Configuration space topology and quantum internal symmetries

Fractional quantum Hall effect revisited

“…) [13][14][15][16]. Note that it is impossible to associate CFs with the 1DURs of the full braid group, because 1DURs are periodic with a period of 2π , whereas the CFs require the statistics phase shift qπ , q-odd integer.…”

“…The role of topology in the strongly correlated state creation was noticed [13][14][15] in the context of exceptional topological properties of a plane and locally 2D manifolds like sphere or torus. This unique topology of planar systems is linked with an exceptionally rich structure of their braid groups in comparison to ones of higher dimensional spaces (R d , d 2 > ) [16].…”

Spinning particles, braid groups and solitons

“…If the particle-like objects live on a manifold M (which for the remainder of this section is assumed to be closed, orientable and of dimension d ≥ 3), then one may have guessed that there exist half-integral spin quantizations if and only if M is a spin manifold. However in previous 13 One source of this difference is the fact that…”

“…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].…”

Application of the Mathematical Methods of the Braid Group Theory to Quantization of Many-Particle Systems

“…[14,15]). In general, φ can be multivalued: ,,,, m = 1, 2, ..., L. The quantum theory with one-component wave function Ψ is called the scalar quantum theory.…”