Isotropic random spin weighted functions on 𝑆² vs isotropic random fields on 𝑆³

Abstract

We show that an isotropic random field on S U ( 2 ) SU(2) is not necessarily isotropic as a random field on S 3 S^3 , although the two spaces can be identified. The ambiguity is due to the fact that the notion of isotropy on a group and on a sphere are different, the latter being much stronger. We show that any isotropic random field on S 3 S^3 is necessarily a superposition of uncorrelated random harmonic homogeneous polynomials, such that the one of degree d d is necessarily a superposition of uncorrelated random spin weighted functions of every possible spin weight in the range { − d 2 , … , d 2 } \bigl \{-\frac {d}{2},\dots ,\frac {d}{2}\bigr \} , each of which is isotropic in the sense of S U ( 2 ) SU(2) . Moreover, for a random field of fixed degree, each spin weight appears with the same magnitude, in a sense to be specified.

In addition we will give an overview of the theory of spin weighted functions and Wigner D D -matrices, with the purpose of gathering together many different points of view and adding ours. As a byproduct of this survey we will prove some new properties of the Wigner matrices and a formula relating the operators ð ð ¯ \eth \overline {\eth } and the horizontal Laplacian of the Hopf fibration S 3 → S 2 S^3\to S^2 , in the sense of [Bérard Bergery and Bourguignon, Illinois J. Math. 26 (1982), no. 2, 181–200.]