Some properties of Wigner 3 j coefficients: non-trivial zeros and connections to hypergeometric functions
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Regular Article - Theoretical Physics
Some properties of Wigner 3 j coefficients: non-trivial zeros and connections to hypergeometric functions A tribute to Jacques Raynal Jean-Christophe Pain1,2,a 1 2
CEA, DAM, DIF, 91297 Arpajon, France Université Paris-Saclay, CEA, Laboratoire Matière sous Conditions Extrêmes, 91680 Bruyères-le-Châtel, France
Received: 1 October 2020 / Accepted: 6 November 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020 Communicated by Nicolas Alamanos
Abstract The contribution of Jacques Raynal to angularmomentum theory is highly valuable. In the present article, I intend to recall the main aspects of his work related to Wigner 3 j symbols. It is well known that the latter can be expressed with a hypergeometric series. The polynomial zeros of the 3 j coefficients were initially characterized by the number of terms of the series minus one, which is the degree of the coefficient. A detailed study of the zeros of the 3 j coefficient with respect to the degree n for J = a + b + c ≤ 240 (a, b and c being the angular momenta in the first line of the 3 j symbol) by Raynal revealed that most zeros of high degree had small magnetic quantum numbers. This led him to define the order m to improve the classification of the zeros of the 3 j coefficient. Raynal did a search for the polynomial zeros of degree 1 to 7 and found that the number of zeros of degree 1 and 2 are infinite, though the number of zeros of degree larger than 3 decreases very quickly as the degree increases. Based on Whipple’s symmetries of hypergeometric 3 F2 functions with unit argument, Raynal generalized the Wigner 3 j symbols to any arguments and pointed out that there are twelve sets of ten formulas (twelve sets of 120 generalized 3 j symbols) which are equivalent in the usual case. In this paper, we also discuss other aspects of the zeros of 3 j coefficients, such as the role of Diophantine equations and powerful numbers, or the alternative approach involving Labarthe patterns.
1 Introduction The theory of angular momentum has always been an important research topic for me, for practical applications of course, but also for the elegance of the formalism, its surprising peculiarities and unsolved mysteries. Jacques Raynal’s works are a e-mail:
[email protected] (corresponding author)
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among those which most caught my attention. I never met him, and a few years ago, I questioned colleagues from CEA Saclay to know more about him, but in vain. A few months ago, I learned by chance from Eric Bauge that Raynal had been a regular collaborator of the nuclear physics division of the CEA center where I work, in Bruyères-le-Châtel. Unfortunately Jacques died shortly after, before I could meet him. In quantum mechanics, Clebsch-Gordan coefficients describe how individual angular momentum states may be coupled to yield the total angular momentum state of a system [1]. In the literature, Clebsch–Gordan coefficients are sometimes also known as Wigner coefficien
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