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

Abstract

Parametrization of $4\times 4$-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 $4\times 4$ 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 $G_1$, $G_2$, $G_3$ - 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 ${\Lambda }_k$, being of Gell-Mann type, substantially differs from the basis ${\lambda }_i$ 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 $\{{\Lambda }_k\}=\{{\alpha }_i\oplus {\beta }_j\oplus ({\alpha }_{i}V{\beta }_j=\mathbf{K}\oplus \mathbf{L}\oplus \mathbf{M})\}$, which permit to factorize SU(4) transformations according to $S = e^{i\vec{a}\vecα} e^{i\vec{b}\vecβ}} e^{i{\boldsymbol k}{\boldsymbol K}} e^{i{\boldsymbol l}{\boldsymbol L}} e^{i\boldsymbol m}{\boldsymbol M}}$, 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.

This is a contribution to the Proc. of the Seventh International Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007, Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

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