  We show that an isotropic random field on $SU(2)$ is not necessarily
isotropic as a random field on $S^3$, 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 $S^3$ is necessarily a superposition of
uncorrelated random harmonic homogeneous polynomials, such that the one of
degree $d$ is necessarily a superposition of uncorrelated random spin weighted
functions of every possible spin weight in the range
$\{-\frac{d}{2},\dots,\frac{d}{2}\}$, each of which is isotropic in the sense
of $SU(2)$. 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 $D$-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
$\eth\overline{\eth}$ and the horizontal Laplacian of the Hopf fibration
$S^3\to S^2$.
