Supersymmetric Mechanics Research Articles

The classical supersymmetric Coulomb problem

After setting up a general model for supersymmetric classical mechanics in more than one dimension we describe systems with centrally symmetric potentials and their Poisson algebra. We then apply this information to the investigation and solution of the supersymmetric Coulomb problem, specified by an 1/|x| repulsive bosonic potential.

A hidden supersymmetry of supersymmetric mechanics on the sphere S6

In the supersymmetric classical or quantal mechanics of particle motion on the sphere S 6, an SO(7)-invariant theory, the existence of a totally antisymmetric tensor e ijk (defined by the product law of octonions) invariant under the G 2 sub-group of SO(7) allows the construction of a Killing–Yano tensor and the supercharge Q′ of a hidden supersymmetry of the theory. The canonical bracket { Q′, Q′} (or quantally the corresponding anticommutator) yields not the Hamiltonian of the theory but instead an operator determined by the quadratic Casimir operator of G 2. The situation is compared with the theory of a spin- 1 2 particle moving in the field of a Dirac monopole. Use of the tensor e ijk to break the SO(7)-invariance of the original theory of motion on S 6 to G 2 invariance is discussed.

Note onN=4supersymmetric mechanics on Kähler manifolds

The geometric models of N=4 supersymmetric mechanics with $(2d.2d)_{\DC}$-dimensional phase space are proposed, which can be viewed as one-dimensional counterparts of two-dimensional N=2 supersymmetric sigma-models by Alvarez-Gaum\'e and Freedman. The related construction of supersymmetric mechanics whose phase space is a K\ahler supermanifold is considered. Also, its relation with antisymplectic geometry is discussed.

Integrable supersymmetric fluid mechanics from superstrings

Following the construction of a model for the planar supersymmetric Chaplygin gas, supersymmetric fluid mechanics in (1+1) dimensions is obtained from the light-cone parametrized Nambu–Goto superstring in (2+1) dimensions. The lineal model is completely integrable and can be formulated neatly using Riemann coordinates. Infinite towers of conserved charges and supercharges are exhibited. They form irreducible representations of a dynamical (hidden) SO(2,1) symmetry group.

Quantization of Paraclassical Systems and Parabose- Parafermi Supersymmetry

A convenient formalism is developed to treat classical dynamical systems involving (p = 2) parafermionic and parabosonic dynamical variables. This is achieved via the introduction of a parabracket which summarizes the para-commutation relations of the corrosponding Green components in a unified manner. The formalism is used to investigate the supersymmetries of a system consisting of a (p= 2) parabose and a (p=2) parafermi oscillator. This suggests a definition for supersymmetric paraquantum mechanics of order p=2. For such systems the degeneracy strucure of the spectrum is also analyzed.

On q-deformed supersymmetric classical mechanical models

Based on the idea of quantum groups and para-Grassmannian variables, we present a generalization of supersymmetric classical mechanics with a deformation parameter q=exp(2πi/k) dealing with the k=3 case. The coordinates of the q-superspace are a commuting parameter t and a para-Grassmannian variable θ, where θ3=0. The generator and covariant derivative are obtained, as well as the action for some possible superfields.

PARABOSE-PARAFERMI SUPERSYMMETRY

The (p=2) parabose–parafermi supersymmetry is studied in general terms. It is shown that the algebraic structure of the (p=2) parastatistical dynamical variables allows for (symmetry) transformations which mix the parabose and parafermi coordinate variables. The example of a simple parabose-parafermi oscillator is discussed and its symmetries investigated. It turns out that this oscillator possesses two parabose-parafermi supersymmetries. The combined set of generators of the symmetries forms the algebra of supersymmetric quantum mechanics supplemented with an additional central charge. In this sense there is no relation between the parabose–parafermi supersymmetry and the parasupersymmetric quantum mechanics. A precise definition of a quantum system involving this type of parabose-parafermi supersymmetry is offered, thus introducing (p=2) supersymmetric paraquantum mechanics. The spectrum degeneracy structure of general (p=2) supersymmetric paraquantum mechanics is analyzed in detail. The energy eigenvalues and eigenvectors for the parabose–parafermi oscillator are then obtained explicitly. The latter confirms the validity of the results obtained for general supersymmetric paraquantum mechanics.

The role of Killing-Yano tensors in supersymmetric mechanics on a curved manifold

The supersymmetric extension of the motion of a charged point particle is applied to investigate symmetries of gravitational fields and electromagnetic fields. We mainly focus on the role of the Killing-Yano tensors of both usual and generalized types. Results obtained by systematic analysis strengthen the connection of the Killing-Yano tensor and superinvariants (functions commuting with the supercharge).

Supersymmetric classical mechanics

We study the classical properties of a supersymmetric system which is often used as a model for supersymmetric quantum mechanics. It is found that the classical dynamics of the bosonic as well as the fermionic degrees of freedom is fully described by a so--called quasi--classical solution. We also comment on the importance of this quasi--classical solution in the semi--classical treatment of the supersymmetric quantum model.

Supersymmetric mechanics with an odd action functional

We give an odd Lagrangian formulation of models yielding the Poisson-Buttin bracket in the graded phase space. Such a generalization of the usual supersymmetric mechanics allows the introduction of an n (for n-extended algebra, n=2k) even parameters of deformation of the geometry of an n-extended super-time space.

FRACTIONAL SUPERSPACE FORMULATION OF GENERALIZED MECHANICS

Supersymmetric (pseudoclassical) mechanics has recently been generalized to fractional supersymmetric mechanics. In such a construction, the action is invariant under fractional supersymmetry transformations, which are the F th roots at time translations ( with F = 1, 2,…). Associated with these symmetries, there are conserved charges with fractional canonical dimension 1+1/F. Using paragrassmann variables satisfying θF=0, we present a fractional-superspace formulation of this construction.

The massless superparticle as Chern-Simons mechanics

We show that the locally worldline supersymmetric action of Sorokin, Tkach and Volkov for the massless superparticle in d=3, 4, 6 and 10 spacetime dimensions is an example of supersymmetric Chern-Simons mechanics. For d=4, 6, 10 there are respectively 1, 3, 7 additional local supersymmetries which we express in a common form.

Supersymmetry of the photon

The supersymmetric massless states of a relativistic extension of Witten's supersymmetric quantum mechanics are shown to correspond to Abelian gauge fields realised by antisymmetric tensors. The massive states, which are only partially supersymmetric, include spin-1 and spin-0 particles. In particular the Maxwell and Proca fields are obtained from N=2 supersymmetric relativistic mechanics by first quantisation in much the same way as the Dirac theory arises in the N=1 case. The particles may be coupled supersymmetrically to external scalar and complex Hermitian tensor fields. The latter reduce in special cases to the Riemannian metric of external gravitation and the Kahler metric implied by a vector field coupling. All the couplings exhibit a quadrupole characteristic of the particles. In particular supersymmetry requires their electric charge and magnetic dipole moment to vanish, and there is no coupling to torsion in Riemann-Cartan spacetime. Quantisation in external fields yields covariant generalisations of the classical tensor field equations. Finally classical equations of translational and spin motion involving only real quantities are obtained from the Heisenberg equations.

Supersymmetric classical mechanics

The purpose of the paper is to construct a supersymmetric Lagrangian within the framework of classical mechanics which would be regarded as a candidate for passage to supersymmetric quantum mechanics.

Supersymmetric mechanics: A new look at the equivalence of quantum systems

Supersymmetric mechanics: A new look at the equivalence of quantum systems