2007, Vol.14, pp.31-45

Parametrization of 4 x 4 - matrices G of the complex linear group GL(4.C) in terms of four complex vector-parameters G=G(k,m,n,l) is investigated. Additional restrictions separating some sub-groups of GL(4.C) are given explicitly. In the given parametrization, the problem of inverting any 4 x 4 matrix G is solved. Expression for determinant of any matrix G is found: det G = F(k,m,n,l). Unitarity conditions in the theory of 4 x 4 -matrices on the base of complex vector parametrization in the theory of the group GL(4.C) is investigated. Unitarity conditions have been formulated in the form of non-linear cubic algebraic equations including complex conjugation. Two simplest types of solutions have been constructed: 1-parametric Abelian sub-group G₀ of 4 x 4 unitary matrices; three 2-parametric sub-groups G₁,G₂,G₃; one 4-parametric unitary sub-group. Curvilinear coordinates to cover these sub-groups have been found.
Key words: Dirac matrices, unitary group

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