Numerical study of the Lorentzian Engle-Pereira-Rovelli-Livine spin foam amplitude
Résumé
The Lorentzian Engle-Pereira-Rovelli-Livine (EPRL) spin foam amplitude for loop quantum gravity is a multidimensional noncompact integral of highly oscillating functions. Using a method based on the decomposition of Clebsch-Gordan coefficients for the unitary infinite-dimensional representations of SL(2,C) in terms of those of SU(2), we are able to provide for the first time numerical evaluations of the vertex amplitude. The values obtained support the asymptotic formula obtained by Barrett and collaborators with a saddle point approximation, showing, in particular, a power-law decay and oscillations related to the Regge action. The comparison offers a test of the efficiency of the method. Truncating the decomposition to the first few terms provides a qualitative matching of the power-law decay and oscillations. For vector and Euclidean Regge boundary data, a qualitative matching is obtained with just the first term, which corresponds to the simplified EPRL model. We comment on future developments for the numerics and extension to higher vertices. We complete our work with some analytic results: We provide an algorithm and explicit configurations for the different geometries that can arise as boundary data, and explain the geometric consequences of the decomposition used.
Mots clés
String theory
quantum gravity
gauge/gravity duality
spin: foam
representation: SL(2
action: Regge
quantum gravity: loop space
oscillation
Clebsch-Gordan coefficients
saddle-point approximation
higher-dimensional
EPRL model
noncompact
geometry: Euclidean
representation: unitarity
SU(2)
geometry: Regge
geometry
numerical methods
numerical calculations
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