DIRAC – KÄHLER PARTICLE IN THE EXTERNAL MAGNETIC FIELD, CYLINDRICAL TETRAD AND FEDOROV – GRONSKIY METHOD
##plugins.themes.bootstrap3.article.sidebar##
##plugins.themes.bootstrap3.article.main##
Анатацыя
16-component system of equations for the Dirac – Kähler particle in the presence of the external uniform magnetic field has been studied. This equation describes a multi-spin boson field equivalent to the scalar, pseudoscalar, vector, pseudovecto, and antisymmetric tensor. On the searched solutions, we diagonalize operators of the energy, the third projection of the total angular momentum, and the third projection of the linear momentum. After separating the variables, we derive the system of sixteen first order differential equations in polar coordinate. To resolve this system, we apply the method by Fedorov – Gronskiy based on projective operators constructed from generator j12 for the 2-rank bispinor. According to this approach, we decompose the complete wave function into three 16-dimensional projective constituents, each expressed trough only one corresponding function of the polar coordinate. There are imposed additional differential constraints which permit us to transform all equations to algebraic form. Three basic variables are constructed in terms of the confluent hypergeometric functions, at this a quantization rule arises due to the presence of the external magnetic field. The 16-dimensional structure of solutions is determined by the linear algebraic system of equations. We have found five lineraly independent solutions.
##plugins.themes.bootstrap3.article.details##
Бібліяграфічныя спасылкі
1. Dirac, P. A. M. The quantum theory of the electron. Part I / P. A. M. Dirac // Proc. Roy. Soc. A. – 1928. – Vol. 117. – P. 610–624; The quantum theory of the electron. Part II // Proc. Roy. Soc. A. – 1928. – Vol. 118. – P. 351–361.
2. Darwin, C. G. The electron as a vector wave / C. G. Darwin // Proc. Roy. Soc. Lond. A. – 1927. – Vol. 116. – P. 227–253; The wave equations of the electron // Proc. Roy. Soc. Lond. A. – 1928. – Vol. 118, nr 780. – P. 654–680.
3. Ivanenko, D. Zur theorie des magnetischen electrons / D. Ivanenko, L. Landau // Zeit. Phys. – 1928. – Vol. 48, nr 8. – P. 340–348.
4. Kähler, E. Der innere differentialkalkül / E. Kähler // Rendiconti di Mat. (Roma). – 1962. – Vol. 21. – nr 3, 4. – P. 425–523.
5. Red’kov, V. M. On equations for the Dirac – Kähler field and bosons with different parities in the Riemannian space / V. M. Red’kov // Proceedings of the National Academy of Sciences of Belarus. Ser. fiz.-mat. – 2000. – Nr. 1. – 90–95.
6. Red’kov, V. M. Dirac-Kähler field, spinor technique, and 2-potential approach to electrodynamics with two charges / V. M. Red’kov // Nonlinear Dynamics and Applications. – 2008. – Vol. 15. – P. 147–163.
7. Red’kov, V.M. Field particles in Riemannian space and the Lorentz group / V. M. Red’kov. – Minsk : Belarussian Science, 2009. – 486 p. (in Russian).
8. Red’kov, V. M. Tetrad formalism, spherical symmetry and Schrödinger basis group / V. M. Red’kov. – Minsk : Belarussian Science, 2011. – 339 p. (in Russian).
9. V. M. Red’kov. Dirac – Kähler field and 2-potential approach to electrodynamics with two сharges. Chapter I / V. M. Red’kov // Progress in Relativity, Gravitation, Cosmology. – New York :Nova Science Publishers. Inc., 2012. – P. 23–37.
10. Ovsiyuk, E. Some Consequences from the Dirac-Kähler Theory: on Intrinsic Spinor Substructure of the Different Boson Wave Functions / E. Ovsiyuk, O. Veko, V. Red’kov // Nonlinear Phenomena in Complex Systems. – 2013. – Vol. 16, nr 1. – P. 13–23.
11. Ovsiyuk, E. M. Maxwell Electrodynamics and Boson Fields in Spaces of Constant Curvature / E. M. Ovsiyuk, V. V. Kisel, V. M. Red’kov. – New York : Nova Science Publishers Inc., 2014. – 486 p.
12. Veko, O. V. Lounesto classification and the theory of the Dirac-Kähler field: on intrinsic spinor substructure of the different boson wave functions / O. V. Veko, E. M. Ovsiyuk, V. M. Red’kov // Relativity, Gravitation, Cosmology: Foundations. – New York : Nova Science Publishers, Inc, 2015. – P. 75–88.
13. Dirac – Kähler particle in Riemann spherical space: boson interpretation / A. M. Ishkhanyan [et al.] // Canad. J. Phys. – 2015. – Vol. 93. – P. 1427–1433.
14. The Dirac – Kähler field in spherical Riemann space: boson interpretation, exact solutions / A. M. Ishkhanyan [et al.] // Proceedings of the Brest State Pedagogical University. Ser. Physics. Mathematics. – 2015. – Nr. 1. – P. 15–26.
15. Ovsiyuk, E. M. The Dirac – Kähler particle in Lobachevsky space, nonrelativistic approximation, exact solutions / E. M. Ovsiyuk, A. N. Red’ko, V. M. Red’kov // Proceedings of the National Academy of Sciences of Belarus. Ser. fiz.-mat. – 2015. – Nr 4. – P. 61–70.
16. Chichurin, A. V. Dirac – Kähler particle in Riemann spherical space: analytical and numerical study, visualization / A. V. Chichurin, E. M. Ovsiyuk, V. M. Red’kov // Studia i Materialy. – 2015. – Vol. 1, nr 9. – P. 41–54.
17. Nonrelativistic description of the Dirac – Kähler particle on the background of curved space-time / K. V. Kazmerchuk [et al.] // Relativity, Gravitation, Cosmology: Foundations. – New York : Nova Science Publishers, Inc., 2015. – P. 59–74.
18. Veko, O. V. Lounesto classification and the theory of the Dirac – Kähler field on intrinsic spinor sub-structure of the different boson wave functions / O. V. Veko, E. M. Ovsiyuk, V. M. Red’kov // Relativity, Gravitation, Cosmology: Foundations. – New York : Nova Science Publishers, Inc., 2015. – P. 75–88.
19. Pletukhov, V. A. Relativistic wave equations and intrinsic degrees of freedom / V. A. Pletukhov, V. M. Red’kov, V. I. Strazhev. – Minsk : Belarussian Science, 2015. – 328 p. (in Russian).
20. Elementary Particles with Internal Structure in External Fields. Vol I. General Theory / V. V. Kisel [et al.]. – New York : Nova Science Publishers Inc., 2018. – 404 p.
21. Elementary Particles with Internal Structure in External Fields. Vol II. Physical Problems / V. V. Kisel [et al.]. – New York : Nova Science Publishers Inc., 2018. – 402 p.
22. Gronskiy, V. K. Magnetic properties of a particle with spin 3/2 / V. K. Gronskiy, F. I. Fedorov // Doklady of the National Academy of Sciences of Belarus. – 1960. – Vol. 4, № 7. – P. 278–283.