Quantum Theories Associated to Increasing Hilbert Space Filtrations and Generalized Jacobi 3–Diagonal Relation

We prove that the quantum decomposition of a classical random variable, or random field, is a very general phenomenon involving only an increasing filtration of Hilbert spaces and a family of Hermitean operators increasing by 1 the filtration. The creation, annihilation and preservation operators (CAP operators), defining the quantum decomposition of these Hermitean operators, satisfy commutation relations that generalize those of usual quantum mechanics. In fact there are two types of commutation relations (Type I and Type II). In Type I commutation relations the commutator is given by an operator–valued sesqui–linear form. The case when this operator–valued sesqui–linear form is scalar valued (multiples of the identity) characterizes the non–relativistic free Bose field and the associated commutation relations reduce to the Heisenberg ones. Type II commutation relations did not appear up to now because they are identically satisfied when the probability distribution of the random field is a product measure. In this sense they encode information on the self–interaction of the random field.