N-body relativistic systems

Droz-Vincent, Ph.

Annales de l'institut Henri Poincaré. Section A, Physique Théorique, Tome 32 (1980) no. 4, pp. 377-389.

[1] An exhaustive list of references is by now impossible. See for instance

R.N. Hill, J. Math. Phys., t. 8, 1967, p. 201.

J.G. Wray, Phys. Rev., t. D 1, n° 8, 1970, p. 2212.

L. Bel, Ann. Inst. Henri Poincaré, t. 12, 1970, p. 307. | Numdam | MR

R. Arens, Arch. for Rat. Mech. and Analysis, t. 47, 1972, p. 255. | MR | Zbl

C. Fronsdal, Phys. Rev., t. D 4, 1971, p. 1689.

I.T. Todorov, Phys. Rev., t. D 3, 1971, p. 2351.

H. Leutwyler and J. Stern, Nucl. Phys., t. B 133, 1978, p. 115.

T. Takabayashi, Prog. Theor. Phys., t. 54, n° 2, 1975, p. 563.

D. Dominici, J. Gomis, G. Longhi, Nuovo Cimento, t. 48 A, 1978, p. 257; Nuovo Cimento, t. 48 B, 1978, p. 152.

And also references [2-4] and [8].

Quoted below.

[2] Ph. Droz-Vincent, Lett. Nuovo Cim., t. 1, 1969, p. 839; Physica Scripta, t. 2, 1970, p. 129. | Zbl

[3] Ph. Droz-Vincent, Reports on Math. Phys., t. 8, n° 1, 1975, p. 79. | MR

[4] Ph. Droz-Vincent, Ann. Inst. Henri Poincaré, t. 27, 1977, p. 407. | Numdam | MR

[5] G. Preparata and K. Szego, Phys. Letters, t. 68 B, 1977, p. 239.

T. Takabayachi, Progr. Theor. Phys., t. 57, 1977, p. 331; t. 58, 1977, p. 1229; D. P. N. U. Report 15-78, 1978.

H.W. Crater, Phys. Rev., t. D 18, n° 8, 1978.

[6] L. Bel and X. Fustero, Ann. Inst. Henri Poincaré, t. 24, 1976, p. 411. See also

L. Bel, Phys. Rev., t. D 18, n° 12, 1979, p. 4770. In their case, classical field theory automatically provides N-body difference-differential equations, as usual. Then they reduce these equations to a predictive differential system by a series expansion method. In our case one wishes to ignore field theory from the outset.

[7] The spirit of our formulation is similar to that of P.A.M. Dirac, Commun. Dublin Inst. Adv. Studies, A, n° 2, 1943. But of course we take into account the facts implied by Currie's No-Go Theorem.

[8] D.G. Currie, J. Math. Phys., t. 4, 1963, p. 1470; Phys. Rev., t. 142, 1966, p. 817. | MR | Zbl

D.G. Currie, T.F. Jordan and E.C.C. Sudarshan, Rev. Mod. Phys., t. 35, 1963, p. 350. | MR

H. Leutwyler, Nuovo Cim., t. 37, 1965, p. 556.

[9] Ph. Droz-Vincent, C. R. Acad. Sc. Paris, t. A 182, 1979.

[10] Trivial for a single particle. For N = 2 see ref. [4] and DROZ-VINCENT, in Volume in the honor of A. Lichnerowicz, Cahen and Flato, Ed. D. Reidel, Dordrecht. The argument holds for any N. It is based upon the « individuality » property expressed in eq. (1.4).

[11] Ph. Droz-Vincent, Lett. Nuovo Cim., t. 23, n° 5, 1978, p. 184. | MR

[12] Ph. Droz-Vincent, Phys. Rev., t. D 19, n° 2, 1979, p. 702. | MR

[13] Note that the sign of the potential depends on the space time signature.

[14] For N = 2, see for example:

R.P. Feynman, M. Kislincer and R. Ravndal, Plays. Rev., t. D 3, 1971, p. 2706.

Y.S. Kim and M.L. Noz, Phys. Rev., t. D 15, 1977, p. 335.

J.F. Gunion and L.F. Li, Phys. Rev., t. D 12, 1975, p. 3583.

H.W. Crater, Phys. Rev., t. D 16, 1977, p. 1580. For N = 3, see ref. [5].

[15] Note that abandoning the single-potential assumption will only introduce interaction terms in the « subsidiary » equations (4.6).