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On Parametrization of the Linear $\mathrm{GL}(4,C)$ and Unitary $\mathrm{SU}(4)$ Groups in Terms of Dirac
Matrices
Victor M. Red'kov,
Andrei A. Bogush,
Natalia G. Tokarevskaya B. I. Stepanov Institute of Physics, National Academy of Sciences of Belarus, Minsk, Belarus
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
$4\times4$ 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
$SU(3,1)$.
Keywords:
Dirac matrices; linear group; unitary group; Gell-Mann basis; parametrization.
MSC: 20C35;
20G45;
22E70;
81R05 Received: September 19, 2007; in final form
January 24, 2008; Published online
February 19, 2008
Language: English
DOI:
10.3842/SIGMA.2008.021
Fulltext:
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