Supersymmetry without Fermions

Abstract

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.