  We derive exact expressions for the scalar and electromagnetic self-forces
and self-torques acting on arbitrary static extended bodies in arbitrary static
spacetimes with any number of dimensions. Non-perturbatively, our results are
identical in all dimensions. Meaningful point particle limits are quite
different in different dimensions, however. These limits are defined and
evaluated, resulting in simple "regularization algorithms" which can be used in
concrete calculations. In these limits, self-interaction is shown to be
progressively less important in higher numbers of dimensions; it generically
competes in magnitude with increasingly high-order extended-body effects.
Conversely, we show that self-interaction effects can be relatively large in
$1+1$ and $2+1$ dimensions. Our motivations for this work are twofold: First,
no previous derivation of the self-force has been provided in arbitrary
dimensions, and heuristic arguments presented by different authors have
resulted in conflicting conclusions. Second, the static self-force problem in
arbitrary dimensions provides a valuable testbed with which to continue the
development of general, non-perturbative methods in the theory of motion.
Several new insights are obtained in this direction, including a significantly
improved understanding of the renormalization process. We also show that there
is considerable freedom to use different "effective fields" in the laws of
motion---a freedom which can be exploited to optimally simplify specific
problems. Different choices give rise to different inertias, gravitational
forces, and electromagnetic or scalar self-forces, but there is a sense in
which none of these quantities are individually accessible to experiment.
Certain combinations are observable, however, and these remain invariant under
all possible field redefinitions.
