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Effective Field Theory from Spin Foam Models :3D Example

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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 gmapseometries. N(quantaodes and lofink s3din tvolume)he spin network evolve into 1-dimensional edges and faces. Area(! H) = A New links are created and spins are reassigned at(2ver2)texes (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 Schr¨odinger’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.
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