  We investigate the SO(4,1) gauge-invariant theory of the Dirac fermions in
the external field of the Kaluza-Klein monopole, pointing out that the quantum
modes can be recovered from a Klein-Gordon equation analogous to the Schr\"
odinger equation in the Taub-NUT background. Moreover, we show that there is a
large collection of observables that can be directly derived from those of the
scalar theory. These offer many possibilities of choosing complete sets of
commuting operators which determine the quantum modes. In addition there are
some spin- like and Dirac-type operators involving the covariantly constant
Killing-Yano tensors of the hyper-K\" ahler Taub-NUT space. The energy
eigenspinors of the central modes in spherical coordinates are completely
evaluated in explicit, closed form.
