Radiation Back-Reaction and Renormalization in Classical Field Theory with Singular Sources

We propose a general covariant method for regularizing the radiation back-reaction in linear and nonlinear field theory models with singular sources. Typical examples of such sources are the currents produced by extended relativistic objects (branes). As an illustration, we consider the models of minimal and nonminimal coupling of a brane to an $n$-form gauge field, a scalar field, and the Einstein gravity field. We find the structure of divergent and finite contributions due to the radiation back-reaction and obtain relations for the parameters of the theory ensuring cancellation of divergences. We prove that the divergences are Lagrangian n the case where the metric induced on the brane surface is nondegenerate. We find special types of a (nonminimal) coupling leading to local and Lagrangian effective equations of motion of the brane. We show that the requirement for classical renormalizability imposes strong restrictions on the self-coupling vertices of the field, similar to the quantum renormalizability conditions. In particular, we establish the nonrenormalizability of the gravitational self-coupling of a codimension-($k>2$) brane, whereas for $k\leq2$, the theory becomes not only renormalizable but also finite.