Exact and Asymptotic Computations of Elementary Spin Networks: Classification of the Quantum–Classical Boundaries

Abstract

Increasing interest is being dedicated in the last few years to the issues of exact computations and asymptotics of spin networks. The large-entries regimes (semiclassical limits) occur in many areas of physics and chemistry, and in particular in discretization algorithms of applied quantum mechanics. Here we extend recent work on the basic building block of spin networks, namely the Wigner 6j symbol or Racah coefficient, enlightening the insight gained by exploiting its self-dual properties and studying it as a function of two (discrete) variables. This arises from its original definition as an (orthogonal) angular momentum recoupling matrix. Progress also derives from recognizing its role in the foundation of the modern theory of classical orthogonal polynomials, as extended to include discrete variables. Features of the imaging of various regimes of these orthonormal matrices are made explicit by computational advances -based on traditional and new recurrence relations-which allow an interpretation of the observed behaviors in terms of an underlying Hamiltonian formulation as well. This paper provides a contribution to the understanding of the transition between two extreme modes of the 6j, corresponding to the nearly classical and the fully quantum regimes, by studying the boundary lines (caustics) in the plane of the two matrix labels. This analysis marks the evolution of the turning points of relevance for the semiclassical regimes and puts on stage an unexpected key role of the Regge symmetries of the 6j. V 2 = 1 288 0 J 2 3 J 2 J 2 23 1 J 2 3 0 J 2 12 J 2 2 1 J 2 J 2 12 0 J 2 1 1 J 2 23 J 2 2

The Screen Representation of Spin Networks: Images of 6j Symbols and Semiclassical Features

“…This section reports and discusses a series of graphs where the plots of caustics and ridges published in Ref. [15] are superimposed on x − y color plots of true U . The following features are common to all of the plots except…”

“…3. fig.1d in [15], where only the caustic and ridge curves were given. The values spanned by x and y are both 2a + 1.…”

“…Note for comparison to the previous case that now r = −10 :the lower difference between sums of columns shows qualitative shape changes for mismatch between columns. in [15], where only the caustic and ridge curves were given. Now r = v = 0, and the two Regge conjugates are again identical, but the coalescence is now of both the North and South gates with the West gate, on the full line from the lower left to the upper left corners of the screen, As noted in [15], since a, b and x are smaller than c, d and y, we can regard this plot as that of a 3j symbol, ( : : : ) where the entries in the upper row are the angular momenta 100, 100, x and the corresponding projections in the lower row are y -1000, 1000y, 0.…”

“…After a presentation of the general case in Section II, we illustrate symmetric and limiting cases in Section III (an important case being that of the Clebsch-Gordan coefficients, also known as Wigners 3-j symbols, see [17] ). In the order of presentation, we are closely following the previous classification of the classical-quantum boundaries [15]. In section IV, we provide additional and concluding remarks.…”

“…In this account, we present for the first time illustrations from the two-dimensional perspective, which is naturally based on the view [4], [9], [10] of the 6-j symbols as matrix elements enjoying a self dual property. The basic ideas of this approach are referred to in [9], [10] as the 4-j model : accordingly here plots as a function of two discrete variables are given in a square screen (see [15]), in a sense generalizing the traditional presentations in square numerical tables [16]. After a presentation of the general case in Section II, we illustrate symmetric and limiting cases in Section III (an important case being that of the Clebsch-Gordan coefficients, also known as Wigners 3-j symbols, see [17] ).…”