Boosting Wigner's nj-symbols

Simone Speziale (1, 2)

1 CPT - Centre de Physique Théorique - UMR 7332
2 CPT - E4 Gravité quantique

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.

Mots clés: spin foam models; loop quantum gravity; Clebsch-Gordan coefficients; SL(2; C) unitary irreps

Domaines: Relativité Générale et Cosmologie Quantique [gr-qc]; Physique mathématique [math-ph]; Théorie des groupes [math.GR]

Simone Speziale. Boosting Wigner's nj-symbols. Journal of Mathematical Physics, 2017, 58 (3), pp.032501 ⟨10.1063/1.4977752⟩. ⟨hal-01476640⟩