On parametrization of the finite transformations of the unitary group SU(4)
The Dirac matrix basis is taken to construct the theory of the unitary group SU(4). Parametrization of finit matrices G of the complex linear group GL(4,C) in terms of four complex vector-parameters G=G(k,m,n,l) is proposed and investigated. Additional restrictions separating some sub-groups of GL(4,C) are given explicitly. In the given parametrization, the problem of inverting any matrix G is solved. Expression for determinant of G is found: det G = F(k,m,n,l). Unitarity conditions in the theory on SU(4) group on the base of complex vector parametrization are investigated. Unitarity conditions have been formulated in the form of non-linear cubic algebraic equations for 16 complex-valued pavariables. Several simplest types of solutions have been constructed: 1-parametric Abelian sub-group; three 2-parametric sub-groups ; one 4-parametric unitary sub-group. Curvilinear coordinates to cover these sub-groups have been found.
Joint work with A.A. Bogush and N.G. Tokarevskaya (B.I. Stepanov Institute of Physics, Minsk, Belarus).