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On a family of quasi-invariant measures and related representations of a diffeomorphisms group

Keywords:

E.D. Romanov - Post-graduate Student, Department of Mathematical Analysis, Faculty of Mechanics and Mathematics, Lomonosov Moscow State University. E-mail: romanoved@yandex.ru


A family of quasi-invariant measures on the special functional space of curves with respect to the action of diffeomorphisms is constructed. An explicit expression for the Radon-Nikodym derivative of the transformed measure relative to the original one is presented. The stochastic Ito integral allows to express the result in an invariant form for a wider class of diffeomorphisms and to simplify calculations related with unitary representation. A general structure of the unitary representations based on such quasi-invariant measure described. The irreducibility proved in the case of the special measure choice. A geometric interpretation of the action considered together with a generalization to the multidimensional case makes such representations applicable to problems of quantum mechanics.
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