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

M{\o}ller maps for Dirac fields in external backgrounds