Graviton propagator from LQG

Transcription

1 Graviton propagator from LQG Carlo Rovelli International Loop Quantum Gravity Seminar from Marseille, September 2006

2 4d: 1. Particle scattering in loop quantum gravity Leonardo Modesto, CR PRL, ,2005, gr-qc/ general idea 2. Graviton propagator from background-independent quantum gravity CR PRL, to appear, gr-qc/ st calulation 3. Graviton propagator in loop quantum gravity Eugenio Bianchi, Leonardo Modesto, Simone Speziale, CR CQG, to appear, gr-qc/ detailed discussion and 2nd order results 4. Group Integral Techniques for the Spinfoam Graviton Propagator Etera Livine, Simone Speziale gr-qc/ improved boundary state 3d: 5. Towards the graviton from spinfoams: The 3-D toy model Simone Speziale JHEP 0605:039,2006, gr-qc/ Towards the graviton from spinfoams: Higher order corrections in the 3-D toy model Etera Livine, Simone Speziale, Joshua L. Willis gr-qc/ ( 6. From 3-geometry transition amplitudes to graviton states Federico Mattei, Simone Speziale, Massimo Testa, CR Nucl.Phys.B739: ,2006, gr-qc/ )

3 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 A vertex e is Regge + e is Regge + D good bad Barrett, Williams, Baez, Christensen, Egan, Freidel, Louapre 3

4 A key open issue Low energy limit Newton s law Computing scattering amplitudes 4

5 The problem W (x, y) = Dφ φ(x) φ(y) e is[φ] if measure and action are diff invariant, then immediately W (x, y) = W (f(x), f(y)) 5

6 Idea for a solution: define the boundary functional ϕ φ int y W [ϕ, Σ] = φ int Σ =ϕ Dφ int e is[φ int] 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) 6

7 What happens in a diff invariant theory? Clearly Therefore W [ϕ, Σ] = W [ϕ] 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) Ψ q [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 7

8 Particle detector Spacetime region Particle detector Distance and time measurements In GR distance and time measurements are field measurements like the other ones: they determine the boundary data of the problem. 8

9 Give meaning to the expression W (x, y; q) = Dϕ ϕ(x) ϕ(y) W [ϕ] Ψ q [ϕ] (, q) φ int x y Dφ 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. W abcd (x, y; q) = N ss W [s ] s h ab (x)h cd (y) s Ψ q [s] Modesto, CR, PRL 05 9

10 W [s] : Group field theory (here GFT/B): W [s] = Dφ f s (φ) e φ 2 λ 5! φ 5 The Feynman expansion in λ gives a sum over spinfoams W [s] = A faces A vertex σ=s faces vertices which has a nice interpretation as a discretization of the Misner-Hawking sum over geometries W ( 3 g) = g= 3 g Dg e is Einstein Hilbert[g] with background triangulations summed over as well. 10

11 To first order in λ, the only nonvanishing connected term in W[s] is for s = j 51 j 12 j 52 j 14 j 13 j j24 35 j 45 j 23 j 34 And the dominant contribution for large j is given by the spinfoam σ dual to a single four-simplex. This is W [s] = λ 5! ( n<m dim(j nm ) ) A vertex (j nm ) 11

12 The boundary state Ψ q (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=j L. Ψ q (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: { Ψ q [s] = exp α 2 (j nm j (0) nm )2 + i n<m n<m Φ (0) nm j nm } Ψ q (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. 12

13 j 12 ϕ 12 L 13

14 The field operator h ab ( x) = g ab ( x) δ ab = E ai ( x)e bi ( x) δ ab 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 E Ii (n)e I i (n) s = (8π G)2 j I (j I + 1) s Define j 51 j 12 j 52 j 14 j 13 j j24 35 j 45 j 23 x j 34 n m y W (L) = W abcd (x, y; q) n a n b m c m d Standard perturbative theory gives W (L) = i 8π 1 = i 8π G 1 4π 2 x y 2 q 4π 2 L 2 14

15 m y n x 15

16 The expression for the propagator is then well defined: W (L) = W abcd (x, y; q)n a n b m c m d = N λ( G)4 5! j nm (j 12 (j ) j 2 L ) (j 34(j ) j 2 L ) A vertex(j nm ) e α 2 n,m (j nm j L ) 2 iφ n,m j nm A vertex e is Regge + e is Regge + D rapidly oscillating phase But since S Regge (j nm ) = n<m Φ nm (j) j nm and S Regge (j nm ) Φ nm j nm G (mn)(kl)δj mn δj kl only the good component of A vertex survives! This is the forward propagating (Oriti, Livine) component of A vertex cfr. Colosi, CR 16

17 S Regge (j nm ) Φ nm j nm G (mn)(kl)δj mn δj kl The gaussian integration gives finally W (L) = 4i α 2 G (12)(34) Where G (mn)(kl) is the ( discrete ) derivative of the dihedral angle, with respect to the area (the spin). G (mn)(kl) = Φ mn(j ij ) j kl jij =j L It can be computed from geometry, giving, L where k is a numerical factor ~ l 2 G (12)(34) = 8π Gk 17

18 A 34 =8πħG j 34 Φ12 18

19 A 34 =8πħG j 34 Φ12 19

20 A 34 =8πħG j 34 Φ12 Cfr the nutshell dynamics in 3d gravity Colosi, Doplicher, Fairbairn, Modesto, Noui, CR 20

21 A 34 =8πħG j 34 Φ12 Cfr the nutshell dynamics in 3d gravity Colosi, Doplicher, Fairbairn, Modesto, Noui, CR 21

22 Adjusting numerical factors α 2 = 16π 2 k, this gives W (L) = i 8π G 4π 2 1 L 2 = i 8π 4π 2 1 x y 2 q CR, PRL 06 which is the correct graviton-propagator (component). This is only valid for L 2 >> 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 22

23 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! 23

24 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?... 24

25 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. 25