Consider a system of n undistinguishable particles moving in d-dimensional Euclidean space Ed. The configuration space is M=((Ed)n∖Δ)/Sn where Δ is the diagonal (subspace where at least 2 particles have coincidental positions) and Sn is the group permuting the particles
Quantization of this system yields superselection sectors corresponding to unitary irreducible representations of π1(M): Sn for d>2, Bn for d=2. The trivial representation yields bosonic statistics, the sign representations yield fermionic statistics. For d>2 there are no other 1-dimensional repsentations. For d=2 there are other 1-dimensional representations in which switching two particles generates an arbitrary phase. These yield anyonic statistics.
What about higher dimensional irreducible representations? These correspond to "parastatistics". It is said that for d>2 we can safely ignore them because in some sense they are equivalent to ordinary bosons/fermisons. However for d=2 this is not the case. Why?
Why is parastatistics redundant for d>2 but not for d=2?
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