Asymptotics of $\mathrm{SL}(2,\mathbb{C})$ coherent invariant tensors

TL;DR

This paper analyzes the semiclassical limit of invariant tensors for SL(2,C) representations, revealing geometric configurations related to 3D polygons and providing asymptotic formulas relevant for spin foam models in quantum gravity.

Contribution

It introduces a saddle point approximation for SL(2,C) invariant tensors with n≥3 legs, connecting algebraic invariants to geometric 3D polygons and their Lorentz deformations.

Findings

Critical configurations exhibit power-law decay of invariants.

Configurations correspond to deformable 3D polygons under Lorentz transformations.

Provides a universal power-law decay for minimal SU(2) spins.

Abstract

We study the semiclassical limit of a class of invariant tensors for infinite-dimensional unitary representations of $\mathrm{SL}(2,\mathbb{C})$ of the principal series, corresponding to generalized Clebsch-Gordan coefficients with $n\geq3$ 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.