The Group Field Theory Approach to Quantum Gravity Or: Quantum Gravity As a Quantum ﬁeld Theory of Simplicial Geometry?A

(TALKING TO HIS BRAIN) “Ok, brain, let’s face it: I don’t like you and you don’t like me... but let’s get through this thing...... and then I can continue killing you with beer...”

Homer Simpson

The Group Field Theory approachto Quantum Gravity – p. 1/3 The Group Field Theory approach to Quantum Gravity or: Quantum Gravity as a quantum ﬁeld theory of simplicial geometry?a

Daniele Oriti

Department of Applied Mathematics and Theoretical Physics University of Cambridge

General reviews: J. Baez, Lect. Notes Phys. 543, 25-94 (2000), gr-qc/9905087; D. Oriti, Rept. Prog. Phys. 64, 1489 (2001), gr-qc/0106091; A. Perez, Class. Quant. Grav. 20, R43 (2003), gr-qc/0301113; D. Oriti, gr-qc/0311066; C. Rovelli, Quantum Gravity, CUP (2005), L. Freidel, hep-th/0505016

The Group Field Theory approachto Quantum Gravity – p. 2/3 Plan of the talk

Past dreams: Sum-over-histories formulation of Quantum Gravity and 3rd Quantization Recent struggles: Matrix Models, Dynamical Triangulations, Spin Foam Models Present hopes: the Group Field Theory formalism, what it is and how to interpret it Future struggles, or how to make the dream come true and the hopes fulﬁlled

The Group Field Theory approachto Quantum Gravity – p. 3/3 Past dreams: ———- sum-over-histories formulation of Quantum Gravity and the idea of “third quantization”

The Group Field Theory approachto Quantum Gravity – p. 4/3 QG from sum-over-geometries

histories: 4-geometries gµν Partition function:

iSgr(g,M) ZQG = Dge {g(M)}

States/boundary data: 3-geometries hab Transition amplitudes for boundary data: ( M) Z(h(S),h(S )) = DgeiSgr g, {g|h,h}

Cosmological setting...... physical meaning? Can be made local? The Group Field Theory approachto Quantum Gravity – p. 5/3 Generalization

Why not to consider also topology as a dynamical variable?

+ + +......

The Group Field Theory approachto Quantum Gravity – p. 6/3 Generalization

Why not to consider also topology as a dynamical variable?

+ + +......

simply: ( M) Z(h(S),h(S )) = w(M) DgeiSgr g, ? M {g|h,h }

The Group Field Theory approachto Quantum Gravity – p. 6/3 3rd quantization?

The Group Field Theory approachto Quantum Gravity – p. 7/3 3rd quantization?

push for more: a “3rd quantized”formulation, with the topology changing processes described dynamically as an “interaction of quantum universes”(S. Coleman, S. Giddings, A. Strominger, M. McGuigan, ...) deﬁne a scalar ﬁeld theory in superspace (space of all 3-geometries) for φ(hij), by the action: 1 ( )=− D ( )∇ ( )+ D n S φ 2 hij φ hij φ hij λ hij φ , where ∇ is the Wheeler-DeWitt operator, with a (non-local) interaction term in superspace

The Group Field Theory approachto Quantum Gravity – p. 7/3 3rd quantization?

and study the quantum theory deﬁned by: Z = DφeiS(φ)

The Group Field Theory approachto Quantum Gravity – p. 8/3 3rd quantization?

and study the quantum theory deﬁned by: Z = DφeiS(φ) through its perturbative expansion in Feynman diagrams

The Group Field Theory approachto Quantum Gravity – p. 8/3 3rd quantization?

and study the quantum theory deﬁned by: Z = DφeiS(φ) through its perturbative expansion in Feynman diagrams the (perturbative) 3rd quantized vacuum of the theory will be the “no-spacetime state” the amplitude for the free propagation will be given by the usual quantum gravity path integral for trivial topology, and the other Feynman graphs of the theory will describe topology-changing processes

The Group Field Theory approachto Quantum Gravity – p. 8/3 Recent struggles: ——– Matrix Models, Dynamical Triangulations, Spin Foam Models

The Group Field Theory approachto Quantum Gravity – p. 9/3 Matrix Models for 2d quantum gravity

deﬁne for an N × N hermitian matrix Mij the action 1 λ S(M)= tr(M 2) − √ tr(M 3) 2 3 N

The Group Field Theory approachto Quantum Gravity – p. 10/3 Matrix Models for 2d quantum gravity

deﬁne for an N × N hermitian matrix Mij the action 1 λ S(M)= tr(M 2) − √ tr(M 3) 2 3 N and the quantum theory via Z = dM e− S(M), deﬁned as a Feynman series in λ

The Group Field Theory approachto Quantum Gravity – p. 10/3 Matrix Models for 2d quantum gravity

deﬁne for an N × N hermitian matrix Mij the action 1 λ S(M)= tr(M 2) − √ tr(M 3) 2 3 N and the quantum theory via Z = dM e− S(M), deﬁned as a Feynman series in λ Feynman expansion produces trivalent (fat) graphs of ALL topologies

M ij

M ij M ji j

i j

k The Group Field Theory approachto Quantum Gravity – p. 10/3

M jk M ki Matrix Models for 2d quantum gravity

these graphs are dual to TRIANGULATIONS of orientable closed surfaces M ij i

M ij M ji j

i j

M jk M ki

The Group Field Theory approachto Quantum Gravity – p. 11/3 Matrix Models for 2d quantum gravity

these graphs are dual to TRIANGULATIONS of orientable closed surfaces M ij i

M ij M ji j

i deﬁnition of 2d quantum gravity asj a sum over ALL 2d k triangulations of ALL topologies M jk M ki

The Group Field Theory approachto Quantum Gravity – p. 11/3 Matrix Models for 2d quantum gravity

these graphs are dual to TRIANGULATIONS of orientable closed surfaces M ij i

M ij M ji j

i deﬁnition of 2d quantum gravity asj a sum over ALL 2d k triangulations of ALL topologies M jk M ki that means: 1 Z = dM e− S(M) = λn2(T )N χ(T ) sym(T ) T

The Group Field Theory approachto Quantum Gravity – p. 11/3 Dynamical triangulations

idea: triangulate a given continuum manifold → spacetime obtained by gluing D-simplices

The Group Field Theory approachto Quantum Gravity – p. 12/3 Dynamical triangulations

idea: triangulate a given continuum manifold → spacetime obtained by gluing D-simplices assume all edge lengths are equal to a; geometric degrees of freedom are encoded in the combinatorics of the triangulation theory (both Riemannian and Lorentzian) deﬁned by: 1 ( Λ) Z(Λ,a)= eiSRegge T,a, sym(T ) T where T are triangulations (ﬁxed topology), and SRegge is Regge action for simplicial gravity

The Group Field Theory approachto Quantum Gravity – p. 12/3 Dynamical triangulations

in Lorentzian case one distinguishes between spacelike edges (length square a2) and timelike ones (length square −a2), and imposes some additional restrictions...

The Group Field Theory approachto Quantum Gravity – p. 13/3 Dynamical triangulations

in Lorentzian case one distinguishes between spacelike edges (length square a2) and timelike ones (length square −a2), and imposes some additional restrictions... then one looks for continuum limit (smooth phase, with correct dimensionality) as a → 0 and the triangulation is reﬁned (and Λ gets renormalised)

The Group Field Theory approachto Quantum Gravity – p. 13/3 Dynamical triangulations

The Group Field Theory approachto Quantum Gravity – p. 13/3 Spin Foam models: the general idea

spacetime manifold → combinatorial 2-complex σ (vertices, links, faces)

The Group Field Theory approachto Quantum Gravity – p. 14/3 Spin Foam models: the general idea

spacetime manifold → combinatorial 2-complex σ (vertices, links, faces) histories → 2-complexes with representations J of Lorentz group assigned to faces

The Group Field Theory approachto Quantum Gravity – p. 14/3 Spin Foam models: the general idea

spacetime manifold → combinatorial 2-complex σ (vertices, links, faces) histories → 2-complexes with representations J of Lorentz group assigned to faces states/boundary data → spin networks Ψ=graphs with links labeled by irreps J combinatorial and representation data encode geometric degrees of freedom Spin Foam model (M. Reisenberger, J. Baez): Z = w(σ) Af (Jf ) Ae(Jf|e) Av(Jf|v) σ|Ψ,Ψ {J} f e v The Group Field Theory approachto Quantum Gravity – p. 14/3 Sf

q t e 1

b a e 2 p d v v 2 1 c f S i e

e 3 a spin foam a spin network purely combinatorial and algebraic version of a path integral for quantum geometry, about which lots of results have been obtained recently...... The Group Field Theory approachto Quantum Gravity – p. 15/3 Present hopes: ———- the Group Field Theory formalism

The Group Field Theory approachto Quantum Gravity – p. 16/3 General structure of GFT consider a (real) scalar ﬁeld over D copies of a group manifold G (Lorentz group, for quantum gravity)

The Group Field Theory approachto Quantum Gravity – p. 17/3 General structure of GFT consider a (real) scalar field over D copies of a group manifold G (Lorentz group, for quantum gravity) field action: 1 ( )= ˜ ( )K( ˜−1) (˜ )+ SD φ, λ 2 dgidgiφ gi gigi φ gi i=1,..,D D+1 λ −1 + φ(g1 )...φ(g +1 ) V(g g ) (D +1)! j D j ij ji i=j=1 where the exact choice of the kinetic and interaction operator defines the model

The Group Field Theory approachto Quantum Gravity – p. 17/3 General structure of GFT symmetries: invariance of the ﬁeld under global shifts of the arguments by a group element, and under their permutations

The Group Field Theory approachto Quantum Gravity – p. 18/3 General structure of GFT symmetries: invariance of the ﬁeld under global shifts of the arguments by a group element, and under their permutations the quantum theory is deﬁned by the partition function, in terms of its Feynman expansion: λN = D −S[φ] = (Γ) Z φe [Γ] Z Γ sym

The Group Field Theory approachto Quantum Gravity – p. 18/3 General structure of GFT

Feynman graphs Γ are cellular complexes toplogically dual to D-dimensional triangulated (pseudo-)manifolds of ALL topologies

The Group Field Theory approachto Quantum Gravity – p. 19/3 General structure of GFT

Feynman graphs Γ are cellular complexes toplogically dual to D-dimensional triangulated (pseudo-)manifolds of ALL topologies Feynman amplitudes are given, in momentum space, by Spin Foam models

The Group Field Theory approachto Quantum Gravity – p. 19/3 General structure of GFT

Feynman graphs Γ are cellular complexes toplogically dual to D-dimensional triangulated (pseudo-)manifolds of ALL topologies Feynman amplitudes are given, in momentum space, by Spin Foam models observables are given by functionals of the field operators, that can be expanded in spin networks one can define transition amplitudes by inserting field operators in the partition function, which give spin network states on the boundaries

The Group Field Theory approachto Quantum Gravity – p. 19/3 General structure of GFT geometric interpretation: 1) a ﬁeld represent a (2nd quantized) (D-1)-simplex, 2) its interactions generate a D-dimensional simplicial spacetime of arbitrary topology (depending on the particular interaction process) and arbitrary complexity (depending on the complexity of the states involved)

The Group Field Theory approachto Quantum Gravity – p. 20/3 General structure of GFT geometric interpretation: 1) a ﬁeld represent a (2nd quantized) (D-1)-simplex, 2) its interactions generate a D-dimensional simplicial spacetime of arbitrary topology (depending on the particular interaction process) and arbitrary complexity (depending on the complexity of the states involved) the amplitude for each process (Feynman diagram) can be related to a discretization of the gravity action for the given simplicial spacetime

The Group Field Theory approachto Quantum Gravity – p. 20/3 Speciﬁc example: 3d Riemannian QG

3 Consider the ﬁeld: φ(g1,g2,g3):(SU(2)) →R

The Group Field Theory approachto Quantum Gravity – p. 21/3 Speciﬁc example: 3d Riemannian QG

3 Consider the ﬁeld: φ(g1,g2,g3):(SU(2)) →R with the symmetry: φ(g1g,g2g,g3g)=φ(g1,g2,g3) imposed through the projector: φ(g1,g2,g3)=Pgφ(g1,g2,g3)= dg φ(g1g,g2g,g3g)

The Group Field Theory approachto Quantum Gravity – p. 21/3 Speciﬁc example: 3d Riemannian QG

3 Consider the ﬁeld: φ(g1,g2,g3):(SU(2)) →R with the symmetry: φ(g1g,g2g,g3g)=φ(g1,g2,g3) imposed through the projector: φ(g1,g2,g3)=Pgφ(g1,g2,g3)= dg φ(g1g,g2g,g3g) and the symmetry: φ(g1,g2,g3)=φ(gπ(1),gπ(2),gπ(3)), with π a permutation of the arguments

The Group Field Theory approachto Quantum Gravity – p. 21/3 Speciﬁc example: 3d Riemannian QG

The Group Field Theory approachto Quantum Gravity – p. 21/3 Classical and quantum GFT

Deﬁne the classical theory by the action: 1 [ ]= [ ( )]2 − S φ 2 dg1..dg3 Pgφ g1,g2,g3 λ − dg1..dg6[P φ(g1,g2,g3)][P φ(g3,g5,g4)] 4! h1 h2 [ ( )][ ( )] Ph3 φ g4,g2,g6 Ph4 φ g6,g5,g1

The Group Field Theory approachto Quantum Gravity – p. 22/3 Classical and quantum GFT

Deﬁne the classical theory by the action: 1 [ ]= [ ( )]2 − S φ 2 dg1..dg3 Pgφ g1,g2,g3 λ − dg1..dg6[P φ(g1,g2,g3)][P φ(g3,g5,g4)] 4! h1 h2 [ ( )][ ( )] Ph3 φ g4,g2,g6 Ph4 φ g6,g5,g1

(...geometric interpretation...)

The Group Field Theory approachto Quantum Gravity – p. 22/3 Classical and quantum GFT

Deﬁne the classical theory by the action: 1 [ ]= [ ( )]2 − S φ 2 dg1..dg3 Pgφ g1,g2,g3 λ − dg1..dg6[P φ(g1,g2,g3)][P φ(g3,g5,g4)] 4! h1 h2 [ ( )][ ( )] Ph3 φ g4,g2,g6 Ph4 φ g6,g5,g1

(...geometric interpretation...) and the quantum theory by the partition function: Z = Dφe−S[φ]

The Group Field Theory approachto Quantum Gravity – p. 22/3 GFT feynmanology

In fact, the quantum theory is deﬁned in terms of perturbative expansion in Feynman graphs:

The Group Field Theory approachto Quantum Gravity – p. 23/3 GFT feynmanology

In fact, the quantum theory is deﬁned in terms of perturbative expansion in Feynman graphs: N = −S[φ] = λ (Γ) Z dφ e [Γ] Z Γ sym

The Group Field Theory approachto Quantum Gravity – p. 23/3 GFT feynmanology

In fact, the quantum theory is deﬁned in terms of perturbative expansion in Feynman graphs: N = −S[φ] = λ (Γ) Z dφ e [Γ] Z Γ sym we need propagator, vertex, and quantum amplitude Z(Γ) for each Feynman graph Γ 1 [ ]= ˜ ( )K( ˜ ) (˜ ) S φ 2 dgidgj φ gi gi, gj φ gj − λ V( ) ( ) ( ) ( ) ( ) 4! dgij gij φ g1j φ g2j φ g3j φ g4j

The Group Field Theory approachto Quantum Gravity – p. 23/3 propagator, vertex for the propagator one gets: P = ¯ ( ¯−1˜−1 ) ( ¯−1˜−1 ) ( ¯−1˜−1 ) dgdgδg1gg gπ(1) δ g2gg gπ(2) δ g3gg gπ(3) π

The Group Field Theory approachto Quantum Gravity – p. 24/3 propagator, vertex for the propagator one gets: P = ¯ ( ¯−1˜−1 ) ( ¯−1˜−1 ) ( ¯−1˜−1 ) dgdgδg1gg gπ(1) δ g2gg gπ(2) δ g3gg gπ(3) π while the vertex is given by: −1 −1 −1 −1 −1 −1 V = dhi δ(g1h1h3 g˜1 )δ(g2h1h4 g˜2 )δ(g3h1h2 g˜3 )

−1 −1 −1 −1 −1 −1 δ(g4h2h4 g˜4 )δ(g5h2h3 g˜5 )δ(g6h3h4 g˜6 )

= + +

The Group Field Theory approachto Quantum Gravity – p. 24/3 propagator interaction vertex Feynman graphs and amplitudes

How do these Feynman graph look like? each line in a propagator goes through several vertices and (for closed graphs) comes back, thus identiﬁes a 2-cell;

The Group Field Theory approachto Quantum Gravity – p. 25/3 Feynman graphs and amplitudes

How do these Feynman graph look like? each line in a propagator goes through several vertices and (for closed graphs) comes back, thus identiﬁes a 2-cell; faces, lines and vertices identify a 2-complex, i.e. GFT Feynman graphs are 2-complexes the sum over Feynman graphs, and the sum over permutations, generate a sum over 2-complexes of all topologies

The Group Field Theory approachto Quantum Gravity – p. 25/3 Feynman graphs and amplitudes

The Group Field Theory approachto Quantum Gravity – p. 25/3 tetrahedron tetra + dual dual For each Feynman graph/2-complex/triangulation, Γ, the quantum amplitude Z(Γ) turns out to be: Z(Γ) = dge δ( ge) e∈Γ f e∈∂f

The Group Field Theory approachto Quantum Gravity – p. 26/3 Feynman amplitudes

( )= in momentum space: φ g1,g2,g3 j1j2j3 j1 j2 j3 ( 1) ( 2) ( 3) j1,j2,j3 φm1n1m2n2m3n3 Dm1n1 g Dm2n2 g Dm3n3 g

The Group Field Theory approachto Quantum Gravity – p. 27/3 Feynman amplitudes

( )= in momentum space: φ g1,g2,g3 j1j2j3 j1 j2 j3 ( 1) ( 2) ( 3) j1,j2,j3 φm1n1m2n2m3n3 Dm1n1 g Dm2n2 g Dm3n3 g the propagator, vertex and amplitude are:

P = ˜ ˜ ˜ δj1j1 δm1m˜ 1 δj2j2 δm2m˜ 2 δj3j3 δm3m˜ 3 V = δ ˜ δm1m˜ 1 δ ˜ δm2m˜ 2 δ ˜ δm3m˜ 3 δ ˜ δm4m˜ 4 j1j1 j2j2 j3j3 j4j4 j1 j2 j3 ˜ ˜ δj5j5 δm5m˜ 5 δj6j6 δm6m˜ 6 j4 j5 j6 (Γ) = ∆ j1 j2 j3 Z jf j4 j5 j6 {jf } f v

Ponzano-Regge spin foam model The Group Field Theory approachto Quantum Gravity – p. 27/3 in the end..... for the full theory we have: N = λ ∆ j1 j2 j3 Z jf [Γ] 4 5 6 Γ sym j j j {jf } f v v

The Group Field Theory approachto Quantum Gravity – p. 28/3 in the end..... for the full theory we have: N = λ ∆ j1 j2 j3 Z jf [Γ] 4 5 6 Γ sym j j j {jf } f v v this is a purely combinatorial-algebraic sum-over-geometries and over topologies .....about which a lot more is known.... interpretation: QFT of simplicial geometry; classical dynamical objects: triangles; quantum dynamical objects: φ(g1,g2,g3); histories/Feynman graphs: 3d triangulations.

The Group Field Theory approachto Quantum Gravity – p. 28/3 GFT: the general picture

GFT are a LOCAL, DISCRETE, ALGEBRAIC and COMBINATORIAL 3rd Quantization of Gravity

The Group Field Theory approachto Quantum Gravity – p. 29/3 GFT: the general picture

GFT are a LOCAL, DISCRETE, ALGEBRAIC and COMBINATORIAL 3rd Quantization of Gravity both geometry and topology are dynamical

The Group Field Theory approachto Quantum Gravity – p. 29/3 GFT: the general picture

GFT are a LOCAL, DISCRETE, ALGEBRAIC and COMBINATORIAL 3rd Quantization of Gravity both geometry and topology are dynamical spacetime emerges via creation/annihilation of “chunks”of it, of spacetime quanta spacetime is purely virtual in the quantum theory Quantum Gravity as an (almost) ordinary QFT (with a background spacetime given by the group manifold)

The Group Field Theory approachto Quantum Gravity – p. 29/3 GFT: the general picture

The Group Field Theory approachto Quantum Gravity – p. 29/3 Future struggles: ———– what is left to do (and starts to be done)......

The Group Field Theory approachto Quantum Gravity – p. 30/3 Future... analysis of the classical GFT, classical equations of motion, classical solutions, symmetries,...

The Group Field Theory approachto Quantum Gravity – p. 31/3 Future... analysis of the classical GFT, classical equations of motion, classical solutions, symmetries,... Fock structure:

The Group Field Theory approachto Quantum Gravity – p. 31/3 Future... analysis of the classical GFT, classical equations of motion, classical solutions, symmetries,... Fock structure: rigorous deﬁnition of creation/annihilation operators analysis of 3rd quantized vacuum (the absolute nothing) symmetries: full classiﬁcation of classical and quantum symmetries of the theory

renormalization group for GFT The Group Field Theory approachto Quantum Gravity – p. 31/3 Future... relation with canonical quantization of GR:

The Group Field Theory approachto Quantum Gravity – p. 32/3 Future... relation with canonical quantization of GR: Hamiltonian for GFT

The Group Field Theory approachto Quantum Gravity – p. 32/3 Future... relation with canonical quantization of GR: Hamiltonian for GFT GFT derivation of canonical Hamiltonian constraint (non-linear modification of WdW equation) GFT definition of canonical inner product Non-perturbative analysis: GFT instantons GFT statistical mechanics, vacua and phases → continuum/classical limit and emerging GR coupling of matter and gauge fields → unification?

GFT can represent uniﬁed framework for various approaches (but details to be understood):

The Group Field Theory approachto Quantum Gravity – p. 33/3 Convergence of approaches..

GFT can represent uniﬁed framework for various approaches (but details to be understood): LQG: boundary data are SpinNets, but full Lorentz group, no clear relation with canonical theory

The Group Field Theory approachto Quantum Gravity – p. 33/3 Convergence of approaches..

GFT can represent uniﬁed framework for various approaches (but details to be understood): LQG: boundary data are SpinNets, but full Lorentz group, no clear relation with canonical theory Spin Foams: GFT Feynman amplitudes are SF Quantum Regge Calculus: if amplitude for Feynman graphs is exp of Regge action, but may use different geometric variables Dynamical Triangulations: if amplitude for Feynman graphs is exp of Regge action, triangulations are summed over, but need to check details Causal Sets: GFT sums over directed graphs and provides quantum amplitude, but need to impose absence of closed timelike loops The Group Field Theory approachto Quantum Gravity – p. 33/3 Those are my principles, and if you don’t like them...... well, I have others.

Groucho Marx

The Group Field Theory approachto Quantum Gravity – p. 34/3