On parametrization of the finite transformations of the unitary group SU(4)
Abstract:
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).