Asymptotics of lowest unitary SL(2,C) invariants on graphs

Abstract

We describe a technique to study the asymptotics of SL(2, C) invariant tensors associated to graphs, with unitary irreps and lowest SU(2) spins, and apply it to the Lorentzian EPRL-KKL (Engle, Pereira, Rovelli, Livine; Kaminski, Kieselowski, Lewandowski) model of quantum gravity. We reproduce the known asymptotics of the 4-simplex graph with a different perspective on the geometric variables and introduce an algorithm valid for any graph. On general grounds, we find that critical configurations are not just Regge geometries, but a larger set corresponding to conformal twisted geometries. These can be either Euclidean or Lorentzian, and include curved and flat 4d polytopes as subsets. For modular graphs, we show that multiple pairs of critical points exist, and there exist critical configurations of mixed signature, Euclidean and Lorentzian in different subgraphs, with no 4d embedding possible. To Jurek Lewandowski, for his 60th birthday 8 Conclusions 39 A Vectorial representation of SL(2, C) 41 B Orientation conditions imply Euclidean geometry 41 C Evaluating the on-shell action 42 D Spherical and hyperbolic trigonometry 43 E Degrees of freedom of polytopes 43 1 Introduction The spin foam formalism provides transition amplitudes for loop quantum gravity. The state of the art is the Lorentzian Engle-Pereira-Rovelli-Livine (EPRL) model [1, 2], which is based on the group SL(2, C) and its infinite-dimensional, unitary irreducible representations of the principal series. Support in favour of this model comes from the emergence of Regge geometries and the Regge action in the asymptotics of the 4-simplex vertex amplitude for large quantum numbers. This result was obtained by Barrett and collaborators [3], building on previous work [ 4,5,6,7,8], and has been used in a number of applications of the model, e.g. [9,10,11,12,13,14,15,16,17]. The 4-simplex vertex amplitude is sufficient if one restricts attention to spin foams which are dual to triangulations of spacetime, and is the building block for transition amplitudes of 4-valent, simplicial spin network states. But from a canonical perspective, more general vertex amplitudes are to be included in order to provide transition amplitudes to all spin network states, and not just simplicial ones. One such generalization has been proposed by Kaminski, Kieselowski and Lewandwski (KKL) [18], see also [19], and has been applied to cosmological models and studies of spin foam renormalization [20,21,22,23,24,25]. It is however not known what are the dominant geometric configurations and the asymptotic behaviour of the EPRL-KKL Lorentzian amplitude on general vertices. These are the open questions that we answer in this paper. To do so, we introduce a novel technique to study the saddle point approximation of the SL(2, C) amplitudes. The technique and results presented here, while motivated by the quantum gravity applications, are of more general interest for any situation in which asymptotics of unitary SL(2, C) Clebsch-Gordan coefficients are needed. The derivations of 4-simple...

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