Deformed Heisenberg Algebra and Fractional Spin Field in 2+1 Dimensions

With the help of the deformed Heisenberg algebra involving Klein operator, we construct the minimal set of linear differential equations for the (2+1)-dimensional relativistic field with arbitrary fractional spin, whose value is defined by the deformation parameter. MIRAMARE–TRIESTE September 1993 Permanent address: IHEP, Protvino, Moscow Region, 142284 Russia, e-mail: [email protected] The considerable interest to the (2+1)-dimensional particles with fractional spin and statistics (anyons) [1] is conditioned nowadays by their applications to the theory of planar physical phenomena: fractional quantum Hall effect and high-Tc superconductivity [2]. Moreover, anyons attract a great attention due to their relationship with different theoretical fields of research such as conformal field theories, braid groups and deformed theories (see, e.g., refs. [3]–[6]). From the field-theoretical point of view, such particles can be described in two, possibly related, ways. The first way consists in organizing a statistical interaction of the scalar or fermionic field with the Chern-Simons U(1) gauge field, that changes spin and statistics of the matter field. In this approach, the gauge field can be formally excluded from the theory due to their Lagrange equations (or corresponding Hamiltonian constraints), resulting in the anyonic permutation relations of the redefined matter field [7]. But such an exclusion was intensively criticized, and, till now it is not clear whether the role of the Chern-Simons gauge field is reduced only to the change of the spin and statistics of the matter field, or there is some relic of the statistical gauge field in the theory [8]. Another, less developed way consists in attempting to describe anyons within the group-theoretical approach analogously to the case of integer and half-integer spin fields. The present paper is devoted to further development of this approach, whose program consists in constructing equations for (2+1)-dimensional fractional spin field, subsequent identifying corresponding field action and, finally, in realizing a quantization of the theory to reveal a fractional statistics. Within this approach, there are, in turn, two related possibilities: to use many-valued representations of the SO(2, 1) group, or to work with the infinite-dimensional unitary representations of its universal covering group, SO(2, 1) (or SL(2, R), isomorphic to it) [9]. Up to now, only the equations were proposed for the fractional spin field within the former possibility [9]-[11], whereas different variants of the equations and field actions were constructed with the use of the unitary infinitedimensional representations of SL(2, R) [9], [12]-[17]. At the same time, the problem of quantizing the theory is still open here. The main difficulty in quantization consists in the infinite component nature of the fractional spin field which is used to describe in a covariant way one-dimensional irreducible representations of the (2+1)-dimensional quantum mechanical Poincaré group ISO(2, 1) specified by the values of mass and arbitrary (fixed) spin. Due to this fact, an infinite set of the corresponding Hamiltonian constraints must be present in the theory to exclude an infinite number of the ‘auxiliary’ field degrees of freedom [9]. This infinite set of constraints should appropriately be taken into account. But on the other hand, the infinite component nature of the fractional spin field indicates on the hidden nonlocal nature of the theory, and, therefore, can be considered in favour of the existence of the anyonic spin-statistics relation for the fractional spin fields within the framework of the group-theoretical approach [4, 18]. Relativistic field with arbitrary fractional spin s = εα, α > 0, ε± 1, can be described in this approach by the Klein-Gordon equation (P 2 +m)Ψ = 0 (1) and (2+1)-dimensional analog of the Majorana equation [19] (PJ − εαm)Ψ = 0. (2) Here it is supposed that the field Ψ = Ψ(x) transforms according to the infinitedimensional unitary irreducible representation (UIR) of the discrete series D α or D − α