On Parametrization of the Linear GL(4,C) and Unitary SU(4) Groups in Terms of Dirac Matrices
Parametrization of 4 × 4-matrices G of the complex linear group GL(4,C) in terms of four complex 4-vector parameters (k,m,n,l) is investigated. Additional restrictions separating some subgroups of GL(4,C) are given explicitly. In the given parametrization, the problem of inverting any 4 × 4 matrix G...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2008 |
| Автори: | , , |
| Формат: | Стаття |
| Мова: | English |
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Інститут математики НАН України
2008
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/148991 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On Parametrization of the Linear GL(4,C) and Unitary SU(4) Groups in Terms of Dirac Matrices / V.M. Red'kov, A.A. Bogush, N.G. Tokarevskaya // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 75 назв. — англ. |
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Red'kov, V.M. Bogush, A.A. Tokarevskaya, N.G. 2019-02-19T12:46:41Z 2019-02-19T12:46:41Z 2008 On Parametrization of the Linear GL(4,C) and Unitary SU(4) Groups in Terms of Dirac Matrices / V.M. Red'kov, A.A. Bogush, N.G. Tokarevskaya // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 75 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 20C35; 20G45; 22E70; 81R05 https://nasplib.isofts.kiev.ua/handle/123456789/148991 Parametrization of 4 × 4-matrices G of the complex linear group GL(4,C) in terms of four complex 4-vector parameters (k,m,n,l) is investigated. Additional restrictions separating some subgroups of GL(4,C) are given explicitly. In the given parametrization, the problem of inverting any 4 × 4 matrix G is solved. Expression for determinant of any matrix G is found: det G = F(k,m,n,l). Unitarity conditions G⁺ = G⁻¹ have been formulated in the form of non-linear cubic algebraic equations including complex conjugation. Several simplest solutions of these unitarity equations have been found: three 2-parametric subgroups G₁, G₂, G₃ - each of subgroups consists of two commuting Abelian unitary groups; 4-parametric unitary subgroup consisting of a product of a 3-parametric group isomorphic SU(2) and 1-parametric Abelian group. The Dirac basis of generators Λk, being of Gell-Mann type, substantially differs from the basis λi used in the literature on SU(4) group, formulas relating them are found - they permit to separate SU(3) subgroup in SU(4). Special way to list 15 Dirac generators of GL(4,C) can be used {Λk} = {αiÅβjÅ(αiVβj = KÅL ÅM )}, which permit to factorize SU(4) transformations according to S = eiaα eibβeikKeilLeimM, where two first factors commute with each other and are isomorphic to SU(2) group, the three last ones are 3-parametric groups, each of them consisting of three Abelian commuting unitary subgroups. Besides, the structure of fifteen Dirac matrices Λk permits to separate twenty 3-parametric subgroups in SU(4) isomorphic to SU(2); those subgroups might be used as bigger elementary blocks in constructing of a general transformation SU(4). It is shown how one can specify the present approach for the pseudounitary group SU(2,2) and SU(3,1). This paper is a contribution to the Proceedings of the Seventh International Conference “Symmetry in Nonlinear Mathematical Physics” (June 24–30, 2007, Kyiv, Ukraine). Authors are grateful to participants of the seminar of Laboratory of Physics of Fundamental Interaction, National Academy of Sciences of Belarus for discussion. Authors are grateful to the anonymous reviewer for many comments and advice improving the paper. This work was supported by Fund for Basic Research of Belarus F07-314. We wish to thank the Organizers of the Seventh International Conference “Symmetry in Nonlinear Mathematical Physics” (June 24–30, 2007, Kyiv) and ICTP Of fice of External Activities for having given us the opportunity to talk on this subject as well as for local and travel support. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On Parametrization of the Linear GL(4,C) and Unitary SU(4) Groups in Terms of Dirac Matrices Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
On Parametrization of the Linear GL(4,C) and Unitary SU(4) Groups in Terms of Dirac Matrices |
| spellingShingle |
On Parametrization of the Linear GL(4,C) and Unitary SU(4) Groups in Terms of Dirac Matrices Red'kov, V.M. Bogush, A.A. Tokarevskaya, N.G. |
| title_short |
On Parametrization of the Linear GL(4,C) and Unitary SU(4) Groups in Terms of Dirac Matrices |
| title_full |
On Parametrization of the Linear GL(4,C) and Unitary SU(4) Groups in Terms of Dirac Matrices |
| title_fullStr |
On Parametrization of the Linear GL(4,C) and Unitary SU(4) Groups in Terms of Dirac Matrices |
| title_full_unstemmed |
On Parametrization of the Linear GL(4,C) and Unitary SU(4) Groups in Terms of Dirac Matrices |
| title_sort |
on parametrization of the linear gl(4,c) and unitary su(4) groups in terms of dirac matrices |
| author |
Red'kov, V.M. Bogush, A.A. Tokarevskaya, N.G. |
| author_facet |
Red'kov, V.M. Bogush, A.A. Tokarevskaya, N.G. |
| publishDate |
2008 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
Parametrization of 4 × 4-matrices G of the complex linear group GL(4,C) in terms of four complex 4-vector parameters (k,m,n,l) is investigated. Additional restrictions separating some subgroups of GL(4,C) are given explicitly. In the given parametrization, the problem of inverting any 4 × 4 matrix G is solved. Expression for determinant of any matrix G is found: det G = F(k,m,n,l). Unitarity conditions G⁺ = G⁻¹ have been formulated in the form of non-linear cubic algebraic equations including complex conjugation. Several simplest solutions of these unitarity equations have been found: three 2-parametric subgroups G₁, G₂, G₃ - each of subgroups consists of two commuting Abelian unitary groups; 4-parametric unitary subgroup consisting of a product of a 3-parametric group isomorphic SU(2) and 1-parametric Abelian group. The Dirac basis of generators Λk, being of Gell-Mann type, substantially differs from the basis λi used in the literature on SU(4) group, formulas relating them are found - they permit to separate SU(3) subgroup in SU(4). Special way to list 15 Dirac generators of GL(4,C) can be used {Λk} = {αiÅβjÅ(αiVβj = KÅL ÅM )}, which permit to factorize SU(4) transformations according to S = eiaα eibβeikKeilLeimM, where two first factors commute with each other and are isomorphic to SU(2) group, the three last ones are 3-parametric groups, each of them consisting of three Abelian commuting unitary subgroups. Besides, the structure of fifteen Dirac matrices Λk permits to separate twenty 3-parametric subgroups in SU(4) isomorphic to SU(2); those subgroups might be used as bigger elementary blocks in constructing of a general transformation SU(4). It is shown how one can specify the present approach for the pseudounitary group SU(2,2) and SU(3,1).
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/148991 |
| citation_txt |
On Parametrization of the Linear GL(4,C) and Unitary SU(4) Groups in Terms of Dirac Matrices / V.M. Red'kov, A.A. Bogush, N.G. Tokarevskaya // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 75 назв. — англ. |
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