no. 2
Predictive relativistic mechanics of systems of N particles with spin. II. The electromagnetic interaction
Annales de l'institut Henri Poincaré. Section A, Physique Théorique, Tome 34 (1981) no. 2, pp. 231-252.
partie précédente Predictive relativistic mechanics of systems of N particles with spin 
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     author = {Bel, L. and Martin, J.},
     title = {Predictive relativistic mechanics of systems of {N} particles with spin. {II.} {The} electromagnetic interaction},
     journal = {Annales de l'institut Henri Poincar\'e. Section A, Physique Th\'eorique},
     pages = {231--252},
     publisher = {Gauthier-Villars},
     volume = {34},
     number = {2},
     year = {1981},
     mrnumber = {610866},
     language = {en},
     url = {http://www.numdam.org/item/AIHPA_1981__34_2_231_0/}
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Bel, L.; Martin, J. Predictive relativistic mechanics of systems of N particles with spin. II. The electromagnetic interaction. Annales de l'institut Henri Poincaré. Section A, Physique Théorique, Tome 34 (1981) no. 2, pp. 231-252. http://www.numdam.org/item/AIHPA_1981__34_2_231_0/

[1] L. Bel and J. Martin, Ann. Institut H. Poincaré, t. 23, 1980, p. 409.

[2] V. Bargmann, L. Michel and V.L. Telegdi, Phys. Rev. Lett., t. 2, 1959, p. 435.

[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] 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. | JFM

[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] 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] This condition eliminates the presence of an electric dipolar moment.

[8] We use the convention η0123 = + 1, and consequently η0123 = - 1.

[9] We here consider the advanced propagator as an open possibility.

[10] See for example: J. Cohn and H. Wiebe, J. Math. Phys., t. 17, 1976, p. 1496.

[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] See for example: L. BEL and X. FUSTERO (ref. 3).

[13] See for example: Y. Choquet-Bruhat, Géométrie Différentielle et Systèmes Extérieurs, éd. Dunod, Paris, 1968. | MR | Zbl

[14] In particular formulae (4.14 b) and (4.34) of BM.

[15] For the case of spinless particles consult ref. 12.

[16] For more details on Exterior Calculus techniques consult ref. 13.

[17] This part is already obtained from Darwin's well-known Lagrangian: C.G. Darwin, Phil. Mag., t. 39, 1920, p. 537.

[18] This term, which contains a Dirac « delta », is purely quantum mechanical. Consult the refs. at 4.

[19] L. Bel, Contribution to Differential Geometry and Relativity, Cahen and Flato (eds.), D. Reidel Publishing Co., Dordrecht, Holland, 1976. | MR