A METHOD OF INTEGRATION FOR CLASSICAL AND QUANTUM EQUATIONS BASED ON THE CONNECTION BETWEEN CANONICAL TRANSFORMATIONS AND IRREDUCIBLE REPRESENTATIONS OF LIE GROUPS

Magazev A. A., Shirokov I. V.

Information About Author:

We propose a method for integrating the right-invariant geodesic flows on Lie groups based on the use of a special canonical transformation in the cotangent bundle of group. We also describe an original method of constructing exact solutions for the Klein-Gordon equation on unimodular Lie groups. Finally, we formulate a theorem which establishes a connection between the special canonical transformation and irreducible representations of Lie group. This connection allows us to consider the proposed methods of integrating for classical and quantum equations in the framework of a unified approach.

Keywords: geodesic flow, the Klein-Gordon equation, canonical transformation, irreducible representation, integrability

References:

[1] Abraham R. et al. 1978 Foundations of mechanics (Addison-Wesley Publishing Company, Inc.).

[2] Miller Jr W. 1977 Symmetry and separation of variables (Addison-Wesley, Reading, Massachusetts).

[3] Bagrov V. G. and Gitman D. 1990 Exact solutions wave equations (Springer).

[4] Arnold V. I. 1966 Annales de l’institut Fourier 16(1) 319.

[5] Manakov S. 1976 Functional Analysis and Its Applications 10(4) 328.

[6] Mishchenko A. S. and Fomenko A. T. 1978 Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 42(2) 396.

[7] Magazev A. A. and Shirokov I. V. Theoretical and Mathematical Physics 136(3) 1212.

[8] Shapovalov A. V. and Shirokov I. V. Theoretical and Mathematical Physics 104(2) 921.

[9] Kirillov A. A. 1976 Elements of the Theory of Representations (Berlin: Springer-Verlag).

[10] Shirokov I. V. 2000 Theoretical and Mathematical Physics 123(3) 754.

[11] Dixmier J. 1977 Enveloping algebras (Newnes).

( 579.41 kB ) ( 464.27 kB )

Issue: 12, 2014

Series of issue: Issue 12

Pages: 152 — 157

Downloads: 626