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

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, ap peared in the context of integrable systems and dynamical symmetries. We sh ow that the minimally bosonized SUSYQM can be obtained from Witten's SUSYQM by applying to it a non-local unitary transformation with a subsequent red uction to one of the eigenspaces of the total reflection operator. The tran sformation depends on the parity operator, and the deformed Heisenberg alge bra with reflection, intimately related to parabosons and parafermions, eme rges 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 i nterpretation, the bosonization construction is generalized to the case of N = 1 supersymmetry in two dimensions. The same special unitary transformat ion diagonalizes the Hamiltonian operator of the 2D massive free Dirac theo ry. The resulting Hamiltonian is not a square root like in the Foldy-Wouthu ysen case, but is linear in spatial derivative. Subsequent reduction to the 'up' or 'down' field component gives rise to a linear differential equatio n with reflection whose 'square' is the massive Klein-Gordon equation. In t he massless limit this becomes the self-dual Weyl equation. The linear diff erential equation with reflection admits generalisations to higher dimensio ns and can be consistently coupled to gauge fields. The bosonized SUSYQM ca n also be generated applying the non-local unitary transformation to the Di rac field in the background of a non-linear scalar field in a kink configur ation. (C) 1999 Elsevier Science B.V.