Bosons, fermions and anyons in the plane, and supersymmetry

Abstract

Universal vector wave equations allowing for a unified description of anyons, and also of usual bosons and fermions in the plane are proposed. The existence of two essentially different types of anyons, based on unitary and also on non-unitary infinite-dimensional half-bounded representations of the (2+1)D Lorentz algebra is revealed. Those associated with non-unitary representations interpolate between bosons and fermions. The extended formulation of the theory includes the previously known Jackiw-Nair (JN) and MajoranaDirac (MD) descriptions of anyons as particular cases, and allows us to compose bosons and fermions from entangled anyons. The theory admits a simple supersymmetric generalization, in which the JN and MD systems are unified in N = 1 and N = 2 supermultiplets. Two different non-relativistic limits of the theory are investigated. The usual one generalizes Lévy-Leblond's spin 1/2 theory to arbitrary spin, as well as to anyons. The second, "Jackiw-Nair" limit (that corresponds to Inönü-Wigner contraction with both anyon spin and light velocity going to infinity), is generalized to boson/fermion fields and interpolating anyons. The resulting exotic Galilei symmetry is studied in both the non-supersymmetric and supersymmetric cases.

Three-dimensional fractional-spin gravity

“…Later, a group-theory approach based on wave-functions was developed in 1 [9][10][11][12]. Later, a group-theory approach based on wave-functions was developed in [9][10][11][12]; see [13] for a summary and extensions of these models.…”

“…For ν < −1, but non-critical, these representations are still of a discrete type, but non-unitary. In reference [13] it was shown that the latter were essential for the construction of anyon wave equations possessing standard boson/fermion limits.…”

“…12 The structure coefficients of osp(N |2) can be found using the realization M αβ = q (α q β) , Q i α = ξ i qα and T ij = iξ i ξ j where qα obey (2.3) with ν = 0 and ξ i are external operators that obey {ξ i , ξ j } = 2δ ij . 13 The Lorentz generators…”

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

“…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].…”

Deformed su(1,1) Algebra as a Model for Quantum Oscillators

“…Having two parameters available in a mathematical model for the quantum oscillator, opens the way to more flexibility in applications. For possible applications (of the deformed algebra and its representations) in quantum field theory, see the ideas presented in [13].…”