Anyonic Statistics, Quantum Configuration Spaces, and Diffeomorphism Group Representations

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We begin with a fundamental approach to quantum mechanics based on the unitary representations of the group of diffeomorphisms of physical space (and correspondingly, self-adjoint representations of a local current algebra). From these, various classes of quantum configuration spaces arise naturally. One obtains in addition the usual exchange statistics for spatial dimension d >2, induced by representations of the symmetric group, while for d = 2, the approach led to an early prediction of intermediate or “anyonic” statistics induced by unitary representations of the braid group. After reviewing these ideas, which are based on joint work with R. Menikoff and D. H. Sharp at Los Alamos National Laboratory, we shall discuss briefly some analogous possibilities for infinite-dimensional configuration spaces, including anyonic statistics for extended objects in 3-dimensional space.