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Emergence of Spin Foam in Feynman graphs

Explore the formulation of perturbative quantum gravity theories, topological limits, and the detection of spin foam traces in Feynman graphs. Study classical solutions, gauge symmetries, and the extension of invariant measures in various dimensions. Investigate the relationship between Feynman graphs and spin foam models, emphasizing the background-independent nature of 4D formulations.

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Emergence of Spin Foam in Feynman graphs

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

  2. Introduction • 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 • 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 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?

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

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

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

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

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

  8. 4d case: emergence of the model 1 • The key identity: 4 2 Identified as gauge-fixed identities (2,4), (3,3) and (1,5) 3 0 5 Move (2,4) 1 new edge, 4 new faces Z Topological spin foam model with boundaries: State sum version of the invariant exhibited by I.Korepanov, 2002 +Gauge fixing

  9. 4d case: symmetries and gauge fixing • Study of the action • Classical solutions: flatness condition and s_F=A_F area (Schlafli) • The study ofKernel 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. 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 Can one detect traces of a topological spin foam model? • Topological model based on the Poincare group • 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?

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