Identical particles, exotic statistics and braid groups

Spinning particles, braid groups and solitons

“…In the associated quantum theory, the two particles obey a type of "half Bose -half Fermi" statistics which has been called ambistatistics. In the past, ambistatistics has been obtained for structureless particles either on nonsimply connected spaces [5] [7][10], or on simply connected spaces in the presence of spectators which effectively create noncontractible loops [21] [22]. Similar quantizations have also been found for extended objects such as identical geons in quantum gravity [23], topological solitons in certain nonlinear sigma models [24] [25], and strings in mechanics [26] and certain gauge theories [22].…”

“…Moreover, using techniques similar to those developed for the structureless case [5], one can show that the extension (2.5) splits if d ≥ 3. Thus, in this case, B E n (M ) can be viewed as a semidirect product of π 1 (E) n by S n ; the group S n acts by permuting the n factors of π 1 (E).…”

“…considers only scalar quantum theories (those associated with the one-dimensional IUR's of B n (M )), then it has been shown that the only allowed statistics for the particles are Bose and Fermi if dim M ≥ 3 (both always being possible) [5]. By contrast, for open 2-manifolds (like IR 2 ) one obtains the full range of fractional (or θ-) statistics 3 for any n ≥ 2 [7].…”

“…By contrast, for open 2-manifolds (like IR 2 ) one obtains the full range of fractional (or θ-) statistics 3 for any n ≥ 2 [7]. On the 2-sphere S 2 only a finite (n-dependent) subset of statistical angles θ are allowed [8], while on all other closed surfaces only Bose and Fermi statistics (θ = 0 and π) are available [5]. However, by looking at nonscalar quantum theories (higher-dimensional IUR's of B n (M )), one can regain all the rational values of θ/π for n identical particles on any closed, orientable 2-manifold [9][10] [11].…”

“…However, by looking at nonscalar quantum theories (higher-dimensional IUR's of B n (M )), one can regain all the rational values of θ/π for n identical particles on any closed, orientable 2-manifold [9][10] [11]. Further, nonscalar quantum theories allow for the well-known parastatistics when n ≥ 3 (and any M ), as well as a complex generalization of them for three or more particles in two dimensions [5][12] [13]. If M is not simply connected, then there may be even more exotic possibilities such as ambistatistics where the superselection rule between bosons and fermions is effectively broken [5].…”