On Parametrization of the Linear $\mathrm{GL}(4,C)$ and Unitary $\mathrm{SU}(4)$ Groups in Terms of Dirac

Victor M. Red'kov ; Andrei A. Bogush ; Natalia G. Tokarevskaya

Symmetry, integrability and geometry: methods and applications, Tome 4 (2008)

Résumé

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 $4\times4$ 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 $S=e^{i\vec{a}\vec{\alpha}}e^{i\vec{b}\vec{\beta}}e^{i{\mathbf k}{\mathbf K}}e^{i{\mathbf l}{\mathbf L}}e^{i{\mathbf m}{\mathbf M}}$, 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. Besides, the structure of fifteen Dirac matrices $\Lambda_k$ 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 $SU(3,1)$.

Détail

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

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