Symbol Correspondences for Spin Systems by Pedro de M. Rios & Eldar Straume

Author:Pedro de M. Rios & Eldar Straume
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

We now turn to the formal proof of Proposition 6.2.2 of Sect. 6.2.1.

Proof of Proposition 6.2.2.

To see why the choice c l n  ≡ c l in (6.12) yields an operator kernel K given by the formula (6.16), we apply formula (6.13) with P = e(l, 0) and g = e, and use the fact that Y l, 0 takes the value at the north pole n 0 = (0, 0, 1). The second part of the proposition is immediate (cf. Corollary 6.2.15). □

Remark 6.2.26.

The numbers c l  ≡ c l n of Proposition 6.2.2 are precisely the characteristic numbers of W in the sense of Definition 6.2.22. The notation c l n instead of c l is to emphasize their dependence on n = 2j, whenever this dependence is an important issue. We also denote by the n-string , as in (6.12), and by the n-string . The notation for W K will be heavily used in the sequel. In particular, the dual of is .

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