KEK素粒子原子核研究所・理論セミナー（Prof. Luigi Accardi）
For more than one century quantum mechanics has been considered a singular theory, uniquely related to quantum physics. In the past few years it has become clear that all the basic structures of quantum theory naturally emerge from a combination of classical probability with the theory of orthogonal polynomials. In fact any classical random variable has a canonical quantum decomposition as a sum of 3 linear operators called respectively: generalized creation, annihilation and preservation (CAP) operators. These operators satisfy generalized commutation relations (GCR) that are natural extensions of Heisenberg commutation relations and characterize the given random variable in the sense of moments. The Heisenberg commutation relations characterize the Gaussian class which is included in the larger class of measures ”linearly equivalent” to product measures. This larger class is characterized by the property that CAP operators associated to different degrees of freedom commute. Thus usual quantum mechanics belongs to this class. For this class the theory of multi—dimensional orthogonal polynomials is essentially reduced to the tensor product of 1-dimensional cases. For truly interacting random variables (or fields) new commutations relations arise from the commutativity of the multiplication operators associated to different components of the random variable. In this sense non-commutativity arises from commutativity. The construction has functorial properties that generalize the usual Fock functor.