  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 Witten's SUSYQM by applying to
it a nonlocal 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
supersymmetric two-fermion system. With this interpretation, the bosonization
construction is generalized to the case of N=1 supersymmetry in 2 dimensions.
The same special unitary transformation diagonalises 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 `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 nonlocal
unitary transformation to the Dirac field in the background of a nonlinear
scalar field in a kink configuration.
