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Approximate relativistic quantum Hamiltonians for N interacting particle systems
1.L. Bel and E. Ruiz, J. Math. Phys. 28, 18 (1988).
2.Summation over repeated upper and lower indices is assumed. Nevertheless we sometimes make explicit the summation symbol, mainly in the last sections.
3.This Hamiltonian is the sum of N standard one particle Hamiltonians. See, for instance, S. Schweber, An Introduction to Relativistic Quantum Field Theory (Harper & Row, New York, 1961).
4.The action of the operator is perfectly well denned in the momentum‐space. See, for instance, Ref. 3.
5.These calculations are more clearly and rigorously made using the momentum‐space representation [BR equation (5.19)]. This last procedure requires solving the Schrödinger equation in the momentum‐space the general solution of this equation is
6.See, for instance, S. Schweber (Ref. 3).
7.L. L. Foldy and S. A. Wouthuysen, Phys. Rev. 78, 29 (1950);
7.J. Bjorken and S. Drell, Relativistic Quantum Mechanics (McGraw‐Hill, New York, 1965).
8.We refer to the work on these classical systems by L. Bel and J. Martin, Ann. Inst. Henri Poincaré A 33, 409 (1980).
8.The description of the system is made by means of an extension to N particles of the one‐particle symplectic form worked out by J. M. Souriau, Structure des Systèmes Dynamiques (Dunod, Paris, 1970).
9.M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).
10.It amounts to saying that the system will be accurately described by some few terms of the series whenever the energies involved are much smaller than the rest masses of the particles. It means that we are dealing with true massive particles, for zero‐mass particles are excluded from our analysis.
11.See, for instance, K. Gotfried, Quantum Mechanics (Benjamin, London, 1974).
12.Even though contact terms can obviously be included in we have preferred to omit them, since their usual expressions come from spin‐ particle calculations and perhaps they may be rather different for some other kind of particles.
13.H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One and Two Electron Atoms (Springer‐Verlag, Berlin, 1957);
13.V. Berestetski, E. Lifchitz, and L. Pitayevski, Théorie Quantique Relativiste (Mir, Moscow, 1972);
13.a classical version of the Breit Hamiltonian that allows arbitrary values of the intrinsic angular momenta (classical spin) can be found in L. Bel and J. Martín, Ann. Inst. Henri Poincaré A 34, 235 (1981).
14.For gravitational interaction in theories other than general relativity (PPN with parameters β and γ), see B. M. Barker and R. F. O’Connell, Phys. Rev. D 14, 861 (1976). The two‐particle Hamiltonian that is given in this paper can be achieved by making the numerical constant that precedes each multiple sum in the expression of the operator corresponding to the Barker‐O’Connell Hamiltonian (5.3) into a simple function of the parameters β and γ.
15.B. M. Barker and R. F. O’Connell, Phys. Rev. D 12, 329 (1975);
15.B. M. Barker and R. F. O’Connell, Gen. Relativ. Gravit. 11, 149 (1979).
16.L. Landau and E. Lifchitz, Théorie du Champ (Mir, Moscow, 1966).
17.J. Ibáñez and J. Martín, Gen. Relativ. Gravit. 14, 439 (1982).
18.See the first source in Ref. 15.
19.S. Bažański, in Recent Developments in General Relativity (Pergamon, New York, 1962);
19.B. M. Barker and R. F. O’Connell, J. Math. Phys. 18, 1818 (1977).
20.J. Martín and J. L. Sanz, J. Math. Phys. 19, 780 (1978);
20.J. Martín and J. L. Sanz, 20, 25 (1979)., J. Math. Phys.
21.The electromagnetic Lagrangian up to order is discussed in B. M. Barker and R. F. O’Connell, Ann. Phys. (NY) 129, 358 (1980);
21.the gravitational Lagrangian in general relativity up to the same order is discussed in T. Damour, in Gravitational Radiation, edited by N. Deruelle and T. Piran (North‐Holland, Amsterdam, 1983).
22.X. Jaén, J. Llosa, and A. Molina, Phys. Rev. D 34, 2302 (1986).
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