Abstract
A fundamental approach to quantum mechanics is 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, as well as the usual exchange statistics for point particles in spatial dimensions \( d\,\,\geq\, 3\), induced by representations of the symmetric group. For \( d\,\,=\, 3\), this approach led to an early prediction of intermediate or “anyon” statistics induced by unitary representations of the braid group. I review these ideas, and discuss briefly some analogous possibilities for infinite-dimensional configuration spaces, includinga nyonic statistics for extended objects in three-dimensional space.
Mathematics Subject Classification (2010). Primary 81R10; Secondary 81Q70.
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Presented at the Felix Berezin Memorial Session, XXX Workshop on Geometric Methods in Physics, Białowieża, Poland, and dedicated also to my colleagues Bogdan Mielnik and Stanisław Woronowicz
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Goldin, G.A. (2013). Quantum Configuration Spaces of Extended Objects, Diffeomorphism Group Representations and Exotic Statistics. In: Kielanowski, P., Ali, S., Odzijewicz, A., Schlichenmaier, M., Voronov, T. (eds) Geometric Methods in Physics. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0448-6_19
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