| Abstract: | For particles in three or more dimensions the forms of quantum statistics of indistinguishable particles are either Bose-Einstein or Fermi-Dirac corresponding to the two abelian representations of the first homology group of the configuration space. Restricting particles to the plane the fundamental group of the configuration space is the braid group and a new form of particle statistics corresponding to its abelian representations appears, anyon statistics. Restricting the dimension of the space further to a quasi-one-dimensional quantum graph opens new forms of statistics determined by the connectivity of the graph.
We develop a full characterization of abelian quantum statistics on graphs which leads to an alternative proof of the structure theorem for the first homology group of the n-particle configuration space. For two connected graphs the statistics are independent of the particle number. On three connected non-planar graphs particles are either bosons or fermions while in three connected planar graphs they are anyons. Graphs with more general connectivity exhibit interesting mixtures of these behaviors which we illustrate. For example, a graph can be constructed where particles behave as bosons, fermions and anyons depending on the region of the graph that they inhabit. An advantage of this direct approach to analysis of the first homology group is that it makes the physical origin of these new forms of statistics clear. This is work with Jon Keating, Jonathan Robbins and Adam Sawicki at Bristol, arXiv:0809.3476. |
