Born-series approach to the calculation of Casimir forces

Abstract: 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.