Composite boson fields from local and nonlocal fermionic Lagrangians

Path-integral methods are used to reformulate nonlocal effective fermion Lagrangians in terms of composite bilocal boson variables. The bosons' equations of motion are derived and some inconsistencies in previous treatments are circumvented. When linearized, the equations are of the Bethe-Salpeter type. The local four-fermion interaction is treated as a limiting case. A generalization of the usual $\delta$ distribution is constructed and it is applied to the renormalization of the theory. With its help one can also establish the formal equivalence between a four-fermion local Lagrangian and other Lagrangians containing bosons and fermions with Yukawa interactions.