2-vertex Lorentzian Spin Foam Amplitudes for Dipole Transitions

摘要

We compute transition amplitudes between two spin networks with dipolegraphs, using the Lorentzian EPRL model with up to two (non-simplicial)vertices. We find power-law decreasing amplitudes in the large spin limit,decreasing faster as the complexity of the foam increases. There are nooscillations nor asymptotic Regge actions at the order considered, nonethelessthe amplitudes still induce non-trivial correlations. Spin correlations betweenthe two dipoles appear only when one internal face is present in the foam. Wecompute them within a mini-superspace description, finding positivecorrelations, decreasing in value with the Immirzi parameter. The paper alsoprovides an explicit guide to computing Lorentzian amplitudes using thefactorisation property of SL(2,C) Clebsch-Gordan coefficients in terms of SU(2)ones. We discuss some of the difficulties of non-simplicial foams, and providea specific criterion to partially limit the proliferation of diagrams. Wesystematically compare the results with the simplified EPRLs model, much fasterto evaluate, to learn evidence on when it provides reliable approximations ofthe full amplitudes. Finally, we comment on implications of our results for thephysics of non-simplicial spin foams and their resummation.