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Noncommutative planar particles higher order versus first order formalism and supersymmetri.pdf

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Noncommutative planar particles higher order versus first order formalism and supersymmetri

a r X i v : h e p - t h / 0 2 1 0 1 1 2 v 1 1 1 O c t 2 0 0 2 Noncommutative Planar Particles: Higher Order Versus First Order Formalism and Supersymmetrization J. Lukierski? Institute for Theoretical Physics, University of Wroc law, pl. Maxa Borna 9, 50-204 Wroc law, Poland e-mail:lukier@ift.uni.wroc.pl P. Stichel An der Krebskuhle 21, D-33619 Bielefeld, Germany e-mail:pstichel@gmx.de W.J. Zakrzewski Department of Mathematical Sciences, Science Laboratories, University of Durham, South Road, Durham DH1 3LE, UK e-mail: W.J.Zakrzewski@durham.ac.uk Abstract We describe the supersymmetrization of two formulations of free noncommutative planar particles – in coordinate space with higher or- der Lagrangian [1] and in the framework of Faddeev and Jackiw [2,3], with first order action. In nonsupersymmetric case the first formula- tion after imposing subsidiary condition eliminating internal degrees of freedom provides the second formulation. In supersymmetric case one can also introduce the split into “external” and “internal” degrees of freedom both describing supersymmetric models. ?Talk given by J. Lukierski 1 1 Introduction In [1] the present authors introduced the following nonrelativistic higher order action for D=2 (planar) particle: L (0) 1 = mx?21 2 ? k?ij x?ix?j . (1) The canonical quantization of (1) implies the consideration of xi, x?i as in- dependent degrees of freedom (see e.g. [4]) with the following canonically conjugated two momenta [1,4] pi = ?L(0) ?x?i ? d dt ?L(0) ?x?i = mx?i ? 2k?ijx?j , (2) p?i = k?ij x?j . (3) The relation (3) introduces a second class constraint, i.e. after the introduc- tion of Dirac brackets the Lagrangian system (1) is described by six degrees of freedom YA = (xi, pi, vi = x?i). One gets the following set of Dirac brackets [1]: {YA, YB} = ? ? 0 12 0?12 0 0 0 0 ? 1 2k ? ? ? (4) In order to get the first order formulation of the action (1) one can use the technique proposed by Faddeev and Jackiw [2,3]. The action (1) is equiv