×

Generalized Foldy-Wouthuysen transformation and pseudodifferential operators. (English. Russian original) Zbl 1274.81069

Theor. Math. Phys. 167, No. 2, 547-566 (2011); translation from Teor. Mat. Fiz. 167, No. 2, 171-192 (2011).
Summary: We show that the Foldy-Wouthuysen transformation and its generalizations are simplified if the methods of pseudodifferential operators are used, which also allow estimating the exactness of the transition from the Dirac equation to the reduced equations for electrons and positrons. The methods and techniques used can be useful not only in studying asymptotic solutions of the Dirac equation but also in other problems.

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35S05 Pseudodifferential operators as generalizations of partial differential operators
35J10 Schrödinger operator, Schrödinger equation
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] L. L. Foldy and S. A. Wouthuysen, Phys. Rev., 78, 29–36 (1950). · Zbl 0039.22605
[2] A. J. Silenko, J. Math. Phys., 44, 2952–2966 (2003); arXiv:math-ph/0404067v1 (2004); V. P. Neznamov, Dokl. Phys., 43, 531-533 (1998); Phys. Part. Nucl., 37, 86–103 (2006); arXiv:hep-th/0411050v2 (2004); V. P. Neznamov, ”The necessary and sufficient conditions for transformation from Dirac representation to Foldy-Wouthuysen representation,” arXiv:0804.0333v2 [math-ph] (2008); K. Yu. Bliokh, Europhys. Lett., 72, 7–13 (2005); arXiv:quant-ph/0501183v4 (2005). · Zbl 1062.81154
[3] V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Course of Theoretical Physics [in Russian], Vol. 4, Part 1. Relativistic Quantum Theory, Nauka, Moscow (1968); English transl., Pergamon, Oxford (1971); J. D. Bjorken and S. D. Drell, Relativistic Quantum Theory, Vol. 1, Relativistic Quantum Mechanics, McGraw-Hill, New York (1964); A. Messiah, Quantum Mechanics (Chap. 20, ”Dirac Equation”), Vol. 2, Wiley, New York (1976); V. G. Levich, Yu. A. Vdovin, and V. A. Mjamlin, Course of Theoretical Physics [in Russian] (Chap. 13, ”Relativistic quantum mechanics”), Vol. 2, Nauka, Moscow (1971).
[4] V. P. Maslov, Operator Methods [in Russian], Nauka, Moscow (1973); English transl.: Operational Methods, Mir, Moscow (1976); M. V. Karasev and V. P. Maslov, Nonlinear Poisson Brackets: Geometry and Quantization [in Russian], Nauka, Moscow (1991); English transl. (Transl. Math. Monogr., Vol. 119), Amer. Math. Soc., Providence, R. I. (1993).
[5] S. Yu. Dobrokhotov, Sov. Phys. Dokl., 28, 229–231 (1983); L. V. Berljand and S. Yu. Dobrokhotov, Sov. Phys. Dokl., 32, 714–716 (1987).
[6] V. V. Belov, S. Yu. Dobrokhotov, and T. Ya. Tudorovskii, Theor. Math. Phys., 141, 1562–1592 (2004); V. V. Belov, S. Yu. Dobrokhotov, and T. Ya. Tudorovskiy, J. Engrg. Math., 55, 183–237 (2006); arXiv:math-ph/0503041v1 (2005). · Zbl 1178.81080
[7] L. D. Landau and E. M. Lifshits, Course of Theoretical Physics [in Russian], Vol. 9, Statistical Physics: Part 2. Theory of the Condensed State, Nauka, Moscow (1978); English transl., Pergamon, Oxford (1980).
[8] I. M. Gel’fand, Lectures on Linear Algebra [in Russian], Nauka, Moscow (1971); English transl., Dover, New York (1989).
[9] M. V. Karasev and V. P. Maslov, J. Soviet Math., 15, 273–368 (1981). · Zbl 0482.58029
[10] V. P. Maslov, Théorie des perturbations et méthodes asymptotiques [in Russian], Moscow State Univ., Moscow (1965); French transl., Dunod, Paris (1972).
[11] V. P. Maslov and M. V. Fedoryuk, Semiclassical Approximation for Equations of Quantum Mechanics [in Russian], Nauka, Moscow (1976); English transl.: Semi-classical Approximation in Quantum Mechanics, (Math. Phys. Appl. Math., Vol. 7), Reidel, Dordrecht (1981). · Zbl 0458.58001
[12] G. Panati, H. Spohn, and S. Teufel, Comm. Math. Phys., 242, 547–578 (2003); arXiv:math-ph/0212041v2 (2002). · Zbl 1058.81020
[13] V. V. Grushin and S. Yu. Dobrokhotov, Math. Notes, 87, 521–536 (2010). · Zbl 1197.35086
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.