LPT Strasbourg 95-20 Local Fractional Supersymmetry for Alternative Statistics

A group theory justification of one dimensional fractional supersymmetry is proposed using an analogue of a coset space, just like the one introduced in 1D supersymmetry. This theory is then gauged to obtain a local fractional supersymmetry i.e. a fractional supergravity which is then quantized à la Dirac to obtain an equation of motion for a particle which is in a representation of the braid group and should describe alternative statistics. A formulation invariant under general reparametrization is given, by means of a curved fractional superline. With string theory[1], a new approach came out in the description of space-time symmetry. Indeed, by studying the symmetries on the world sheet of the string one can get the space-time properties of the string states, i.e. the particles (representations of the gauge group are controlled by Kac-Moody algebra[2] and of the Poincaré one by (super)conformal invariance[3]). However, all those results were anticipated and an alternative formulation of relativistic wave equations[4, 5, 6] and quantum field theory can be obtained with the study of physics on the world line of the particle. Particles with spin N/2 could be described by an N− extented supersymmetry[6] on the world line, and gauge symmetries by the introduction of internal Grassmann variables[7]. All this was recently promoted into an alternative and efficient description of field theory using the world-line formalism[8], introducing 1D Feyman rules and appropriate one dimensional Green functions[9]. However, the spin statistics theorem and the Haag, Lopuszanski and Sohnius no-go theorem[10] tell us that supersymmetry is the more general non-trivial symmetry that one can consider; as soon as we are in a D ≤ 3 dimensional space-time one can find statistics that are neither fermions nor bosons, but anyons[11] or particles which admit fractional statistics. Technically the former particles are in the representation of the permutation group and the latter of the braid group. In the meantime some extensions of 1D supersymmetry have been considered, for instance parasupersymmetry[12, 13] or fractional supersymmetry[14, 15, 16, 17, 18]. It has been proved that 1D parasupersymmetry of order p could be equivalent to p−extended world-line supersymmetry and describes particles of spin p2 [13]. Fractional supersymmetry has been recently the subject of intensive studies[14, 15, 16, 17, 18]. Following the way which leads from 1D supersymmetry to the Dirac equation, applied in the context of fractional supersymmetry, we get a new equation acting on states which are in the representation of the braid group. This equation can be seen as an extension of the Dirac equation in the sense that the n−th power of the field operator is equal to the Klein-Gordon one. In this paper we particularize the case n = 3. In a first step we define, in analogy with the superspace, the fractional superspace as some kind of coset space reobtaining all what has been done in the framework of fractional susy. In a second step, we construct a local fractional supersymmetry i.e a fractional supergravity by using two one dimensional gauge fields: the einbein and a field which can be compared to the 1D gravitino and that we call the fractional gravitino. A formulation, in a curved fractional superline, which is invariant under general coordinate transformations is then given. The second part is devoted to the quantization of the theory, taking under consideration the first and second class constraints[19]. After having constructed the Fock space with the help of the q− deformed oscillators[20] we obtain a new equation, that we call the fractional Dirac equation. I.Fractional Superspace and Fractional Supersymmetry Historically, 4D supersymmetry has been built explicitly, components by components (see for example [21]). However it was understood later that this symmetry is just a consequence of a symmetry in a so-called superspace which can be seen as the coset space of the Superpoincaré group by the Lorentz group[21]. The superspace is just the 8-fold space (x, θ), where x is the space-time components and θ its spinor partner. Because we are studying physics on the world line we just particularize the 1D case. Noting H the generator of the time translation, and Q the generator of the