Legendre transforms of the generating functionals in quantum field theory

A study is made of the Legendre transforms [1, 2] (the first and the second) of the generating functional of connected Green's functions. Equations of motion and constraint equations (equations in variational derivatives) are written down for these transforms. Anatysis of the iterative solution of these equations shows that the first (second) Legendre transform can be represented by a sum of single-particle-irreducible (respectively, two-particle-irreducible) Feynman graphs.