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SIGMA 4 (2008), 021, 46 pages arXiv:0802.2634
https://doi.org/10.3842/SIGMA.2008.021
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics
On Parametrization of the Linear GL(4,C) and Unitary SU(4) Groups in Terms of Dirac Matrices
Victor M. Red'kov, Andrei A. Bogush and Natalia G. Tokarevskaya
B.I. Stepanov Institute of Physics, National Academy of Sciences of Belarus, Minsk, Belarus
Received September 19, 2007, in final form January 24, 2008; Published online February 19, 2008
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 G1, G2, G3 -
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 Λk, being of
Gell-Mann type, substantially differs from the basis
λi 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
{Λk} = {αiÅβjÅ(αiVβj = KÅL ÅM )}, which permit to factorize SU(4) transformations
according
to
S = eiaα eibβeikKeilLeimM, 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 Λk 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).
Key words:
Dirac matrices; linear group; unitary group; Gell-Mann basis; parametrization.
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