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Introduction to Group Field Theory
Sylvain Carrozza
University of Bordeaux, LaBRI
The Helsinki Workshop on Quantum Gravity, 01/06/2016
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
1 / 21
Group Field Theory: what is it?
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
2 / 21
Group Field Theory: what is it?
It is an approach to quantum gravity at the crossroad of loop quantum gravity (LQG)
and matrix/tensor models.
A simple definition:
A Group Field Theory (GFT) is a non-local quantum field theory defined on
a group manifold.
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
2 / 21
Group Field Theory: what is it?
It is an approach to quantum gravity at the crossroad of loop quantum gravity (LQG)
and matrix/tensor models.
A simple definition:
A Group Field Theory (GFT) is a non-local quantum field theory defined on
a group manifold.
The group manifold is auxiliary: should not be interpreted as space-time!
Rather, the Feynman amplitudes are thought of as describing space-time
processes → QFT of space-time rather than on space-time.
Specific non-locality: determines the combinatorial structure of space-time
processes (graphs, 2-complexes, triangulations...).
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
2 / 21
Group Field Theory: what is it?
It is an approach to quantum gravity at the crossroad of loop quantum gravity (LQG)
and matrix/tensor models.
A simple definition:
A Group Field Theory (GFT) is a non-local quantum field theory defined on
a group manifold.
The group manifold is auxiliary: should not be interpreted as space-time!
Rather, the Feynman amplitudes are thought of as describing space-time
processes → QFT of space-time rather than on space-time.
Specific non-locality: determines the combinatorial structure of space-time
processes (graphs, 2-complexes, triangulations...).
Recommended reviews:
L. Freidel, ”Group Field Theory: an overview”, 2005
D. Oriti, ”The microscopic dynamics of quantum space as a group field theory”, 2011
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
2 / 21
From Loop Quantum Gravity to Group Field Theory
1
From Loop Quantum Gravity to Group Field Theory
2
Group Field Theory Fock space and physical applications
3
Group Field Theory renormalization programme
4
Summary and outlook
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
3 / 21
Loop Quantum Gravity proposes kinematical states describing (spatial) quantum
geometry [Ashtekar, Rovelli, Smolin, Lewandowski... ’90s; Dittrich, Geiller, Bahr ’15]:
Dynamics? Define the (improper) projector P : Hkin → Hphys on physical states
Hphys 3 |siphys ≡ P|si ,
hs|s 0 iphys ≡ hs|P|s 0 i
Spin Foams [Reisenberger, Rovelli... ’00s] are a path-integral formulation of the dynamics
→ amplitudes As,C associated to a 2-complex C with boundary spin-network state s.
XY Y Y
As,C =
Af
Ae
Av
j
Sylvain Carrozza (Univ. Bordeaux)
f
Introduction to GFT
e
v
Univ. Helsinki, 01/06/2016
4 / 21
Structural incompleteness of Spin Foams:
How one should interpret and organize the 2-complexes?
How to extract As from the family {As,C | ∂C = s}?
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
5 / 21
Structural incompleteness of Spin Foams:
How one should interpret and organize the 2-complexes?
How to extract As from the family {As,C | ∂C = s}?
Three interpretations of C found in the literature:
(i) a convenient way of writing up the amplitudes, but amplitudes independent of it
from the outset: As = As,C ;
(ex: Turaev-Viro model)
(ii) a regulator, analogous to the lattice of lattice gauge theory;
(iii) a specific quantum history compatible with the boundary state, analogous to a
Feynman diagram in QFT.
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
5 / 21
Structural incompleteness of Spin Foams:
How one should interpret and organize the 2-complexes?
How to extract As from the family {As,C | ∂C = s}?
Three interpretations of C found in the literature:
(i) a convenient way of writing up the amplitudes, but amplitudes independent of it
from the outset: As = As,C ;
(ex: Turaev-Viro model)
(ii) a regulator, analogous to the lattice of lattice gauge theory;
(iii) a specific quantum history compatible with the boundary state, analogous to a
Feynman diagram in QFT.
First interpretation seems very hard to realize in 4d (→ construction of 4d invariants of
manifolds), and the other two hinge on renormalization theory:
1
Lattice interpretation: refining and coarse-graining C (and s)
⇒ As ≡ lim As,C
[Dittrich, Bahr, Steinhaus, Martin-Benito...
C →∞
2
’10s]
QFT interpretation:
amplitudes of a Group Field Theory, to be summed over
P
⇒ As ≡
wC As,C
[De Pietri, Rovelli, Freidel, Oriti... ’00s, ’10s]
C |∂C =s
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
5 / 21
In the two interpretations, renormalization is central and allows in principle to address
some other open challenges:
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
6 / 21
In the two interpretations, renormalization is central and allows in principle to address
some other open challenges:
1
consistency of the quantum dynamics under coarse-graining?
2
quantization / discretization ambiguities inherent to spin-foams: what are the
universal features of the known models?
[EPRL, DL, BO, ...]
3
macro-physics from microscopic dynamics: how do we extract the low-energy limit
of LQG? are there several quantum phases? compatibility with general relativity?
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
6 / 21
In the two interpretations, renormalization is central and allows in principle to address
some other open challenges:
1
consistency of the quantum dynamics under coarse-graining?
2
quantization / discretization ambiguities inherent to spin-foams: what are the
universal features of the known models?
[EPRL, DL, BO, ...]
3
macro-physics from microscopic dynamics: how do we extract the low-energy limit
of LQG? are there several quantum phases? compatibility with general relativity?
Refining framework ⇒ background independent generalization of direct space
renormalization methods:
scale = lattice itself
consistency over scales ⇔ dynamical cylindrical consistency
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
6 / 21
In the two interpretations, renormalization is central and allows in principle to address
some other open challenges:
1
consistency of the quantum dynamics under coarse-graining?
2
quantization / discretization ambiguities inherent to spin-foams: what are the
universal features of the known models?
[EPRL, DL, BO, ...]
3
macro-physics from microscopic dynamics: how do we extract the low-energy limit
of LQG? are there several quantum phases? compatibility with general relativity?
Refining framework ⇒ background independent generalization of direct space
renormalization methods:
scale = lattice itself
consistency over scales ⇔ dynamical cylindrical consistency
Summing framework ⇒ background independent generalization of momentum shell
renormalization methods:
scale = spectrum of a specific 1-particle operator (e.g. spin labels)
consistency over scales ⇔ renormalization group flow of a (non-local) field theory
defined on internal space (e.g. SU(2)).
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
6 / 21
General structure of a GFT and long-term objectives
Typical form of a GFT: field ϕ(g1 , . . . , gd ), g` ∈ G , with partition function


Z
X
X Y
n
V
tV V · ϕ  =
(tVi )kVi {SF amplitudes}
Z = [Dϕ]Λ exp −ϕ · K · ϕ +
{V}
Sylvain Carrozza (Univ. Bordeaux)
kV1 ,...,kV
Introduction to GFT
i
i
Univ. Helsinki, 01/06/2016
7 / 21
General structure of a GFT and long-term objectives
Typical form of a GFT: field ϕ(g1 , . . . , gd ), g` ∈ G , with partition function


Z
X
X Y
n
V
tV V · ϕ  =
(tVi )kVi {SF amplitudes}
Z = [Dϕ]Λ exp −ϕ · K · ϕ +
{V}
kV1 ,...,kV
i
i
Main objectives of the GFT research programme:
1
Model building: define the theory space.
e.g. spin foam models + combinatorial considerations (tensor models) → d, G , K,
{V} and [Dϕ]Λ .
2
Perturbative definition: prove that the spin foam expansion is consistent in some
range of Λ.
e.g. perturbative multi-scale renormalization.
3
Systematically explore the theory space: effective continuum regime reproducing
GR in some limit?
e.g. functional RG, constructive methods, condensate states...
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
7 / 21
Group Field Theory Fock space and physical applications
1
From Loop Quantum Gravity to Group Field Theory
2
Group Field Theory Fock space and physical applications
3
Group Field Theory renormalization programme
4
Summary and outlook
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
8 / 21
GFT Hilbert space
No embedding in a continuum manifold and no cylindrical consistency imposed.
Instead: Fock construction through decomposition of spin network states in terms
of elementary building blocks.
h1
g1
g2
g1 h−1
1
h4
h2
h3
g3
g4
Elementary excitations over a vacuum |0i interpreted as a ’no-space vacuum’.
Creation/annihilation operators ϕ(g
b i )† /ϕ(g
b i ).
HGFT = Fock(Hv ) =
+∞
M
Sym Hv(1) ⊗ · · · ⊗ Hv(n)
Hv = L2 (G ×d /G )
with
n=0
(rem: bosonic statistics, arbitrary at this stage)
ϕ̂(g1 , g2 , g3 , g4 )|0i = 0 ,
Sylvain Carrozza (Univ. Bordeaux)
†
g1
ϕ̂ (g1 , g2 , g3 , g4 )|0i = |
Introduction to GFT
g4
g2
g3
i,
...
Univ. Helsinki, 01/06/2016
9 / 21
Dynamics
Dynamics expressed as a projection in the Fock Hilbert space
b |Ψi ≡ P
b − 1l |Ψi = 0
F
It turns out that current GFT models do not correspond to a ’micro-canonical’
ensemble
X
b )|si
Z=
hs|δ(F
s
but a kind of ’grand-canonical’ ensemble
X
b b
Z=
hs|e−β(F −µN ) |si
[Oriti ’13]
s
⇒ the GFT genuinely contains more information than the LQG projector on
physical states [Freidel ’05]
Open questions: how to extract the LQG physical projector? what is the role of
topology changing processes?
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
10 / 21
Physical applications
The Fock representation permits the construction of simple condensate states e.g.
Z
|σi ∝ exp
[dgi ]4 σ(g1 , g2 , g3 , g4 )ϕ̂† (g1 , g2 , g3 , g4 ) |0i
→ arbitrary number of spin-network vertices excited with the same 1-particle
wave-function σ(g1 , g2 , g3 , g4 ).
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
11 / 21
Physical applications
The Fock representation permits the construction of simple condensate states e.g.
Z
|σi ∝ exp
[dgi ]4 σ(g1 , g2 , g3 , g4 )ϕ̂† (g1 , g2 , g3 , g4 ) |0i
→ arbitrary number of spin-network vertices excited with the same 1-particle
wave-function σ(g1 , g2 , g3 , g4 ).
Such states have been successfully used to describe symmetric quantum geometries
directly at the GFT level, hence without recourse to classical symmetry reduction:
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
11 / 21
Physical applications
The Fock representation permits the construction of simple condensate states e.g.
Z
|σi ∝ exp
[dgi ]4 σ(g1 , g2 , g3 , g4 )ϕ̂† (g1 , g2 , g3 , g4 ) |0i
→ arbitrary number of spin-network vertices excited with the same 1-particle
wave-function σ(g1 , g2 , g3 , g4 ).
Such states have been successfully used to describe symmetric quantum geometries
directly at the GFT level, hence without recourse to classical symmetry reduction:
Cosmology:
[Gielen, Oriti, Sindoni, Calcagni, Wilson-Ewing, Pithis,...]
EPRL model coupled to a scalar field −→ condensate in the hydrodynamic
approximation −→ Friedmann equations with quantum gravity corrections −→
bounce at the Planck scale.
[Oriti, Sindoni, Wilson-Ewing ’16]
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
11 / 21
Physical applications
The Fock representation permits the construction of simple condensate states e.g.
Z
|σi ∝ exp
[dgi ]4 σ(g1 , g2 , g3 , g4 )ϕ̂† (g1 , g2 , g3 , g4 ) |0i
→ arbitrary number of spin-network vertices excited with the same 1-particle
wave-function σ(g1 , g2 , g3 , g4 ).
Such states have been successfully used to describe symmetric quantum geometries
directly at the GFT level, hence without recourse to classical symmetry reduction:
Cosmology:
[Gielen, Oriti, Sindoni, Calcagni, Wilson-Ewing, Pithis,...]
EPRL model coupled to a scalar field −→ condensate in the hydrodynamic
approximation −→ Friedmann equations with quantum gravity corrections −→
bounce at the Planck scale.
[Oriti, Sindoni, Wilson-Ewing ’16]
Black Holes:
[Pranzetti, Sindoni, Oriti ’15]
Condensates encoding spherically symmetric quantum geometry −→ reduced
density matrix associated to a horizon −→ horizon entanglement entropy −→
Bekenstein-Hawking entropy formula for any value of the Immirzi parameter.
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
11 / 21
Summary up to now
GFT can be understood as a version of LQG with:
no embedding in a continuous manifold;
organization of LQG states in ’space atoms’;
b
a new fundamental observable: N.
Provides statistical techniques to explore the many-body sector of quantum
geometry: condensate states used for e.g. quantum cosmology and black holes
The construction seems quite general ⇒ other choices of ’building blocks’ ?
Useful for construction of GFT analogues of new kinematical vacua?
[Dittrich, Geiller ’15 ’16]
Quantization ambiguities are encoded in free coupling constants for the various
spin foam vertices compatible with the dynamics one would like to implement ⇒
renormalization has to tell us which of these are more relevant.
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
12 / 21
Group Field Theory renormalization programme
1
From Loop Quantum Gravity to Group Field Theory
2
Group Field Theory Fock space and physical applications
3
Group Field Theory renormalization programme
4
Summary and outlook
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
13 / 21
Importance of combinatorics
Mathematical objective: step-by-step generalization of standard renormalization
techniques, until we are able to tackle 4d quantum gravity proposals.
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
14 / 21
Importance of combinatorics
Mathematical objective: step-by-step generalization of standard renormalization
techniques, until we are able to tackle 4d quantum gravity proposals.
Two main aspects in the definition of a group field theory:
Algebraic content and type of dynamics implemented: from LQG and Spin Foams
Combinatorial structures:
Which types of spin-network boundary states? In general, restriction on the valency.
Which type of spin foam vertices? In general, restriction on the valency too.
Which types of 2-complexes are summed over? Local restrictions on gluing rules to
avoid too pathological topologies.
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
14 / 21
Importance of combinatorics
Mathematical objective: step-by-step generalization of standard renormalization
techniques, until we are able to tackle 4d quantum gravity proposals.
Two main aspects in the definition of a group field theory:
Algebraic content and type of dynamics implemented: from LQG and Spin Foams
Combinatorial structures:
Which types of spin-network boundary states? In general, restriction on the valency.
Which type of spin foam vertices? In general, restriction on the valency too.
Which types of 2-complexes are summed over? Local restrictions on gluing rules to
avoid too pathological topologies.
Requirement: the GFT theory space should be stable enough under renormalization / coarse-graining.
We currently know of only one such combinatorial structure: tensorial interactions
initially introduced in the context of tensor models.
[Gurau, Bonzom, Rivasseau, Ben Geloun... ’11...]
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
14 / 21
Trace invariants
Trace invariants of fields ϕ(g1 , g2 , . . . , gd ) labelled by d-colored bubbles b:
3
Z
1
2
2
1
Trb (ϕ, ϕ) =
[dgi ]6 ϕ(g6 , g2 , g3 )ϕ(g1 , g2 , g3 )
ϕ(g6 , g4 , g5 )ϕ(g1 , g4 , g5 )
3
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
15 / 21
Trace invariants
Trace invariants of fields ϕ(g1 , g2 , . . . , gd ) labelled by d-colored bubbles b:
3
Z
1
2
1
2
Trb (ϕ, ϕ) =
[dgi ]6 ϕ(g6 , g2 , g3 )ϕ(g1 , g2 , g3 )
ϕ(g6 , g4 , g5 )ϕ(g1 , g4 , g5 )
3
···
(d = 2)
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
15 / 21
Trace invariants
Trace invariants of fields ϕ(g1 , g2 , . . . , gd ) labelled by d-colored bubbles b:
3
Z
1
2
1
2
Trb (ϕ, ϕ) =
[dgi ]6 ϕ(g6 , g2 , g3 )ϕ(g1 , g2 , g3 )
ϕ(g6 , g4 , g5 )ϕ(g1 , g4 , g5 )
3
···
(d = 2)
···
(d = 3)
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
15 / 21
Trace invariants
Trace invariants of fields ϕ(g1 , g2 , . . . , gd ) labelled by d-colored bubbles b:
3
Z
1
2
1
2
Trb (ϕ, ϕ) =
[dgi ]6 ϕ(g6 , g2 , g3 )ϕ(g1 , g2 , g3 )
ϕ(g6 , g4 , g5 )ϕ(g1 , g4 , g5 )
3
···
(d = 2)
···
(d = 3)
···
(d = 4)
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
15 / 21
Feynman amplitudes of TGFTs
Perturbative expansion in the bubble coupling constants tb :
!
X Y
nb (G)
Z=
(−tb )
AG
G
b∈B
Feynman graphs G:
g1
g2
=
g3
g
g2
g̃
g1
g̃1
g3
g̃3
R
dg1 dg2 dg3 . . .
= δ(gg̃ −1 )
g̃2
= C(g1 , g2 , g3 ; g̃1 , g̃2 , g̃3 )
Covariances associated to the dashed, color-0 lines.
Face of color ` = connected set of (alternating) color-0 and color-` lines.
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
16 / 21
Perturbative renormalization: overview
Goal: check that the perturbative expansion - and henceforth the connection to
spin foam models - is consistent.
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
17 / 21
Perturbative renormalization: overview
Goal: check that the perturbative expansion - and henceforth the connection to
spin foam models - is consistent.
Types of models considered so far:
’combinatorial’ models on G = U(1)D :
C =(
X
∆` )-1 ,
CΛ (g` ; g`0 ) =
`
Z
+∞
dα
Λ−2
d
Y
KαG (g` g`0−1 )
`=1
[Ben Geloun, Rivasseau ’11; Ben Geloun, Ousmane Samary ’12; Ben Geloun, Livine ’12...]
models with ’gauge invariance’ on G = U(1)D or SU(2):
Z +∞
Z
d
X
Y
C = P(
∆` )-1 P ,
CΛ (g` ; g`0 ) =
dα
dh
KαG (g` hg`0−1 )
Λ−2
`
G
`=1
[SC, Oriti, Rivasseau ’12 ’13; Ousmane Samary, Vignes-Tourneret ’12; SC ’14 ’14; Lahoche, Oriti,
Rivasseau ’14...]
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
17 / 21
Perturbative renormalization: overview
Goal: check that the perturbative expansion - and henceforth the connection to
spin foam models - is consistent.
Types of models considered so far:
’combinatorial’ models on G = U(1)D :
C =(
X
∆` )-1 ,
CΛ (g` ; g`0 ) =
`
Z
+∞
dα
Λ−2
d
Y
KαG (g` g`0−1 )
`=1
[Ben Geloun, Rivasseau ’11; Ben Geloun, Ousmane Samary ’12; Ben Geloun, Livine ’12...]
models with ’gauge invariance’ on G = U(1)D or SU(2):
Z +∞
Z
d
X
Y
C = P(
∆` )-1 P ,
CΛ (g` ; g`0 ) =
dα
dh
KαG (g` hg`0−1 )
Λ−2
`
G
`=1
[SC, Oriti, Rivasseau ’12 ’13; Ousmane Samary, Vignes-Tourneret ’12; SC ’14 ’14; Lahoche, Oriti,
Rivasseau ’14...]
Methods:
multiscale analysis: allows to rigorously prove renormalizability at all orders in
perturbation theory;
Connes–Kreimer algebraic methods [Raasakka, Tanasa ’13; Avohou, Rivasseau, Tanasa ’15].
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
17 / 21
Quasi-locality of divergences
The divergent subgraphs must be quasi-local, i.e. look like trace invariants at
high scales. Always the case in known models, but non–trivial!
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
18 / 21
Quasi-locality of divergences
The divergent subgraphs must be quasi-local, i.e. look like trace invariants at
high scales. Always the case in known models, but non–trivial!
ϕ(g1 )
3
h1 , α1
1
ϕ(g3 )
ϕ(g2 )
h2 , α2
Z
dα1 dα2
ϕ(g3 )
∼ K×
2
Z
ϕ(g1 )
ϕ(g4 )
2
dh1 dh2 Kα1 +α2 (h1 h2 )
ϕ(g2 )
+ ···
ϕ(g4 )
Z Y
−1
−1
[
dgij ] Kα1 (g11 h1 g31 )Kα2 (g21 h2 g41 )
i<j
−1
−1
−1
−1
δ(g12 g22 )δ(g13 g22 )δ(g42 g32 )δ(g43 g33 ) ϕ(g1 ) ϕ(g2 ) ϕ(g3 ) ϕ(g4 )
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
18 / 21
Quasi-locality of divergences
The divergent subgraphs must be quasi-local, i.e. look like trace invariants at
high scales. Always the case in known models, but non–trivial!
ϕ(g1 )
3
h1 , α1
1
ϕ(g3 )
ϕ(g2 )
h2 , α2
Z
dα1 dα2
ϕ(g3 )
∼ K×
2
Z
ϕ(g1 )
ϕ(g4 )
2
dh1 dh2 Kα1 +α2 (h1 h2 )
ϕ(g2 )
+ ···
ϕ(g4 )
Z Y
−1
−1
[
dgij ] Kα1 (g11 h1 g31 )Kα2 (g21 h2 g41 )
i<j
−1
−1
−1
−1
δ(g12 g22 )δ(g13 g22 )δ(g42 g32 )δ(g43 g33 ) ϕ(g1 ) ϕ(g2 ) ϕ(g3 ) ϕ(g4 )
This property is not generic in TGFTs → ”traciality” criterion.
Nice interplay between structure of divergences and topology → renormalizable
interactions are spherical.
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
18 / 21
Current developments
1
Non-perturbative renormalization:
Wetterich equation applied to:
matrix and tensor models;
TGFT without gauge-invariance;
gauge-invariant models.
[Eichhorn, Koslowski ’13 ’14]
[Benedetti, Ben Geloun, Oriti ’14]
[Lahoche, Benedetti ’15; Lahoche, SC wip]
Polchinski equation
[Krajewski, Toriumi ’15]
Constructive methods such as the loop-vertex expansion (intermediate field) applied
to:
tensor models;
TGFTs without gauge invariance;
TGFTs with gauge invariance.
Sylvain Carrozza (Univ. Bordeaux)
[Gurau ’11 ’13; Delepouve, Gurau, Rivasseau ’14...]
[Delepouve, Rivasseau ’14...]
[Lahoche, Oriti, Rivasseau ’15]
Introduction to GFT
Univ. Helsinki, 01/06/2016
19 / 21
Current developments
1
Non-perturbative renormalization:
Wetterich equation applied to:
matrix and tensor models;
TGFT without gauge-invariance;
gauge-invariant models.
[Eichhorn, Koslowski ’13 ’14]
[Benedetti, Ben Geloun, Oriti ’14]
[Lahoche, Benedetti ’15; Lahoche, SC wip]
Polchinski equation
[Krajewski, Toriumi ’15]
Constructive methods such as the loop-vertex expansion (intermediate field) applied
to:
tensor models;
TGFTs without gauge invariance;
TGFTs with gauge invariance.
[Gurau ’11 ’13; Delepouve, Gurau, Rivasseau ’14...]
[Delepouve, Rivasseau ’14...]
[Lahoche, Oriti, Rivasseau ’15]
Lesson: non-trivial fixed points seem generic. Phase transition to a condensed
phase?
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
19 / 21
Current developments
1
Non-perturbative renormalization:
Wetterich equation applied to:
matrix and tensor models;
TGFT without gauge-invariance;
gauge-invariant models.
[Eichhorn, Koslowski ’13 ’14]
[Benedetti, Ben Geloun, Oriti ’14]
[Lahoche, Benedetti ’15; Lahoche, SC wip]
Polchinski equation
[Krajewski, Toriumi ’15]
Constructive methods such as the loop-vertex expansion (intermediate field) applied
to:
tensor models;
TGFTs without gauge invariance;
TGFTs with gauge invariance.
[Gurau ’11 ’13; Delepouve, Gurau, Rivasseau ’14...]
[Delepouve, Rivasseau ’14...]
[Lahoche, Oriti, Rivasseau ’15]
Lesson: non-trivial fixed points seem generic. Phase transition to a condensed
phase?
2
Towards renormalizable models with simplicity constraints:
GFT on SU(2)/U(1);
[Lahoche, Oriti ’15]
4d GFT on Spin(4) with Barrett-Crane simplicity constraints.
CΛ (g` ; g`0 ) =
Sylvain Carrozza (Univ. Bordeaux)
Z
+∞
Z
Z
dα
Λ−2
Z
dh
Spin(4)
dk
SU(2)
Introduction to GFT
Hk
[dl` ]
d
Y
[Lahoche, Oriti, SC wip]
Spin(4)
Kα
(g` hl` g`0−1 ) .
`=1
Univ. Helsinki, 01/06/2016
19 / 21
Summary and outlook
1
From Loop Quantum Gravity to Group Field Theory
2
Group Field Theory Fock space and physical applications
3
Group Field Theory renormalization programme
4
Summary and outlook
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
20 / 21
Summary and outlook
GFT is a QFT completion of spin foam models.
It allows to (define and) explore the many-body sector of LQG.
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
21 / 21
Summary and outlook
GFT is a QFT completion of spin foam models.
It allows to (define and) explore the many-body sector of LQG.
Two parallel lines of investigations:
Construction of effective geometries from condensate states and approximations of
the full GFT dynamics → some aspects of quantum cosmology and black holes
recovered from 4d quantum gravity models!
See talks by Wilson-Ewing and Pithis
Development of suitable renormalizable tools to check the overall consistency of
GFTs and explore more systematically their phase diagrams → applicable to
simplified toy-models, not yet to 4d quantum gravity.
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
21 / 21
Summary and outlook
GFT is a QFT completion of spin foam models.
It allows to (define and) explore the many-body sector of LQG.
Two parallel lines of investigations:
Construction of effective geometries from condensate states and approximations of
the full GFT dynamics → some aspects of quantum cosmology and black holes
recovered from 4d quantum gravity models!
See talks by Wilson-Ewing and Pithis
Development of suitable renormalizable tools to check the overall consistency of
GFTs and explore more systematically their phase diagrams → applicable to
simplified toy-models, not yet to 4d quantum gravity.
Can we define a renormalizable 4d quantum gravity model and prove the existence
of a condensed phases with the right properties?
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
21 / 21
Summary and outlook
GFT is a QFT completion of spin foam models.
It allows to (define and) explore the many-body sector of LQG.
Two parallel lines of investigations:
Construction of effective geometries from condensate states and approximations of
the full GFT dynamics → some aspects of quantum cosmology and black holes
recovered from 4d quantum gravity models!
See talks by Wilson-Ewing and Pithis
Development of suitable renormalizable tools to check the overall consistency of
GFTs and explore more systematically their phase diagrams → applicable to
simplified toy-models, not yet to 4d quantum gravity.
Can we define a renormalizable 4d quantum gravity model and prove the existence
of a condensed phases with the right properties?
Thank you for your attention
Sylvain Carrozza (Univ. Bordeaux)
Introduction to GFT
Univ. Helsinki, 01/06/2016
21 / 21