Supersymmetries in pure parabosonic systems (original) (raw)
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Deformed Heisenberg algebra with reflection, anyons and supersymmetry of parabosons
AIP Conference Proceedings, 2000
Deformed Heisenberg algebra with reflection appeared in the context of Wigner's generalized quantization schemes underlying the concept of parafields and parastatistics of Green, Volkov, Greenberg and Messiah. We review the application of this algebra for the universal description of ordinary spin-j and anyon fields in 2+1 dimensions, and discuss the intimate relation between parastatistics and supersymmetry.
Hidden Nonlinear Supersymmetries in Pure Parabosonic Systems
International Journal of Modern Physics A, 2000
The existence of intimate relations between generalized statistics and supersymmetry is established by the observation of hidden supersymmetric structure in pure parabosonic systems. This structure is characterized generally by a nonlinear superalgebra. The nonlinear supersymmetry of parabosonic systems may be realized, in turn, by modifying appropriately the usual supersymmetric quantum mechanics. The relation of nonlinear parabosonic supersymmetry to the Calogero-like models with exchange interaction and to the spin chain models with inverse-square interaction is pointed out.
Modern Physics Letters A, 1999
We show that the single-mode parafermionic type systems possess supersymmetry, which is based on the symmetry of characteristic functions of the parafermions related to the generalized deformed oscillator of Daskaloyannis et al. The supersymmetry is realized in both unbroken and spontaneously broken phases. As in the case of parabosonic supersymmetry observed recently by one of the authors, the form of the associated superalgebra depends on the order of the parafermion and can be linear or nonlinear in the Hamiltonian. The list of supersymmetric parafermionic systems includes usual parafermions, finite-dimensional q-deformed oscillator, q-deformed parafermionic oscillator and parafermionic oscillator with internal Z2 structure.
Graded structure and Hopf structures in parabosonic algebra. An alternative approach to bosonisation
Arxiv preprint arXiv:0706.2825, 2007
Parabosonic algebra in infinite degrees of freedom is presented as a generalization of the bosonic algebra, from the viewpoints of both physics and mathematics. The notion of super-Hopf algebra is shortly discussed and the super-Hopf algebraic structure of the parabosonic algebra is established (without appealing to its Lie superalgebraic structure). Two possible variants of the parabosonic algebra are presented and their (ordinary) Hopf algebraic structure is estabished: The first is produced by "bosonising" the original super-Hopf algebra, while the second is constructed via a slightly different path.
Superconformal mechanics and nonlinear supersymmetry
Journal of High Energy Physics, 2003
We show that a simple change of the classical boson-fermion coupling constant, 2alphato2alphan2\alpha \to 2\alpha n 2alphato2alphan, ninNn\in \NninN, in the superconformal mechanics model gives rise to a radical change of a symmetry: the modified classical and quantum systems are characterized by the nonlinear superconformal symmetry. It is generated by the four bosonic integrals which form the so(1,2) x u(1) subalgebra, and by the 2(n+1) fermionic integrals constituting the two spin-n/2 so(1,2)-representations and anticommuting for the order n polynomials of the even generators. We find that the modified quantum system with an integer value of the parameter alpha\alphaalpha is described simultaneously by the two nonlinear superconformal symmetries of the orders relatively shifted in odd number. For the original quantum model with ∣alpha∣=p|\alpha|=p∣alpha∣=p, pinNp\in \NpinN, this means the presence of the order 2p nonlinear superconformal symmetry in addition to the osp(2|2) supersymmetry.
q-graded Heisenberg algebras and deformed supersymmetries
Journal of Mathematical Physics, 2010
The notion of q-grading on the enveloping algebra generated by products of q-deformed Heisenberg algebras is introduced for q complex number in the unit disc. Within this formulation, we consider the extension of the notion of supersymmetry in the enveloping algebra. We recover the ordinary Z 2 grading or Grassmann parity for associative superalgebra, and a modified version of the usual supersymmetry. As a specific problem, we focus on the interesting limit q → −1 for which the Arik and Coon deformation of the Heisenberg algebra allows to map fermionic modes to bosonic ones in a modified sense. Different algebraic consequences are discussed.
Deformed Heisenberg Algebra, Fractional Spin Fields, and Supersymmetry without Fermions
Annals of Physics, 1996
Within a group-theoretical approach to the description of (2+1)-dimensional anyons, the minimal covariant set of linear differential equations is constructed for the fractional spin fields with the help of the deformed Heisenberg algebra (DHA), [a − , a + ] = 1 + νK, involving the Klein operator K, {K, a ± } = 0, K 2 = 1. The connection of the minimal set of equations with the earlier proposed 'universal' vector set of anyon equations is established. On the basis of this algebra, a bosonization of supersymmetric quantum mechanics is carried out. The construction comprises the cases of exact and spontaneously broken N = 2 supersymmetry allowing us to realize a Bose-Fermi transformation and spin-1/2 representation of SU(2) group in terms of one bosonic oscillator. The construction admits an extension to the case of OSp(2|2) supersymmetry, and, as a consequence, both applications of the DHA turn out to be related. A possibility of 'superimposing' the two applications of the DHA for constructing a supersymmetric (2+1)-dimensional anyon system is discussed. As a consequential result we point out that osp(2|2) superalgebra is realizable as an operator algebra for a quantum mechanical 2-body (nonsupersymmetric) Calogero model.
Three aspects of bosonized supersymmetry and linear differential field equation with reflection
Nuclear Physics B, 1999
Recently it was observed by one of the authors that supersymmetric quantum mechanics (SUSYQM) admits a formulation in terms of only one bosonic degree of freedom. Such a construction, called the minimally bosonized SUSYQM, appeared in the context of integrable systems and dynamical symmetries. We show that the minimally bosonized SUSYQM can be obtained from Witten's SUSYQM by applying to it a nonlocal unitary transformation with a subsequent reduction to one of the eigenspaces of the total reflection operator. The transformation depends on the parity operator, and the deformed Heisenberg algebra with reflection, intimately related to parabosons and parafermions, emerges here in a natural way. It is shown that the minimally bosonized SUSYQM can also be understood as supersymmetric two-fermion system. With this interpretation, the bosonization construction is generalized to the case of N = 1 supersymmetry in 2 dimensions. The same special unitary transformation diagonalises the Hamiltonian operator of the 2D massive free Dirac theory. The resulting Hamiltonian is not a square root like in the Foldy-Wouthuysen case, but is linear in spatial derivative. Subsequent reduction to 'up' or 'down' field component gives rise to a linear differential equation with reflection whose 'square' is the massive Klein-Gordon equation. In the massless limit this becomes the self-dual Weyl equation. The linear differential equation with reflection admits generalizations to higher dimensions and can be consistently coupled to gauge fields. The bosonized SUSYQM can also be generated applying the nonlocal unitary transformation to the Dirac field in the background of a nonlinear scalar field in a kink configuration.
Journal of Physics A: Mathematical and Theoretical, 2008
Parabosonic algebra in finite or infinite degrees of freedom is considered as a Z 2 -graded associative algebra, and is shown to be a Z 2graded (or: super) Hopf algebra. The super-Hopf algebraic structure of the parabosonic algebra is established directly without appealing to its relation to the osp(1/2n) Lie superalgebraic structure. The notion of super-Hopf algebra is equivalently described as a Hopf algebra in the braided monoidal category CZ 2 M. The bosonisation technique for switching a Hopf algebra in the braided monoidal category H M (where H is a quasitriangular Hopf algebra) into an ordinary Hopf algebra is reviewed. In this paper we prove that for the parabosonic algebra P B , beyond the application of the bosonisation technique to the original super-Hopf algebra, a bosonisation-like construction is also achieved using two operators, related to the parabosonic total number operator. Both techniques switch the same super-Hopf algebra P B to an ordinary Hopf algebra, producing thus two different variants of P B , with ordinary Hopf structure.