Axioms for quantum theory

The first three of these axioms describe quantum theory and classical mechanics as statistical theories from the very beginning. With these, it can be shown in which sense a more general than the conventional measure theoretic probability theory is used in quantum theory. One gets this generalization defining transition probabilities on pairs of events (not sets of pairs) as a fundamental, not derived, concept. A comparison with standard theories of stochastic processes gives a very general formulation of the non existence of quantum theories with hidden variables. The Cartesian product of probability spaces can be given a natural algebraic structure, the structure of an orthocomplemented, orthomodular, quasi-modular, not modular, not distributive lattice, which can be compared with the quantum logic (lattice of all closed subspaces of an infinite dimensional Hubert space). It is shown how our given system of axioms suggests generalized quantum theories, especially Schrödinger equations, for phase space amplitudes.