The Kirillov character formula gives a striking correspondence between the unitary irreducible representations of a compact semisimple Lie group and its set of integral orbits on the dual of its Lie algebra. In this thesis, the same correspondence is derived without the use of character theory. It is shown to be related to the convexity properties of the support of the Weyl functional calculus of the infinitesimal generators of the representation. This result uses Edward Nelson's theory of "operants" in a fundamental way. This had been developed to put Feynman's operator calculus on a...