Scattering states of plektons (particles with braid group statistics) in 2+1 dimensional quantum field theory

Abstract

A Haag-Ruelle scattering theory for particles with braid group statistics is developed, and the arising structure of the Hilbert space of multiparticle states is analyzed.

The Spin-Statistics Theorem for Anyons and Plektons in d = 2+1

“…But our framework, using a restricted notion of charged fields, can be set up starting from the standard assumptions [12] of local relativistic quantum theory on the observabels plus weak Haag Duality, together with our assumptions on the mass spectrum. For the convenience of the reader, we sketch in Appendix A how this may be done and indicate the relation with the notions used in the literature [3,5,8]. 4 Two other possibilities are: To introduce a reference space-like cone from which all allowed localization cones have to keep space-like separated (this cone playing the role of a "cut" in the context of multivalued functions) [3]; or a cohomology theory of nets of operator algebras as introduced by Roberts [19][20][21].…”

“…(47). 8 We use a superscript λ instead ofλ, which causes no confusion since we have a one-to-one correspondence between U12 andŨ12.…”

“…The notion of localized generalized field operators has been introduced in [5] in the case of permutation group statistics. The extension to the case of braid group statistics needs a refinement, which has been introduced in [9], see also [8]. There, K denotes the class of space-like cones or causal complements thereof, and a path in K is a finite sequence (I 0 , .…”

Quantum group symmetry and particle scattering in (2+1)-dimensional quantum gravity

“…Hence there are natural flat P B n (B n ) bundles over M n (N n ). The result of [6], [7] and [19] is that n-anyon Hilbert spaces are the spaces of L 2 -sections of associated vector bundles. For our purposes it is useful to recast the construction in the language of ribbon Hopf algebras.…”

“…19) for the momentum in terms of the parametrisation (A.14) of the Lorentz transformation L. The formula for j is…”