Introduction to Group Field Theory

Transcription

1 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/ / 21

2 Group Field Theory: what is it? Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/ / 21

3 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/ / 21

4 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/ / 21

5 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/ / 21

6 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/ / 21

7 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 : H kin H phys on physical states H phys s phys P s, s s phys s P s Spin Foams [Reisenberger, Rovelli... 00s] are a path-integral formulation of the dynamics amplitudes A s,c associated to a 2-complex C with boundary spin-network state s. A s,c = j A f A e f e v A v Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/ / 21

8 Structural incompleteness of Spin Foams: How one should interpret and organize the 2-complexes? How to extract A s from the family {A s,c C = s}? Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/ / 21

9 Structural incompleteness of Spin Foams: How one should interpret and organize the 2-complexes? How to extract A s from the family {A s,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: A s = A s,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/ / 21

10 Structural incompleteness of Spin Foams: How one should interpret and organize the 2-complexes? How to extract A s from the family {A s,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: A s = A s,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) A s lim A s,c [Dittrich, Bahr, Steinhaus, Martin-Benito... 10s] C 2 QFT interpretation: amplitudes of a Group Field Theory, to be summed over A s w C A s,c [De Pietri, Rovelli, Freidel, Oriti... 00s, 10s] C C=s Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/ / 21

11 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/ / 21

12 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/ / 21

13 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/ / 21

14 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/ / 21

15 General structure of a GFT and long-term objectives Typical form of a GFT: field ϕ(g 1,..., g d ), g l G, with partition function Z = [Dϕ] Λ exp t V V ϕ n V = (t Vi ) k V i {SF amplitudes} ϕ K ϕ + {V} k V1,...,k Vi i Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/ / 21

16 General structure of a GFT and long-term objectives Typical form of a GFT: field ϕ(g 1,..., g d ), g l G, with partition function Z = [Dϕ] Λ exp t V V ϕ n V = (t Vi ) k V i {SF amplitudes} ϕ K ϕ + {V} k V1,...,k Vi 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/ / 21

17 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/ / 21

18 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. g 1 g 2 h 1 h 2 h 3 h 4 g 1h 1 1 g 3 g 4 Elementary excitations over a vacuum 0 interpreted as a no-space vacuum. Creation/annihilation operators ϕ(g i ) / ϕ(g i ). H GFT = Fock(H v ) = + n=0 ( Sym H v (1) ) H v (n) with H v = L 2 (G d /G) (rem: bosonic statistics, arbitrary at this stage) ˆϕ(g 1, g 2, g 3, g 4) 0 = 0, ˆϕ g1 g2 (g 1, g 2, g 3, g 4) 0 =,... g4 g3 Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/ / 21

19 Dynamics Dynamics expressed as a projection in the Fock Hilbert space ) F Ψ ( P 1l Ψ = 0 It turns out that current GFT models do not correspond to a micro-canonical ensemble Z = s δ( F ) s s but a kind of grand-canonical ensemble [Oriti 13] Z = s s e β( F µ N) 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/ / 21

20 Physical applications The Fock representation permits the construction of simple condensate states e.g. ( ) σ exp [dg i ] 4 σ(g 1, g 2, g 3, g 4) ˆϕ (g 1, g 2, g 3, g 4) 0 arbitrary number of spin-network vertices excited with the same 1-particle wave-function σ(g 1, g 2, g 3, g 4). Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/ / 21

21 Physical applications The Fock representation permits the construction of simple condensate states e.g. ( ) σ exp [dg i ] 4 σ(g 1, g 2, g 3, g 4) ˆϕ (g 1, g 2, g 3, g 4) 0 arbitrary number of spin-network vertices excited with the same 1-particle wave-function σ(g 1, g 2, g 3, g 4). 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/ / 21

22 Physical applications The Fock representation permits the construction of simple condensate states e.g. ( ) σ exp [dg i ] 4 σ(g 1, g 2, g 3, g 4) ˆϕ (g 1, g 2, g 3, g 4) 0 arbitrary number of spin-network vertices excited with the same 1-particle wave-function σ(g 1, g 2, g 3, g 4). 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/ / 21

23 Physical applications The Fock representation permits the construction of simple condensate states e.g. ( ) σ exp [dg i ] 4 σ(g 1, g 2, g 3, g 4) ˆϕ (g 1, g 2, g 3, g 4) 0 arbitrary number of spin-network vertices excited with the same 1-particle wave-function σ(g 1, g 2, g 3, g 4). 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/ / 21

24 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 ; 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/ / 21

25 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/ / 21

26 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/ / 21

27 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/ / 21

28 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 ] Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/ / 21

29 Trace invariants Trace invariants of fields ϕ(g 1, g 2,..., g d ) labelled by d-colored bubbles b: Tr b (ϕ, ϕ) = [dg i ] 6 ϕ(g 6, g 2, g 3)ϕ(g 1, g 2, g 3) ϕ(g 6, g 4, g 5)ϕ(g 1, g 4, g 5) Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/ / 21

30 Trace invariants Trace invariants of fields ϕ(g 1, g 2,..., g d ) labelled by d-colored bubbles b: Tr b (ϕ, ϕ) = [dg i ] 6 ϕ(g 6, g 2, g 3)ϕ(g 1, g 2, g 3) ϕ(g 6, g 4, g 5)ϕ(g 1, g 4, g 5) (d = 2) Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/ / 21

31 Trace invariants Trace invariants of fields ϕ(g 1, g 2,..., g d ) labelled by d-colored bubbles b: Tr b (ϕ, ϕ) = [dg i ] 6 ϕ(g 6, g 2, g 3)ϕ(g 1, g 2, g 3) ϕ(g 6, g 4, g 5)ϕ(g 1, g 4, g 5) (d = 2) (d = 3) Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/ / 21

32 Trace invariants Trace invariants of fields ϕ(g 1, g 2,..., g d ) labelled by d-colored bubbles b: Tr b (ϕ, ϕ) = [dg i ] 6 ϕ(g 6, g 2, g 3)ϕ(g 1, g 2, g 3) ϕ(g 6, g 4, g 5)ϕ(g 1, g 4, g 5) (d = 2) (d = 3) (d = 4) Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/ / 21

33 Feynman amplitudes of TGFTs Perturbative expansion in the bubble coupling constants t b : Z = ( ) ( t b ) n b(g) A G G Feynman graphs G: b B g 1 g 2 g 3 = dg 1 dg 2 dg 3... g g = δ(g g 1 ) g 1 g 1 g 2 g 2 = C(g1, g 2, g 3 ; g 1, g 2, g 3 ) g 3 g 3 Covariances associated to the dashed, color-0 lines. Face of color l = connected set of (alternating) color-0 and color-l lines. Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/ / 21

34 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/ / 21

35 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 = ( l + l ) -1, C Λ (g l ; g l ) = dα Λ 2 d l=1 K G α (g lg 1 l ) [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): C = P( + d l ) -1 P, C Λ (g l ; g l ) = dα dh l Λ 2 G l=1 K G α (g lhg 1 l ) [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/ / 21

36 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 = ( l + l ) -1, C Λ (g l ; g l ) = dα Λ 2 d l=1 K G α (g lg 1 l ) [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): C = P( + d l ) -1 P, C Λ (g l ; g l ) = dα dh l Λ 2 G l=1 K G α (g lhg 1 l ) [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/ / 21

37 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/ / 21

38 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(g 1 ) 1 h 1, α 1 ϕ(g 3 ) ϕ(g 1 ) ϕ(g 3 ) 3 2 K + ϕ(g 2 ) h 2, α 2 ϕ(g 4 ) ϕ(g 2 ) ϕ(g 4 ) dα 1dα 2 [ dh 1dh 2 Kα1 +α 2 (h ] 2 1h 2) [ dg ij ] K α1 (g 11h 1g 1 31 )Kα 1 (g 2 21 h2g41) i<j δ(g 12g )δ(g13g22 )δ(g42g32 )δ(g43g33 ) ϕ(g 1) ϕ(g 2 ) ϕ(g 3 ) ϕ(g 4 ) Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/ / 21

39 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(g 1 ) 1 h 1, α 1 ϕ(g 3 ) ϕ(g 1 ) ϕ(g 3 ) 3 2 K + ϕ(g 2 ) h 2, α 2 ϕ(g 4 ) ϕ(g 2 ) ϕ(g 4 ) dα 1dα 2 [ dh 1dh 2 Kα1 +α 2 (h ] 2 1h 2) [ dg ij ] K α1 (g 11h 1g 1 31 )Kα 1 (g 2 21 h2g41) i<j δ(g 12g )δ(g13g22 )δ(g42g32 )δ(g43g33 ) ϕ(g 1) ϕ(g 2 ) ϕ(g 3 ) ϕ(g 4 ) 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/ / 21

40 Current developments 1 Non-perturbative renormalization: Wetterich equation applied to: matrix and tensor models; [Eichhorn, Koslowski 13 14] TGFT without gauge-invariance; [Benedetti, Ben Geloun, Oriti 14] gauge-invariant models. [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; [Gurau 11 13; Delepouve, Gurau, Rivasseau 14...] TGFTs without gauge invariance; [Delepouve, Rivasseau 14...] TGFTs with gauge invariance. [Lahoche, Oriti, Rivasseau 15] Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/ / 21

41 Current developments 1 Non-perturbative renormalization: Wetterich equation applied to: matrix and tensor models; [Eichhorn, Koslowski 13 14] TGFT without gauge-invariance; [Benedetti, Ben Geloun, Oriti 14] gauge-invariant models. [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; [Gurau 11 13; Delepouve, Gurau, Rivasseau 14...] TGFTs without gauge invariance; [Delepouve, Rivasseau 14...] TGFTs with gauge invariance. [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/ / 21

42 Current developments 1 Non-perturbative renormalization: Wetterich equation applied to: matrix and tensor models; [Eichhorn, Koslowski 13 14] TGFT without gauge-invariance; [Benedetti, Ben Geloun, Oriti 14] gauge-invariant models. [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; [Gurau 11 13; Delepouve, Gurau, Rivasseau 14...] TGFTs without gauge invariance; [Delepouve, Rivasseau 14...] TGFTs with gauge invariance. [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. [Lahoche, Oriti, SC wip] + d C Λ (g l ; g l ) = dα dh dk [dl l ] Λ 2 Spin(4) SU(2) H k l=1 K Spin(4) α (g l hl l g 1 l ). Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/ / 21

43 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/ / 21

44 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/ / 21

45 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/ / 21

46 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/ / 21

47 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/ / 21