Potential-moving, Migdal Recursion Formula, Differential Renormalization and Duality

Migdal's original recursion formula is rederived as a low-temperature approximation by an isotropic type of potential-moving. For self-dual spin or gauge systems this transformation is shown to be differentiably conjugate to another one, which is obtained as a high-temperature approximation. The conjugation relation is established through the duality mapping. This explains the mechanism leading to some exact results obtained with Migdal's differential renormalization equation. The last equation is also explicitly rederived as the result of potential-moving approximations inspired by the methods of differential renormalization in real space. Some applications and extensions of the above results are finally considered in connection with an approach, which was recently proposed for systematically improving Migdal's approximation.