  We compare the quantisation of linear systems of bosons and fermions. We
recall the appearance of projectively flat connection and results on parallel
transport in the quantisation of bosons. We then discuss pre-quantisation and
quantisation of fermions using the calculus of fermionic variables. We then
define a natural connection on the bundle of Hilbert spaces and show that it is
projectively flat. This identifies, up to a phase, equivalent spinor
representations constructed by various polarisations. We introduce the concept
of metaplectic correction for fermions and show that the bundle of corrected
Hilbert spaces is naturally flat. We then show that the parallel transport in
the bundle of Hilbert spaces along a geodesic is the rescaled projection or the
Bogoliubov transformation provided that the geodesic lies within the complement
of a cut locus. Finally, we study the bundle of Hilbert spaces when there is a
symmetry.
