Identical particles, exotic statistics and braid groups

Karl-Henning Rehren

Dorota Jacak, Lucjan Jacak, Konrad Wieczorek

The problem of the non-standard statistics for one-, two-and three--dimensional systems of N identical particles on various manifolds is reviewed in terms of the braid group theory.The braid groups together with their unitary representations are studied for the line, circle, plane, sphere, torus and the three-dimensional Euclidean space.Nonequivalent quantizations of several physical systems are presented.

Jon Magne Leinaas

Hanyu Xue

Braiding statistics of $p$- and $q$-dimensional topological excitations is conventionally defined in $p+q+2$ spatial dimensions. We find a novel statistical process $W_N(X,Y)=(Y^{-1}X^{-1})^N(YX)^N$ for two order-$N$ excitations in $p+q+1$ dimensions, detecting the Bockstein response $A\smile β(B)$. This new statistics and fermionic loop statistics exhaust all loop statistics in three dimensions whose fusion rules form an Abelian group $G$, classified by $H^5(B^2G,U(1))$. Surprisingly, conventional three-loop braiding goes beyond this classification, so it must have non-Abelian fusion rules. We suggest viewing three-loop braiding as particle-loop braiding together with exotic fusion rules between loops and point-like defects. We also try to clarify the relationship between statistics and symmetry anomaly.

R. Aldrovandi

Identical impenetrable particles in a 2-dimensional configuration space obey braid statistics, intermediate between bosons and fermions. This statistics, based on braid groups, is introduced as a generalization of the usual statistics founded on the symmetric groups. The main properties of an ideal gas of such particles are presented. They do interpolate the properties of bosons and fermions but include classical particles as a special case. Restriction to 2 dimensions precludes lambda points but originates a peculiar symmetry, responsible in particular for the identity of boson and fermion specific heats.

Jürg Martin Fröhlich, Thomas Kerler