Supersymmetry without Fermions

The simplest N = 2 supersymmetric quantum mechanical system is realized in terms of the bosonic creation and annihilation operators obeying either ordinary or deformed Heisenberg algebra involving Klein operator. The construction comprises both the exact and spontaneously broken supersymmetry cases with the scale of supersymmetry breaking being governed by the deformation parameter. Proceeding from the broken supersymmetry case, we realize the Bose-Fermi transformation and obtain spin-1/2 representation of SU(2) group in terms of one bosonic oscillator. We demonstrate that the constructions can be generalized to the more complicated N = 2 supersymmetric systems, in particular, corresponding to the Witten supersymmetric quantum mechanics. 1 E-mail: [email protected] It was demonstrated recently [1] that the classical analog of the quantum massless superparticle can be constructed without incorporating odd Grassmann variables being the classical analogs of the corresponding fermionic operators. Such “bosonization” was realized with the help of topologically nontrivial classical variables, which after quantization supply the system with the necessary fermionic degrees of freedom. But, moreover, the next step can be carried out in the direction of bosonization of the supersymmetry. In recent paper [2], devoted to the generalized harmonic oscillator systems, Brzezinski, Egusquiza and Macfarlane have pointed out on a remarkable possibility of uncovering supersymmetry structure totally described within the Fock space of the bosonic oscillator. In the present letter we shall investigate such a possibility of realizing N = 2 supersymmetry for the case of quantum mechanical bosonic oscillator revealing similarity of the constructions with the standard ones comprising fermionic degrees of freedom. Certainly, the bosonization mechanisms for the supersymmetry could find their concrete interesting applications similarly to the Bose-Fermi transmutaion constructions, known since very long time ago [3], which nowdays together with the anyonic generalizations are applied, e.g., to the description of the planar physical phenomena, viz. high-Tc superconductivity and fractional quantum Hall effect [4]. We shall start with the case of the ordinary bosonic oscillator and realize N = 2 supesymmetry in terms of its creation and annihilation operators demonstrating that there are two possibilities corresponding to the cases of exact and spontaneuosly broken supersymmetry. Proceeding from the broken supersymmetry case, we shall construct the fermionic oscillator variables, i.e. realize a Bose-Fermi transformation in terms of only one bosonic oscillator. The fermionic oscillator variables will permit us to represent the Hamiltonian for both cases in the form similar to the standard supersymmetric form and construct spin-1/2 representation of the SU(2) group on the one bosonic oscillator Fock space. Then we shall generalize all the constructions to the case of the bosonic oscillator operators obeying the deformed Heisenberg algebra involving the Klein operator [5, 2]. The interesting feature of such generalization is that in the case of the spontaneously broken supersymmetry the scale of supersymmetry breaking is defined by the deformation parameter of the system. After that we shall show how the described constructions can be generalized to the more complicated N = 2 supersymmetric systems, in particular, corresponding to the Witten supersymmetric quantum mechanics.