  The Casimir force between two objects is notoriously difficult to calculate
in anything other than parallel-plate geometries due to its non-additive
nature. This means that for more complicated, realistic geometries one usually
has to resort to approaches such as making the crude proximity force
approximation (PFA). Another issue with calculation of Casimir forces in
real-world situations (such as with realistic materials) is that there are
continuing doubts about the status of the standard Lifshitz treatment as a true
quantum theory. Here we demonstrate an alternative approach to calculation of
Casimir forces for arbitrary geometries which sidesteps both these problems.
Our calculations are based upon a Born expansion of the Green's function of the
quantised electromagnetic vacuum field, interpreted as multiple scattering,
with the relevant coupling strength being the difference in the dielectric
functions of the various materials involved. This allows one to consider
arbitrary geometries in single or multiple scattering simply by integrating
over the desired shape, meaning that extension beyond the PFA is trivial. This
work is mostly dedicated to illustration of the method by reproduction of known
parallel-slab results -- a process that turns out to be non-trivial and
provides several useful insights. We also present a short example of
calculation of the Casimir energy for a more complicated geometry, namely that
of two finite slabs.
