Predictive relativistic mechanics of systems of N particles with spin. II. The electromagnetic interaction

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1. [1] L. Bel and J. Martin, Ann. Institut H. Poincaré, t. 23, 1980, p. 409. 
2. [2] V. Bargmann, L. Michel and V.L. Telegdi, Phys. Rev. Lett., t. 2, 1959, p. 435. 
3. [3] See for example: L. Bel, A. Salas and J.M. Sanchez-Ron, Phys. Rev., t. D 7, 1973, p. 1099; L. Bel and J. Martin, Phys. Rev., t. D 8, 1973, p. 4347; L. Bel, in Journées Relativistes de Toulouse, Université de Toulouse, Département de Mathé- matiques, 1974 ; L. Bel et X. Fustero, Ann. Inst. H. Poincaré, t. 25, 1976, p. 411. 
4. [4] G. Breit, Phys. Rev., t. 34, 1929, p. 553; Phys. Rev., t. 36, 1930, p. 383. A deduction of this Hamiltonian by Quantum Electrodynamic procedures may be found, in : LANDAU et LIFCHITZ, Théorie Quantique Relativiste (première partie), éditions Mir, Moscou, 1972. JFM55.0527.03
5. [5] A different classical derivation of the equations of motion and lagrangians relative to this Hamiltonian may be found, in : J. Llosa, Tesis doctoral, Universitat de Barcelona, 1978; X. Fustero and E. Verdaguer, preprint, Universitat Autonoma de Barcelona, Spain, 1979. 
6. [6] It is assumed that the curve is time-like and future oriented. We take signature + 2 for M4. Einstein's summation convention will be utilized for all kinds of indices; these will always be placed in the appropriate position (« covariant » or « contravariant ») to respect the said convention. 
7. [7] This condition eliminates the presence of an electric dipolar moment. 
8. [8] We use the convention η0123 = + 1, and consequently η0123 = - 1. 
9. [9] We here consider the advanced propagator as an open possibility. 
10. [10] See for example: J. Cohn and H. Wiebe, J. Math. Phys., t. 17, 1976, p. 1496. 
11. [11] Actually BARGMANN, MICHEL and TELEGDI only use these equations for the case of a homogeneous electromagnetic field. It should also be pointed out that similar Eqs. appear in the following: L.H. Thomas, Phil. Mag., t. 3, 1927, p. 1; J. Frenkel, Z. Physik, t. 37, 1926, p. 243; H.A. Kramers, Quantum Mechanics, North Holland Publishing Co., Amsterdam, 1957. In Kramer's equations there appear, nevertheless, certain inconsistencies, as is pointed out in Bargmann, Michel and Teledgi. 
12. [12] See for example: L. BEL and X. FUSTERO (ref. 3). 
13. [13] See for example: Y. Choquet-Bruhat, Géométrie Différentielle et Systèmes Extérieurs, éd. Dunod, Paris, 1968. Zbl0164.22001MR236824
14. [14] In particular formulae (4.14 b) and (4.34) of BM. 
15. [15] For the case of spinless particles consult ref. 12. 
16. [16] For more details on Exterior Calculus techniques consult ref. 13. 
17. [17] This part is already obtained from Darwin's well-known Lagrangian: C.G. Darwin, Phil. Mag., t. 39, 1920, p. 537. 
18. [18] This term, which contains a Dirac « delta », is purely quantum mechanical. Consult the refs. at 4. 
19. [19] L. Bel, Contribution to Differential Geometry and Relativity, Cahen and Flato (eds.), D. Reidel Publishing Co., Dordrecht, Holland, 1976. MR430976