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

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

2008-02-19

arxiv.org/abs/0802.2634v1

SIGMA 4 (2008), 021, 46 pages

Physics

Mathematical Physics

This is a contribution to the Proc. of the Seventh International Conference ''Symmetry in Nonlinear Mathematical Physics'' (Ju

Scientific paper

10.3842/SIGMA.2008.021

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 $\{\Lambda_k\} = \{\alpha_i\oplus\beta_j\oplus(\alpha_iV\beta_j = {\boldsymbol K} \oplus {\boldsymbol L}\oplus{\boldsymbol M})\}$, which permit to factorize SU(4) transformations according to $S = e^{i\vec{a}\vec{\alpha}} e^{i\vec{b}\vec\beta}} e^{i{\boldsymbol k}{\boldsymbol K}} e^{i{\boldsymbol l}{\boldsymbol L}} e^{i\boldsymbol m}{\boldsymbol M}}$, 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.

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