  The paper contains a differential-geometric foundations for an attempt to
formulate Lagrangian (canonical) quantum field theory on fibre bundles. In it
the standard Hilbert space of quantum field theory is replace with a Hilbert
bundle; the former playing a role of a (typical) fibre of the letter one.
Suitable sections of that bundle replace the ordinary state vectors and the
operators on the system's Hilbert space are transformed into morphisms of the
same bundle. In particular, the field operators are mapped into corresponding
field morphisms.
