Geometry dependence of Casimir forces beyond the proximity approximation

Casimir interactions between macroscopic objects axe strongly influenced by their geometrical features as shape and orientation as well as by their material properties. The effect of geometry is commonly obtained from the proximity approximation (PA). Here we present a path integral quantization for the electromagnetic field in the presence of deformed metallic surfaces. From the resulting effective action the Casimir force between the surfaces can be calculated without the PA. For corrugated surfaces the force is obtained both perturbatively for small deformations and by a numerical approach for general deformation amplitudes. For general dielectric materials with flat surfaces a path integral based derivation of the Lifshitz theory is outlined, pointing towards a possible approach to study the combined effect of deformations and material properties.