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

Abstract

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 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 Λ k , being of Gell-Mann type, substantially differs from the basis λ 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 usedwhich permit to factorize SU (4) transformations according to S = e i a α e i b β e ikK e ilL e imM , 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 Λ 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).

Parabolic Coordinates and the Hydrogen Atom in Spaces H_{3} and S_{3}

“…Such a possibility evidently will extend the class of integrable problems in these spaces -see [53]. Also, complex coordinates in 3D-spaces of constant curvature can be of interest in the context of the theory of the Lorentz group SO(3, 1) -see in [59,60].…”

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“…(37) in the φ representation, we introduce the following representation of the SU(4) group [50]: U iα = cos (λ) − i sin (λ)φ iα · , where λ is a group parameter and are the usual Dirac gamma matrices. Here, we need a set of five matrices and among the possible choices [50], we use 1,2,3 = σ x,y,z ⊗ σ y , 4 = σ 0 ⊗ σ x , and 5 = σ 0 ⊗ σ z . Also, notice that these matrices satisfy the anticommuting relation { μ , ν } = 2δ μν and 5 = − 1 2 3 4 .…”

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“…In our version of the approach, the starting point of the method of Fedorov and collaborators, eqs. (248), (249), is Z 2 -invariant for the elements of SU(4). On the other hand, eq.…”