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Graviton Propagator from LQG

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Graviton Propagator from LQG Graviton propagator from LQG Carlo Rovelli International Loop Quantum Gravity Seminar from Marseille, September 2006 4d: 1. Particle scattering in loop quantum gravity general idea Leonardo Modesto, CR PRL, 191301,2005, gr-qc/0502036 2. Graviton propagator from background-independent quantum gravity 1st calulation CR PRL, to appear, gr-qc/0508124 3. Graviton propagator in loop quantum gravity detailed discussion Eugenio Bianchi , Leonardo Modesto , Simone Speziale , CR and 2nd order results CQG, to appear, gr-qc/0604044 4. Group Integral Techniques for the Spinfoam Graviton Propagator improved boundary state Etera Livine, Simone Speziale gr-qc/0608131 3d: 5. Towards the graviton from spinfoams: The 3-D toy model Simone Speziale JHEP 0605:039,2006, gr-qc/0512102 6. Towards the graviton from spinfoams: Higher order corrections in the 3-D toy model Etera Livine , Simone Speziale , Joshua L. Willis gr-qc/0605123 ( 6. From 3-geometry transition amplitudes to graviton states Federico Mattei, Simone Speziale, Massimo Testa , CR Nucl.Phys.B739:234-253,2006 , gr-qc/0508007 ) Where we are in LQG • Kinematics well-defined: + Separable kinematical Hilbert space (spin networks, s-knots) + Geometrical interpretation (area and volume operators) - Euclidean/Lorentzian ? Immirzi parameter ? • Dynamics: • Hamiltonian operator (in various formulations) • Spinfoam formalism (several models) - triangulation independence: Group Field Theory (+ good finiteness, - λ ?) - Barrett-Crane vertex, or 10j symbol iS iS A e Regge + e− Regge + D vertex ∼ good bad Barrett, Williams, Baez, Christensen, Egan, Freidel, Louapre carlo rovelli 3 graviton propagator A key open issue • Low energy limit • Newton’s law • Computing scattering amplitudes carlo rovelli 4 graviton propagator The problem W (x, y) = Dφ φ(x) φ(y) eiS[φ] ! if measure and action are diff invariant, then immediately W (x, y) = W (f(x), f(y)) carlo rovelli 5 graviton propagator Idea for a solution: define the boundary functional y ϕ • iS[φint] φint W [ϕ, Σ] = Dφint e φ =ϕ ! int|Σ x• Σ then W (x, y; Σ, Ψ) = Dϕ ϕ(x) ϕ(y) W (ϕ, Σ) Ψ[ϕ] ! cfr: R Oeckl (see also Conrady, Doplicher, Mattei) carlo rovelli 6 graviton propagator What happens in a diff invariant theory? Clearly W [ϕ, Σ] = W [ϕ] Therefore W (x, y; Ψ) = Dϕ ϕ(x) ϕ(y) W (ϕ) Ψ[ϕ] ! But in GR the information on the geometry of a surface is not in ∑ It is in the state of the (gravitational) field on the surface ! Hence: choose Ψ to be a state picked on a given geometry q of ∑ ! W (x, y; q) = Dϕ ϕ(x) ϕ(y) W (ϕ) Ψq[ϕ] ! Distance and time separations between x and y are now well defined with respect to the mean boundary geometry q. Conrady ,Doplicher, Oeckl, Testa, CR carlo rovelli 7 graviton propagator Particle detector Spacetime region Distance and time measurements Particle detector In GR distance and time measurements are field measurements like the other ones: they determine the boundary data of the problem. carlo rovelli 8 graviton propagator y (∑, q) • Give meaning to the expression W (x, y; q) = Dϕ ϕ(x) ϕ(y) W [ϕ] Ψq[ϕ] x• ! • φ ∫intDφ ➞ ∑s-knots from LQG • W [φ] ➞ W [s] defined by GFT spinfoam model • Ψq ➞ a suitable coherent state on the geometry q • φ(x) ➞ graviton field operator from LQG. abcd ab cd W (x, y; q) = N W [s!] s! h (x)h (y) s Ψ [s] " | | # q !ss! Modesto, CR, PRL 05 carlo rovelli 9 graviton propagator W [s] : Group field theory (here GFT/B): φ2 λ φ5 W [s] = Dφ fs(φ) e− −5! ! " " The Feynman expansion in λ gives a sum over spinfoams W [s] = Afaces Avertex ∂!σ=s f"aces ve"rtices which has a nice interpretation as a discretization of the Misner-Hawking sum over geometries 3 iSEinstein Hilbert[g] W ( g) = Dg e − 3 !∂g= g with background triangulations summed over as well. carlo rovelli 10 graviton propagator To first order in λ, the only nonvanishing connected term in W[s] is for ¢BQ j51 ¢ t Q j B Q 12 ¢ B Q ¢ Q B QQ l B # C l ¢ j £ s = tC l ¢ 52 B # £ t j j13B # C l ¢ 14 # £ C ¢ j35j24# £ C ¢ cc # B £ j C ¢ c B £ j23 45 c C ¢ c B £ C ¢ c B £ C¢ cB£ t j34 t And the dominant contribution for large j is given by the spinfoam σ dual to a single four-simplex. This is λ W [s] = dim(jnm) Avertex(jnm) 5! !n<m # " carlo rovelli 11 graviton propagator The boundary state Ψq(s) • Choose a boundary geometry q: let q be the geometry of the 3d boundary (∑,q) of a spherical 4d ball, with linear size L >> √ħG. • Interpret s as the (dual) of a triangulation of this geometry. Choose a regular triangulation of (∑, q); interpret the spins as the areas of the corresponding triangles, using the standard LQG interpretation of spin networks. • This determine the “background” spins j(0)nm=jL. Ψq(s) must be picked on these values. Choose a Gaussian state around these values with with α, to be determined. • A Gaussian can have an arbitrary phase: α Ψ [s] = exp (j j(0) )2 + i Φ(0) j q − nm − nm nm nm ! 2 n<m n<m # " " ➞ Ψq(s) must be a coherent state, determined by coordinate and momentum, namely by intrinsic 3-geometry and extrinsic 3-geometry q !! ➞ The Φ (0)nm = Φ are the background dihedral angles. carlo rovelli 12 graviton propagator ϕ12 j12 L carlo rovelli 13 graviton propagator The field operator hab(!x) = gab(!x) δab = Eai(!x)Ebi(!x) δab − − Choose x to be on the nodes and contract the indices with two parallel vectors along the links. Then we have the standard action on boundary spin networks, well known from LQG Ii I 2 E (n)E (n) s = (8π!G) jI(jI + 1) s i | ! | ! Define ¢BQ abcd j51 ¢ t Q j W (L) = W (x, y; q) n n m m B Q 12 a b c d ¢ B Q ¢ Q B QQm l B # C l ¢ j £ tC l ¢ 52 B # £ t y j j13B # C l ¢ 14 # £ C ¢ j35j24# £ Standard perturbative theory gives C ¢ cc # B £ j C ¢ c B £ j23 45 c C ¢ c B £ C ¢ c B £ C¢ cB£ 8π 1 8π!G 1 x t j t W (L) = i = i n 34 4π2 x y 2 4π2 L2 ( q ) | − | carlo rovelli 14 graviton propagator m y n x carlo rovelli 15 graviton propagator The expression for the propagator is then well defined: abcd W (L) = W (x, y; q)nanbmcmd = 4 λ(!G) α 2 2 2 2 n,m(jnm jL) iΦ n,m jnm N (j12(j12 + 1) jL) (j34(j34 + 1) jL) Avertex(jnm) e− − − 5! − − " " !jnm rapidly oscillating phase But since SRegge(jnm) = Φnm(j) jnm n<m ! 1 S (j ) Φ j + G δj δj and Regge nm ∼ nm (mn)(kl) mn kl nm 2 ! iS iS A e Regge + e− Regge + D only the “good” component ofve r texAver∼tex survives ! This is the “forward propagating” (Oriti, Livine) component of Avertex cfr. Colosi, CR carlo rovelli 16 graviton propagator The gaussian “integration” gives finally 4i W (L) = G(12)(34) α2 Where G ( mn )( k l ) is the (“discrete”) derivative of the dihedral angle, with respect to the area (the spin). 1 S (j ) Φ j + G δj δj Regge nm ∼ nm (mn)(kl) mn kl nm 2 ! ∂Φmn(jij) G(mn)(kl) = ∂j kl !jij=jL ! ! ! 8π!Gk It can be computed from geometry, giving G ( 12 )(3 4) = , L2 where k is a numerical factor ~ l carlo rovelli 17 graviton propagator A34=8πħG j34 Φ12 carlo rovelli 18 graviton propagator A34=8πħG j34 Φ12 carlo rovelli 19 graviton propagator A34=8πħG j34 Φ12 Cfr the “nutshell’ dynamics in 3d gravity Colosi, Doplicher, Fairbairn, Modesto, Noui, CR carlo rovelli 20 graviton propagator A34=8πħG j34 Φ12 Cfr the “nutshell’ dynamics in 3d gravity Colosi, Doplicher, Fairbairn, Modesto, Noui, CR carlo rovelli 21 graviton propagator Adjusting numerical factors α 2 = 16 π 2 k , this gives 8π!G 1 8π 1 W (L) = i = i 4π2 L2 4π2 x y 2 CR, PRL 06 | − |q which is the correct graviton-propagator (component). ➞ This is only valid for L2 >> ħG. For small L, the propagator is affected by quantum gravity effects, and is given by the 10j symbol combinatorics. ➞ This is equivalent to the Newton law carlo rovelli 22 graviton propagator So far • Second order terms have been computed, and do not spoil the result (Bianchi Modesto Speziale CR) • Better definition of the bondary state (Livine Speziale) • A detailed analysis in 3d has been made, including the computation of Planck scale corrections to the propagator (Livine Speziale Willis) • Numerical investigations (Dan Christensen) Do not miss Simone’s talk next week ! carlo rovelli 23 graviton propagator Open issues • Other components? Full tensorial structure (Alesci) (intertwiners) • Better definition of the bondary state (Livine Speziale, Yongge Ma) • Higher order terms in λ (Mamone) • Other models (GFT/C seems to give the same result (Modesto)) • n - point functions ? • Matter • Computing the undetermined constants of the non-renormalizable perturbative QFT ? • ... carlo rovelli 24 graviton propagator Conclusion i. Low energy limit. (One component of) the graviton propagator (or the Newton law) appears to be correct, to first orders in λ. ii. Barrett-Crane vertex. Only the “good” component of the 10j symbol survives, because of the phase of the state, given by the extrinsic geometry of the boundary state. The BC vertex works. iii. Scattering amplitudes. A technique to compute n-point functions within a background-independent formalism exists.
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