A unified treatment of Wigner D functions, spin‐weighted spherical harmonics, and monopole harmonics
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12.Peter Batenburg, Doctoraalscriptie (in English), University of Utrecht, 1984 (unpublished).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∨. The interpretation of o∨ 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 but only a dense subspace of ).
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.
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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 can be thought of as the lens spaces In particular, are isomorphic and the Hopf bundle thought of in this way is isomorphic to
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21.J. F. Adams, Lectures on Lie Groups (Benjamin, New York, 1969).
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23.M. Eastwood and P. Tod, Math. Proc. Cambridge Philos. Soc. 92, 317 (1982).
24.We could just as well have used 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 i.e., as the operator so that the connection is and the curvature is
26.Note that the coordinates (α,β,γ) and (θ,φ,ψ) are not well defined at the poles and respectively.
27.This follows immediately from the definition of as the usual rotation matrices.
28.Note that is a section, whereas is a function.
29.These agree with (24) of Ref. 11 with and 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 However, if we interpret (3.4) of Ref. 6 (see also our Ref. 16) as defining then we are forced to conclude that (3.9) of Ref. 6 is missing a factor Finally, note that although no explicit expression analogous to (165) is given in Ref. 4 the functions defined there are identical to our functions
31.This agrees with (19) of Ref. 11, where and We have omitted the hats for simplicity.
32.Note that the section of the Hopf bundle (139) induced by is defined everywhere on except at the poles whereas the sections induced by are each defined everywhere on except at one pole (namely and respectively).
33.This agrees with the, of Ref. 11 in the standard spin gauge but differs from Ref. 6 by a factor
34.The factor in (28) of Ref. 11 is thus incorrect.
35.This agrees with both Ref. 8 and Ref. 11.
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