Three aspects of bosonized supersymmetry and linear differential field equation with reflection Original Research Article

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 Wittenʹs SUSYQM by applying to it a non-local 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 a supersymmetric two-fermion system. With this interpretation, the bosonization construction is generalized to the case of N = 1 supersymmetry in two dimensions. The same special unitary transformation diagonalizes 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 the ‘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 non-local unitary transformation to the Dirac field in the background of a non-linear scalar field in a kink configuration.