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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?

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-timeprocesses → QFT of space-time rather than on space-time.

Specific non-locality: determines the combinatorial structure of space-timeprocesses (graphs, 2-complexes, triangulations...).

Recommended reviews:L. Freidel, ”Group Field Theory: an overview”, 2005D. Oriti, ”The microscopic dynamics of quantum space as a group field theory”, 2011

a group manifold.

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

Loop Quantum Gravity proposes kinematical states describing (spatial) quantumgeometry [Ashtekar, Rovelli, Smolin, Lewandowski... ’90s; Dittrich, Geiller, Bahr ’15]:

Dynamics? Define the (improper) projector P : Hkin → Hphys on physical states

Hphys 3 |s〉phys ≡ P|s〉 , 〈s|s ′〉phys ≡ 〈s|P|s ′〉

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.

As,C =∑j

��

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 itfrom 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 aFeynman diagram in QFT.

First interpretation seems very hard to realize in 4d (→ construction of 4d invariants ofmanifolds), and the other two hinge on renormalization theory:

1 Lattice interpretation: refining and coarse-graining C (and s)⇒ As ≡ lim

C→∞As,C [Dittrich, Bahr, Steinhaus, Martin-Benito... ’10s]

2 QFT interpretation: amplitudes of a Group Field Theory, to be summed over⇒ As ≡

∑C |∂C=s

wCAs,C [De Pietri, Rovelli, Freidel, Oriti... ’00s, ’10s]

��

∑C |∂C=s

��

∑C |∂C=s

In the two interpretations, renormalization is central and allows in principle to addresssome other open challenges:

1 consistency of the quantum dynamics under coarse-graining?

2 quantization / discretization ambiguities inherent to spin-foams: what are theuniversal features of the known models? [EPRL, DL, BO, ...]

3 macro-physics from microscopic dynamics: how do we extract the low-energy limitof LQG? are there several quantum phases? compatibility with general relativity?

Refining framework ⇒ background independent generalization of direct spacerenormalization methods:

scale = lattice itself

consistency over scales ⇔ dynamical cylindrical consistency

Summing framework ⇒ background independent generalization of momentum shellrenormalization methods:

scale = spectrum of a specific 1-particle operator (e.g. spin labels)

consistency over scales ⇔ renormalization group flow of a (non-local) field theorydefined on internal space (e.g. SU(2)).

General structure of a GFT and long-term objectives

Typical form of a GFT: field ϕ(g1, . . . , gd), g` ∈ G , with partition function

∫[Dϕ]Λ exp

−ϕ · K · ϕ+∑{V}

tV V · ϕnV

kV1,...,kVi

(tVi )kVi {SF amplitudes}

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 somerange of Λ.e.g. perturbative multi-scale renormalization.

3 Systematically explore the theory space: effective continuum regime reproducingGR in some limit?e.g. functional RG, constructive methods, condensate states...

General structure of a GFT and long-term objectives

Typical form of a GFT: field ϕ(g1, . . . , gd), g` ∈ G , with partition function

∫[Dϕ]Λ exp

−ϕ · K · ϕ+∑{V}

tV V · ϕnV

kV1,...,kVi

(tVi )kVi {SF amplitudes}

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 somerange of Λ.e.g. perturbative multi-scale renormalization.

3 Systematically explore the theory space: effective continuum regime reproducingGR in some limit?e.g. functional RG, constructive methods, condensate states...

Group Field Theory Fock space and physical applications

GFT Hilbert space

No embedding in a continuum manifold and no cylindrical consistency imposed.

Instead: Fock construction through decomposition of spin network states in termsof elementary building blocks.

g1h−11

Elementary excitations over a vacuum |0〉 interpreted as a ’no-space vacuum’.Creation/annihilation operators ϕ̂(gi )

†/ϕ̂(gi ).

HGFT = Fock(Hv ) =+∞⊕n=0

Sym(H(1)

v ⊗ · · · ⊗ H(n)v

)with Hv = L2(G×d/G)

(rem: bosonic statistics, arbitrary at this stage)

ϕ̂(g1, g2, g3, g4)|0〉 = 0 , ϕ̂†(g1, g2, g3, g4)|0〉 = |g1 g2

g3g4 〉 , . . .

Dynamics

Dynamics expressed as a projection in the Fock Hilbert space

F̂ |Ψ〉 ≡(P̂ − 1l

)|Ψ〉 = 0

It turns out that current GFT models do not correspond to a ’micro-canonical’ensemble

Z =∑s

〈s|δ(F̂ )|s〉

but a kind of ’grand-canonical’ ensemble [Oriti ’13]

Z =∑s

〈s|e−β(F̂−µN̂)|s〉

⇒ the GFT genuinely contains more information than the LQG projector onphysical states [Freidel ’05]

Open questions: how to extract the LQG physical projector? what is the role oftopology changing processes?

Physical applications

The Fock representation permits the construction of simple condensate states e.g.

|σ〉 ∝ exp

(∫[dgi ]

4 σ(g1, g2, g3, g4)ϕ̂†(g1, g2, g3, g4)

)|0〉

→ arbitrary number of spin-network vertices excited with the same 1-particlewave-function σ(g1, g2, g3, g4).

Such states have been successfully used to describe symmetric quantum geometriesdirectly 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 hydrodynamicapproximation −→ 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 −→ reduceddensity matrix associated to a horizon −→ horizon entanglement entropy −→Bekenstein-Hawking entropy formula for any value of the Immirzi parameter.

|σ〉 ∝ exp

(∫[dgi ]

4 σ(g1, g2, g3, g4)ϕ̂†(g1, g2, g3, g4)

)|0〉

|σ〉 ∝ exp

(∫[dgi ]

4 σ(g1, g2, g3, g4)ϕ̂†(g1, g2, g3, g4)

)|0〉

|σ〉 ∝ exp

(∫[dgi ]

4 σ(g1, g2, g3, g4)ϕ̂†(g1, g2, g3, g4)

)|0〉

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: N̂.

Provides statistical techniques to explore the many-body sector of quantumgeometry: 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 variousspin foam vertices compatible with the dynamics one would like to implement ⇒renormalization has to tell us which of these are more relevant.

Group Field Theory renormalization programme

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 toavoid too pathological topologies.

�Requirement: the GFT theory space should be stable enough under renormaliza-

tion / coarse-graining.

We currently know of only one such combinatorial structure: tensorial interactionsinitially introduced in the context of tensor models.

[Gurau, Bonzom, Rivasseau, Ben Geloun... ’11...]

Trace invariants

Trace invariants of fields ϕ(g1, g2, . . . , gd) labelled by d-colored bubbles b:

Trb(ϕ,ϕ) =

∫[dgi ]

6 ϕ(g6, g2, g3)ϕ(g1, g2, g3)

ϕ(g6, g4, g5)ϕ(g1, g4, g5)

(d = 2) · · ·

(d = 3) · · ·

(d = 4) · · ·

Trace invariants

Trb(ϕ,ϕ) =

∫[dgi ]

6 ϕ(g6, g2, g3)ϕ(g1, g2, g3)

ϕ(g6, g4, g5)ϕ(g1, g4, g5)

(d = 2) · · ·

(d = 3) · · ·

(d = 4) · · ·

Trace invariants

Trb(ϕ,ϕ) =

∫[dgi ]

6 ϕ(g6, g2, g3)ϕ(g1, g2, g3)

ϕ(g6, g4, g5)ϕ(g1, g4, g5)

(d = 2) · · ·

(d = 3) · · ·

(d = 4) · · ·

Trace invariants

Trb(ϕ,ϕ) =

∫[dgi ]

6 ϕ(g6, g2, g3)ϕ(g1, g2, g3)

ϕ(g6, g4, g5)ϕ(g1, g4, g5)

(d = 2) · · ·

(d = 3) · · ·

(d = 4) · · ·

Feynman amplitudes of TGFTs

Perturbative expansion in the bubble coupling constants tb:

Z =∑G

(∏b∈B

(−tb)nb(G)

Feynman graphs G:

∫dg1 dg2 dg3 . . .

= δ(gg̃−1)g g̃

g1 g̃1g2

g3 g̃3

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.

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 = (∑`

∆`)-1 , CΛ(g`; g

′`) =

∫ +∞

Λ−2dα

d∏`=1

KGα (g`g

′−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):

C = P(∑`

∆`)-1P , CΛ(g`; g

′`) =

∫ +∞

Λ−2dα

∫Gdh

d∏`=1

KGα (g`hg

′−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 inperturbation theory;

Connes–Kreimer algebraic methods [Raasakka, Tanasa ’13; Avohou, Rivasseau, Tanasa ’15].

C = (∑`

∆`)-1 , CΛ(g`; g

′`) =

∫ +∞

Λ−2dα

d∏`=1

KGα (g`g

′−1` )

C = P(∑`

∆`)-1P , CΛ(g`; g

′`) =

∫ +∞

Λ−2dα

∫Gdh

d∏`=1

KGα (g`hg

′−1` )

Rivasseau ’14...]

Methods:

C = (∑`

∆`)-1 , CΛ(g`; g

′`) =

∫ +∞

Λ−2dα

d∏`=1

KGα (g`g

′−1` )

C = P(∑`

∆`)-1P , CΛ(g`; g

′`) =

∫ +∞

Λ−2dα

∫Gdh

d∏`=1

KGα (g`hg

′−1` )

Rivasseau ’14...]

Methods:

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)

ϕ(g2)

ϕ(g3)

ϕ(g4)

∼ϕ(g3)

ϕ(g4)

ϕ(g1)

ϕ(g2)

K× + · · ·

h1 , α1

h2 , α2

∫dα1dα2

∫dh1dh2

[Kα1+α2

(h1h2)]2∫

[∏i<j

dgij ] Kα1(g11h1g

−131 )Kα2

(g−121 h2g41)

δ(g12g−122 )δ(g13g

−122 )δ(g42g

−132 )δ(g43g

−133 )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 → renormalizableinteractions are spherical.

ϕ(g1)

ϕ(g2)

ϕ(g3)

ϕ(g4)

∼ϕ(g3)

ϕ(g4)

ϕ(g1)

ϕ(g2)

K× + · · ·

h1 , α1

h2 , α2

∫dα1dα2

∫dh1dh2

[Kα1+α2

(h1h2)]2∫

[∏i<j

dgij ] Kα1(g11h1g

−131 )Kα2

(g−121 h2g41)

δ(g12g−122 )δ(g13g

−122 )δ(g42g

−132 )δ(g43g

−133 )ϕ(g1)ϕ(g2)ϕ(g3)ϕ(g4)

ϕ(g1)

ϕ(g2)

ϕ(g3)

ϕ(g4)

∼ϕ(g3)

ϕ(g4)

ϕ(g1)

ϕ(g2)

K× + · · ·

h1 , α1

h2 , α2

∫dα1dα2

∫dh1dh2

[Kα1+α2

(h1h2)]2∫

[∏i<j

dgij ] Kα1(g11h1g

−131 )Kα2

(g−121 h2g41)

δ(g12g−122 )δ(g13g

−122 )δ(g42g

−132 )δ(g43g

−133 )ϕ(g1)ϕ(g2)ϕ(g3)ϕ(g4)

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) appliedto:

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 condensedphase?

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]

CΛ(g`; g′`) =

∫ +∞

Λ−2dα

∫Spin(4)

∫SU(2)

[dl`]d∏`=1

KSpin(4)α (g`hl`g

′−1` ) .

CΛ(g`; g′`) =

∫ +∞

Λ−2dα

∫Spin(4)

∫SU(2)

[dl`]d∏`=1

KSpin(4)α (g`hl`g

′−1` ) .

CΛ(g`; g′`) =

∫ +∞

Λ−2dα

∫Spin(4)

∫SU(2)

[dl`]d∏`=1

KSpin(4)α (g`hl`g

′−1` ) .

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 ofthe full GFT dynamics → some aspects of quantum cosmology and black holesrecovered from 4d quantum gravity models!

See talks by Wilson-Ewing and Pithis

Development of suitable renormalizable tools to check the overall consistency ofGFTs and explore more systematically their phase diagrams → applicable tosimplified 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

Summary and outlook

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