  We perform the momentum-space quantization of a spin-less particle moving on
the $SU(2)$ group manifold, that is, the three-dimensional sphere $S^{3}$, by
using a non-canonical method entirely based on symmetry grounds. To achieve
this task, non-standard (contact) symmetries are required as already shown in a
previous article where the configuration-space quantization was given. The
Hilbert space in the momentum space representation turns out to be made of a
subset of (oscillatory) solutions of the Helmholtz equation in four dimensions.
The most relevant result is the fact that both the scalar product and the
generalized Fourier transform between configuration and momentum spaces deviate
notably from the naively expected expressions, the former exhibiting now a
non-trivial kernel, under a double integral, traced back to the non-trivial
topology of the phase space, even though the momentum space as such is flat. In
addition, momentum space itself appears directly as the carrier space of an
irreducible representation of the symmetry group, and the Fourier transform as
the unitary equivalence between two unitary irreducible representations.
