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The principles of the electromagnetic fluctuation-induced phenomena such asCasimir forces are well understood. However, recent experimental advancesrequire universal and efficient methods to compute these forces. While severalapproaches have been proposed in the literature, their connection is often notentirely clear, and some of them have been introduced as purely numericaltechniques. Here we present a unifying approach for the Casimir force and freeenergy that builds on both the Maxwell stress tensor and path integralquantization. The result is presented in terms of either bulk or surfaceoperators that describe corresponding current fluctuations. Our surfaceapproach yields a novel formula for the Casimir free energy. The path integralis presented both within a Lagrange and Hamiltonian formulation yieldingdifferent surface operators and expressions for the free energy that areequivalent. We compare our approaches to previously developed numerical methodsand the scattering approach. The practical application of our methods isexemplified by the derivation of the Lifshitz formula.

Unifying Theory for Casimir Forces: Bulk and Surface Formulations