  In this paper we study the foundations of the algebraic treatment of
classical and quantum field theories for Dirac fermions under external
backgrounds following the initial contributions made by several collegues. The
treatment is restricted to contractible spacetimes of globally hyperbolic
nature in dimensions $d\ge 4$. In particular, we construct the classical
M{\o}ller maps intertwining the configuration spaces of charged and uncharged
fermions. In the last part, as a first step towards a quantization of the
theory, we explore the combination of the classical M{\o}ller maps with
Hadamard bidistributions and prove that they are involutive isomorphisms
(algebraically and topologically) between suitable (formal) algebras of
functionals (observables) over the configuration spaces of charged and
uncharged Dirac fields.
