We present a procedure of differential renormalization at the one loop level which avoids introducing unnecessary renormalization constants and automatically preserves abelian gauge invariance. The amplitudes are expressed in terms of a basis of singular functions. The local terms appearing in the renormalization of these functions are determined by requiring consistency with the propagator equation. Previous results in abelian theories, with and without supersymmetry, are discussed in this context.Comment: 13 pages, LaTeX. Some equations corrected and a reference added. Complete ps paper also available at http://www-ftae.ugr.es/papiros.html or ftp://ftae3.ugr.es/pub/rmt/ugft73.p

We describe in detail the constrained procedure of differential renormalization and develop the techniques required for one-loop calculations. As an illustration we renormalize Scalar QED and show that the two-, three-and four-point Ward identities are automatically satisfied.

The method of differential renormalization is extended to the calculation of the one-loop graviton and gravitino corrections to (g−2) l in unbroken supergravity. Rewriting the singular contributions of all the diagrams in terms of only one singular function, U (1) gauge invariance and supersymmetry are preserved. We compare this calculation with previous ones which made use of momentum space regularization (renormalization) methods.