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

Pietro Dona, Marco Fanizza, Pierre Martin-Dussaud, Simone Speziale

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 $γ$-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 $γ$-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.

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

Abstract

We study the semiclassical limit of a class of invariant tensors for infinite-dimensional unitary representations of of the principal series, corresponding to generalized Clebsch-Gordan coefficients with 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 -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 -dependent otherwise. As a side result, we provide a compact formula for -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.

Paper Structure

This paper contains 22 sections, 142 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $\mathrm{SL}(2,{\mathbb C})$ unitary irreps of the principal series
  4. SU(2) Coherent states
  5. $\gamma$-simplicity and Lorentz transformations
  6. Coherent invariant tensors
  7. Saddle point analysis
  8. Spinorial gradient and the boosted orientation equations
  9. Gradient in the group variable and the closure condition
  10. Action at the critical points
  11. Solutions and asymptotic formulas
  12. Lowest case
  13. Half-lowest case
  14. General case
  15. Boosted coherent states
  16. ...and 7 more sections

Figures (3)

  • Figure 1: Numerical evaluation of the absolute value of the coherent invariant with $n=3$, for boundary data corresponding to two an equilateral triangle with $l_a=k_a=1$, boosted to an isosceles triangle with $l_a=(1,2,2)$, with $\gamma=1.2$. The plot is consistent with the expected $\lambda^{-\frac{7}{2}}$ decay -- see asymptotic formula at the end of this Section, added with a ad hoc coefficient to help the eye, and with the presence of distinct critical points, because of the oscillations. Without explicitly computing the Hessian determinant to compare quantitatively the slope coefficents, the numerics alone cannot tell us the precise number of distinct configurations. The expectation is that there are two distinct critical points, just like for non-coplanar configurations with $n\geq 4$.
  • Figure 2: Numerical test of the asymptotic behaviour. The data points are the absolute value of the exact numerical evaluations of the coherent invariant for the configuration of Section \ref{['SecEx1']}. The computation was performed with $\gamma=1.2$, $k=1$ and $j= 2$, using the numerical library sl2cfoam-nextGozzini-to-appear?. The straight line is $\propto \lambda^{-4}$, and it is reported to guide the eye of the reader. The plot confirms the expected power law and absence of oscillations, as explained in the main text due to a numerical coincidence for the symmetry of the boundary data. Picture courtesy of Francesco Gozzini.
  • Figure 3: Numerical test of the asymptotic behaviour. The data points are the absolute value of the exact numerical evaluations of the coherent invariant for the configuration of Section \ref{['SecEx2']}. The computation was performed with $\gamma=1.2$, $k=1$ and $j= 2$, using the numerical library sl2cfoamDona:2018nev. The straight line is $\propto \lambda^{-15/4}$, and it is added to guide the eye of the reader. The plot confirms the expected power law and presence of oscillations. Picture courtesy of Giorgio Sarno.