Stochastic Analysis and Quantum Field Theory

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We have investigated several current topics in stochastic calculus and quantum dynamics. To fix the idea, we explain one of the most important topics of what we have studied.White noise analysis may be viewed as the harmonic analysis arising from the infinite dimensional rotation group 0*.Since the white noise measure, which is introduced on the space of the generalized functions, is invariant under the action of the rotation group, it is natural to think of an infinite dimensional analogue of the harmonic analysis on a finite dimensional sphery on which the finite dimensional rotation group acts.There are several steps for our harmonic analysis depending on the choice of subgroup of 0*.1. Inductive limit of finite dimensional rotation groups, denoted by G*. The Fock space <symmetry>H_n, the Laplace-Beltrami operator DELTA*, class-one irreducible unitary representations of G* on H_n, and so forth, are well investigated.2. There is a subgroup Y, called Levy group, which contains infinit … More e dimensional rotations, far from the finite dimensional ones, defined by permutations of coordinates of the basic function space. To carry on a similar calculus as in 1, associated with the Levy group, we need to go one step ahead. Namely, a space of generalized white noise functionals should naturally be introduced, then we are led to discuss the Levy's Laplacian operator that acts effectively on the space of the generalized white noise functionals.3. We then come to a very interesting subgroup W consisting of all the transformations that come from diffeomorphism of the parameter space. The W is quite different, in character, from G* and the Levy group. To introduce some analytic structure, We consider continuous one-parameter subgroups (called whiskers) of W. Then we take a family of random variables {X(C)}, where the parameter C runs through a certain subclass of analytic, closed, simple manifolds, and where C is deformed by actions of the subgroup of W. A general theory is not yet established, however we can find many interesting questions in analysis as well as several applications. Less