Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 4 (2008), 021, 46 pages      arXiv:0802.2634      http://dx.doi.org/10.3842/SIGMA.2008.021
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

On Parametrization of the Linear GL(4,C) and Unitary SU(4) Groups in Terms of Dirac Matrices

Victor M. Red'kov, Andrei A. Bogush and Natalia G. Tokarevskaya
B.I. Stepanov Institute of Physics, National Academy of Sciences of Belarus, Minsk, Belarus

Received September 19, 2007, in final form January 24, 2008; Published online February 19, 2008

Abstract
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} = {αiβjiVβj = KL M )}, 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).

Key words: Dirac matrices; linear group; unitary group; Gell-Mann basis; parametrization.

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