  A reformulation of fermionic QFT in electromagnetic backgrounds is presented
which uses methods analogous to those of conventional multiparticle quantum
mechanics. Emphasis is placed on the (Schr\"odinger picture) states of the
system, described in terms of Slater determinants of Dirac states, and not on
the field operator $\hat{\psi}(x)$ (which is superfluous in this approach). The
vacuum state `at time $\tau$' is defined as the Slater determinant of a basis
for the span of the negative spectrum of the `first quantized' Hamiltonian
$\hat{H}(\tau)$, thus providing a concrete realisation of the Dirac Sea. The
general S-matrix element of the theory is derived in terms of time-dependent
Bogoliubov coefficients, demonstrating that the S-matrix follows directly from
the definition of inner product between Slater determinants. The process of
`Hermitian extension', inherited directly from conventional multiparticle
quantum mechanics, allows second quantized operators to be defined without
appealing to a complete set of orthonormal modes, and provides an extremely
straightforward derivation of the general expectation value of the theory. The
concept of `radar time', advocated by Bondi in his work on k-calculus, is used
to generalise the particle interpretation to an arbitrarily moving observer. A
definition of particle results, which depends {\it only} on the observer's
motion and the background present, not on any choice of coordinates or gauge,
or of the particle detector. We relate this approach to conventional methods by
comparing and contrasting various derivations. Our particle definition can be
viewed as a generalisation to arbitrary observers of Gibbons' approach.
