmetry reduction, many technical
complications typical of LQG can be avoided, and the quantization program can be carried out beyond
what has been achieved so far in the full theory of LQG.
Because of the well-defined structures of LQG, LQC allows a rigorous analysis of issues that could
not be addressed within the Wheeler-DeWitt quantization used in conventional quantum cosmology,
such as a definition of the physical inner product. A significant achievement of LQC is the development
of a well-defined quantum theory for cosmological models where the classical singularity is absent. In
some models, the Big Bang is replaced by a quantum bounce, referred as the Big Bounce, essentially
due to the repulsive quantum effects at the Planck scale.
The resolution of the classical singularity is a robust feature of LQC. However, the quantization
of a symmetry-reduced field theory has no physical reason to display faithfully the behaviour of the
true theory. This procedure violates the uncertainty principle, since degrees of freedom are neglected
together with corresponding momenta. Although it does not rule out all what we have learnt from
LQC, the right procedure should be the following: first quantize the full theory, then reduce the phase
space at the quantum level.
The recent spinfoam cosmology program follows this path and tries to identify the relevant d.o.f.
at the quantum level. The main idea is to identify some simple states in spinfoam theory and use
coherent states techniques to approximate homogeneous (and isotropic) geometries. LQC has been a
very active topic of research since the seminal work by Bojowald [22, 23].
1.4.2
Group field cosmology
Our approach is to use the GFT formalism as a second quantization formulation of LQG to describe
cosmologically relevant universes. One has a Fock space of LQG spin network vertices or tetrahedra,
as building blocks of a simplicial complex, respectively annihilated and created by the field operator
φ and its hermitian conjugate φ† . φ† acts on the Fock vacuum, the "no-space" state, to create a GFT
particle, i.e. in 4d a 4-valent spin network or a dual tetrahedron:
φ̂† (g1 , g2 , g3 , g4 )|∅i = |
i
(1.20)
Interestingly, and similarly to spinfoam cosmology, in GFT we identify the relevant degrees of freedom
at the quantum level. We need to identify some states that capture the relevant information about the
regime that we want to explore.
11
CHAPTER 1. GROUP FIELD THEORY: FROM QUANTUM GRAVITY TO COSMOLOGY
Group field theory condensates. The quantum counterpart of the classical homogeneity condition
becomes the requirement that the GFT many-particle state has a structure in which the same wavefunction σ is assigned to each GFT quantum tetrahedron. We can build superpositions of such Nparticle states, with N arbitrarily large: we get a GFT condensate state [24]. The appearance of
smooth and homogeneous macroscopic geometries is described by a process similar to Bose-Einstein
condensation of the fundamental quanta, leading to the idea of spacetime as a condensate. Despite
the simplified form of the condensate, the wave-function still contains an infinite number of degrees of
freedom.
The strategy adopted for extracting information about the dynamics of these states is the use
of Schwinger-Dyson equations of a given GFT model, following the procedure for quantum fluids.
These equations give constraints on the n-point correlations functions of the theory evaluated in a
given condensate state approximating a non-perturbative vacuum. In a mean-field approximation,
the equations can be translated into non-linear integro-differential equations for the condensate wavefunction σ used in the definition of the state. In many-body quantum condensate physics, this is
analogous to the simple case where the dynamics of the condensate wave-function is ruled by the
Gross-Pitaevskii equation. The effective dynamical equations obtained can be viewed as defining a
quantum cosmology model, with the condensate wave-function interpreted as a quantum cosmology
wave-function. This provides a general procedure for deriving an effective cosmological dynamics
directly from the underlying microscopic theory of spacetime.
It has been shown that, in a semiclassical WKB limit, this type of quantum cosmology equation
can reduce to the classical Friedmann equation of homogeneous and isotropic empty universes, with
quantum corrections [25, 2]. Even if a correspondence with Bianchi models for anisotropic universes
has been pointed out, the general reconstruction of the effective dynamics of the complete family of
anisotropic models that can be treated by the condensates is still an open problem.
GFT minisuperspace. Condensate states are determined by a wave-function σ, which is a complexvalued function on the space of four group elements for a given gauge group G. The wave-function
encodes the geometric data of a tetrahedron. However, in general, it stores more information that we
need in order to build homogeneous geometries. If we just impose the usual gauge invariance on the
right (equation 1.11), the wave-functions specify not only the edge lengths, but also the orientation of
the tetrahedron itself. However, this is too much, since it does not reflect the gauge invariance under
arbitrary change of reference frame that we expect at the classical level. Therefore, we further restrict
the wave-function σ to be invariant under a local rotation of the frame2 . The wave-function σ becomes
simultaneously invariant under left and right translation of all its arguments:
σ(hg1 k, hg2 k, hg3 k, hg4 k) = σ(g1 , g2 , g3 , g4 ),
∀h, k ∈ G
(1.21)
The quotient space G\G4 /G is a smooth manifold with boundary, without a group structure. It is the
reduced configuration space of the geometric data associated to a tetrahedron. When the effective quantum dynamics of GFT condensate states is reinterpreted as quantum cosmology equations, G\G4 /G
therefore becomes a minisuperspace of spatially homogeneous but generally anisotropic geometries.
Some open issues. GFT can be seen as the second quantization of LQG. In the same spirit, we may
try to extract a cosmology from a field theory quantizing a LQC model. This approach, considered
in [26], allows to define a field theory on minisuperspace. Following this is idea, is it possible to derive
a field theory on minisuperspace from the GFT condensate approach?
A second issue resides in the identification of an isotropic limit for the GFT condensate. What further requirements on the condensate wave-function are needed in order to identify the macroscopic state
as an isotropic geometry? What are the repercussions on the dynamics of the isotropic condensate?
2
A rotation of the frame is characterized by the adjoint action of SU(2). The invariance of the wave-function under
local rotation of the frame means that ∀h ∈ SU(2), σ(hg1 h−1 , hg2 h−1 , hg3 h−1 , hg4 h−1 ) = σ(g1 , g2 , g3 , g4 ). Combining
this equation with the usual gauge invariance on the right, we get the condition 1.21
12
Chapter 2
Isotropic sector of Group Field Theory
condensates
Up to now, we introduced the GFT formalism for quantum gravity and presented a way of understanding spacetime as a quantum fluid of GFT particles. In the cosmological context, a homogeneous
simplicial spacetime is seen as the result of the condensation of its building blocks, whose quantum
geometry is described by a wave-function interpreted as the cosmological wave-function. This wavefunction encodes the symmetry reduction leading to a minisuperspace of homogeneous (but generally
anisotropic) geometries.
Our work is focused on the isotropic sector of such condensates. We look for a geometric variable
that would be an analogue of the Hubble rate in the GFT microscopic construction based on discrete holonomy variables and their associated group representations. Then, it is natural to study the
possibility of describing the dynamics by using this variable in the isotropic limit.
In order to better grasp the motivations of the present work, we present a simple analogy. The
Laplace operator ∆ is a differential operator given by the divergence of the gradient of a function on
Euclidean space. In cartesian coordinates, the Laplace operator of a function V on R3 reads:
2
∂
∂2
∂2
∆V (x, y, z) =
+
+
V (x, y, z)
(2.1)
∂x2 ∂y 2 ∂z 2
In spherical coordinates, the Laplacian reads:
1 ∂
1
∂2
1
∂2
2 ∂
∆V (r, θ, φ) =
r
+ 2 2
+
V (r, θ, φ)
(2.2)
r2 ∂r
∂r
r sin φ ∂θ2 r2 sin2 φ ∂φ2
p
The Laplacian applied on a spherically symmetric function V (x, y, z) = V ( x2 + y 2 + z 2 ) is still given
by 2.1 in the cartesian coordinates system. No further simplification can be made. More interestingly,
in the spherical coordinates system, when the potential is spherically symmetric, i.e. V (r, θ, φ) = V (r),
the equation 2.2 reduces to:
1 d
dV
∆V (r) = 2
r2
(2.3)
r dr
dr
In GFT condensates, identifying the isotropic limit equals to singling out the relevant geometrical
variable which characterizes the condensate wave-function. Once we found such a variable, we want
to express the dynamics of the isotropic wave-function explicitly using this variable. In our analogy,
the procedure consists of identifying r as the relevant variable in a system with a spherical symmetry.
Then, the second step is to give the explicit expression for the Laplacian of a spherically symmetric
function only in terms of its derivatives with respect to r (equation 2.3).
In section 2.1, we motivate our interest in the GFT classical dynamics in the cosmological context.
In section 2.2, we show in a toy model of 2d GFT that, in the reduced configuration space G/G2 \G,
we are able to extract one single relevant geometrical variable; then we give a procedure to simplify
the dynamics. In section 2.3, we show that, starting from the configuration space G/G3 \G, one has
to impose further constraints on the field to be able to identify one geometric relevant variable. We
propose such a restriction on the field interpreted as an isotropic restriction. Then, we extend the
procedure introduced in the toy model to express the isotropic dynamics in a model for 3d GFT.
Finally, we provide in section 2.2 an outlook on the isotropic sector of 4d GFT condensates.
13
CHAPTER 2. ISOTROPIC SECTOR OF GROUP FIELD THEORY CONDENSATES
2.1
Classical dynamics of the GFT condensates
Classical equations of motion in GFT. The role of the GFT classical equations of motion is
prominent. From the point of view of GFT per se, they define the classical dynamics of the field theory,
and in the quantum theory they may allow the identification of classical background configurations
around which to expand in a semi-classical perturbation expansion. From the point of view of quantum
gravity, solving the GFT equations of motion amounts to identifying non-trivial quantum gravity wavefunctions satisfying all the quantum gravity constraints, an important and still unachieved goal of
canonical quantum gravity. For the time being, we have a poor understanding of the solutions of the
classical equations of motion in GFT.
Dynamics of the GFT condensates. As was explained in section 1.4.2, the quantum dynamics
of the condensates is ruled by the Schwinger-Dyson equations of the corresponding GFT. In general,
such condensate states are not solutions of these quantum equations of motion. One possibility is
to impose the dynamics in a weaker form, imposing the equations of motion only on average. This
essentially leads to the classical equations of motion GFT. That is why we focus, in our case, on the
classical dynamics of the isotropic condensates. On the long run, the more ambitious purpose would
be to extract the effective quantum equation ruling the dynamics of the single microscopic geometrical
relevant variable, that would be the analogue of the Wheeler-de-Witt equation in the homogeneous and
isotropic vacuum universes 1.19 and, eventually, derive a relation between this microscopic variable
and the Hubble rate.
2.2
A toy model: 2d GFT in reduced configuration space
GFT in two dimensions was put into correspondence with matrix models [27]: the theory can be
thought as a collection of non-interacting matrix models, one spin representation j being associated
with a matrix model with N × N matrices, provided that N = 2j + 1. We present here a GFT model
in 2d which will serve as a guideline for the search of isotropic GFT condensates in higher dimensions.
The field. φ is a real scalar field defined on two copies on G = SU (2). By projection, we impose the
field to be invariant under left and right translation of all its arguments:
Z
φ(g1 , g2 ) =
dhdkφ(hg1 k, hg2 k)
⇒
∀(h, k) ∈ SU(2), φ(hg1 k, hg2 k) = φ(g1 , g2 )
(2.4)
SU(2)
Let ϕ be the real function on G defined by ϕ(g) = φ(g, id). ϕ obeys two important properties:
• φ(g1 , g2 ) = ϕ(g1 g2−1 ). To show this, take k = g2−1 and h = id in equation 2.4.
• ϕ(hgh−1 ) = ϕ(g). ϕ is a class function (cf. equation A.24). Take g1 = g, g2 = id and k = h−1 .
φ generally depends on two group elements, so six angles. As φ can be expressed as a function ϕ of one
group element, the dependence is reduced to three angles. Moreover, as discussed in appendix A.3, the
Peter-Weyl decomposition of a class function shows that ϕ(g) actually depends on one single angle θ,
called the polar angle of the group element g. We write:
ϕ(g) =
Xp
Xp
dj ϕj χj (g) ≡
dj ϕj ρj (θ) ≡ ξ(θ)
j
(2.5)
j
where dj = 2j + 1 is the dimension of the spin-j representation. The polar angle θ12 of the group
element g1 g2−1 is thus interpreted as the single relevant geometric variable for the pair (g1 , g2 ) in the
reduced configuration space G/G2 \G. A basis of the space L2 (G/G2 \G, R) of integrable real functions
on this configuration space is formed by the characters {χj , j ∈ N2 }.
14
Isotropic restriction in Group Field Theory condensates
Mathieu Martin
Dynamics. The action of our model is:
Z
Z
λ
1
dg1 dg2 φ(g1 , g2 )φ(g2 , g1 ) −
dg1 dg2 dg3 φ(g1 , g2 )φ(g2 , g3 )φ(g3 , g1 )
(2.6)
S2d =
2
3!
Z
Z
λ
1
[dg]4 φ(g1 , g2 )K(g1 , g2 , g3 , g4 )φ(g3 , g4 ) −
[dg]6 φ(g1 , g2 )φ(g3 , g4 )φ(g5 , g6 )V(g1 , . . . , g6 )
=
2
3!
(2.7)
where we identify the kinetic and interaction kernels:
K(g1 , g2 , g3 , g4 ) = δ(g1 g4−1 )δ(g2 g3−1 ) and V(g1 , g2 , g3 , g4 , g5 , g6 ) = δ(g1 g6−1 )δ(g2 g3−1 )δ(g4 g5−1 ) (2.8)
The equation of motion derived from the action 2.6 reads:
Z
λ
φ(g2 , g1 ) −
dg3 φ(g2 , g3 )φ(g3 , g1 ) = 0
2
(2.9)
Our goal here is to simplify the dynamics making explicit the dependence in the polar angle θ. To that
aim, we use the symmetries of the field φ. We first write the action in terms of the class function ϕ.
The key step is to use the invariance of the integration measure under left and right shift to project the
kinetic and vertex kernels onto their gauge-invariant part. Then, we decompose the projected kernels
in SU(2) representations. We are then able to express the dynamics with respect to the geometric
variable θ, and notably to write down the dependence of the kernels on this variable. See appendix
C.1 for details of the calculations. We find that the action takes the form:
Z
Z
1
λ
S2d =
dθ1 dθ2 ξ(θ1 )ξ(θ2 )K(θ1 , θ2 ) −
dθ1 dθ2 dθ3 ξ(θ1 )ξ(θ2 )ξ(θ3 )V (θ1 , θ2 , θ3 )
(2.10)
2
3!
with
K(θ1 , θ2 ) =
X
ρj (θ1 )ρj (θ2 )
V (θ1 , θ2 , θ3 ) =
and
j
X
ρj (θ1 )ρj (θ2 )ρj (θ3 )
(2.11)
j
Then, varying the action with respect to ξ(θ), we derive the following equation of motion:
Z
Z
λ
dθ1 ξ(θ1 )K(θ, θ1 ) −
dθ1 dθ2 ξ(θ1 )ξ(θ2 )V (θ, θ1 , θ2 ) = 0
2
(2.12)
Before summarizing what we have achieved here, one comment needs to be made. If one leaves the
kinetic and interaction kernels K and V completely unspecified (without even requiring a condition
on their combinatorics), the same procedure will lead to the same result. Actually, we can show that
equations 2.10 and 2.12 remain unchanged. The only difference will lie in the form of the kernels in
equation 2.11. And still, one can show that the kernels remain factorized.
Summary. We showed that, in the gauge-invariant configuration space G/G2 \G ∼ U(1), there is
one single relevant geometric variable: it is an angle. Then, we succeeded in expressing the dynamics of
the field using this angular variable. It is important that no further assumption was needed to perform
this analysis. We now increase the level of difficulty by tackling the problem in higher dimensions.
In 3d, the reduced configuration space G/G3 \G does not allow to identify one single geometrically
relevant variable. Still, we follow the same strategy than in the 2d toy model: we look for functions in
the reduced configuration space which depend on one single relevant geometric parameter, that would
be the analogue of the scale factor (or the Hubble rate), and seek to express the dynamics in terms of
this parameter.
2.3
Isotropic restriction for 3d GFT condensates
The model introduced in section 1.3.1 was defined on the configuration space G3 \G. We present
here the same model, but restraining the configuration space to G/G3 \G. Then, we explain how to
perform an isotropic reduction.
15
CHAPTER 2. ISOTROPIC SECTOR OF GROUP FIELD THEORY CONDENSATES
The field. Let φ be a real field defined on three copies of SU(2) that is left and right gauge-invariant:
∀(h, k) ∈ SU(2),
φ(hg1 k, hg2 k, hg3 k) = φ(g1 , g2 , g3 )
The field decomposes in Fourier modes (see appendix C.2) as:
X
Aj1 j2 j3 Ωj1 j2 j3 (g1 , g2 , g3 )
φ(g1 , g2 , g3 ) =
with:
(2.13)
(2.14)
{j}
Ω
j1 j2 j3
X j
p
j1 j2 j3
j1 j2 j3
j3
j2
1
Da1 b1 (g1 )Da2 b2 (g2 )Da3 b3 (g3 )
(g1 , g2 , g3 ) = dj1 dj2 dj3
a1 a2 a3
b1 b2 b3
{a,b}
Ωj1 j2 j3
L2 (G/G3 \G, R),
(2.15)
the space
The functions
are real and form an orthonormal basis of Wgauge =
of real integrable functions on G/G3 \G.
Z
Z
0 0 0
3 j1 j2 j3
j10 j20 j30
[dg] Ω
(g1 , g2 , g3 )Ω
(g1 , g2 , g3 ) = [dg]3 Ωj1 j2 j3 (g1 , g2 , g3 )Ωj1 j2 j3 (g1 , g2 , g3 ) = δj1 j10 δj2 j20 δj3 j30
(2.16)
The orthonormality of these functions is essential since it enforces the decoupling of their dynamics. In
2d, imposing the left and right gauge invariance was enough extract one relevant geometric variable.
Here, the case is more complex. The field is a function of three group elements, so nine angles. The
right gauge invariance eliminates at least three d.o.f. Indeed, taking k = g3−1 in equation 2.13, we have
that φ(g1 , g2 , g3 ) = φ(g1 g3−1 , g2 g3−1 , id), so the field is actually a function of only two group elements, i.e.
six angles. Left gauge invariance further reduces this number. Our goal is to identify one microscopic
geometric relevant variable describing isotropic states. We must then elaborate a strategy to identify
this variable.
The Ωj1 j2 j3 assign a shape to a triangle. We can think of each j as assigning a length to each side
of the triangle. Roughly speaking, Ωjjj represents an equilateral triangle. The key intuition is that,
with equilateral structures, no anisotropy should appear. Then, we adopt the requirement that the
field should be a functional f of the Ωj1 j2 j3 for which j1 = j2 = j3 . This requirement identifies the
"weak" isotropic sector of our model.
φ(g1 , g2 , g3 ) = f [Ωj ](g1 , g2 , g3 ),
with
Ωj , Ωjjj .
(2.17)
We note that Ωj vanishes when j is not an integer, because a 3j-symbol vanishes when j1 + j2 + j3 is
not an integer. Ωj is a symmetric function of the nine angles {αi , βi , γi , i = 1, 2, 3} parameterizing its
three arguments. Locally, Ωj (g1 , g2 , g3 ) is characterized by one intricate combination Ψj of these nine
angles:
(2.18)
Ωj (g1 , g2 , g3 ) = ζ j Ψj ({αi , βi , γi , i = 1, 2, 3}) ≡ ζ j Ψj123
where ζ j is a function of the combination Ψj which labels the level surfaces of Ωj . The level surfaces of
Ωj for different j’s are generally distinct1 . Therefore, Ψj effectively depends on j. As a consequence,
we further restrict the field φ to be a functional f of one single of the functions Ωj . The simplest
non-trivial possibility is to take j = 1. Moreover, we assume that f is expandable in Taylor series.
X
n X
n
φ(g1 , g2 , g3 ) = f Ω1 (g1 , g2 , g3 ) =
an Ω1 (g1 , g2 , g3 ) =
an ζ 1 (Ψ1123 )
(2.19)
n
n
The isotropic field φ is then characterized in each point (g1 , g2 , g3 ) by the single combination Ψ1123 .
For convenience, we drop the indices corresponding to j = 1. We note Ω , Ω1 , Ψ , Ψ1 and ζ , ζ 1 .
Equation 2.19 becomes :
X
X
φ(g1 , g2 , g3 ) =
an (Ω(g1 , g2 , g3 ))n =
an (ζ(Ψ123 ))n
(2.20)
n
n
1
To show this, we compared the gradients of Ωj in some randomly chosen points of G3 for different j’s with Mathematica. We found that, generally, the gradients are not proportional, which implies that the level surfaces are distinct.
16
Isotropic restriction in Group Field Theory condensates
Mathieu Martin
In this way, we have a wave-function that depends on one single geometric variable Ψ, which is a good
analogue of the angle θ in the toy model. In our construction, Ψ becomes the microscopic version of
the Hubble rate in terms of holonomy variables.
Dynamics. The action considered is the same as defined in equation 1.14:
Z
Z
1
λ
S3d =
[dg]3 φ(g1 , g2 , g3 )φ(g3 , g2 , g1 ) −
[dg]6 φ(g1 , g2 , g3 )φ(g3 , g4 , g5 )φ(g5 , g2 , g6 )φ(g6 , g4 , g1 )
2
4!
(2.21)
We want to express the dynamics in terms of Ψ. In order to do so, we follow the same procedure as
the in the toy model. In appendix C.2, we write down the action with the gauge-invariant projection
of both the kinetic and interaction kernels K ∗ and V ∗ expressed in terms of the functions Ωj1 j2 j3 . We
use the notation φ(h11 , h12 , h13 ) = φ(h1k ).
Z
Z
1
λ
6
∗
S3d =
[dh] φ(h1k )φ(h2k )K (h1k , h2k ) −
[dh]12 φ(h1k )φ(h2k )φ(h3k )φ(h4k )V ∗ (h1k , h2k , h3k , h4k )
2
4!
(2.22)
with:
K ∗ (h1k , h2k ) =
X
Ωj1 j2 j3 (h1k )Ωj3 j2 j1 (h2k )
(2.23a)
{j}
X j1 j2 j3 2
Ωj1 j2 j3 (h1k )Ωj3 j4 j5 (h2k )Ωj5 j2 j6 (h3k )Ωj6 j4 j1 (h4k ) (2.23b)
V (h1k , h2k , h3k , h4k ) =
j4 j5 j6
∗
{j}
The equation of motion reads:
Z
Z
λ
4
∗
0 = [dh] φ(h1k )K (gk , h1k ) −
[dh]9 φ(h1k )φ(h2k )φ(h3k )V ∗ (gk , h1k , h2k , h3k )
3!
(2.24)
We now restrict the field to satisfy the isotropic restriction given by equation 2.20. In the equation of
motion 2.24, one then has to evaluate integrals having the form:
Z
[dg]3 Ωj1 j2 j3 (g1 , g2 , g3 ) (Ω(g1 , g2 , g3 ))n
(2.25)
The {(Ω(g1 , g2 , g3 ))n , n ∈ N} generate the infinite-dimensional space Wiso of isotropic real functions on
the configuration space G/G3 \G. Wiso is thus a linear subspace of the space Wgauge = L2 (G/G3 \G, R)
spanned by the {Ωj1 j2 j3 }. In order to evaluate these integrals, we project the functions Ωj1 j2 j3 onto
Wiso :
Ωj1 j2 j3 = T j1 j2 j3 + Rj1 j2 j3
(2.26)
such that the scalar product between Rj1 j2 j3 and (Ω)n vanishes for all n:
Z
[dg]3 Rj1 j2 j3 (g1 , g2 , g3 )(Ω(g1 , g2 , g3 ))n = 0
(2.27)
T j1 j2 j3 is thus the orthogonal projection of Ωj1 j2 j3 on Wiso (see figure 2.1). T j1 j2 j3 is then a functional
τ j1 j2 j3 of Ω defined by a linear decomposition of T on W .
X
T j1 j2 j3 =
bjn1 j2 j3 (Ω)n ≡ τ j1 j2 j3 [Ω]
(2.28)
n
Note that the decomposition 2.28 is not necessarily unique, because {(Ω(g1 , g2 , g3 ))n , n ∈ N} is not a set
of orthogonal functions. However, we can still use the Gram-Schmidt process to identify an orthogonal
basis of Wiso . Using equation 2.20, we see that T j1 j2 j3 (g1 , g2 , g3 ) can be written as a function η j1 j2 j3
of the microscopic geometrical variable Ψ123 :
X
T j1 j2 j3 (g1 , g2 , g3 ) = η j1 j2 j3 (Ψ123 )
where
η j1 j2 j3 (Ψ123 ) =
bjn1 j2 j3 (ζ(Ψ123 ))n
(2.29)
n
17
CHAPTER 2. ISOTROPIC SECTOR OF GROUP FIELD THEORY CONDENSATES
Ωj1 j2 j3
Wgauge
T j1 j2 j3
Rj1 j2 j3
Wiso
Figure 2.1: Projection on the isotropic sector. K ∗ and
V ∗ live in the space Wgauge spanned by the {Ωj1 j2 j3 }.
We project each Ωj1 j2 j3 onto Wiso . The isotropic projections of K ∗ and V ∗ are noted K and V .
Once the isotropic projection is performed, the equation of motion 2.24 simplifies to:
X Z
λ X
an dΨ321 ζ(Ψ321 )K(Ψ123 , Ψ321 ) −
an an0 an00 ×
3! 0 00
n
n,n ,n
Z
(dΨ)3 ζ(Ψ1k )ζ(Ψ2k )ζ(Ψ3k )V (Ψ123 , Ψ1k , Ψ2k , Ψ3k ) = 0
where we introduced the projected kernels K and V :
X
η j1 j2 j3 (Ψ123 )η j3 j2 j1 (Ψ321 )
K(Ψ123 , Ψ321 ) =
(2.30)
(2.31a)
{j}
X j1 j2 j3 2
η j1 j2 j3 (Ψ123 )η j3 j4 j5 (Ψ1k )η j5 j2 j6 (Ψ2k )η j6 j4 j1 (Ψ3k )
V (Ψ123 , Ψ1k , Ψ2k , Ψ3k ) =
j4 j5 j6
{j}
(2.31b)
We achieved to write down the classical equation of motion as a field equation on minisuperspace.
Discussion. In the reduced configuration space G/G3 \G, we identified a class of functions Ωj that are
candidates for isotropic wave-functions, i.e. wave-functions attached to equilateral triangles. However,
the level surfaces of these functions are distinct for different j’s, which causes an ambiguity in the
choice of a single geometrical relevant variable. We fixed this ambiguity by considering the field as a
functional of the simplest of these functions, Ω1 ≡ Ω. We could then identify a geometrical variable Ψ
interpreted as the microscopic version of the Hubble rate in holonomy variables.
The ambiguity was not unexpected: the translation from the classical phase space of GR to the
holonomy-flux algebra leads to ordering ambiguities. Interestingly, and despite these ambiguities, it
turns out that the level surfaces of the Ωj (for j > 0) coincide at the lowest order, in an expansion
near the identity2 . Holonomies close to the identity are identified with the near-flatness of the triangle.
This condition is often used in a semi-classical approximation, where the connection is almost constant
along a small path dual to the boundary of the building blocks. Therefore, our analysis shows that
wave-functions verifying equation 2.17 are actually controlled in the low-curvature regime by one single
combination Υ123 of the nine angles. This suggests that some corrections to our construction could
appear in regimes involving higher curvatures.
Then, we expressed the dynamics in terms of the geometrical variable Ψ (equation 2.30). The
procedure is based on two steps. First, and similarly to the 2d toy model, we project of the kernels on
the gauge-invariant space Wgauge = L2 (G/G3 \G, R). Second, we project the basis elements Ωj1 j2 j3 of
Wgauge onto the isotropic subspace Wiso . Only the isotropic part of Ωj1 j2 j3 enters the dynamics of the
isotropic field.
Once again, the model presented here involved trivial kernels for simplicity, but the whole procedure
is clearly independent of the kernels involved, and easily extends to any kernels and, most significantly,
to any dimension.
2
With Mathematica, we expanded Ωj (g1 , g2 , g3 ) near the identity using an angular parametrization for the group
elements. For small j’s, we can extract a combination Υ123 of the nine angles independent of j which controls the
behaviour of Ωj (g1 , g2 , g3 ). See appendix C.2.
18
Isotropic restriction in Group Field Theory condensates
2.4
Mathieu Martin
Isotropic restriction for 4d GFT condensates: an outlook
GFT in four dimensions. Real world is four-dimensional. We introduced GFT for 3d Riemannian
quantum gravity. 3d Riemannian gravity is topological and can been recast as a BF theory with
internal gauge group SU(2).
A direct higher-dimensional generalization of the Boulatov model was realized by Ooguri [28]. The
Feynman amplitudes of the Ooguri GFT model reproduce spinfoam partition functions for 4d BF theory
with internal gauge group SU(2). 4d Lorentzian (Riemannian) gravity can be recast as a constraint
BF theory with internal gauge group SL(2,C) (Spin(4)=SU(2)×SU(2)). In both the Lorentzian and
Riemannian cases, the representation theory of SU(2) is essential and justifies a posteriori the interest
of our work with the gauge group SU(2).
In the spinfoam theory, the cumbersome issue is to find the appropriate way to impose the constraints at the quantum level. Different paths were explored, leading to the EPRL/FK models which
are the ones currently used in the community [11]. A full model taking into account the cosmological
constant is still under construction. More intricate kinetic and interaction kernels must be involved in
GFT to reproduce the corresponding spinfoam amplitudes, and deal with renormalization issues.
Geometry of the tetrahedron. In 4d, the GFT field is associated with a tetrahedron. A tetrahedron is fully determined by the data of 3 vectors, so 9 numbers. The shape of the tetrahedron, invariant
under global rotation, is only grasped by 6 numbers (the 6 edge lengths). An equilateral tetrahedron
is fully determined by 1 edge length, or equivalently by its volume.
Quantum tetrahedron. In 4d GFT, the field is a function of 4 group elements, each described by
3 angles. The field has therefore 12 degrees of freedom. The right gauge invariance gives to the field
the geometrical interpretation of a tetrahedron. As in 3d, the right gauge invariance removes 3 degrees
of freedom. The field is thus left with 9 d.o.f. It is consistent with the 9 data that are necessary to
describe a tetrahedron. After imposition of the left gauge invariance, we can show that only 6 degrees
of freedom remain. It is consistent with the 6 edges lengths that describe the shape of a tetrahedron.
We expect that 1 degree of freedom should characterize an equilateral tetrahedron. The concept of
quantum tetrahedron is presented in [29].
Isotropic limit in 4d GFT condensates. A work explaining how to build isotropic condensate
wave-functions in 4d is to be published [3]. We provide a sketch of its construction.
The procedure is based on the Lie algebra representation of GFT (cf. section 1.3.2). In 4d,
Lie algebra variables have a very clear geometrical interpretation in terms of metric variables. The
isotropic limit is taken by considering only wave-functions which describe the geometry of an equilateral
tetrahedron. They only encode the information of their total volume.
In the isotropic limit, performing a non-commutative group Fourier transform to work back in
group variables, the authors find that the condensate wave-functions describing an isotropic universe
must obey the relation 1.21, as it was expected. Moreover, the authors specify additional requirements
on the general form of the wave-function, and derive its effective quantum dynamics following the
procedure described in section 1.4.2. The dynamics in presence of matter fields is also considered.
Dynamics of the 4d isotropic condensates. The present work is complementary to the latter
approach. In 4d, the relevant isotropic variable is available without ambiguity: it is the volume of the
tetrahedron. A well-defined minisuperspace is available. What has not been achieved yet is to express
the dynamics of the 4d GFT condensate in this minisuperspace.
Although some complications of technical order may arise, the recipe that we presented in the previous section can be extended to 4d. An extension of the present work will tackle the issue of explicitly
rewriting the classical equations of motion as the equations of a field theory on minisuperspace.
Once the effective quantum dynamics in minisuperspace is revealed, one may hope to make explicit
the link between our microscopic variable and the Hubble rate. That would be the last main stage
towards the elaboration of a quantum cosmology model with the ansatz that homogeneous and isotropic
empty universes are described by such GFT condensates.
19
Chapter 3
Conclusion
The ambitious program of the group field theory approach to quantum gravity is to provide a conceptually new understanding of the fundamental nature of space and time. In the description given in
this report, spacetime emerges in a continuum approximation as a quantum fluid of GFT particles.
In order to address the challenging goal to extract physical predictions from GFTs, we presented
a way to derive an effective quantum cosmology from the microscopic dynamics of spacetime. This
path is in nature very different from all the other approaches to quantum cosmology. In our approach,
homogeneous geometries emerge from the condensation of a large number of building blocks of space
in the same quantum state.
In four dimensions, it is possible to identify a geometrical variable encoding the information of
an isotropic GFT condensate [3]. This variable is the volume of the tetrahedron interpreted as the
building block of space.
What has not been achieved yet is to express the dynamics of the condensate wave-function using
this variable. In the present work, we provide a general procedure which will allow to address this
issue. We presented the procedure in a simple case of 3d GFT. This procedure will be explicitly applied
in a future work to the case of four-dimensional isotropic GFT condensates.
The present work is a concrete achievement of the idea introduced in [26], where the authors define
a field theory on minisuperspace as describing the cosmological degrees of freedom of the quantum
universe. However, our strategy differs from the original derivation of such a theory. The procedure
presented in [26] is to quantize the degrees of freedom of loop quantum cosmology that were classically
truncated. In our approach, a minisuperspace naturally emerges from the identification of a class of
states, within the full GFT Fock space, having the macroscopic interpretation of homogeneous and
isotropic geometries.
In the same spirit as group field theory, such a field theory on minisuperspace is a realization of the
"third quantization" of gravity. In LQG, the canonical quantization is carried out at fixed topology of
spacetime. GFT, seen as the second quantization of LQG, allows topology changes via the interaction
of its building blocks. The application of our procedure to a four-dimensional GFT model would result
in a field theory on minisuperspace describing the interaction of homogeneous and isotropic quantum
universes.
Several issues need to be further investigated. At the fundamental level, an extensive analysis of
the symmetries present in GFT must be carried out. That would enlighten our understanding of some
issues linked to renormalization and help gaining control over the GFT dynamics.
GFT was born from the insights of many approaches to quantum gravity, involving a wide range
of physical ideas and mathematical tools. Although GFT offers its own description of spacetime,
contributions from other areas of research, such as loop quantum gravity and simplicial gravity, should
not be neglected too.
At the phenomenological level, the cosmological sector of GFT seems to be the most relevant in
order to make physical predictions, and will be further developed. GFT condensates form a seducing
basis for studying effective cosmology, and need to be further investigated. However, GFT offers a wide
range of directions to follow in order to recover spacetime physics. Difficulties arising are mainly of
technical order. That is why investigations at the fundamental level need to be carried out in parallel.
Some alternative approaches to cosmology in the GFT context will probably be explored in the future.
20
Appendix A
About SU(2)
We assume basic notions about the representation theory of SU(2). We give in this appendix a list of
properties which are of interest in the frame of this work and more generally which allow to understand
and perform basic calculations in group field theory.
A.1
Definition and representations
Definition. The special unitary group of degree two SU(2) is the set of 2×2 unitary matrices with
determinant 1 equipped with the matrix multiplication:
α −β̄
2
2
SU(2) =
, α, β ∈ C, |α| + |β| = 1
(A.1)
β ᾱ
Writing α = a + ib and β = c + id, a,b,c and d being real numbers, the last condition reduces to:
a2 + b2 + c2 + d2 = 1
(A.2)
One can use this equation to show that SU(2) is diffeomorphic to the unit 3-sphere S 3 . That implies
that SU(2) is a compact and connected Lie group. Every element g ∈ SU(2) can be written as:
g = exp(θi τi )
,
i = 1, 2, 3
(A.3)
where the {τi } are the generators of the Lie algebra su(2) and obey the Lie bracket: [τi , τj ] = iijk τk .
~
One can take τi = 2i σi , the {σi } being the Pauli matrices. Using the notations θ~ = (θ1 , θ2 , θ3 ), Θ = |θ|
and ~n =
θ~
Θ,
one shows that:
Θ
Θ
id − i sin
~n · ~σ
g = cos
2
2
(A.4)
Where id stands for the 2 × 2 identity matrix. Writing ~n = (sin(ϕ) cos(ψ), sin(ϕ) sin(ψ), cos(ϕ)), we
see that g is characterized three angles Θ, ψ and ϕ. Another parametrization can be given in terms of
the so-called Euler angles.
Wigner matrices. A representation of a group G associates to each element g of G a linear automorphism D(g) of a vector space, in such a way that:
∀g1 , g2 ∈ G,
D(g1 )D(g2 ) = D(g1 g2 )
(A.5)
j
Wigner matrices are unitary matrices in an irreducible representation of SU(2). We note Dmn
(g) the
matrix elements of the spin-j representation Dj of a group element g ∈ SU(2). In the bra-ket notation:
j
Dmn
(g) = hj, m|Dj (g)|j, ni
(A.6)
where j, m, n are the usual (quantum) numbers labelling the vectors in the complete set of kets which
diagonalizes one generator and the Casimir invariant of the Lie algebra su(2). The dimension of a
spin-j representation is dj = 2j + 1. Wigner matrices obey several relations:
j
j
j †
(g † ) = Dmn
(g −1 ) = (Dj )−1
Dmn
mn (g) = (D ) mn (g) = Dnm (g)
(A.7)
j
j
D−m−n
(g) = (−1)m+n Dmn
(g)
(A.8)
21
APPENDIX A. ABOUT SU(2)
Character. The character χ of a group representation is a function on the group which associates
to each group element the trace of its matrix representation. In the SU(2) case:
j
χj (g) = Dmm
(g)
(A.9)
Following from the properties of the trace, and the reality of the trace of Wigner matrices, we have:
χj (gh) = χj (hg)
χj (h−1 gh) = χj (g)
χj (g −1 ) = χj (g)
1
(A.10)
Θ
From equation A.4, we get in the fundamental representation: χ 2 (g) = 2 cos 2 . Furhtermore, we
can obtain the character of any representation from the character of the fundemental representation
thanks to the relation:
jX
1 +j2
j1
j2
χj (g)
(A.11)
χ (g)χ (g) =
j=|j1 −j2 |
For instance, this formula applied with j1 = j2 = 12 , and writing θ = Θ2 , yields to: χ1 (g) = 4 cos2 (θ)−1.
The character of a group element in any representation is actually a function of one single angle, which
moreover is the same angle for all the representations:
χj (g) = U2j (cos (θ)) =
sin(dj θ)
≡ ρj (θ)
sin(θ)
(A.12)
Where Un is the n-ieth Chebyshev polynomials of the second kind, and we recall that dj = 2j + 1 is
the dimension of the spin-j representation. We call θ the polar angle of the group element g.
A.2
Recoupling theory
Clebsch-Gordan coefficients (CGCs). The CGCs appear when considering the addition of two
angular momenta (or equivalently the tensorial product of irreducible representations). They are the
elements of the (unitary) matrix of transformation between two complete orthonormal sets of wavefunctions which diagonalize two different complete sets of commuting operators. They are noted
j1 j2 j
Cm
1 m2 m or in the bra-ket notation hm1 m2 |jmi.
Wigner matrices and the CGCs. The action of a rotation represented by the group element
g ∈ SU(2) on a tensor state |1i ⊗ |2i = |j1 , n1 i ⊗ |j2 , n2 i is given by:
Dj (g) (|1i ⊗ |2i) = Dj1 (g)|1i ⊗ Dj2 (g)|2i
Contracting the previous equation with hj1 , m1 | ⊗ hj2 , m2 |, one finds that:
X
j1
j2
j1 j2 j
j
Dm
(g)D
(g)
=
Cm
C j1 j2 j Dmn
(g)
m2 n2
1 n1
1 m2 m n1 n2 n
(A.13)
(A.14)
j
j1 j2 j3
Wigner 3j-symbols. The 3j-symbols
appear when one seeks to build a function
m1 m2 m3
of three group elements invariant under rotation. 3j-symbols are related to the CGCs via the relation:
(−1)j1 −j2 +m3
j1 j2 j3
hm1 m2 |j3 − m3 i
(A.15)
= √
m1 m2 m3
2j3 + 1
3j-symbols are invariant under cyclic permutations. They are non-vanishing only when the condition
m1 + m2 + m3 = 0 holds. Moreover, they obey the following relations:
j1 j2 j3
j1 j3 j2
j1
j2
j3
j1 +j2 +j3
(−1)
=
=
(A.16)
m1 m2 m3
m1 m3 m2
−m1 −m2 −m3
X j1 j2 j3 2
=1
(A.17)
m1 m2 m3
m1 ,m2 ,m3
22
Isotropic restriction in Group Field Theory condensates
Mathieu Martin
6j-symbol. We want to introduce scalar quantities only built up with 3j-symbols. The scalar must
depend only on the number j and not on the numbers m, which are altered by rotations and thus
summed over in the definition of the 6j-symbol. We need at least four 3j-symbols to give birth to a
non-trivial scalar. The 6j-symbol is defined in our convention by:
P
j1 j2 j3
j3
j4 j5
j1 j2 j3
mk
j2 +j3 +j5 +j6
k
= (−1)
(−1)
m1 m2 m3
−m3 m4 m5
j4 j5 j6
j5
j2
j6
j6
j4
j1
(A.18)
−m5 −m2 m6
−m6 −m4 −m1
A geometrical interpretation can be given to this writing. The four 3j-symbol can be attached to the
four faces of a tetrahedron represented by the 6j-symbol. The ji label the six edges of the tetrahedron.
3nj-symbols. We can go further and look for scalar quantities built up with more 3j-symbols. These
quantities are represented by the 3nj-symbols. The contraction of six 3j-symbols gives a 9j-symbol.
For n > 3, there exists different kinds of 3nj-symbols.
A.3
Haar measure
Definition. The compactness of SU(2) allows to define a measure on SU(2) and perform integration
of functions over this group. We can write the Haar measure in terms of Euler angles (α, β, γ) as:
Z
Z 2π
Z
Z
Z π
Z 2π
1
1
dµ(g) =
dβ
du
with
du =
dγ sin γ
dα
(A.19)
2π 0
4π 0
SU(2)
S2
S2
0
R
The Haar measure thus defined is normalized, dµ(g) = 1, and invariant under left and right shift:
dµ(g) = dµ(gh) = dµ(hg)
, and also verifies
dµ(g) = dµ(g −1 )
(A.20)
For simplicity, we will adopt the usual notation dg for the Haar measure instead of dµ(g). In the
purpose of our work, the explicit formula A.19 will be barely used in our calculations. Integration over
the group is often used to build functions invariant under a kind of transformations. For exemple, the
rotational-invariant part f˜ of a generic function f on the 3-sphere S 3 can be extract by projection:
Z
f˜(~x) =
dRf (R~x)
⇒
∀R ∈ SU(2), f˜(R~x) = f˜(~x)
(A.21)
SU(2)
Dirac delta function. For all function f on SU(2), the Dirac distribution verifies:
Z
∀g ∈ SU(2)
dhδ(hg −1 )f (h) = f (g)
(A.22)
It can be defined by its expansion in characters:
δ(g) =
X
dj χj (g)
(A.23)
j
P
This formula is the SU(2)-equivalent of the well-known formula "δ(φ) = n exp(inφ)", where n labels
irreducible representations of U(1). A formula of this type holds for every compact group.
Class functions. A class function is a function defined on a group G that is constant on the conjugacy
classes of G:
f is a class function
⇔
∀h ∈ G,
f (g) = f (hgh−1 )
(A.24)
Given any function f on a compact group G, there exists a unique decomposition f = f˜ + fˆ such as:
Z
∀h ∈ G, f˜(g) = f˜(hgh−1 )
and
dhfˆ(hgh−1 ) = 0
(A.25)
G
23
APPENDIX A. ABOUT SU(2)
Its class-invariant part f˜ can be extracted by projection P :
Z
˜
dhf (hgh−1 )
f (g) = P [f ](g) ,
(A.26)
G
If G = SU(2) and if f is class invariant, then f (g) depends only on one out of the three angles
characterizing g, the polar angle θ (cf. the discussion in the last paragraph of the following section).
A.4
Harmonic anaylsis
The Peter-Weyl theorem applied to the compact group SU(2) states that the coefficients of the
Wigner matrices form an orthogonal basis of the space L2 (SU(2), C) of integrable complex functions
on SU(2). Every function f ∈ L2 (SU(2), C) can be expanded as:
Xp
j
j
f (g) =
dj fmn
Dmn
(g)
(A.27)
j,m,n
For a function f of N variables gi , i = 1..N , noting f (g1 , ..gN ) = f (gi ), the decomposition reads:
X
f (g1 , ..., gN ) =
j1 ...jN
fm
1 ...mN ;n1 ...nN
N
Y
p
ji
dji Dm
(g)
i ni
(A.28)
i=1
{j},{m},{n}
The coefficients of the Wigner matrices satisfy the following orthogonalization relations:
Z
1
j
j0
dgDmn
(g)Dm
δjj 0 δmm0 δnn0
0 n0 (g) =
dj
SU(2)
j
as follows:
which allows to express the coefficients fmn
Z
p
j
j
fmn
= dj
dgDmn
(g)f (g)
(A.29)
(A.30)
SU(2)
Starting from equation A.14 that links the CGCs and the Wigner matrices, multiplying by another
j3
Wigner matrix element Dm
3 n3 (g), integrating over the group element, and using the relation between
the CGCs and the 3j-symbols A.15 as well as the properties of the Wigner matrices A.8 and A.29, one
finds an important relation between Wigner matrices and the 3j-symbols:
Z
j1 j2 j3
j1 j2 j3
j1
j2
j3
dgDm
(g)D
(g)D
(g)
=
(A.31)
m2 n2
m3 n3
1 n1
m1 m2 m3
n1 n2 n3
SU(2)
We will also need the following relation:
Z
1
j0
j
dgDmn
(g)Dm
δjj 0 δm−m0 δn−n0 (−1)m+n
0 n0 (g) =
d
j
SU(2)
(A.32)
obtained starting from the left-hand side, and using A.8 and A.29.
Harmonic analysis of class functions. The characters form a basis of the class functions on a
SU(2). A class function f expands as:
Xp
Xp
j
f (g) =
dj f j χj (g) =
dj f j Dmm
(g)
(A.33)
j
j
We recall that the character of a group element in any representation is a function of one single angle
which does not depend on the representation considered (as was discussed in section A.1). Therefore
a class function of a group element also depends on one single angle, the polar angle. Explicitly:
Xp
f (g) =
dj f j U2j (cos(θ)) ≡ ξ(θ)
(A.34)
j
24
Appendix B
Complements in LQG and simplicial
geometry
B.1
A graphical representation for spin networks and spin foams
We give here a graphical interpretation of (abstract) spin networks and spin foams.
j1
Figure B.1: An (abstract) spin network is an oriented graph
which is (not) embedded in a manifold, with irreducible reprei1
i2
j2
sentations jl attached to its links and intertwiners in attached
to its nodes.
j3
Figure B.2: Left: a spin foam with two vertices. Right: a four-valent spinfoam vertex. A spin foam is
a 2-complex with representations jf attached to its faces and intertwiners ie attached to its edges. A
spin foam is intuitively the world history of a spin network. The individual steps of this history are
the split or the joining of the edges of the spin network, that locally changes the number of nodes.
Adapted from [4].
B.2
Simplicial geometry
We give here some complements to the notions of simplicial geometry introduced in the main text.
The definitions are not completely rigorous ones, the aim being to give an intuitive introduction to
some concepts. See [7] for more details.
Simplex and triangulation. We recall that, in dimension D ≥ n, a n-simplex is the convex hull of
its n + 1 vertices which are points in a manifold connected by n(n + 1)/2 segments. See table B.1. A
subsimplex of a given simplex is a subset of points of this simplex.
n
2d
3d
4d
4
3
4-simplex
tetrahedron
tetrahedron
2
triangle
triangle
triangle
1
segment
segment
segment
0
point
point
point
Table B.1: The different possible n-simplices in 2d, 3d and 4d. The simplex carrying the curvature in
a simplicial manifold, the hinge, appears in bold.
25
APPENDIX B. COMPLEMENTS IN LQG AND SIMPLICIAL GEOMETRY
Out of a collection of simplices one can construct a simplicial complex, by gluing simplices along
some of their subsimplices. A D-dimensional simplicial manifold (or triangulation) is a simplicial
complex such that the neighborhood of each point is homeomorphic to the D-dimensional ball.
A D-dimensional simplicial manifold can be obtained by successively gluing pairs of D-simplices
along some of their (D − 1)-faces. However, when performing such an operation, one does not obtain in
general a simplicial manifold, but only a so-called pseudo-manifold. Only the one- or two-dimensional
complexes obtained by gluing lines at end-points or triangles along edges are indeed simplicial manifolds. This fact makes discretized models for 2d gravity much easier to handle. Nevertheless, this is
an issue that has to be addressed in higher-dimensional GFT, and that is not discussed here.
The simplicial manifold structure corresponds to a discretization of ordinary manifolds, but there
are no notions of distance between points, nor of volume element, which would allow to define angles
or scalar products between vector fields, as well as to integrate functions over the manifold. One can
embody a simplicial manifold with a Riemannian metric by specifying that:
1) inside each D-simplex, the metric is flat (i.e. the extrinsic curvature vanishes) and fully determined
by the length of the D(D+1)
edges of the simplex.
2
2) the metric is continuous when one crosses the faces ((D − 1)-simplices).
3) each face of a simplex is flat (the extrinsic curvature also vanishes).
While the metric is continuous when crossing the faces, it is discontinuous across the lower dimensional
simplices. In particular, the curvature is concentrated along the (D − 2)-dimensional "hinges".
Figure B.3: A 2d triangulation (full line) and its associated
dual complex (dashed line). The dual of a point is the face of
a polygon. The dual of the side of a triangle is the side of a
polygon. The dual to a triangle is a vertex. Adapted from [7].
Dual complex. Here is the intuitive procedure to follow to obtain the dual of a triangulation ∆. To
any given p-dimensional subsimplex X of ∆, associate a flat (D − p)-dimensional dual polytope X ∗ .
The dual polytopes of the simplicial complex ∆ form a polytopial (but not simplicial) complex, ∆∗ .
We give in figure B.3 a triangulation and its dual complex.
In the spinfoam theory, we discretize the manifold on a triangulation ∆, but it is more convenient
to work with its dual 2-complex Γ, which is the 2-skeleton of the complex ∆∗ dual to the triangulation
∆. The n-skeleton of a complex is the union of its cells of dimension inferior or equal to n. Γ is thus
formed by a collection of vertices, edges and faces (or 2-cells). See table B.2.
In spinfoam models, the fact that the 2-complex is taken to be dual to a simplicial complex imposes
a constraint on the valency of the nodes of the spin networks slicing a given spin foam. In this
construction, in dimension D, all the nodes of the spin networks will be D-valent. However, in the
canonical theory, there is no simplicial restriction.
In GFT, one usually only generates spin foams dual to simplicial complexes. In order to work in a
more general framework, one should consider more than one single possible interaction process in the
GFT action. Work is currently being done in that direction [30].
∆2
triangle
segment
point
Γ2
vertex
edge
face, or 2-cell
∆3
tetrahedron
triangle
segment
point
Γ3
vertex
edge
face, or 2-cell
(3-cell)∈ ∆∗3
∆4
4-simplex
tetrahedron
triangle
segment
point
Γ4
vertex
edge
face
(3-cell)∈ ∆∗4
(4-cell)∈ ∆∗4
Table B.2: Relation between a triangulation ∆ and the 2-skeleton of its dual Γ.
26
Appendix C
Some intermediate calculations
C.1
Simplification of the action in the reduced configuration space of
a two-dimensional group field theory
In this section, we detail how we get to the simplified 2.10 starting from the general action 2.6.
First, we write the action as a functional of the class-invariant field ϕ (equation C.1). Second, we make
the kinetic and interaction kernels appear (equation C.2). Third, we explicitly use the class invariance
of ϕ (equation C.3). Fourth, we use the invariance under left and right shift of the Haar measure on
SU(2) to project the kernels on their gauge-invariant components (equation C.4).
S2d
Z
Z
1
λ
−1
−1
2
=
[dg] ϕ(g1 g2 )ϕ(g2 g1 ) −
[dg]3 ϕ(g1 g2−1 )ϕ(g2 g3−1 )ϕ(g3 g1−1 )
(C.1)
2
3!
Z
1
−1 −1
=
[dg]2 [dh]2 ϕ(h1 )ϕ(h2 )δ(g1 g2−1 h−1
1 )δ(g2 g1 h2 )−
2
Z
λ
−1 −1
−1 −1
(C.2)
[dg]3 [dh]3 ϕ(h1 )ϕ(h2 )ϕ(h3 )δ(g1 g2−1 h−1
1 )δ(g2 g3 h2 )δ(g3 g1 h3 )
3!
Z
1
−1 −1
=
[dg]2 [dh]2 dxdyϕ(xh1 x−1 )ϕ(yh2 y −1 )δ(g1 g2−1 h−1
1 )δ(g2 g1 h2 )−
2
Z
λ
−1 −1
−1 −1
[dg]3 [dh]3 dxdydzϕ(xh1 x−1 )ϕ(yh2 y −1 )ϕ(zh3 z −1 )δ(g1 g2−1 h−1
1 )δ(g2 g3 h2 )δ(g3 g1 h3 )
3!
(C.3)
Z
1
−1 −1
−1
−1
=
[dg]2 [dh]2 dxdyϕ(h1 )ϕ(h2 )δ(g1 g2−1 xh−1
1 x )δ(g2 g1 yh2 y )−
2
Z
λ
−1 −1
−1
−1 −1
−1
−1
[dg]3 [dh]3 dxdydzϕ(h1 )ϕ(h2 )ϕ(h3 )δ(g1 g2−1 xh−1
1 x )δ(g2 g3 yh2 y )δ(g3 g1 zh3 z )
3!
(C.4)
We now integrate over the variables x and y in the kinetic term, and over x, y, and z in the interaction
term, using the expansion of the Dirac distribution in characters A.23.
Z
dxδ(G1 xG2 x
−1
)=
X
Z
dj
j
dhχ (G1 xG2 x
−1
)=
j
=
X
Z
dj
j
j
j
j
dxDab
(G1 )Dbc
(x)Dcd
(G2 )Dda
(x−1 )
j
X
j
j
j
dj Dab
(G1 )Dcd
(G2 )
X
1
δab δcd =
χj (G1 )χj (G2 )
dj
(C.5)
j
where we used the property A.7 and the orthogonality of the Wigner matrices elements A.29. We get:
S2d
1X
=
2 0
Z
0
0
−1 j
−1
j
[dg]2 [dh]2 ϕ(h1 )ϕ(h2 )χj (g1 g2−1 )χj (h−1
1 )χ (g2 g1 )χ (h2 )−
j,j
Z
λ X
−1 j −1 j 00
−1 j 00 −1
j0
[dg]3 [dh]3 ϕ(h1 )ϕ(h2 )ϕ(h3 )χj (g1 g2−1 )χj (h−1
1 )χ (g2 g3 )χ (h2 )χ (g3 g1 )χ (h3 )
3! 0 00
j,j ,j
(C.6)
27
APPENDIX C. SOME INTERMEDIATE CALCULATIONS
Then, we integrate over g1 , g2 in the kinetic term, and over g1 , g2 , g3 in the interaction term, using
equation A.29 again. We find that:
Z
Z
0
0
00
[dg]2 χj (g1 g2−1 )χj (g2 g1−1 ) = [dg]3 χj (g1 g2−1 )χj (g2 g3−1 )χj (g3 g1−1 ) = 1
(C.7)
Relabeling hi → gi in the integrals, the action becomes:
Z
Z
1
λ
∗
S2d =
dg1 dg2 ϕ(g1 )ϕ(g2 )K (g1 , g2 ) −
dg1 dg2 dg3 ϕ(g1 )ϕ(g2 )ϕ(g3 )V ∗ (g1 , g2 , g3 )
2
3!
(C.8)
with:
K ∗ (g1 , g2 ) =
X
χj (g1 )χj (g2 )
V ∗ (g1 , g2 , g3 ) =
and
X
j
χj (g1 )χj (g2 )χj (g3 )
(C.9)
j
The action can be greatly simplified using the Peter-Weyl decomposition of the field. However, this is
not the purpose of our work. It is rather to show that we can explicitly express the dynamics in terms
of the single relevant geometric variable in our reduced configuration space, the polar angle θ. Writing
χj (g) = ρj (θ) and ϕ(g) = ξ(θ), and recalling that the integration over SU(2) can be decomposed
as an integration over the polar angle and an integration over the 2-sphere as in equation A.19, we
immediately get:
Z
Z
1
λ
S2d =
dθ1 dθ2 ξ(θ1 )ξ(θ2 )K(θ1 , θ2 ) −
dθ1 dθ2 dθ3 ξ(θ1 )ξ(θ2 )ξ(θ3 )V (θ1 , θ2 , θ3 )
(C.10)
2
3!
with
K(θ1 , θ2 ) =
X
ρj (θ1 )ρj (θ2 )
and
V (θ1 , θ2 , θ3 ) =
X
j
C.2
ρj (θ1 )ρj (θ2 )ρj (θ3 )
(C.11)
j
Three-dimensional group field theory in reduced configuration
space
We provide here some details of the calculations leading to the results presented in section 2.3.
The field. We decompose the field in SU(2) representations using its gauge invariance.
Z
φ(g1 , g2 , g3 ) = dhdkφ(hg1 k, hg2 k, hg3 k)
(C.12)
Z
X
p
j1
j2
j3
=
Ajm1 1j2mj32 m3 ;n1 n2 n3 dj1 dj2 dj3 dhdkDm
(hg1 k)Dm
(hg2 k)Dm
(hg3 k) (C.13)
1 n1
2 n2
3 n3
{z
}
|
{j}
j j j
Ωm112m23 m3 ;n1 n2 n3 (g1 ,g2 ,g3 )
=
X
Aj1 j2 j3 Ωj1 j2 j3 (g1 , g2 , g3 )
(C.14)
{j}
where
A
j1 j2 j3
X
=
{m},{n}
Ajm1 1j2mj32 m3 n1 n2 n3
j1 j2 j3
m1 m2 m3
j1 j2 j3
n1 n2 n3
(C.15)
and
Ω
j1 j2 j3
X p
j1 j2 j3
j1 j2 j3
j1
j2
j3
(g1 , g2 , g3 ) =
dj1 dj2 dj3 Da1 b1 (g1 )Da2 b2 (g2 )Da3 b3 (g3 )
a1 a2 a3
b1 b2 b3
{a},{b}
(C.16)
There is a relation between Ωjm1 1j2mj32 m3 ;n1 n2 n3 and Ωj1 j2 j3 :
j1 j2 j3
j1 j2 j3
j1 j2 j3
j1 j2 j3
Ωm1 m2 m3 ;n1 n2 n3 (g1 , g2 , g3 ) = Ω
(g1 , g2 , g3 )
m1 m2 m3
n1 n2 n3
28
(C.17)
Isotropic restriction in Group Field Theory condensates
Mathieu Martin
The Ωj1 j2 j3 are real and form an orthonormal set of functions:
Z
Z
0 0 0
j10 j20 j30
3 j1 j2 j3
(g1 , g2 , g3 )Ω
(g1 , g2 , g3 ) = [dg]3 Ωj1 j2 j3 (g1 , g2 , g3 )Ωj1 j2 j3 (g1 , g2 , g3 ) = δj1 j10 δj2 j20 δj3 j30
[dg] Ω
(C.18)
We used A.29 and A.17. Due to the reality of the field, the Fourier coefficients obey the relation:
Aj1 j2 j3 = Aj1 j2 j3 .
Expansion of Ωj around the identity. We now focus on the functions Ωj = Ωjjj which are
argued to be the only ones on which an isotropic wave-function should depend. A group element g is
characterized by three Euler angles (α, β, γ). The Wolfram Language uses phase conventions where:
j
j
Dm
(α, β, γ) = exp(im1 α + im2 γ)Dm
(0, β, 0)
1 m2
1 m2
(C.19)
Even for j = 1, Ωj (g1 , g2 , g3 ) is a complex trigonometric sum involving nine angles (αi , βi , γi , i = 1, 2, 3).
Taking the angles to be close to zero amounts to consider group elements close to the identity. With
Mathematica, we could expand in series the expression of the Ωj at the fourth order in , where the
three group elements gi are parametrized by (αi , βi , γi ), for j = 1, 2, 3, 4. We find that:
j
Ω (g1 , g2 , g3 ) '
1−W2j 2
X
3
3
3
X
X
2
2
2
(αk αl + βk βl + γk γl ) −
(αi + βi + γi ) +
αk γl +4 W4j ({αi , βi , γi })
i=1
k6=l=1
|
k,l=1
{z
Υ123
}
(C.20)
where W2j is a constant depending on j; Υ123 is a combination on the nine angles which does not
depends on j; W4j is a combination on the nine angles which depends on j. Υ123 is then the relevant
combination describing the behaviour of every Ωj for group arguments close to the identity matrix.
Dynamics. The action of our model is:
Z
Z
1
λ
3
S3d =
[dg] φ(g1 , g2 , g3 )φ(g3 , g2 , g1 ) −
[dg]6 φ(g1 , g2 , g3 )φ(g3 , g4 , g5 )φ(g5 , g2 , g6 )φ(g6 , g4 , g1 )
2
4!
(C.21)
We will use the notation: φ(h11 , h12 , h13 ) = φ(h1k ). We insert a bunch of Dirac’s distributions in
order to make appear the interaction kernel, and follow the same strategy than in 2d: we use the left
and right gauge invariance of the field to project the kernel on their gauge-invariant parts.
Z
λ
3d
Sint
=−
[dg]6 φ(g1 , g2 , g3 )φ(g3 , g4 , g5 )φ(g5 , g2 , g6 )φ(g6 , g4 , g1 )
(C.22)
4!
Z
λ
[dg]6 [dh]12 φ(h11 , h12 , h13 )φ(h21 , h22 , h23 )φ(h31 , h32 , h33 )φ(h41 , h42 , h43 )×
=−
4!
δ(h11 g1−1 )δ(h12 g2−1 )δ(h13 g3−1 )δ(h21 g3−1 )δ(h22 g4−1 )δ(h23 g5−1 )×
δ(h31 g5−1 )δ(h32 g2−1 )δ(h33 g6−1 )δ(h41 g6−1 )δ(h42 g4−1 )δ(h43 g1−1 )
Z
λ
=−
[dg]6 [dh]12 [dx]4 [dy]4 φ(x1 h1k y1 )φ(x2 h2k y2 )φ(x3 h3k y3 )φ(x4 h4k y4 )×
4!
δ(h11 g1−1 )δ(h12 g2−1 )δ(h13 g3−1 )δ(h21 g3−1 )δ(h22 g4−1 )δ(h23 g5−1 )×
(C.23)
δ(h31 g5−1 )δ(h32 g2−1 )δ(h33 g6−1 )δ(h41 g6−1 )δ(h42 g4−1 )δ(h43 g1−1 )
(C.24)
Z
λ
=−
[dg]6 [dh]12 [dx]4 [dy]4 φ(h1k )φ(h2k )φ(h3k )φ(h4k )×
4!
δ(x1 h11 y1 g1−1 )δ(x1 h12 y1 g2−1 )δ(x1 h13 y1 g3−1 )δ(x2 h21 y2 g2−1 )δ(x2 h22 y2 g4−1 )δ(x2 h23 y2 g5−1 )×
δ(x3 h31 y3 g5−1 )δ(x3 h32 y3 g2−1 )δ(x3 h33 y3 g6−1 )δ(x4 h41 y4 g6−1 )δ(x4 h42 y4 g4−1 )δ(x4 h43 y4 g1−1 ) (C.25)
29
APPENDIX C. SOME INTERMEDIATE CALCULATIONS
Then, we use the decomposition (equation A.23) of the Dirac’s delta in characters:
X
X
X
j
j
j
δ(xhyg −1 ) =
dj Daa
(xhyg −1 ) =
dj
Dad
(xhy)Dda
(g −1 )
j
j
(C.26)
{a},{d}
The integration over the gi identifies the numbers labeling the Wigner matrices elements of the group
elements that are pairwise linked in the action. The integration over the xi and yi involves the functions
Ωaj11ja22ja33 ;d1 d2 d3 defined in equation C.13. We get:
3d
Sint
λ
=−
4!
Z
[dh]12 φ(h11 , h12 , h13 )φ(h21 , h22 , h23 )φ(h31 , h32 , h33 )φ(h41 , h42 , h43 )×
j22 j23
j12 j13
(h21 , h22 , h23 )×
(h11 , h12 , h13 )Ωja21
Ωja11
21 a22 a23 ;d21 d22 d23
11 a12 a13 ;d11 d12 d13
j42 j43
j32 j33
(h41 , h42 , h43 )×
(h31 , h32 , h33 )Ωja41
Ωja31
41 a42 a43 ;d41 d42 d43
31 a32 a33 ;d31 d32 d33
(−1)d11 +d12 +d13 +d22 +d23 +d33 +a11 +a12 +a13 +a22 +a23 +a33 δj11 j43 δj12 j32 δj13 j21 δj22 j42 δj23 j31 δj33 j41 ×
δa11 −a43 δa12 −a32 δa13 −a21 δa22 −a42 δa23 −a31 δa33 −a41 δd11 −d43 δd12 −d32 δd13 −d21 δd22 −d42 δd23 −d31 δd33 −d41
(C.27)
Z
X
λ
=−
[dh]12 φ(h1k )φ(h2k )φ(h3k )φ(h4k )
(−1)d1 +d2 +d3 +d4 +d5 +d6 +a1 +a2 +a3 +a4 +a5 +a6 ×
4!
{j},{a},{d}
j1 j2 j3
Ω
j3 j4 j5
(h1k )Ω
j5 j2 j6
(h2k )Ω
j6 j4 j1
(h4k )Ω
(h4k )×
j1 j2 j3
j3 j4 j5
j5
j2 j6
j6
j4
j1
×
a1 a2 a3
−a3 a4 a5
−a5 −a2 a6
−a6 −a4 −a1
j1 j2 j3
j3 j4 j5
j5
j2 j6
j6
j4
j1
d1 d2 d3
−d3 d4 d5
−d5 −d2 d6
−d6 −d4 −d1
2 Z
Z
λ j1 j2 j3
[dh]3 Ωj1 j2 j3 (h1k )φ(h1k ) [dh]3 Ωj3 j4 j5 (h2k )φ(h2k )
=−
4! j4 j5 j6
Z
Z
3 j5 j2 j6
[dh] Ω
(h3k )φ(h3k ) [dh]3 Ωj6 j4 j1 (h4k3 )φ(h4k )
We can show with a very similar construction that:
Z
Z
1
3 j1 j2 j3
3d
[dh] Ω
(h1k )φ(h1k ) [dh]3 Ωj3 j2 j1 (h2k )φ(h2k )
Skin =
2
(C.28)
(C.29)
(C.30)
Regrouping the kinetic and interaction terms, we can write:
Z
Z
1
λ
6
∗
[dh] φ(h1k )φ(h2k )K (h1k , h2k ) −
[dh]12 φ(h1k )φ(h2k )φ(h3k )φ(h4k )V ∗ (h1k , h2k , h3k , h4k )
S3d =
2
4!
(C.31)
where we identify the kernels
X
K ∗ (h1k , h2k ) =
Ωj1 j2 j3 (h1k )Ωj3 j2 j1 (h2k )
(C.32a)
{j}
X j1 j2 j3 2
V (h1k , h2k , h3k ) =
Ωj1 j2 j3 (h1k )Ωj3 j4 j5 (h2k )Ωj5 j2 j6 (h3k )Ωj6 j4 j1 (h4k )
j4 j5 j6
∗
{j}
Equations C.31 and C.32 are the 3d-analogue of equations C.8 and C.9.
30
(C.32b)
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