Recursive Graphical Construction of Tadpole-Free Feynman Diagrams and Their Weights in ϕ 4 -Theory

In 1982 Hagen Kleinert proposed a program for systematically constructing all Feynman diagrams of a field theorytogether with their proper weights by graphically solving a set of functional differential equations [1]. It relies onconsidering a Feynman diagram as a functional of its graphical elements, i.e., its lines and vertices. Functionalderivatives with respect to these graphical elements are represented by removing lines or vertices of a Feynmandiagram in all possible ways. With these graphical operations, the program proceeds in four steps. First, a nonlinearfunctional differential equation for the free energy is derived as a consequence of the field equations. Subsequently,this functional differential equation is converted to a recursion relation for the loop expansion coefficients of the freeenergy. From its graphical solution, the connected vacuum diagrams are constructed. Finally, all diagrams of n-pointfunctions are obtained by removing lines or vertices from the connected vacuum diagrams. This program was recentlyused to systematically generate all Feynman diagrams of QED [2] and of φ