  In the first-quantised worldline approach to quantum field theory, a
long-standing problem has been to extend this formalism to amplitudes involving
open fermion lines while maintaining the efficiency of the well-tested
closed-loop case. In the present series of papers, we develop a suitable
formalism for the case of quantum electrodynamics in vacuum (part one and two)
and in a constant external electromagnetic field (part three), based on
second-order fermions and the symbol map. We derive this formalism from
standard field theory, but also give an alternative derivation intrinsic to the
worldline theory. In this first part, we use it to obtain a Bern-Kosower type
master formula for the fermion propagator, dressed with $N$ photons, in terms
of the "$N$-photon kernel," where off-shell this kernel appears also in
"subleading" terms involving only $N-1$ of the $N$ photons. Although the
parameter integrals generated by the master formula are equivalent to the usual
Feynman diagrams, they are quite different since the use of the inverse symbol
map avoids the appearance of long products of Dirac matrices. As a test we use
the $N=2$ case for a recalculation of the one-loop fermion self energy, in $D$
dimensions and arbitrary covariant gauge, reproducing the known result. We find
that significant simplification can be achieved in this calculation choosing an
unusual momentum-dependent gauge parameter.
