Predictive relativistic mechanics of systems of N particles with spin

Bel, L.; Martin, J.

Annales de l'institut Henri Poincaré. Section A, Physique Théorique, Tome 33 (1980) no. 4, pp. 409-442.

[1] We use the expression « Classical Mechanics » as opposed to « Quantum Mechanics ».

[2] J.M. Souriau, Structure des Systèmes Dynamiques, Dunod Université (Paris, 1970). | MR | Zbl

[3] R. Arens, Comm. Math. Phys., t. 21, 1971, p. 139. | MR

[4] L. Martinez Alonso, J. Math. Phys., t. 20, 1979, p. 219. | MR | Zbl

[5] L. Bel and J. Martin, Ann. Inst. H. Poincaré, t. 22 A, 1975, p. 173. | Numdam | MR

[6] J. Martin and J.L. Sanz, J. Math. Phys., t. 19, 1978, p. 780.

[7] See for instance : D.G. Currie, Phys. Rev., t. 142, 1966, p. 817; R.N. Hill, J. Math. Phys., t. 8, 1967, p. 201 ; L. Bel, Ann. Inst. H. Poincaré, t. 12, 1970, p. 307; Ph. Droz-Vincent, Physica Scripta, t. 2, 1970, p. 129; L. Bel, Ann. Inst. Poincaré, t. 14, 1970, p. 189 ; R. Arens, Arch. for Rat. Mech. and Analysis, t. 47, 1972, p. 255; L. Bel and X. Fustero, Ann. Inst. H. Poincaré, t. 24, 1979, p. 411. | MR

[8] See for instance : B.M. Barker and R.F. O'Connell, Phys. Rev. D, t. 12, 1975, p. 329; X. Fustero and E. Verdaguer, « Interaction among systems of finite size in Predictive Relativistic Mechanics » Preprint Universidad Autonoma de Barcelona. Spain, 1979; J.M. Gracia-Bondia, Physics Letters, 75 A, t. 4, 1980, p. 262.

[9] We use the sommation convention for all kind of indices: repeated indices, one in a covariant and the other one in a contravariant position, are summed over their range.

[10] We shall us the signature + 2. Therefore uλauaλ = - 1.

[11] ηijl represents the Levi-Civita symbol in three dimensions with η123 = + 1. In M4 we shall assume that η0123 = + 1.

[12] L.P. Eisenhart, Continuous groups of transformations. Dover Publications Inc. (New York, 1961). | MR | Zbl

[13] For systems of particles with no spin the proof of this result can be found in L. BEL, ref. 7, 12 and in L. BEL, Lecciones sobre Mecànica Relativista Predictiva. Departamento de Fisica Teorica, Universidad Autònoma de Barcelona, Spain1976.

[14] In this sub-section we consider an arbitrary initial time. This is more convenient because we shall forget momentarily the Poincaré invariance.

[15] We assume that the vectors πλa are time-like and future oriented (π0a > 0).

[16] See an alternative formulation in : H.P. Kunzle, J. Math. Phys., t. 15, 1974, p. 1033.

[17] See for instance : J.M. Souriau, ref. 2; R. Abraham, Foundation of Mechanics, W. A. Benjamin, 1967; C. Godbillon, « Géométrie différentielle et Mécanique Analytique », Hermann, 1969.

[18] See for instance the proof of ref. 21.

[19] This assumption is the most natural one which is consistent with the fact that the co-phase space for a system of N particles is the Nth cartesian power of the co-phase space for one particle.

[20] We introduce here the modulus of the spin to make eq. (4.15) dimensionally homogeneous. sa has the dimensions of an action.

[21] D.G. Currie, T.F. Jordan and E.C.G. Sudarshan, Rev. of Mod. Phys., t. 35, 1963, p. 530.

[22] This is due to the fact that we use « instantaneous spins » instead of « intrinsic spins » (See Section 2).

[23] L. Bel and X. Fustero, Ann. Inst. H. Poincaré, t. 24, 1979, p. 411.

[24] See for instance : Y. Choquet-Bruhat, Géométrie différentielle et Systèmes extérieurs, Dunod. Paris, 1968. | MR | Zbl

[25] For more details see ref. 23.

[26] Let us remark that this construction defines the quantities Ha unambiguously (there are no more arbitrary additive constants).