Geodesic
Legendre transformations and equations for Green's functions
Teoretičeskaâ i matematičeskaâ fizika, Tome 47 (1981) no. 2, pp. 184-197

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Higher Legendre transformations of the generating function of the Green's functions are used to obtain relations which have the meaning of many-particle equations. The second Legendre transformation gives two relations: the Edwards equation and the Bethe–Salpeter equation. The third Legendre transformation gives a system of three equations corresponding to the processes $3\to 1$, $3\to 2$, $3\to 3$. 
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     author = {V. E. Rochev},
     title = {Legendre transformations and equations for {Green's} functions},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {184--197},
     publisher = {mathdoc},
     volume = {47},
     number = {2},
     year = {1981},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1981_47_2_a3/}
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V. E. Rochev. Legendre transformations and equations for Green's functions. Teoretičeskaâ i matematičeskaâ fizika, Tome 47 (1981) no. 2, pp. 184-197. http://geodesic.mathdoc.fr/item/TMF_1981_47_2_a3/