Recursive graphical construction of Feynman diagrams and their multiplicities in phi(4) and phi(2)A theory

The free energy of a field theory can be considered as a functional of the free correlation function. As such it obeys a nonlinear functional differen tial equation that can be turned into a recursion relation. This is solved order by order in the coupling constant to find all connected vacuum diagra ms with their proper multiplicities. The procedure is applied to a multicom ponent scalar field theory with a phi(4) self-interaction and then to a the ory of two scalar fields phi and A with an interaction phi(2)A. All Feynman diagrams with external lines are obtained from functional derivatives of t he connected vacuum diagrams with respect to the free correlation function. Finally, the recursive graphical construction is automatized by computer a lgebra with the help of a unique matrix notation for the Feynman diagrams.