Group-theoretic treatment of the axioms of quantum mechanics

This axiomatization is based on the observation that if G is the group of automorphisms of the states (induced, e.g., by suitable evolutions), then one can define a spherical function by mapping each element of G to the matrix of its transition probabilities. Starting from five physically conservative axioms, one utilizes the correspondence between spherical functions and representations to apply the structure theory for compact Lie groups and their orbits in representation spaces to arrive at the standard complex Hilbert space structure of quantum mechanics