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Generalized 6j-coefficients and their symmetries

Обобшенные 6J-козффициенты и их симметрии

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Il Nuovo Cimento A (1965-1970)

Summary

A unifying treatment ofSU 2 andSU 1.1 6j-coefficients with discrete unitary representation is realized through a generalizing functionΦ 6j . From the invariance properties ofΦ 6j , as a by-product, the symmetry group of theSU 1.1 6j-coefficient with discrete unitary representations is obtained. The asymptotic connection ofΦ 6j with the generalized 3j-coefficients is pointed out.

Riassunto

Introducendo una funzione generalizzatriceΦ 6j si effettua una trattazione unificata dei coefficienti 6j diSU 2 e diSU 1.1 con parametri corrispondenti a rappresentazioni unitarie discrete. Dalle proprietà di invarianza diΦ 6j si ottiene come conseguenza il gruppo di simmetria dei coefficienti 6j diSU 1.1 relativi a rappresentazioni unitarie discrete. Si studia la connessione asintotica diΦ 6j con i coefficienti 3j generalizzati.

Реэюме

С помошью обобшенной функции Φ6j реалиэуется унифицированное рассмотрениеSU 2 иSU 1,1 6j-козффициентов в случае дискретного унитарного представления. Иэ инвариантных свойств Φ6j получается группа симметрии дляSU 1,1 6j-козффициентов в случае дискретного унитарного представления. Отмечается свяэь Φ6j с обобшенными 3j-козффициентами.

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References

  1. A. D’Adda, R. D’Auria andG. Ponzano:Symmetries of extended 3j-coefficients, preprint, April 1972, Torino, to be published in theJourn. Math. Phys. (hereafter referred to as I). For an account of the main results contained in I and in the present paper see:A. D’Adda, R. D’Auria andG. Ponzano:Generalized 3j and 6j coefficients and their symmetries, Lett. Nuovo Cimento,5, 973 (1972).

    Article  Google Scholar 

  2. A. A. Bandzaitis, A. V. Karosene, A. Yu. Savakinas andA. P. Yucys:Phys. Doklady,9, 139 (1964).

    ADS  Google Scholar 

  3. L. C. Biedenharn:Journ. Math. Phys.,31, 287 (1953);G. Ponzano andT. Regge:Semiclassical limit of Racah coefficients, inSpectroscopic and Group Theoretical Methods in Physics, edited byF. Bloch et al. (Amsterdam, 1968). For a generalization toU(n), see:J. D. Louck:Am. Journ. Phys.,38, 3 (1970).

    MATH  MathSciNet  Google Scholar 

  4. B. M. Minton:Journ. Math. Phys.,11, 3061 (1970).

    Article  MathSciNet  ADS  Google Scholar 

  5. E. Yakimiw:Journ. Math. Phys.,12, 1134 (1971);V. Joshi:Journ. Math. Phys.,12, 1134 (1971).

    Article  MathSciNet  ADS  Google Scholar 

  6. See, for instance:A. R. Edmonds:Angular Momentum in Quantum Mechanics (Princeton, N. J., 1957).

  7. L. A. Shelepin:Sov. Phys. JETP,21, 238 (1965).

    MathSciNet  ADS  Google Scholar 

  8. T. Regge:Nuovo Cimento,11, 116 (1959).

    Article  Google Scholar 

  9. F. J. W. Whipple:Proc. London Math. Soc.,40, 336 (1936).

    Article  MathSciNet  Google Scholar 

  10. L. J. Slater:Generalized Hypergeometric Functions (Cambridge, Mass., 1966).

  11. The functionsG +1(i),G −1(i) coincide respectively withG p (i), G n (i) of ref. (10).

    Article  MathSciNet  Google Scholar 

  12. A. Erdelyi:Math. Rev.,14, No. 3 (1953);M. E. Rose:Multipole Fields (New York, N. Y., 1955).

  13. We do not have a simple recipe to do this, however we remark that the same technique used byG. Racah (Phys. Rev.,62, 438 (1942), Appendix B) to perform closely the summations when expressing the 6j as a sum of products of four 3j’s should apply also to theSU 1,1 case when only d.u.r. are involved.

    Article  ADS  Google Scholar 

  14. A. P. Jucys andA. A. Bandzaitis:The Theory of Angular Momentum in Quantum Mechanics (in Russian), Mintis, Vilnius, Lithuanian SSR (1965).

  15. In ref. (4), {ie94-1} is kept integer, so that the factor {ie94-2} is omitted.

  16. G −1 can be expressed in terms of a7 F 6 in 16 different ways which correspond to the different choices of the setσ in eq. (2.6). To perform the limit {ie97-1} → ∞, the most convenient choices are the ones with |σ|=3, because in this case7 F 6 {ie97-2}3 F 2 and the result can be directly compared with the definition ofF n . If |σ|=1 one getsF n expressed in terms of a series6 F 5 of argument −1 and if |σ|=5 the limit cannot be performed as it leads outside the convergency domain of the series. These statements agree with Sect.5 of ref. (10).

  17. This property was pointed out in ref. (4).

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Authors and Affiliations

  1. Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Italy

    A. D’Adda, G. Ponzano & R. D’Auria

  2. Istituto di Fisica dell’Università, Torino

    G. Ponzano & R. D’Auria

Authors
  1. A. D’Adda
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  2. G. Ponzano
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  3. R. D’Auria
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D’Adda, A., Ponzano, G. & D’Auria, R. Generalized 6j-coefficients and their symmetries. Nuov Cim A 23, 69–102 (1974). https://doi.org/10.1007/BF02748294

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  • DOI: https://doi.org/10.1007/BF02748294

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