  We propose a fibre bundle formulation of the mathematical base of
relativistic quantum mechanics. At the present stage the bundle form of the
theory is equivalent to its conventional one, but it admits new types of
generalizations in different directions. In the bundle description the
wavefunctions are replaced with (state) sections (covariant approach) or
liftings of paths (equivalently: sections along paths) (time-dependent
approach) of a suitably chosen vector bundle over space-time whose (standard)
fibre is the space of the wavefunctions. Now the quantum evolution is described
as a linear transportation of the state sections/liftings in the (total) bundle
space. The equations of these transportations turn to be the bundle versions of
the corresponding relativistic wave equations. Connections between the
(retarded) Green functions of these equations and the evolution operators and
transports are found. Especially the Dirac and Klein-Gordon equations are
considered.
