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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 studyingclassical theta functions. First we translate in knot theoretic languageclassical theta functions, the action of the finite Heisenberg group, and thediscrete Fourier transform. Then we explain how the non-abelian counterparts ofthese arise in the framework of the quantum group quantization of the modulispace of flat $SU(2)$-connections on a surface, in the guise of the non-abeliantheta functions, the action of a skein algebra, and the Reshetikhin-Turaevrepresentation of the mapping class group. We prove a Stone-von Neumann theoremon the moduli space of flat $SU(2)$-connections on the torus, and using it wededuce the existence and the formula for the Reshetikhin-Turaev representationon the torus from quantum mechanical considerations. We show how one can derivein a quantum mechanical setting the skein that allows handle slides, which isthe main ingredient in the construction of quantum $3$-manifold invariants.

Quantum mechanics and non-abelian theta functions for the gauge group $SU(2)$