Boosting Wigner's nj-symbols

Simone Speziale 1, 2
Abstract : We study the SL(2,C) Clebsch-Gordan coefficients appearing in the lorentzian EPRL spin foam amplitudes for loop quantum gravity. We show how the amplitudes decompose into SU(2) nj-symbols at the vertices and integrals over boosts at the edges. The integrals define edge amplitudes that can be evaluated analytically using and adapting results in the literature, leading to a pure state sum model formulation. This procedure introduces virtual representations which, in a manner reminiscent to virtual momenta in Feynman amplitudes, are off-shell of the simplicity constraints present in the theory, but with the integrands that peak at the on-shell values. We point out some properties of the edge amplitudes which are helpful for numerical and analytical evaluations of spin foam amplitudes, and suggest among other things a simpler model useful for calculations of certain lowest order amplitudes. As an application, we estimate the large spin scaling behaviour of the simpler model, on a closed foam with all 4-valent edges and Euler characteristic X, to be N^(X - 5E + V/2). The paper contains a review and an extension of results on SL(2,C) Clebsch-Gordan coefficients among unitary representations of the principal series that can be useful beyond their application to quantum gravity considered here.
Type de document :
Article dans une revue
Journal of Mathematical Physics, American Institute of Physics (AIP), 2017, 58 (3), pp.032501 〈10.1063/1.4977752〉
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Soumis le : mardi 24 avril 2018 - 17:44:24
Dernière modification le : mercredi 2 mai 2018 - 17:21:16


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Simone Speziale. Boosting Wigner's nj-symbols. Journal of Mathematical Physics, American Institute of Physics (AIP), 2017, 58 (3), pp.032501 〈10.1063/1.4977752〉. 〈hal-01476640〉



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