On the Minimal Model of Anyons

The first quantized theory of N=2, D=3 massive superparticles with arbitrary fixed central charge and (half)integer or fractional superspin is constructed. The quantum states are realized on the fields carrying a finite dimensional, or a unitary infinite dimensional representation of the supergroups OSp(2|2) or SU(1, 1|2). The construction originates from quantization of a classical model of the superparticle we suggest. The physical phase space of the classical superparticle is embedded in a symplectic superspace T * (R 1,2 ) × L 1|2 , where the inner Kähler supermanifold L 1|2 ∼ = OSp(2|2)/[U(1) × U(1)] ∼ = SU(1, 1|2)/[U(2|2) × U(1)] provides the particle with superspin degrees of freedom. We find the relationship between Hamiltonian generators of the global Poincaré supersymmetry and the "internal" SU(1, 1|2) one. Quantization of the superparticle combines the Berezin quantization on L 1|2 and the conventional Dirac quantization with respect to space-time degrees of freedom. Surprisingly, to retain the supersymmetry, quantum corrections are required for the classical N=2 supercharges as compared to the conventional Berezin method. These corrections are derived and the Berezin correspondence principle for L 1|2 underlying their origin is verified. The model admits a smooth contraction to the N=1 supersymmetry in the BPS limit.

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“…An adapted reformulation of the canonical model is suggested in Refs. [15,25]. We observe, that the monopole two-form Ω m in Eq.…”

Section: Minimal and Extended Phase Spaces Of A Canonical Model Omentioning

confidence: 84%

Hidden supersymmetry and Berezin quantization of N=2, D=3 spinning superparticles

Gorbunov

Lyakhovich

1999

Journal of Mathematical Physics

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“…We introduce N = 1, D = 3 super Poincaré invariant action for a massive fractional spin superparticle living in R 3|2 × L, where R 3|2 denotes the N = 1, D = 3 flat superspace and L the Lobachevsky plane. This mechanical system is a minimal supersymmetric extension of special anyon model proposed in [27]. Our interest to the latter is caused by the fact that the model proves to be classically equivalent to the formulation based on the monopole-like symplectic two-form [17 -21] and, hence, allows introduction of coupling to arbitrary background fields.…”

Section: Introductionmentioning

confidence: 99%

“…The action of the Poincaré group is obviously well defined on T * (R 1,2 ) × L. At the same time, supersymmetry can not be globally realized on T * (R 1,2 ) × L 1|1 and restores only on the surface of the rest constraints (23). Straightforward calculations of (anti) commutation relations of the generators (25,27), with respect to the Dirac brackets, show that all the brackets (28) remain intact in the strong sense except {Q I α , Q J β } * and {Z , Q I α } * . The latter can be presented in the manner…”

Section: Introductionmentioning

confidence: 99%

N=1,D=3superanyons,osp(2|2), and the deformed Heisenberg algebra

1997

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We introduce N = 1 supersymmetric generalization of the mechanical system describing a particle with fractional spin in D = 1 + 2 dimensions and being classically equivalent to the formulation based on the Dirac monopole two-form. The model introduced possesses hidden invariance under N = 2 Poincaré supergroup with a central charge saturating the BPS bound. At the classical level the model admits a Hamiltonian formulation with two first class constraints on the phase space T * (R 1,2 )×L 1|1 , where the Kähler supermanifold L 1|1 ∼ = OSp(2|2)/U (1|1) is a minimal superextension of the Lobachevsky plane. The model is quantized by combining the geometric quantization on L 1|1 and the Dirac quantization with respect to the first class constraints. The constructed quantum theory describes a supersymmetric doublet of fractional spin particles. The space of quantum superparticle states with a fixed momentum is embedded into the Fock space of a deformed harmonic oscillator.

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“…As it is well known, from the fact that this two-form is globally defined it follows that the spin s is not quantized. For a detailed explanation of this, as well as other aspect of the quantum theory of anyons, we refer the interested reader to [5,6].…”

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confidence: 99%

A Geometrical Particle Model for Anyons

Nersessian

Ramos

1999

Mod. Phys. Lett. A

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We consider the simplest geometrical particle model associated with light-like curves in (2 + 1) dimensions. The action is proportional to the pseudo-arc length of the particle's path. We show that under quantization, it yields the (2 + 1)-dimensional anyonic field equation supplemented with a Majorana-like relation on mass and spin, i.e. mass × spin = α 2 , with α the coupling constant in front of the action.The search for Lagrangians describing spinning particles, both massive and massless, has a long history. The conventional approach is based on an extension of Minkowski space-time by auxiliary Grassmann variables which, after quantization, provide the extra degrees of freedom required for integer or half-integer spin. In contrast, in planar physics, spin is not quantized and particle states can be anyonic, i.e. their spin can take any real value. Thus, in the general case, it is necessary to provide the initial classical model with extra bosonic variables. An interesting, and by now well-explored, possibility is to supply those extra degrees of freedom by Lagrangians depending on higher order geometrical invariants. The requirements of Poincaré and reparametrization invariance restrict the admissible set of such Lagrangians to the ones depending only on curvature and torsion. An extensive study in this direction, performed in the late '80s by several authors (see Ref. 2 and references therein), was inspired by the remarkable work of Polyakov 1 on the CP 1 model minimally coupled to a Chern-Simons gauge field. An evaluation of the *

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On the Minimal Model of Anyons

Abstract: We present new geometric formulations for the fractional spin particle models on the minimal phase spaces. New consistent couplings of the anyon to background fields are constructed. The relationship between our approach and previously developed anyon models is discussed.

Cited by 28 publications

References 31 publications

Hidden supersymmetry and Berezin quantization of N=2, D=3 spinning superparticles

Hidden supersymmetry and Berezin quantization of N=2, D=3 spinning superparticles

N=1,D=3superanyons,osp(2|2), and the deformed Heisenberg algebra

A Geometrical Particle Model for Anyons

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