Relativistic wave equation for anyons

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

“…The fact that Eqs. (11), (13) and (14) also hold in the case of braid group statistics has been shown in [16].…”

“…It has the following properties [1,13]: it commutes with the representation U , and has the value W = −ms1 (23) if, and only if, U contains only irreducible representations whose masses and spins have the product value ms. Considering now the representations U χ and Uχ, we denote their Pauli-Lubanski scalars as W χ and Wχ, respectively.…”

“…The proof requires only a slight modification from that of Buchholz and Epstein. Namely, the role of the square of the Pauli-Lubanski vector as a Casimir operator is, in 2+1 dimensions, played by a scalar operator, the so-called Pauli-Lubanski scalar [1,13] which is defined as follows. Let U be a representation of the universal covering of the Poincaré group in three spacetime dimmensions, let L 0 denote the generator of the rotation subgroup in the representation U , and let L i be the generator of the boosts in direction…”

Duffin–Kemmer–Petiau and Dirac Equations—A Supersymmetric Connection

“…We have demonstrated that certain subsolutions of the free Duffin-Kemmer-Petiau (DKP) and the Dirac equations obey the same Dirac equation with some built-in projection operators [9]. We shall refer to this equation as supersymmetric since it has bosonic (spin 0 and 1) as well as fermionic spin 1 2 degrees of freedom. In the present paper we extend our results to the case of interacting fields.…”

“…Recently, several supersymmetric systems, concerned mainly with anyons in 2 + 1 dimensions [1][2][3][4][5] as well as with the 3+1 dimensional Majorana-Dirac-Staunton theory [6], uniting fermionic and bosonic fields, have been described. Furthermore, bosonic symmetries of the Dirac equation have been found in the massless [7] as well as in the massive case [8].…”