  We utilize an effective field theory approach to calculate Casimir
interactions between objects bound to thermally fluctuating fluid surfaces or
interfaces. This approach circumvents the complicated constraints imposed by
such objects on the functional integration measure by reverting to a point
particle representation. To capture the finite size effects, we perturb the
Hamiltonian by DH that encapsulates the particles' response to external fields.
DH is systematically expanded in a series of terms, each of which scales
homogeneously in the two power counting parameters: \lambda \equiv R/r, the
ratio of the typical object size (R) to the typical distance between them (r),
and delta=kB T/k, where k is the modulus characterizing the surface energy. The
coefficients of the terms in DH correspond to generalized polarizabilities and
thus the formalism applies to rigid as well as deformable objects.
Singularities induced by the point particle description can be dealt with using
standard renormalization techniques. We first illustrate and verify our
approach by re-deriving known pair forces between circular objects bound to
films or membranes. To demonstrate its efficiency and versatility, we then
derive a number of new results: The triplet interactions present in these
systems, a higher order correction to the film interaction, and general scaling
laws for the leading order interaction valid for objects of arbitrary shape and
internal flexibility.
