Probabilistic foundations of general quantum theories and characterizations of representation space

A program for re-structuring the mathematical foundations of general quantum theories along probabilistic lines suggested by Mielnik as opposed to the lattice theoretical approach due to Birkhoff and von Neumann is undertaken. In pursuing this program, some interesting characterizations of inner product spaces, generalized inner product spaces and uniformly convex spaces are given in terms of two-dimensional probability space structures imposed on the unit sphere. Next, an analysis of the variety of two-dimensional probability structures admissible on the unit sphere of a real Hilbert space is given and the difficulties in obtaining probability space structures of dimension greater than two in the normed linear space setting are discussed. Finally, it is shown, by imposing two-dimensional probability structures on abelian groups, that our probabilistic postulates imply a high degree of algebraic and geometrical structure on a given set of quantum states --Abstract, page iii