Casimir and Casimir-Polder interactions for magneto-dielectric materials: Surface scattering expansion

We develop a general multiple scattering expansion (MSE) for computing Casimir forces between magneto-dielectric bodies and Casimir-Polder forces between polarizable particles and magneto-dielectric bodies. The approach is based on fluctuating electric and magnetic surface currents and charges. The surface integral equations for these surface fields can be formulated in terms of surface scattering operators (SSOs). We show that there exists an entire family of such operators. One particular member of this family is only weakly divergent and allows for a MSE that appears to be convergent for general magneto-dielectric bodies. We prove a number of properties of this operator, and demonstrate explicitly convergence for sufficiently low and high frequencies, and for perfect conductors. General expressions are derived for the Casimir interaction between macroscopic bodies and for the Casimir-Polder interaction between particles and macroscopic bodies in terms of the SSO, both at zero and finite temperatures. An advantage of our approach over previous scattering methods is that it does not require the knowledge of the scattering amplitude ($T$ operator) of the bodies. A number of simple examples are provided to demonstrate the use of the method. Some applications of our approach have appeared previously [T. Emig and G. Bimonte, Phys. Rev. Lett. 130, 200401 (2023)]. Here we provide additional technical aspects and details of our approach.