Renormalizing the renormalization group pathologies

We review the status of the “pathologies” of the Renormalization Group (RG) encountered when one tries to define rigorously the RG transformation as a map between Hamiltonians. We explain their origin and clarify their status by relating them to the Griffiths’ singularities appearing in disordered systems; moreover, we suggest that the best way to avoid those pathologies is to use the contour representation rather than the spin representation for lattice spin models at low temperatures. Finally, we outline how to implement the RG in the contour representation.