  The QED four-photon amplitude has been well-studied by many authors, and
on-shell is treated in many textbooks. However, a calculation with all four
photons off-shell is presently still lacking, despite of the fact that this
amplitude appears off-shell as a subprocess in many different contexts, in
vacuum as well as with some photons connecting to external fields. The present
paper is the first in a series of four where we use the worldline formalism to
obtain this amplitude explicitly in terms of hypergeometric functions, and
derivatives thereof, for both scalar and spinor QED. The formalism allows us to
unify the scalar and spinor loop calculations, to avoid the usual breaking up
of the amplitude into three inequivalent Feynman diagrams, and to achieve
manifest transversality as well as UV finiteness at the integrand level by an
optimized version of the integration-by-parts procedure originally introduced
by Bern and Kosower for gluon amplitudes. The full permutation symmetry is
maintained throughout, and the amplitudes get projected naturally into the
basis of five tensors introduced by Costantini et al. in 1971. Since in many
applications of the "four-photon box" some of the photons can be taken in the
low-energy limit, and the formalism makes it easy to integrate out any such
leg, apart from the case of general kinematics (part 4) we also treat the
special cases of one (part 3) or two (part 2) photons taken at low energy. In
this first part of the series, we summarize the application of the worldline
formalism to the N-photon amplitudes and its relation to Feynman diagrams,
derive the optimized tensor-decomposed integrands of the four-photon amplitudes
in scalar and spinor QED, and outline the computational strategy to be followed
in parts 2 to 4.
