Yu. Kurochkin, V. Otchik, Dz. Shoukovy On the magnetic field in the extended Lobachevsky Space B.I. Stepanov Institute of Physics of NAS Minsk 2013

Plan 1. The realization of the three dimensional extended Lobachevsky space in the four dimensional pseudo- E euclidean space. 2. About systems of the coordinates in the three dimensional Lobachevsky space. Hyperbolic system of coordinates. 3.Analog of the uniform magnetic field in the hyperbolic system of coordinates. 4.Charged particle in analog of the uniform magnetic field in the three dimensional extended Lobachevsky space. 5. Symmetry of the quantum mechanical problem. 6. The realization of the three dimensional extended Lobachevsky space in the three dimensional Euclidean space 7. Connection with problem of the magnetic charge. 8. Conclusion

In terms of pseudoeuclidean space, Lobachevsky space is realized on the upper sheet of a two-sheeted hyperboloid and imaginary Lobachevsky space is realized on single-sheet hyperboloid Realization of the Lobachevsky space

Pictures prof. M.A. Cheshkova Realization of the Lobachevsky space

Hyperbolic systems of coordinates Real Lobachevsky space (1) In all the cases, the Maxwell equations and problem of motion of charged particle in the magnetic field are considered in the four dimensional space- time with Robertson –Walker metrics. (2)(2) I. Imaginary Lobachevsky space (1’) z  z+i  /2, R  iR and here the metrics of imaginary Lobachevsky space is

Analogs of the constant homogeneous magnetic field in three dimensional Lobachesky space We will use the analogy with well known case in the Euclidean space and take into account that the vector-potential and magnetic field defined in the Cartesian coordinates as can be written in the cylindrical coordinates in the form (3) The only nonzero component of the electromagnetic field corresponding to the potential (3) is It satisfies the Maxwell equations

Analogs of the constant homogeneous magnetic field in three dimensional Lobachevsky space In this case, the expression for the vector-potential in coordinate system (1), (2) of Lobachevsky space is given by (4) The nonzero component of field which corresponds to this vector-potential is (5) It satisfies the Maxwell equations in the Lobachevsky space:

Hamiltonian of the quantum mechanical problem (6)(6)

Symmetry generators of the quantum mechanical problem of the motion of the charged particle in the uniform magnetic field Commutative relation of the generators (7), (8) forms algebra Lee O(2.1) group (9)(9) (7)(7) (8)(8)

The quantum mechanical problem of the motion of the charged particle in the field Dirac monopole in the Lobachevsky space (10) the Schredinger equation of the problem (11)

Symmetry generators of the quantum mechanical problem of the motion of the charged particle in the field magnetic monopole (12) (13) (14)

Connection with analog uniform magnetic field with field of the magnetic charge (15) (16)

Magnetic fields in the real and imaginary Lobachevsky space (20) (22) (21)

The realization of the three dimensional extended Lobachevsky space in the three dimensional Euclidean space (23) (24) - radius-vector of the points of the three dimensional Euclidean space Realization of the real Lobachevsky space inner three dimensional sphere

The realization of the three dimensional extended Lobachevsky space in the three dimensional Euclidean space (25) (26) - radius-vector of the points of the three dimensional Euclidean space, Realization of the imaginary Lobachevsky space off three dimensional sphere

The realization of the three dimensional extended Lobachevsky space in the three dimensional Euclidean space (27) (28) In the real Lobachevsky space inner three dimensional sphere In the imaginary Lobachevsky space off three dimensional sphere

(30) The expressions of the vector-potential and magnetic fields in terms of variables of Euclidean space (29)

The expressions of the vectors of magnetic field in terms of variables of Euclidean space (31) (32)

References 1. Олевский, М.Н. Триортогональные системы в пространствах постоянной кривизны, в которых уравнение допускает полное разделение переменных / М.Н. Олевский // Мат. Сб. – – Т. 27. – С. 379– A.A.Bogush, V.M. Red’kov, G.G. Krylov. Schrödinger Particle in Magnetic and Electric Fields in Lobachevsky and Rieman Space. NPCS. 11, (2008). 3. V.V. Kudryashov, Yu.A. Kurochkin, E.M. Ovseyuk, V.M. Redkov, The motion in the magnetic fields in the Lobachevsky space. Doklady NAS of Belarus. 53, (2009). 4. Yu Kurochkin, V. Otchik, E. Ovsiyuk, Dz Shoukavy, On some integrable systems in the extended Lobachevsky space. Physics of Atomic Nuclei.74, No. 6, 944–948 (2011). 5. Ю.А. Курочкин, Е.М. Овсиюк. О движении заряженной частицы в магнитном поле в пространстве Лобачевского // Минск, – 28 с. – (Препринт / Национальная акад. наук Беларуси, Ин-т физики; № 747). 6. Yu. Kurochkin, V. Otchik, E. Ovsiyuk. Quantum mechanical particle in magnetic field in the Lobachevsky space. Proceedings of F&ANS 2010 Conference School, 2010, p Yu. A Kurochkin, V.S Otchik, E.M. Ovsiyuk. Magnetic field in the Lobachevsky space and related integrable systems. // Physics of Atomic Nuclei. – – Vol. 75, No. 10. – Р.1245–1249.; Ядерная физика. – – Т. 76, № 10. – С

References 8. Yu. Kurochkin, V. Otchik, E. Ovsiyuk, Dz. Shoukavy Quantum mechanical particle in magnetic field in the extended Lobachevsky space. Talk on International School –Seminar Actual problems of the physics of the microworld. Gomel, 1-12, August,2011 (in press). 9. Сингатулин Р.С., Щиголев В.К. Электрон в поле магнитного монополя на фоне искривленного пространства. В сб. Гравитация и теория относительности. Казань: Издательство Казанского университета. Вып. 16. с Ю.А. Курочкин, В.С. Отчик, В.Н. Терешенков. Электрон в поле магнитного заряда в трехмерном пространстве Лобачевского. Весці Акадэміі навук БССР. Серыя фізіка-матэматычных навук № 1, C.A. Hurst. Charge Quantization and Nonintegreble Lie Algebras / Ann. Phys. –v p В.И. Стражев, Л.М. Томильчик. Электродинамика с магнитным зарядом. Минск: Наука и техника – 334 с. 13. Shin’ichiro Ando, Alexander Kusenko. Evidence for Gamma-Ray Halos Around Active Galactic Nuclei and the First Measurement of Intergalactic Magnetic Fields // arXiv: v2 [astro-ph.HE] 2 Sep 2010.Evidence for Gamma-Ray Halos Around Active Galactic Nuclei and the First Measurement of Intergalactic Magnetic Fields

Conclusions TheT The using of the methods of the non Euclidean geometry very essential increase number of the possibilities of theoretical physics and provides new approaches to the modeling of configurations of the physical fields.