Effective Field Theory from Spin Foam Models :3D Example

Effective field theory from spin Foam models :3d example

Laurent Freidel

Pennstate 2007 Background independent Loop Quantum Gravity in a nutshell Background independence: what the quantum geometry is at Planck scale cannot be postulated its needs to be determined dynamically

•Hamiltonian quantisation: gravity is a gauge theory 2 SU(2) Yang-mills phase space (A,E) + constraints i1 (Γ, j , i ) (15) •Kinematical Hilbert space is spannede v by j1 j5 j2 spin network: graph colored by su(2) rep j6 j4 i4 Wave function i3 Ψ(Γ,je,iv)(A) i2 j3 (16)

•Eigenstates of Geometrical operators, Area, Vol = trj(h (A)) (17) discrete spectra quantized space◦ geometry

s’ 2 •Dynamics:Area !encodedΨΓ,je,i vin= spin8πγ foamlP modelsje(j eallowing+ 1) ΨΓ ,jthee,iv (18) e R Γ + ... computation of transition amplitudes∈!∪ "between Spin networks states: quantum spacetime geometry s 2 Area!Ψ(Γ,je,iv) = 8πγlP je(je + 1) Ψ(Γ,je,iv) (19) # e R Γ ( ∈!∪ " $ %& ' 3 2 VolΨΓ,je,iv = 8πγlP vje,iv ΨΓ,je,iv (20) # v R Γ ( * ∈!∪ ) F (A) E (21) ∝

Area(! H) = A (22)

H = tr([E, E]F (A)) (23)

Hj(i) (24) i !

i S(g) Dg e lP (25) +

K (j) k (l d ) (26) F ∝ F P j

! k [Xi , Xj] = ilP #ijkX (27)

m sin(κm)/κ (28) →

S3 SU(2) (29) ∼

X ∂ (30) ∼ P 2

(Γ, je, iv) (15)

Ψ(Γ,je,iv)(A) (16)

v v Ue Pe + [Ωe, Pe] = 0 (0.32) = trj(h (A)) (17) ◦ ∂!e=v 2 S = tr(XeGe) (0.33) Area!ΨΓ,je,iv = 8πγlP je(je +T1ra)nsΨitioΓn,jaem,ipvlitudes between spin network state(1s a8)re defined by e e R Γ s, s! = A[ ], (11) ∈ ∪ ! "phys F ! " ! F:s→s! v v ! δXe = Ue Φwhvere the[nΩotaeti,onΦanvtic]ip=ates t0he interpretation of such amplitudes as defining the physical (0.34) scala−r product. The domain of the previous sum is left unspecified at this stage. We shall ! 2 v e discuss this question further in Section 6. This last equation is the spin foam counterpart AreaSpinΨ(Γ,je,iv )F=oam8πγlP models:!∈je(je + tro1f )eqansitionuatiΨon ((Γ9),j. eT,ihvi)s defini tiamplitudeon remains formal (1unt9)il we specify what the set of allowed # e R Γ spin0(foams in the sum are and define the corresponding amplitudes. ∈!∪ "Xe = Xe e T (0.35) The dynamics results in a ∈ l l 0p j $ %& ' δ j n p n je,jq e s s succession of evolution moves on a o m o k m 3 Z∆,T k q (0.36) 2 n 2 s canV beolΨ encodedΓ,j ,i = by 8aπ γcoloredl vj ΨΓd,j ,i (20) m e v P e,iv jee v l l e T p j j q p # v R Γ (∈ o singular surface S (2d complex) " k k l * ∈ ∪ → q o ! 2 j interpolating) between initial( +andm final)k (x) = iδ(x) k (0.37) ! F l l j j spin networks F (A) E k (21)k ∝ faces of S are colored by spinsRe ((quantaKFF)(igu rofGe 3area):e)A t=ypical path in a path integral version of loop quantum gravity is given (0.38) by a series of transitions through different spin-network states representing a state of 3- edges of S are colored by invariant maps (quanta of 3d volume) Area(! H) = A geometries. Nodes and links in the spin network e(2volv2)e into 1-dimensional edges and faces. New links are created and spins are reassigned at vertexes (emphasized on the right). The θ =‘topκolmogical’ structure is provided by the underlying 2-complex while the geometric degrees (0.39) Quantum transition amplitude; physicalof freed oscalarm are enco dproducted in the labeling of its elements with irreducible representations and intertwiners. H = tr([E, E]F (A)) (23) AS(Γ0, Γ1) = Af (jf )In standAardeq(ujanftu,miem)echanics Athevpa(tjh fin,teigreal)is used to compute the matrix ele- (0.40) ments of the evolution operator U(t). It provides in this way the solution for dynamics j ,i e v !f e "f since"for any kinematical state"Ψ the state U(t)Ψ is a solution to Schrödinger’s equation. H (i) Analogously, in a generally covariant theory the(2pat4)h integral provides a device for con- 2 complex, initial, final j structing solutions to the quantum constraints. Transition amplitudes represent the matrix triangulation i coloring, intelem ents of thefaceso-ca,edgelled gene rvalerizedt,‘ plocalrojectio n’ operator P (Sections 3.1 and 6.3) such geometr!y geometrythat P Ψ is a physical state for any kinematical state Ψ. As in the case of the vector constraint the solutionamplitudess of the scalar constraint correspond to distributional states (zero is in the continuum part of its spectrum). Therefore, is not a proper subspace of Hphys H and the operator P is not a projector (P 2 is ill defined)8. In Section 4 we give an explicit A(Γ0, Γ1) = AS(Γ0, Γ1) (25) example of this construction. S The background-independent character of spin foams is manifest. The 2-complex can be non perturbative and combinatorial! definition oft hquantumought of as gravityrepresenti namplitudesg ‘space-time’ while the boundary graphs as representing ‘space’. 8 ∗ In the notation of the previous section states in phys are elements of Cyl . H i S(g) Dg e lP 12 (26) +

K (j) k (l d ) (27) F ∝ F P j

! k [Xi , Xj] = ilP #ijkX (28)

m sin(κm)/κ (29) →

S3 SU(2) (30) ∼

3 v v Ue Pe + [Ωe, Pe] = 0 (0.32) ∂!e=v S = tr(XeGe) (0.33) v v e Ue Pe + [Ωe, Pe] = 0 ! (0.32) ∂!e=v δX = U vΦ [Ωv, Φ ] = 0 (0.34) e e v − e v S = tr(X G ) v e (0.33) e e !∈ e 0 ! Xe = Xe e T (0.35) v v ∈ δXe = U Φv [Ω , Φv] = 0 (0.34) e e 0 − δje,j v e e ∈ Z∆,T (0.36) ! d 2 X = X0 e T e T je (0.35) e e "∈ ∈ 2 0 δje,je (! + m )kF (x) = iδ(x) (0.37) Z∆,T (0.36) d 2 e T je "∈ Re(KF)(Ge) = (0.38) Spin Foam2 models: constraint BF (! + m )kF (x) = iδ(x) (0.37) The choice of local amplitudes determines the θdynamics= κm (0.39) Re(KF)(Ge) = (0.38) either by exponentiation of the hamiltonian constraint A (Γ , Γ ) = A (j ) A (j , i ) A (j , i ) (0.40) Or more efficiently via θa =spacetimeκSm 0 1 (feynman integral)f f approache f e (0.39)v f e j ,i e v !f e "f " " One uses the fact that gravity can be written as a constraint AS(Γ0, Γ1) = Af (jf ) Ae(jf , ie) Av(jfi, ie) (0.40) topological BFj theory,i f . SP lebanske i = Bv i F (A), + constr. Bi Bj = αδij (0.41) !f e " " " ∧ ∧ Plebanski, Ooguri, Reisenberger# , L.F, Krasnov, De Pietri, Perez,... i S = B F (A), with+ constr constraints. B B B=i αδBj = αδij (0.41) (0.42) P lebanski i ∧ i ∧ j ∧ ij # Bi Bj = αδij (0.42) We can construct an exact∧ finite path integral representation for the topological theory in terms of a state sum model.

Impose on the amplitudes the constraints that lead to GR •In 3D exact quantisation: Ponzano-regge model •In 4D this leads to the Barrett Crane model G=SO(4), representations (jL,jR) constraint is jL=jR exciting new developpements on his front Engle, Rovelli, Livine, Speziale, L.F, Krasnov,...

3 3 Spin Foam model: Coupling to matter

•Low energy limits: How does a classical spacetime emerge from quantum geometry and a corresponding set of transition amplitude? •We need to construct a sufficient large set of physical observables allowing the interpretation of these amplitudes?

•Technically very hard to construct these observables in pure gravity operationally one needs matters in order to define what is geometry, At the classical level we study the motion of test particles.

At the quantum level we want to know how quantum gravity affects and modify the rules of quantum field theory, what are the QG corrections to usual field theory and usual Feynman rules? Any theory of QG can be view from the point of view of matter as a dimensionfull deformation of Field Theory. Spin Foam model: Coupling to matter Integrate out quantum gravity fluctuation: since spin foam models are combinatorial objects the direct coupling to fields is hard. The strategy is to first construct the coupling of quantum gravity to quantum particles i-e Feynman diagrams and then reconstruct the effective field theory.

iS[φ,g]+ i S[g] Expand D φ in Feynman diagrams Z = DgDφe lp Z

Z i S[g] = C DgI (g)elp ∑ Γ Γ = CΓIΓ(lp) Γ ∑ Z Γ observables of computed using spin foam quantum geometry ! Z = DφeiSe f f [φ] Z • Explicitely realized in 2+1 dimensions Unification: Matter as a charged topological defect! 2+1 Euclidean Gravity

In the ﬁrst order formalism gravity is described in term of: i µ i µ a frame ﬁeld e µ d x a spin connection ωµdx 1 S[e,ω] = ei F(ω) G ∧ i 16π Z e.o.m: pure gravity F =0

particles F(ω)i = 4πG pi δ(x) create a conical singularity with deﬁcit angle θ = κm, κ = 4πG Deser, Jackiw, t hooft 3 m sin(κm)/κ (31) →

S3 SU(2) (32) ∼

X ∂ (33) ∼ P

P 2 1/κ2 (34) ≤

G 0 (35) N →

g = η + h Ponzano-Regge Model(36)

P abcd hab(x)hcd(y) (37) $ First% quantum∼ x y gravity2 model ever written in 1968 • | − | • Background independent and non perturbative finite definition of s = (Γ, j , i ) (38) Euclideane v quantum gravity amplitude in 2+1d

•gIts= kinematics(q, k) is the one of usual loop quantum (3gravity9) • Can be shown to be the continuum limit of a discretization of gravity 2 (a/a˙ ) = 8πpreservingG/3ρ(1 difρ/feoρcrit symmetry) (40) −

Coupling to matter uniquely fixed L.F, Louapre (a/a˙ )2 = 8πG/3(•ρ ρ (v))(ρ (v) ρ)/ρ (41) − 1 2 − crit • Equivalent with (Witten) Chern-Simons quantization when it exists Λ > 0 (42) Ponzano-Regge model • Chose a triangulation ∆ of M dual to a spin foam with boundary spin networks • Color edges of ∆ by SU(2) representations je j j j et1 et2 et3 Z∆( jin, jout) = d je ∑ ∏ ∏ je je je j e t t4 t5 t6 { e} ! " l j = Sum over internal geometries SU(2) 6j symbol lP Z∆ is independent of the internal triangulation finite after proper gauge fixing of diffeo symmetry spin foam formulation of quantum gravity amplitude physical scalar product between spin network states 2

3 ∂∆ = 6 ∅ m sin(κLmaur)/κent Freidel (31) Γ → 3 to vertices of the triangulation, such that the variation i S (p ,x ,λ ) VolΨΓ,je,iv = 8πγlP vje,iv ΨΓ,je,ie3vP e e e (15) IΓ(g) = dxv GF (xse , xte ; g) = DxitreD(λXe eeGSe) iSSPU((2X)e,Pe,λe(0).18) (32) I∆(Γ) = !dXv e "e dgf v dRPΓed!λee e$ P e∼ δX = Φ (1.9)(20) " " ∈ ∪ " e ∇e e f e Γλ#> 0 λ !! ∈ " " "leaves the action (1.3) inXvarian∂P t. The discrete covariant derivative redu(33)ce iS(g) iS(g) i SP (pe,xe,λe) I˜Γ = Dg e IΓ(g) = DgDpeDλe e e e ∼ (0.19) to the usual derivative eΦ Φse Φte when the gauge field is abelian ! ˜ F (A) ! E P ∇ ∼ − (16) I∆(Γ) = dgf KF (Ge) andδ(Gthee)symm= etry is dudjee to theKdFisc(reteje) Bianchi identity. (21) κ∝= 4πG P 2 1/κ2 (34) ! f e Γ e/Γ Since this symmeje etry is≤none Γcompact we needt to gauge fixed it in order " "∈ "∈ # " "∈ " tom κdefine the partition function and expectation values of observables. Area(! H) = A GN 0 (17) (35) ∂∆A =KnatuF (raljegau) ge fixing consis→ ts in choosing a collection of edges T which OΓ = form∅ a tree (no loops) and$which%&is maximal' (connected and which goes(22) Γ dje ∂∂∆∆ == e thΓ rough all vertices). gW=e ηth+enh arbitarily fix the value of Xe for all e(3d6)ges "∅∈ L.F, D. Louapre, Barrett H = tr([E∅, Ee]FT(A. ))In tr(theXeGec)oniSPti(Xneuu,Pe,mλe) this gauge fixing amounts(1to8)choose a vector I∆(Γ) = 2 dXe sidngfκm d2PedλPe eartice le insertion(0.20) iλ P (G) ∈ i i µ ˜ ! e Γ(f κ fielde Γ) v (the tree) and fix the value of e v , that is to chose an ‘axial’ KF (G) = dλe " −Γ" "∈ = abcd µ (23) computation fromab a discretizationcd P2 of SG+SPart “ gauge.” 2 h (x)h (siny) κvm v (37) I (Γ) = dg K˜ (G ) δ(G ) = P (dG) K (j )U P + [Ω2 i,(0$P.21)] = 0 (0.32) !∆ f F e eitr(X G$ )je iS F(Xe %κ,P∼e ,exλ )y e e Takiingtr(tXhiseeGgauee) geiS−fixiPP (ngXeean,Pede,λtheee)−Faddev-Popov determinant into ac- I∆(Γ) = dXe dfgf e Γ Γ d isPH ead e(Feynman/λiΓ)e e jegraphe esupportede Γ ∂e=v t on| −edges| of (19) (20) I∆(Γ) = dXe ! d"gf "•∈ dPejd"∈λe e # " e"∈ " ∆ (20) tr(Gσcouni) t in the derivation!((1.5, 1.10)) we obtain the gauge fixed Ponzano- ee f e Γi 2 2 !! fPi(G)e∈Γ AmplitudeKκ Ffor(Pje) a( ScalarG) particle1s = (ΓS, jcoupled=, i ) tr( toX egravityGe) (38) (24) (0.33) "" "" ""∈•#OΓ = Regge model $ %& ' e v (0.22) ≡ 2iκ dj ≤ e Γ e e ∈ " ! 0 ˜ 2 δje,j j j j iλ P 2(G) sin κm i v e v e1 e2 e3 I∆(Γ) = dgf ˜K˜ F (Ge) δ(G( ieκ ))=Z 0 = dje d KF (je) . (1.10)(21) I∆(Γ) = ˜ dgf KKF (GF)(=Ged)λe Iδ∆((−GΓe)S)(g=) =∆,T,j djδeXge =˜je(Kq, kF)U(ejΦev)2 [(0Ω.23)e, Φv] = 0 (39)(21) (0.34) K(G) = djKF (j)χ“j(G) l djK” F (j2) = sindκmG2 K(G)χ(dj(0G) ) j j j (25) P P (G) i& je − e3 e5 e6 f e Γ ! e/DΓ g e je e jeκ ee Γ ve eT tt # (20) $ ! f e ∈Γ e/Γ je e− {!} "e∈−Γ "∈∈ "t ! " "∈j N "∈∈ # "! "∈ ! " " "∈ " tr(Gparticleσi) 2 #2 insertion"( )" 0 " GF # %Pi(G) As aκ cPon(Gsi)stency1 te2 st it canXbee=shXownethat(0.24)TZ 0 = Z is independent (0.35) ≡ 2iκ ≤(a/a˙ ) = 8πG/3ρ(1 ρ/eρcrit) ∆,T,j ∆ (40) 2 idj (m+i#) 3 − ∈ 0 2iκ e of the choice ofκmaximal tree T and δgauge0 fixing parameter j . ˜ ˜ je,je KFK((Gj)== djKF (j)χj(G) djKF (j) = =dG K(kGF)χ(isjd(G jthe)) Z usual∆,T Feynman(0.25) propagator (26) (0.36) K (j) k (l d ) 2 $ %& ' 2 (21) j cosF κm F dP j 4π dj ∈N j (a/a˙ )!= 8πG/3(ρevaluatedρ1(v))(ρe 2on(Tv) a Planckianeρ)/ρcrit lattice (41) # ∝ − "∈ − 2iκ2 eiκdj (m+i#) κ3 2 KF (j) = = 1.4kFCoupli(κdj) (!ng+ mmatt)kFer(x)to(0=.26)quaniδ(x)tum gravity (0.37) cos κm dj 4π cos κm In order to couple particleΛ >to0matter fields we first construct the cou(4plin2)g gf Ge (0.27) particle=of gra e.vvity ofto anFeynman observableintegral in topologicals since as w estateare goisumng to see there is a 1 −−→G± = 1 (0.28) nate ural and unambiguous wKayF (jtoe) couple the Ponzano-regge model to ∂e=v O(je) = (43) " Feynman integrals. dj i e Γ e ˜ ∈ KF (g1) = We use th2 e fact that F!eynman integrals(0.29)can be written as a worldline P 2(g ) sin κm i& in1 te−gralκ [11],−that is if Γ is a (closed for simplicity) Feynman graph its Feyn(man )integral is given by (0.30) j # i i iSΓ P tr(Geσ ) I (e) = λ (0x .31)p e . (1.11) e ≡ Γ D ΓD ΓD Γ % where2 1 λ S (x , p , λ ) = dτ tr p e e (p2 µ2) . (1.12) Γ Γ Γ Γ 2 e t − 2 e − e e Γ e & ' !∈ % λ is a Lagrange multiplier field which is the worldline frame field and is

2 2 6 Laurent Freidel

to vertices of the triangulation, such that the variation δX = Φ (1.9) e ∇e leaves the action (1.3) invariant. The discrete covariant derivative reduce to the usual derivative Φ Φ Φ when the gauge field is abelian ∇e ∼ se − te and the symmetry is due to the discrete Bianchi identity. Since this symmetry is non compact we need to gauge fixed it in order to define the partition function and expectation values of observables. A natural gau∂ge∆ fix= ing consists in choosing a collection of edges T which ∅ form a tree (no loops) and which is maximal (connected and which goes Γ ∂∆ = through all vertices). We then arbitarily fix the value of Xe for all edges itr(XeGe) iSP (Xe,Pe,λe) I∆(Γ) = ∅ e dXTe . Indgthef condPtienduuλeme this gaugee fixing amounts to choose a vector(20) ∈ ! efield v f(the tereΓe) and fix the value of ei vµ, that is to chose an ‘axial’ Γ " "U vP "∈+ [Ωv, P ] = 0 µ (0.32) gauge. e e e e ∂∆ = I∆(Γ∂)∆= = dgf ∂e=Kv˜F (Ge) δ(Ge) = dj KF (je) (21) ∅ T!akiitrng(XthiseGegau) geiSPfixi(Xnge,Paned,λthee)e Faddev-∅Popov determinant into ac- I∆(Γ) = dXe dgf !dPfedλee eΓ e/Γe je e e Γ t (20) " coun"∈ t in the deriv"∈ ation (1.5,# "1.10) w"∈e obtain th"e gauge fixed Ponzano- Γ S = tr(XeGe) Γ (0.33) ! e f e Γ Regge modelGravity + Particle " " "∈ e KF (je) OΓ = (22) itr(Xe!Ge) iSP (Xe,Pe,λe) $ %& ' itr(XeGe) iSP (Xe,Pe,λe) dj δ 0 I∆(Γ) = dXe dgf dPedIλ∆e(Γe) = v e eΓ dXeve dgjfe,je dPjeed1 λeje2e je3 e (20) (20) ˜ δXe =Z U0e=Φ∈v [Ωed,jΦv] = 0 . (0(1.10).34) I∆(Γ) = dgf KF (Ge) δ(IG∆(eΓ))= ∆,T,j "dje Ke F (je) 2 (21) −2 (d 0 ) j j j e f e Γ 2 sin eκm f je e Γ e3 e5 e6 ! iλ Pv(Ge)!( je) e e T i t # $ ˜ ∈ ∈ − e κ{ } ∈ ∈t ! f " e Γ " K"Fe(/GΓ) = dλe !je "! "e =Γ"" "" 2 (23) ∈ ∈ “ 0 ” ∈ 2 sin κm " " " # " "P (G) "κ i$ GF ! As a conXsiestency= Xeteest itTcan be s−hown that Z−∆,T,j0 = Z∆ is indepe(0nden.35)t I (Γ) = dg K˜ (G ) δ(G ) = d ∈ K˜ (j ) 0 (21) ∆ f F e I∆(Γeof) =the choictr(e ofGdσgmjfe)aximalKtrFFee(TGeande) gaugeδ(fiGxine)g =parameter j d. je KF (je) (21) i δj2 ,j02 ( ) Ge = g f e Γ Pi(Gj) e κeePΓe (G) 1 t ∏ (24) ! f e/Γ eZ≡∆f,T2iκ e Γ ≤ e/Γ je fe e (0e.36)Γ v " "∈ "∈ #! " "d∈∈2 " ∈ " e T "je "∈ # "⊃ " " K˜ (G) = d K (j)χ (G)"∈d K (j) = dG K˜ (G)χ (G) (25) j F j 21.4 Couplij F ng matter to quanj tum gravity j N (! + m )kF (x) = iδ(x!) (0.37) #∈ In order to couple p$article%&to matte' r fields we first construct the coupling of gravity to Feynman integrals since as we are going to see ther1e is a on shell Re(KF)(Ge) = δθ(Ge) Ge = xehθexe (0.38) natural and unambiguous way to couple the Ponzano-regge mo−del to Feynman integrals. Carroll, Matschull,Bais, The momentumWe use th ofe fact theth parat Feticleynman isi ngrtegralsoup cavaluedn be writtenMulleras, Schra oerswor ...ldline integral [11], that is if Γ is a (closed for simplicity) Feynman graph its •Is it Frealleynmyan a inFtegraleynmanis gi vgraen bphy evaluation? •Do we recover QFT in the GN 0 limit ? I (e) = λ x p eiSΓ . (1.11) • Then, what are the quantumΓ D graΓDvityΓD corΓ rections ? % where 1 λ S (x , p , λ ) = dτ tr p e e (p2 µ2) . (1.12) Γ Γ Γ Γ 2 e t − 2 e − e e Γ e & ' !∈ % λ is a Lagrange multiplier field which is the worldline frame field and is

2 2 2 6 Laurent Freidel

δ 0 je,je je1 je2 je3 ˜ 0 I∆(Γ) = dgf KF (Ge) δI(G(Γe) == Z∆,T,j =dj djKe F (je) 2 . (1.10)(21) ∆ e (d 0 ) j j j je e3 e5 e6 je e e T t # $ ! f e Γ e/Γ je e {!} "e Γ "∈ t" " "∈ "∈ # " "∈ " GF As a consistency test it can be shown that Z∆,T,j0 = Z∆ is independent of the choiceAfterof max someimal tr ewe Tork...and gauge ﬁxing parameter j0.

1.4 Coupling matter to quantum gravity In order to couple particle to matter ﬁelds we ﬁrst construct the coupling of gravity to Feynman integrals since as we are going to see there is a natural and unambiguous way to couple the Ponzano-regge model to Feynman integrals. We use the fact that Feynman integrals can be written as a worldline integral [11], that is if Γ is a (closed for simplicity) Feynman graph its Feynman integral is given by

I (e) = λ x p eiSΓ . (1.11) Γ D ΓD ΓD Γ % where 1 λ S (x , p , λ ) = dτ tr p e e (p2 µ2) . (1.12) Γ Γ Γ Γ 2 e t − 2 e − e e Γ e & ' !∈ % λ is a Lagrange multiplier ﬁeld which is the worldline frame ﬁeld and is

2 3

m sin(κm)/κ (31) →

S3 SU(2) (32) ∼

X ∂ 3 (33) ∼ P

m sin(κm)/κ (31) P 2→ 1/κ2 (34) ≤ S3 SU(2) (32) ∼ G 0 (35) N → X ∂ (33) ∼ P v v g = ηU+e Phe + [Ωe, Pe] = 0 (3(0.32)6) P 2 1/κ2 (34) ∂!e=v≤ S = tr(XeGe) (0.33) abcd G 0e P (35) hab(x)hcd(y)N →! (37) $ δX = % ∼U vΦx y[Ω2v, Φ ] = 0 (0.34) e e| v−− | e v g =v ηe+ h (36) !∈ 0 Xe = Xe e T (0.35) s = (Γ, je, iv) ∈ (38) abcd 0 ab cd P δje,je h (x)h (yZ) ∆,T 2 (37) (0.36) $ % ∼ x yd 2 e T je |"∈ − | g = (q, k2) (39) (! + m )kF (x) = iδ(x) (0.37) s = (Γ, je, iv) (38)

Re(KF)(Ge) = (0.38) (a/a˙ )2 = 8πG/3ρ(1 ρ/ρ ) (40) g = (q, k) crit (39) θ −= κm (0.39)

2 AS(Γ0, Γ1) 2= Af (jf ) Ae(jf , ie) Av(jf , ie) (0.40) (a/a˙ ) = 8π(a/aG˙ /)3(=ρ 8πGρ1/(3vρ(1))(ρρ2/(ρvcr)it) ρ)/ρcrit (40) (41) j ,i − e v !f−e "f " − " 2 i (a/aS˙ P lebansk) = 8i π=G/3(Bρi ρF1((vA))(),ρ2(+v)constrρ)/ρ. crBiti Bj = αδij (41) (0.41) v v − − U Pe + [Ω , Pe] = 0 Λ >∧ 0 (0.32) ∧ (42) e e # ∂!e=v QG correction:Bi Bj = GαδNij expansion (0.42) S = tr(XeGe) Λ > 0∧ (0.33) (42) e ! Γ planar and M =S3 δX = U vΦ [Ωv, Φ ] = 0 KF (Djeg) (0.34) (0.43) e e wv −e cane vget rid of the triangulation dependence v e O(je) = (43) !∈ K# (j ) dFje e X = X0 e T O(je) =e Γ (0.35) (43) e e ∈ tr(XvGv) I∆(Γ) = dX!∈v djde ge KF (ge) e (0.44) 0 e Γ δje,je Z∆,T v Γ !∈ e Γ e Γ (0.36) v Γ d 2 # # e T je "∈ "∈ "∈ "∈ "∈ 2 (! + m )kF (x) = iδ(x) 2 sin κm(0.37) 1 iTiT(P(P 2((gg) sin κκm ) ) !P(G) = Tr(G!σ) withKF (gK)F (=g) = dTdeT e −− κ 2iκ (44) (44) Re(KF)(Ge) = (0.38) i " " !X X = X σi θ = κm (0.39) → −→ Gv =−→ gv cyclically order product of (45) Gv = e vge group valued momenta (45) !⊃ e v meeting at the vertex v !⊃

Λ (46)

3 A ! -Product

3 i Idea: non commutative structure on R !X X = X σi → i Tr(Xg ) i Tr(Xg ) i Tr(Xg g ) e2κ 1 ! e2κ 2 = e2κ 1 2 3 Properties: • New Fourier transform R S ↔ 3 i Tr(Xg) f (X) = dge2κ f˜(g) Z L.F, S. majid • generalize Fourier theory to curved momentum space • ! is dual to convolution product on su(2) • 0-Bessel function replace δ ( x ) , it has ﬁnite width κ i j i jk non commutative spacetime [x ,x ] = iκε xk Algebra of derivation of a curved momentum space v v Ue Pe + [Ωe, Pe] = 0 (0.32) ∂!e=v S = tr(XeGe) (0.33) e ! δX = U vΦ [Ωv, Φ ] = 0 (0.34) e e v − e v v e !∈ X = X0 e T (0.35) e e ∈ 0 δje,je Z∆,T (0.36) d 2 e T je "∈ 2 (! + m )kF (x) = iδ(x) (0.37)

Re(KF)(Ge) = (0.38)

θ = κm (0.39)

AS(Γ0, Γ1) = Af (jf ) Ae(jf , ie) Av(jf , ie) (0.40) j ,i e v !f e "f " " S = B F i(A), + constr. B B = αδ (0.41) P lebanski i ∧ i ∧ j ij # B B = αδ (0.42) i ∧ j ij QG amplitude asD Fg eynman diagram (0.43) #

3 tr(XvGv) I∆(Γ) = i d Xv dge Ki F (ge) e (0.44) 2κ Tr(XvGv) ±2κ Tr(XvGe) # veΓ # e Γ !e Γe v Γ "∈ "∈→ ∂e "∈v "∈ ⊃

3 tr(Xvge) I∆(Γ) = d Xv dge KF (ge) %e ve (0.45) ∈ # v Γ # e Γ e Γ v Γ "∈ "∈ "∈ "∈ $ %

2 sin2 κm dT iT P (G) 2 with Kθ(G) = e − κ ! " Z 2π is a Feynman diagram of ...

3 iκp" "σ 2 G = e · = 1 + iκ(p" "σ) + O(κ ) (1) · iκp" "σ 2 G G = e · = 1 + iκ(p" "σ) + O(κ ) (2) 1 2 · 1 Tr(XG) = ip" X" + O(κ) (3) 2κ ·

i iS (φ) iS(φ,g)+ S(g) e eff = ge lp (4) D ! sin d θ χ (h ) = j (5) j θ sin θ eidj θ χ+(h ) = (6) j θ 2i sin θ −→ Xe gf Ge = gf (7) f e "⊃ i X = X σi S = tr(XeGe) (8) e ∆ #∈

itr(XeGe) dXe dgf e (9) ! e f " " 2 itr(XeGe) Z∆(Γo, Γi) = dXe dgf e φΓo (gf¯)φΓ∗ i (gf¯) (10) e ! " "f

Γi,oΓo Γi φΓi,o (gf¯) (11)

3 = djχj(GV)olΨΓ,je,iv = 8πγlP vje,iv ΨΓ,je,i(12)v (15) j N " v R Γ $ #∈ ∈#∪ ! Z∆ = dgf δ(Ge) = dje dgf χje (Ge) (13) ! f e je e ! f e " " {#} " " " F (A) E (16) ∝ i 2 2 i pie + λ(p m ) x(1)=y t − G (x, y; e) = DpDxDλ e ! (14) F ! !x(0)=x Area(H) = A (17) x(1)=y $ %& ' iSP (p,x,λ) GF (x, y; e) = DpDAxD nonλ e commutative field theory(15) !x(0)=x H = tr([E, E]F (A)) (18) S (p, x, λ) = p ei + λ(p2 m2) (16) P i t − ! v i •i Perturbativeµ expansion of massive λ| Γ| et eµx˙ 3 H (i) (17) (19) ≡ λφ coupled to 3d gravity j ∑ I∆(Γ) SΓ i S (ip ,x ,λ ) Γ I (g) = dx G (x , x ; g) = Dx Dλ e e P e e e (18) Γ v F se te is the perturbativee e expansion# of a non commutative field theory v e e P ! " " ! " λ > 0 λ 1 1 1i sinS(g2)mκ λ 3 lP S = d x (∂iφ ! ∂iφ)D(x)g e (φ ! φ)(x) + (φ ! φ ! φ)(x) (20) iS(g) 3 iS(g) i e SP (pe,xe,λe) 2 I˜Γ = Dg e IΓ(g) = DgDpeDλe e e − (19) 8πκ Z !2 2 κ 3! " ! ! P % κ = 4πG mass renormalisation K (j) k (l d ) (21) F ∝ F P j Particles behave as if they propagate on a non commutative space time 1 ! k [Xi , Xj] = ilP #ijkX (22) Properties In momentum space 2 1 2 sin κm 1 λ dg P (g) φ(g)φ(g− ) + dg dg dg δ(g g g )φ(g )φ(g )φ(g ) − 2 1 2 3 1 2 3 1 2 3 2 Z ! κ " 3! Z # # # # # Renormalisation of the mass Deformed conservation law at vertices no deformation of dispersion relation but momenta cut-off 0=!P1 !P2 !P3 ⊕! ⊕! ! 2 =∑Pi κ(P1 P2 + ...) + O(κ ) i − ∧ Symmetric under κ deformed action of the Poincare group: Lorentz part undeformed, the action on one particle state is isomorphic to Poincare, the action of translation is deformed on multiparticle states Satisﬁes the principles of Doubly Special Relativity iκp" "σ 2 G = e · = 1 + iκ(p" "σ) + O(κ ) (1) · iκp" "σ 2 G G = e · = 1 + iκ(p" "σ) + O(κ ) (2) 1 2 · 1 Tr(XG) = ip" X" + O(κ) (3) 2κ ·

i iS (φ) iS(φ,g)+ S(g) e eff = ge lp (4) D ! sin d θ χ (h ) = j (5) j θ sin θ eidj θ χ+(h ) = (6) j θ 2i sin θ 2 −→ Xe gf Ge = gf (7) f e "⊃ 2 2 i 2 X = X σi S = tr(XeGe) (8) e ∆ 3 #∈ VolΨΓ,je,iv = 8πγlP vje,iv ΨΓ,je,iv (15) itr(XeGe) " v R Γ $ dXe dgf e (9) ∈ ∪ 3 # VolΨΓ,j ,i = 8πγl vj ΨΓ,j ,i (15) ! e !v e 3P f e,iv e v VolΨΓ,j ,i = "8"πγl "v R Γ vj $ ΨΓ,j ,i (15) 3 e v P ∈#∪ e,iv e v ! " itr(X$eGe) VolΨΓ,je,iv = 8πγlP F (A) vjEe,iZv (Γ Ψ, ΓΓ),je=,iv dX v RdgΓ e (1φ(16)(5)g ¯)φ∗ (g ¯) (10) ∝ ∆ o i e ∈#∪f Γo f Γi f " v R Γ !$ e F (A)f E (16) ∈#∪ ! " "∝ ! Γ Γ Γ φ (g ¯) (11) Area(! H) = A i,oF (Ao ) i E Γi,o f (17) (16) ! ∝ F (A) E Area(H) = A (16) (17) ∝ = djχj(G) (12) H = tr([E, E]F (A)) HArea(!= tr([HjE,)E=]FA(A)) (18) (18) (17) #∈N Area(! H) = A (17) Z∆ = dgf δ(Ge) = dje dgf χje (Ge) (13) H (i) H = tr([EH, Ej(i])F (A)) (19) (19) (18) j ! f e je e ! f e " " i {#} " " " i # # i 2 2 H = tr([E, E]F (A)) i piet +(1λ8)(p m ) x(1)=y i S(g) − DgHejlP(i) ! (20) (19) i S(gG) F (x, y; e) = DpDxDλ e (14) Dg e lP x(0)=%xi (20) ! # % x(1)=y $ %& ' Hj(i) (19) G (x, y; e)K=F (j) kF (lPDdpj)DxDλ eiSP (p,x,λ) (21) (15) F ∝ i i x(0)=xS(g) lP #KF (j) kF (lP dj) D! g e (21) (20) ∝ [X !, X ] = il # Xk (22) S (p, x%i, λ) j= pP eijik+ λ(p2 m2) (16) Effective geometry from quantumP gravityi t − ! i S(g) k ! [X , Xl ] = il # X (22) Dig e Pj P ijk K m(j) sin(eikκm(eli)/x˙dκµ ) (20) (23) (21) (17) F →∝t ≡F µP j The effect of quantum gravity% is fourfold 3 i e SP (pe,xe,λe) • The mass gets renormalised IΓ(g) = dxv GF (xSse , xtSe ;Ug(2) )= DxeDλe e (24) (18) m sin(κm)/κ ! k (23) → v e[X , X ]∼= il # X e P (22) K (j) k (l d ) ! " " i j P ijk! " (21) F F P j λ > 0 λ • The momentum space acquire a∝ curvature proportional to the PlanckX ∂P (25) S3 SU(2) ∼ (24) iS(g) 3 iS(g) i SP (pe,xe,λe) mass. ∼ I˜Γ = Dg e IΓ(gS) = SDUg(2D)peDλe e e e (23) (19) ! k ∼2 2 P In the euclidean case the[X momentumi , Xj] = il spaceP #ijkX is a! 3 sphere P ! 1/κ (22) (26) ≤κ = 4πG addition of momenta is deformed. All Xmomenta∂P are bounded (25) ∼ X ∂ (24) ∼ P • By duality the position space X ∂ P is non1 commutative (23) ∼ P 2 1 (26) there is a minimal length scale accessible≤ toκ2 the theory Snyder NC 1947!

• The action of the Poincare group and diffeomorphism group is undeformed on one particle state. It is deformed on multiparticle states in order to preserve the existence of a maximal energy scale.

By duality it is related to a minimal length scale or discrete spectra of time intervals 3 g1 g1 space is no longer the flat space but the homogeneously ≡ Km(g1) ≡ δ(g1g2g3) curved space S3 ∼ SO(3). This reflects that the momen- g1 g2 g3 tum is bounded |P | < 1/κ. g1 g2 At the interaction vertex the momentum addition be- ! −1 ! −1 ! −1 ≡ δ(g1g2g1 g2 ) δ(g2g2 ) comes non-linear with a conservation rule P1⊕P2⊕P3 = 0 ! ! g2 g1 which implies a non-conservation of momentum P1 +P2 + P3 %= 0. Intuitively, part of the energy involved in a FIG. 1: Feynman rules for particles propagation in the collision process is absorbed by the gravitational field: Ponzano-Regge model. gravitational effects can not be ignored at high energy. This effect, which is stronger at high momenta and for non-collinear momenta, prevents the total momenta from where λ is a coupling constant. |v | is the number of Γ being larger than the Planck energy. vertices of Γ and S is the symmetry factor of the graph. Γ A last subtlety of the Feynman rules is the evaluation of Remarkably, this sum can be obtained from the pertur- non-planar diagrams. A careful analysis of I shows that bative expansion of a non-commutative field theory given Γ we have a non-trivial braiding: for each crossing of two explicitly by: # −1 # −1 # −1 edges, we associate a weight δ(g1g2g1 g2 ) δ(g2g2 ) d3x 1 1 sin2 mκ (see fig.1). This reflects a non-trivial statistics where the S = (∂iφ & ∂iφ)(x) − (φ & φ)(x) ! 8πκ3 "2 2 κ2 Fourier modes of the fields obey the exchange relation: λ −1 + (φ & φ & φ)(x) (19) φ(g1)φ(g2) = φ(g2)φ(g2 g1g2) (26) 3! # & & & & 3 which is naturally determined by our choice of star pro- where the field φ is in Cκ(R ). Its momentum has sup- −1 duct. Indeed, let us look at the product of two identical port in the ball of radius κ . We can write this action fields: in momentum space 1 tr(Xg1g2) 1 sin2 κm φ & φ (X) = dg1dg2 e 2κ φ(g1)φ(g2), (27) S(φ) = dg P 2(g) − φ(g)φ(g−1) (20) ! 2 ! $ κ2 % & & −1 λ & & Under change of variables (g1, g2) → (g2, g2 g1g2), the + dg1dg2dg3 δ(g1g2g3) φ(g1)φ(g2)φ(g3). 3! ! star product reads & & & 1 This is our effective field theory describing the dynamics tr(Xg1g2) −1 φ&φ (X) = dg1dg2 e 2κ φ(g2)φ(g g1g2). (28) of the matter field after integrating out the gravitational ! 2 sector. This non-commutative field theory action is sym- & & metric under a κ-deformed action of the Poincaré group. The identification of the Fourier modes of φ& φ (X) leads Calling Λ the generators of Lorentz transformations and to the exchange relation (26). This braiding was first proposed in [10] for two particles coupled to 3d QG and T"a the generators of translations, the action of these generators on one-particle states is undeformed: then computed in the Ponzano-Regge model in [5]. It is encoded into a braiding matrix Λ · φ(g) = φ(ΛgΛ−1) = φ(Λ · P (g)), (21) Reconciling QM and GR−1 iP" (g)·"a R · φ(g1)φ(g2) = φ(g2)φ(g2 g1g2). (29) T"a · φ&(g) = e& φ(g). & (22) This is the RQuantumm&atrix Mechanics:&of the κ-&d efor&mation of the Poincaré The non-trivi&al deformation &of the Poincaré group ap- Heisenberggr oduality/democracyup [10]. Such betweenfield th spaceeories andwi tmomentumh non-tri vspaceial b raided pears at the level of multi-particle states and only the statistics are simply called braided non-commutative field action of the translations is deformed : Gravity: space is singled out and curved theories and were first introduced in [11]. 2+1 QG restore the duality by curving momentum space Λ · φ(P1)φ(P2) = φ(Λ · P1)φ(Λ · P2), (23) Finally, the &-product induces a non-commutativity of space-time and a deformation of phase space: iP"1⊕P"2·"a T"a · φ&(P1)φ&(P2) = e& φ&(P1)φ(P2). (24) momentum space : 3sphere of radius Planck mass [X , X ] = iκ( X , It is straig&htforw&ard to derive the F&eynm&an rules from i j ijk k 2 2 the action (21) (see fig.1). The effective Feynman prop- geometrical interpretation[Xi, Pj] = ofi non1 −commutativityκ P δij − iκ(ijkPk. (30) agator is the group Fourier transform of Kθ(g), ) This non-commutativity reflects the fact that momen- 1 tr(Xg) e 2κ & tum space is curved. Indeed the coordinates X are re- Xi : right invariant derivative on S3 Km(X) = i dg 2 . (25) ! P 2(g) − sin κm alized as right invariant derivations on momentum space κ aNCnd geometryderivat iao nbetters of approximationa curved m a....nifold do not commute. ' ( The effect of quantum gravity is two-fold. First the mass Moreover, this non-commutativity being related to hav- gets renormalized m → sin κm/κ. Then the momentum ing bounded momenta implies the existence of a minimal 2

3 VolΨΓ,je,iv = 8πγlP vje,iv ΨΓ,je,iv (15) " v R Γ $ ∈#∪ ! F (A) E (16) ∝

Area(! H) = A (17)

H = tr([E, E]F (A)) (18)

Hj(i) (19) i #

i S(g) Dg e lP (20) % 4d ? K (j) k (l d ) (21) F ∝ F P j

! k Is it possible[Xi , X jto] = extendilP #ijkX this strategy to 4d gravity ? (22) The exact coupling is not yet known. Two key results: m sin(κm)/κ Baratin, L.F, Starodubtsev(23), Kowalski-Glikman → Matter can be described as a topological gravitational defect S3 SU(2) (24) ∼ Usual Feynman diagram in flat space can be effectively written in a purely combinatorialX ∂P manner as some expectation of(2 certain5) ∼ observable in a topological spin foam model. P 2 1/κ2 (26) ≤ This gives a new background perspective on field theory and gives the form of the G 0 limit quantum gravity spin foam model(27) N →

New point of view onP possibleabcd dimensionfull deformation of field hab(x)hcd(y) (28) theories.% & ∼ x y 2 | − |

s = (Γ, je, iv) (29)

g = (q, k) (30)

(a/a˙ )2 = 8πG/3ρ(1 ρ/ρ ) (31) − crit

(a/a˙ )2 = 8πG/3(ρ ρ (v))(ρ (v) ρ)/ρ (32) − 1 2 − crit 3

m sin(κm)/κ (31) →

S3 SU(2) (32) ∼

X ∂ (33) ∼ P

P 2 1/κ2 (34) ≤

G 0 (35) N →

g = η + h (36)

P abcd hab(x)hcd(y) (37) $ % ∼ x y 2 | − |

s = (Γ, je, iv) (38)

g = (q, k) (39)

(a/a˙ )2 = 8πG/3ρ(1 ρ/ρ ) (40) − crit

(a/a˙ )2 = 8πG/3(ρ ρ (v))(ρ (v) ρ)/ρ (41) − 1 2 − crit

Λ > 0 (42)

KF (je) O(je) = (43) dj e Γ e !∈ Conclusion iT (P 2(g) sin κm ) Shown that a NonKF Commutative(g) = dT e Braided− QFTκ gives the effective (44) description of matter" field coupled to 2+1QG Generalisations: Lorentzian gravity: more involved but natural implementation of the G = −→g (45) Feynman propagator, S3 ADS3v v e v Non trivial cosmological constant:! ⊃both x and p space are curved and non- commutative QG affects in a non trivial ( Λ dependent) the density of state of FT (46) A way to tame down the cc problem? Unitarity, conserved charges, renormalisability? Extension to 4D ? New and background independent spin foam perspective on usual field theory in any dimensions Feynman diagrams = e.v of observables in a topological spin foam model.

AdS/CFT: is it possible to do a similar computation? What is the effective quantum geometry