  This paper outlines an approach to the non-abelian theta functions of the
$SU(2)$-Chern-Simons theory with the methods used by A. Weil for studying
classical theta functions. First we translate in knot theoretic language
classical theta functions, the action of the finite Heisenberg group, and the
discrete Fourier transform. Then we explain how the non-abelian counterparts of
these arise in the framework of the quantum group quantization of the moduli
space of flat $SU(2)$-connections on a surface, in the guise of the non-abelian
theta functions, the action of a skein algebra, and the Reshetikhin-Turaev
representation of the mapping class group. We prove a Stone-von Neumann theorem
on the moduli space of flat $SU(2)$-connections on the torus, and using it we
deduce the existence and the formula for the Reshetikhin-Turaev representation
on the torus from quantum mechanical considerations. We show how one can derive
in a quantum mechanical setting the skein that allows handle slides, which is
the main ingredient in the construction of quantum $3$-manifold invariants.
