We study thermal Casimir and quantum nonretarded Lifshitz interactions between dielectrics in general geometries. We map the calculation of the classical partition function onto a determinant, which we discretize and evaluate with the help of Cholesky factorization. The quantum partition function is treated by path integral quantization of a set of interacting dipoles and reduces to a product of determinants. We compare the approximations of pairwise additivity and proximity force with our numerical methods. We propose a factorization approximation that gives rather good numerical results in the geometries that we study. (C) 2008 American Institute of Physics.
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