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Graphical methods for the execution of the γ- or σ-algebra in spinor theories

Графические методы для вычисления γ- или σ-алгебры в спинорных теориях

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Il Nuovo Cimento A (1965-1970)

Summary

The angular momentum decomposition in theories involving fields or many particles with nonzero spin, in particular spinor fields or spin −1/2 particles, soon becomes extremely intricate, since it involves the evaluation of higher products of γ- or σ-matrices contracted with tensors of high rank constructed from various linear 4-momenta. It will be shown that such expressions can be simply represented by graphs which after angular integration go over into graphs corresponding to Wigner 3nj-symbols. For the evaluation of these Wigner 3nj-symbols in terms of products of 6j-symbols graphical reduction formulas are given which not only serve as an intuitive aid in the classification but allow to determine the exact result including the phase factors.

Riassunto

La decomposizione dell'impulso angolare nelle teorie inerenti ai campi o a più particelle con spin diverso da zero, particolarmente dei campi spinoriali o particelle di spin 1/2, diviene presto estremamente complicata, poichè comporta il calcolo di più prodotti di matrici γ o σ contratte con tensori di rango elevato, ricavati da vari quadrimomenti lineari. Si mostrerà come tali espressioni possano essere semplicemente rappresentate tramite grafici, che, dopo integrazione angolare, danno origine a grafici corrispondenti a simboli 3nj di Wigner. Per il calcolo di questi simboli 3nj in funzione dei prodotti di simboli 6j si danno formule di riduzione grafica, che non solo forniscono un aiuto intuitivo nella classificazione ma permettono anche di determinare il risultato esatto compresi i fattori di fase.

Резюме

Разложение по моментам в теориях, включающих поля и много частиц с ненулевым спином, в частности, спинорные поля и частицы со спином 1/2, вскоре станет чрезвычайно затруднительным, так как это разложение включает вычисление более высоких произведений γ- или σ-матриц, полученных из тензоров высшего ранга, сконструированными из различных 4-имнульсов. Будет показано, что такие выражения можно просто представить с помощью графиков, которые после интегрирования по углам переходят в графики, соответствующие 3nj-символам Вигнера. Для вычисления этих 3nj-символов Вигнера через выражения произведения 6j-символов приводятся графические формулы приведения, которые служат не только для интуитивной помощи при классификации, но и позволяют определить точный результат, включая факторы.

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References

  1. P. D. Rumas andA. P. Yutzis:Litv. Fiz. Sbornik:5, 185 (1965) (in Russian). The classification of the 24j-symbol was done by an electronic computer.

    Google Scholar 

  2. A. Yutzis, I. B. Levinson andV. V. Vanagas:Mathematical Apparatus of the Theory of Angular Momentum (Jerusalem, 1962) (translated from Russian).

  3. V. Fano andG. Racah:Irreducible Tensorial Sets (New York, 1959), Appendices;T. Regge andG. Ponzano:Semiclassical Limit of Racah Coefficients, preprint (Princeton, 1967).

  4. An analogous method has been used byYutzis and coworkers for programming on electronic computers.P. Rumas, A. A. Bandzaitis andA. P. Yutzis:Litv. Fiz. Sbornik,5, 185 (1965) (in Russian).

    Google Scholar 

  5. H. Joos:Fortschr. d. Physik,10, 65 (1962).

    Article  ADS  Google Scholar 

  6. H. A. Jähn andJ. Hope:Phys. Rev.:93, 318 (1954);R. J. Ord-Smith:Phys. Rev.,94, 1227 (1954).

    Article  ADS  Google Scholar 

  7. F. R. Innes andC. W. Ufford:Phys. Rev.:111, 194 (1958).

    Article  ADS  Google Scholar 

  8. H. P. Dürr andF. Wagner:Nuovo Cimento, to be published.

  9. A. R. Edmonds:Angular Momentum in Quantum Mechanics (Princeton, 1957).

  10. H. E. Moses:Ann. Phys.,37, 224 (1966).

    Article  ADS  MathSciNet  Google Scholar 

  11. After this paper was finished at the occasion of a lecture given by one of the authors (H.P.D.) in Brussels we were informed about some additional papers related to the methods developed here; 1)F. Dönau andG. Flach:Nucl. Phys.,69, 68 (1965); 2)J. N. Massot, E. El-Baz andJ. Lafoucrière:a) Nucl. Phys.,82, 189 (1966);b) Nucl. Phys. 83, 449 (1966);c) Nucl. Phys.,86, 625 (1966);d) Journ. de Phys., Suppl. Fasc. 3–4.Colloque sur les noyaux légers (Lyon, 1966), C.I. 146. In these papers the graphical notation of ref. (2) is used and enlarged to include the treatment of spherical harmonics. Their notation differs from ours in the interpretation of the «directed line» in the graphs, which in our case is chosen such as to obtain an immediate relationship to CG-coefficients and Pauli matrices.

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Authors and Affiliations

  1. Max-Planck-Institut für Physik und Astrophysik, München

    H. P. Dürr & F. Wagner

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  1. H. P. Dürr
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  2. F. Wagner
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Dürr, H.P., Wagner, F. Graphical methods for the execution of the γ- or σ-algebra in spinor theories. Nuovo Cimento A (1965-1970) 53, 255–285 (1968). https://doi.org/10.1007/BF02824938

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