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A unified treatment of Wigner [script D] functions, spin-weighted spherical harmonics, and monopole harmonics

J. Math. Phys. 27, 781 (1986); doi:10.1063/1.527183

Issue Date: March 1986

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Tevian Dray
School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540
A unified, self-contained treatment of Wigner [script D] functions, spin-weighted spherical harmonics, and monopole harmonics is given, both in coordinate-free language and for a particular choice of coordinates. Journal of Mathematical Physics is copyrighted by The American Institute of Physics.
History: Received 18 April 1985; accepted 30 September 1985
Permalink: http://link.aip.org/link/?JMAPAQ/27/781/1
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KEYWORDS and PACS

Keywords
PACS
  • 03.65.Fd
    Classical and quantum physics: mechanics and fields Quantum theory; quantum mechanics Algebraic methods
  • 02.20.Rt
    Mathematical methods in physics Group theory Discrete subgroups of Lie groups
  • 02.30.Px
    Mathematical methods in physics Function theory, analysis Abstract harmonic analysis
  • YEAR: 1986

PUBLICATION DATA

ISSN:
0022-2488 (print)   1089-7658 (online)
Publisher:
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REFERENCES (35)
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  1. E. P. Wigner, Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren (Vieweg, Braunschweig, 1931). A revised version of this book was later published in English: E. P. Wigner, Group Theory and its Application to the Quantum Mechanics of Atomic Spectra (Academic, New York, 1959).
  2. A. R. Edmonds, Angular Momentum in Quantum Mechanics, (Princeton U.P., Princeton, NJ, 1957).
  3. M. E. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957).
  4. L. C. Biedenharn and J. D. Louck, Angular Momentum in Quantum Physics (Addison-Wesley, Reading, MA, 1981). (Seealso Ref. 9.).
  5. E. T. Newman and R. Penrose, J. Math. Phys. 7, 863 (1966).
  6. J. N. Goldberg, A. J. MacFarlane, E. T. Newman, F. Rohrlich, and E. C. G. Sudanhan, J. Math. Phys. 8, 2155 (1967).
  7. R. Penrose and W. Rindler, Spinors and Space-Time (Cambridge U.P., Cambridge, 1984).
  8. T. T. Wu and C. N. Yang, Nucl. Phys. B 107, 365 (1976).
  9. L. C. Biedenharn and J. D. Louck, The Racah-Wigner Algebra in Quantum Theory (Addison-Wesley, Reading, MA, 1981). (See also Ref. 4.).
  10. W. Greub and H.-R. Petty, J. Math. Phys. 16, 1347 (1975).
  11. T. Dray, J. Math. Phys. 26, 1030 (1985).
  12. Peter Batenburg, Doctoraalscriptie (in English), University of Utrecht, 1984 (unpublished).
  13. The interpretation of spin-weighted function (95) as sections of complex line bundles was given in Ref. 22. The standard definition of spin-weighted spherical harmonics (for integrer spin) is (183), which is given in terms of the differential operator o[logical or]. The interpretation of o[logical or] as an operator on sections of line bundles was given in Ref. 23. However, Ref. 22 does not discuss spin-weighted spherical harmonics at all, and although Ref. 23 does give a precise definition of them it does not discuss them in any detail. The author wishes to thank Ted Newman for providing these two references.
  14. This is to be contrasted with the “standard spin gauge” for spin-weighted spherical harmonics [see Ref. 11 and (184a) below], which amounts to giving only a local trivialization of the complex line bundles (which of course does not cover S2 but only a dense subspace of S2).
  15. The standard procedure (see, e.g., Ref. 9) for half-integer spin is to exponentiate the angular momentum operators.
  16. A previous effort along these lines, namely Ref. 6, unfortunately uses an internally inconsistent choice of conventions.
  17. R. Kuwabara, J. Math. Tokushima Univ. 16, 1 (1982).
  18. V. Guillemin and A. Uribe, “Clustering theorems with twisted spectra,” Princeton Univ. preprint, 1985.
  19. See, e.g. Ref. 10. Equation (40) can be thought of as the definition of this preferred connection, which will be given in coordinates in Sec. VI.
  20. As Riemannian manifolds the spaces En(n[not-equal]0) can be thought of as the lens spaces S3/Z|n|. In particular, E±n are isomorphic and the Hopf bundle thought of in this way is isomorphic to S3.
  21. F. Peter and H. Weyl, Math. Ann. 97, 737 (1927);
    22. J. F. Adams, Lectures on Lie Groups (Benjamin, New York, 1969).
  22. W. D. Curtis and D. E. Lerner, J. Math. Phys. 19, 874 (1978).
  23. M. Eastwood and P. Tod, Math. Proc. Cambridge Philos. Soc. 92, 317 (1982).
  24. We could just as well have used V-bar0 in constructing the bundle (90). Only one of these bundles is strong bundle isomorphic to the Hopf bundle (7) but this does not affect the argument leading up to (92).
  25. The minus sign comes about because one usually writes the momentum as p−eA, i.e., as the operator −i([del]−ieA), so that the connection is −ieA and the curvature is −iedA[equivalent]−ieF.
  26. Note that the coordinates (alpha,beta,gamma) and (theta,phi,psi) are not well defined at the poles beta = 0,pi and theta = 0,pi respectively.
  27. This follows immediately from the definition of alphaa(tau) as the usual rotation matrices.
  28. Note that gamma-hatA is a section, whereas gammaA is a function.
  29. These agree with (24) of Ref. 11 with gammaA[equivalent]−gamma and n[equivalent]2s. We have omitted the hats for simplicity.
  30. This agrees with (4.1.12) and (4.1.15) of Ref. 2 if we note that Edmonds defines

    [dformula [sub E][script D][sub qm][sup l](alpha,beta,gamma) := [script D][sub qm][sup l](beta[sub 3][sup -1](alpha)beta[sub 2][sup -1](beta)beta[sub 3][sup -1](gamma)) [equivalent] [script D][sub qm][sup l](B(gamma,beta,alpha)[sup -1]) [equivalent] [script D][sub qm][sup l](-alpha,-beta,-gamma) [equivalent] [overline [script D][sub qm][sup l](alpha,-beta,gamma)].]

    However, if we interpret (3.4) of Ref. 6 (see also our Ref. 16) as defining

    [dformula [sub G][script D][sub qm][sup l](alpha,beta,gamma) := [script D][sub qm][sup l](B(alpha,beta,gamma)[sup -1]) [equivalent] [script D][sub qm][sup l](-gamma,-beta,-alpha) [equivalent] [overline [script D][sub qm][sup l](alpha,beta,gamma)] [equivalent][sub E][script D][sub mq](alpha,-beta,gamma),]

    then we are forced to conclude that (3.9) of Ref. 6 is missing a factor (−1)m+m[prime]. Finally, note that although no explicit expression analogous to (165) is given in Ref. 4 the functions Dlqm(alpha,beta,gamma) defined there are identical to our functions [script D]<sub>qm</sub><sup>l</sup>(alpha,beta,gamma).

  31. This agrees with (19) of Ref. 11, where gammaA[equivalent]−gamma and n[equivalent]+2s. We have omitted the hats for simplicity.
  32. Note that the section of the Hopf bundle (139) induced by gamma0 is defined everywhere on S2 except at the poles theta = 0,pi, whereas the sections induced by gammaa, gammab are each defined everywhere on S2 except at one pole (namely theta = 0 and theta = pi, respectively).
  33. This agrees with the, qYim of Ref. 11 in the standard spin gauge but differs from Ref. 6 by a factor (−1)q.
  34. The factor (−1)s in (28) of Ref. 11 is thus incorrect.
  35. This agrees with both Ref. 8 and Ref. 11.

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