Abstract
A two-body system with scalar constituents of finite masses is described by a pair of coupled Klein-Gordon equations. The modification required for preserving compatibility in the presence of an external field is nontrivial but can be exactly carried out beyond the static case when the external potential admits a suitable invariance. The conditions ensuring this property are exhibited assuming that the external field is either electromagnetic or a pure spin-two tensor (like a weak gravitational field). Special attention is devoted to plane waves. A suitable superposition of linearly polarized waves permits to apply this method when the field is electromagnetic. Another example is given by an external spin-two field obeying the propagation equation with a mass term.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Relativistic Action-at-a-Distance: Llosa, J. (ed.): Classical and Quantum Aspects (Lecture Notes in Physics, Vol. 162). Berlin-New York: Springer 1982, and references therein; Longhi,G., Lusanna, L: Constraints Theory and Relativistic Dynamics. Singapore: World Scientific 1987
Droz-Vincent, P.: Rep. Math. Phys.8, 79 (1975); Phys. Rev.D19, 702 (1979)
Coupled wave equations have been considered also by Bel L: In: Differential Geometry and Relativity (Cahen, M., Flato, M. eds.), p. 197. Dordrecht: Reidel 1976; Phys. Rev.D28, 1308 (1983); Leutwyler, H., Stern, J.: Ann. Phys. (NY)112, 94 (1978); Phys. Lett.B73, 75 (1978); Crater, H., Van Alstine, P.: Phys. Lett.B100, 166 (1981)
Rizov, V. A., Todorov, I. T., Aneva, B. L.: Nucl. Phys.B98, 447 (1975); Rizov, V. A., Sazdjian, H., Todorov, I. T.: Ann. Phys.165, 59 (1985)
Sazdjian, H.: Phys. Lett.B156, 381 (1985); J. Math. Phys.28, 2618 (1987)
Bijtebier, J.: Nuoyo Cim.A102, 1285 (1989)
Droz-Vincent, P.: Nuovo. Cim.A105, 1103 (1992)
Bijtebier, J., Broekaert, J.: Nuovo Cim.A105, 351 (1992)
According to Bethe, H., Salpeter, E. E.: Phys. Rev.84, 1232 (1951) extension to the case where an external field is present is “straightforward but computational complications can make it prohibitive”. According to Grotch, H., Hegstrom, R. A.: Phys. Rev.A4, 59 (1971) “the BS equation may be minimally coupled provided the external field is time-independent”
Bijtebier, J., Broekaert, J.: Nuovo Cim.A105, 625 (1992)
Parker, L.: Phys. Rev.D22, 1922 (1980); Tourrenc, P., Grossiord, J. L.: Nuovo Cim.32B, 163 (1976)
Fischbach, E. Freeman, B. S., Cheng, W.-K.: Phys. Rev.D23, 2157 (1981)
Bakamjian, B., Thomas, L. H.: Phys. Rev.92, 1300 (1953)
Droz-Vincent, P.: Nuovo Cim. (Sex. X)B45, 245 (1966)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics Vol. 1: Chapt. VIII, Theorem 33; Vol. 2: Fourier Analysis, Self-Adjointness, Chapt. X, Example 3, p. 205. New York: Academic Press 1975
Amrein, W. O.: Non-Relativistic Quantum Dynamics (Mathematical Physics Studies, Vol. 2), Chapt. 2, Proposition 2.11, p. 38. Dordrecht: Reidel 1981
Amrein, W. O.: Ref. [16], Chapt. 1, Problem 1.12 (d), p. 17
Grotch, H., Hegstrom, R. A.: Phys. Rev.A4, 59 (1971)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Droz-Vincent, P. The quantum relativistic two-body problem in time-dependent external potentials. Few-Body Systems 14, 97–115 (1993). https://doi.org/10.1007/BF01076018
Received:
Revised:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01076018