Geodesic
Legendre transformations and equations for Green's functions
Teoretičeskaâ i matematičeskaâ fizika, Tome 47 (1981) no. 2, pp. 184-197 
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/
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     author = {V. E. Rochev},
     title = {Legendre transformations and equations for {Green's} functions},
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     url = {http://geodesic.mathdoc.fr/item/TMF_1981_47_2_a3/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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 $3\to 1$, $3\to 2$, $3\to 3$.

[1] De Dominicis C., “Variational formulations of equilibrium statistical mechanics”, J. Math. Phys., 3 (1962), 983–989 | DOI | MR

[2] De Dominicis C., Martin P. C., “Stationary enthropy principle and renormalization in normal and superfluid systems”, J. Math. Phys., 5 (1964), 14–30 | DOI | MR

[3] Dahmen H. D., Jona-Lasinio G., “Variational formulation of Quantum field theory”, Nuovo Cim., 52A (1967), 807–836 | DOI

[4] Vasilev A. N., Kazanskii A. K., “Preobrazovaniya Lezhandra porozhdayuschikh funktsionalov v kvantovoi teorii polya”, TMF, 12 (1972), 352–360

[5] Vasilev A. N., Kazanskii A. K., “Uravneniya dvizheniya dlya preobrazovaniya Lezhandra proizvolnogo poryadka”, TMF, 14 (1973), 289–305 | MR

[6] Pismak Yu. M., “Dokazatelstvo 3-neprivodimosti tretego preobrazovaniya Lezhandra”, TMF, 18 (1974), 299–309 | MR

[7] Vasilev A. N., Kazanskii A. K., Pismak Yu. M., “Uravneniya dlya vysshikh preobrazovanii Lezhandra v terminakh 1-neprivodimykh vershin”, TMF, 19 (1974), 186–200

[8] Vasilev A. N., Kazanskii A. K., Pismak Yu. M., “Diagrammnyi analiz chetvertogo preobrazovaniya Lezhandra”, TMF, 20 (1974), 181–193

[9] Vasilev A. N., Funktsionalnye metody v kvantovoi teorii polya i statistike, Izd. LGU, L., 1976

[10] Symanzik K., “On the many-particle structure of Green's functions in quantum field theory”, J. Math. Phys., 1 (1960), 249–273 | DOI | MR

[11] Bros J., “Some analyticity properties implied by the two-particle structure of Green's functions in general quantum field theory”, Analytic Methods in Mathematical Physics, eds. R. Gilbert, R. Newton, Gordon and Breach, N. Y., 1970 | MR