The screen representation of vector coupling coefficients or Wigner 3j symbols: exact computation and illustration of the asymptotic behavior

TL;DR

This paper explores the exact computation and asymptotic behavior of Wigner 3j symbols, emphasizing the role of Regge symmetries, recursion relations, and their applications in quantum angular momentum theory.

Contribution

It introduces a detailed analysis of the screen representation of Wigner 3j symbols, highlighting the importance of Regge symmetries and recursion relations for computation and interpretation.

Findings

Identification of the screen projection based on Regge symmetries

Formulation of recursion relations as eigenvalue problems

Insights into the semiclassical behavior of Wigner 3j symbols

Abstract

The Wigner $3j$ symbols of the quantum angular momentum theory are related to the vector coupling or Clebsch-Gordan coefficients and to the Hahn and dual Hahn polynomials of the discrete orthogonal hyperspherical family, of use in discretization approximations. We point out the important role of the Regge symmetries for defining the screen where images of the coefficients are projected, and for discussing their asymptotic properties and semiclassical behavior. Recursion relationships are formulated as eigenvalue equations, and exploited both for computational purposes and for physical interpretations.