The Screen representation of spin networks. Images of 6j symbols and semiclassical features

TL;DR

This paper provides detailed visual and analytical insights into the properties of Racah coefficients (6j symbols) as functions of two variables, revealing symmetries, semiclassical features, and boundary structures through innovative imaging techniques.

Contribution

It introduces a novel visualization method for 6j symbols on a 'screen', systematically classifies their features, and explores the implications of symmetries and semiclassical behavior in quantum mechanics.

Findings

Identification of caustic and ridge curves delineating oscillatory and evanescent regions

Revelation of symmetries influencing the structure of 6j symbols

Enhanced understanding of the amplitudes and phases of discrete wavefunctions

Abstract

This article presents and discusses in detail the results of extensive exact calculations of the most basic ingredients of spin networks, the Racah coefficients (or Wigner 6j symbols), exhibiting their salient features when considered as a function of two variables - a natural choice due to their origin as elements of a square orthogonal matrix - and illustrated by use of a projection on a square "screen" introduced recently. On these screens, shown are images which provide a systematic classification of features previously introduced to represent the caustic and ridge curves (which delimit the boundaries between oscillatory and evanescent behaviour according to the asymptotic analysis of semiclassical approaches). Particular relevance is given to the surprising role of the intriguing symmetries discovered long ago by Regge and recently revisited; from their use, together with other newly discovered properties and in conjunction with the traditional combinatorial ones, a picture emerges of the amplitudes and phases of these discrete wavefunctions, of interest in wide areas as building blocks of basic and applied quantum mechanics.