1. Loops05 1 Emergence of Spin Foam in Feynman graphs Aristide Baratin ENS Lyon (France) and Perimeter Institute (Canada) with Laurent Freidel Potsdam, October 11, 2005

2. Loops05 2Introduction  Formulation of a perturbative theory of 4d quantum gravity (L.Freidel, A.Starodubtsev, Feb 05)  Limit G  0 (« no-gravity limit ») of QG is topological Questions: Can one formulate 4d Feynman graphs in a background independent manner? Can one detect in these graphs traces of a topological spin foam model, without any assumption about QG?  Effective theory for matter in 2+1 dimensions (L.Freidel, E. Livine Feb 05)  Usual 3d Feynman graphs recovered as the no-gravity limit of spin- foams coupled to particles

3. Loops05 3 Feynman diagrams and invariant measure Generic Feynman amplitude: nd variables d(d+1)/2 symmetry parameters nd - d(d+1)/2 VS n(n-1)/2  match for n=d+1: d-simplex d=3, n=4 n=5  for n=d+2, all the distances should be involved, except for one for 3d, John W. Barrett; L.Freidel, D.Louapre

4. Loops05 4 Feynman diagrams and invariant measure General Result: = Triangulation of a d-ball, N vertices on the boundary, no internal (d-2)-faces. Question Question: How can one extend this expression to more general triangulations?

5. Loops05 5 3d Case: a Poincare spin foam model in a 3-ball… :A key « pentagonal » identity : Pachner move (2,3) Building blocks: Poincare 6j-symbols Z Topological spin foam model with boundaries:

6. Loops05 6 …as the limit of Ponzano-Regge gravity Poincare model ~ square of Ponzano Regge

7. Loops05 7 Feynman diagrams and invariant measure General Result: = Triangulation of a d-ball, N vertices on the boundary, no internal (d-2)-faces. Question Question: How can one extend this expression to more general triangulations?

8. Loops05 8 4d case: emergence of the model The key identity: 05 2 3 4 1  Identified as gauge-fixed identities (2,4), (3,3) and (1,5) Move (2,4) 1 new edge, 4 new faces Z Topological spin foam model with boundaries: + Gauge fixing State sum version of the invariant exhibited by I.Korepanov, 2002

9. Loops05 9 4d case: symmetries and gauge fixing Study of the action Classical solutions: flatness condition and s_F=A_F area (Schlafli) The study of Kernel of the Hessian matrix around a classical solution gives infinitesimal transformations corresponding to: - A gauge symmetry acting on vertices (4-vectors) - A (non obvious!) gauge symmetry acting on edges (3-vectors) Fadeev-Popov procedure  choice of edges/faces + determinants

10. Loops05 10 Conclusion and Outlook Can one formulate 4d Feynman graphs in a background independent manner?  Feynman graphs as the expectation value of observables for a spin foam model -Square root of our model? -A deeper understanding of the symmetries and of the underlying algebraic structures (2-category…) is needed -Precise connection with the perturbative formulation of QG? Can one detect traces of a topological spin foam model?  Topological model based on the Poincare group