  The causal perturbation expansion defines the Dirac sea in the presence of a
time-dependent external field. It yields an operator whose image generalizes
the vacuum solutions of negative energy and thus gives a canonical splitting of
the solution space into two subspaces. After giving a self-contained
introduction to the ideas and techniques, we show that this operator is in
general not idempotent. We modify the standard construction by a rescaling
procedure giving a projector on the generalized negative-energy subspace. The
resulting rescaled causal perturbation expansion uniquely defines the fermionic
projector in terms of a series of distributional solutions of the Dirac
equation. The technical core of the paper is to work out the combinatorics of
the expansion in detail. It is also shown that the fermionic projector with
interaction can be obtained from the free projector by a unitary
transformation. We finally analyze the consequences of the rescaling procedure
on the light-cone expansion.
