Asymptotics of $\mathrm {SL}(2,{{\mathbb {C}}})$ coherent invariant tensors
Résumé
We study the semiclassical limit of a class of invariant tensors for infinite-dimensional unitary representations of SL(2,C) of the principal series, corresponding to generalized Clebsch–Gordan coefficients with $n\ge 3$ legs. We find critical configurations of the quantum labels with a power-law decay of the invariants. They describe 3d polygons that can be deformed into one another via a Lorentz transformation. This is defined viewing the edge vectors of the polygons are the electric part of bivectors satisfying a (frame-dependent) relation between their electric and magnetic parts known as $\gamma $-simplicity in the loop quantum gravity literature. The frame depends on the SU(2) spin labelling the basis elements of the invariants. We compute a saddle point approximation using the critical points and provide a leading-order approximation of the invariants. The power-law is universal if the SU(2) spins have their lowest value, and n-dependent otherwise. As a side result, we provide a compact formula for $\gamma $-simplicity in arbitrary frames. The results have applications to the current EPRL model, but also to future research aiming at going beyond the use of fixed time gauge in spin foam models.
Mots clés
spin: foam
representation: unitarity
transformation: Lorentz
gauge: time
quantum gravity: loop space
higher-order: 0
simplex
SU(2)
Clebsch-Gordan coefficients
saddle-point approximation
critical phenomena
semiclassical
deformation
EPRL model
coherence
asymptotic behavior
SL(2
C)
field theory: vector