Baym and Kadanoff showed that we can satisfy the conservation law for any functional Φ Φ [ G ] , thanks to the Noether's theorem. This is followed by the fact that the equation of motion of G responding to one-body external fields apparently satisfies the space- and time- translational symmetries as well as the abelian gauge symmetry (phase symmetry), as long as the equation of motion is given with the derivative of Φ Φ [ G ] . Note that reverse is also true. Based on the diagramatic analysis, what Baym found is that δ δ Σ Σ ( 1 , [ G ] ) δ δ G ( 2 ) = δ δ Σ Σ ( 2 , [ G ] ) δ δ G ( 1 ) is needed to satisfy the conservation law. This is nothing but the completely-integrable condition, implying the existence of Φ Φ [ G ] such that Σ Σ [ G ] = δ δ Φ Φ [ G ] δ δ G (recall the completely-integrable condition for d f = A ( x , y ) d x + B ( x , y ) d y ).