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We present a worldline method for the calculation of Casimir energies forscalar fields coupled to magnetodielectric media. The scalar model we considermay be applied in arbitrary geometries, and it corresponds exactly to onepolarization of the electromagnetic field in planar layered media. Startingfrom the field theory for electromagnetism, we work with the two decoupledpolarizations in planar media and develop worldline path integrals, whichrepresent the two polarizations separately, for computing both Casimir andCasimir-Polder potentials. We then show analytically that the path integralsfor the transverse-electric (TE) polarization coupled to a dielectric mediumconverge to the proper solutions in certain special cases, including theCasimir-Polder potential of an atom near a planar interface, and the Casimirenergy due to two planar interfaces. We also evaluate the path integralsnumerically via Monte-Carlo path-averaging for these cases, studying theconvergence and performance of the resulting computational techniques. Whilethese scalar methods are only exact in particular geometries, they may serve asan approximation for Casimir energies for the vector electromagnetic field inother geometries.

Worldline approach for numerical computation of electromagnetic Casimir energies. I. Scalar field coupled to magnetodielectric media