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

TL;DR

This paper develops a parametrization of 4x4 matrices in GL(4,C) using four complex vectors, explores subgroups and unitarity conditions, and relates Dirac matrices to SU(4) generators, enabling matrix inversion and subgroup separation.

Contribution

Introduces a new parametrization of GL(4,C) matrices with explicit subgroup restrictions and solutions to unitarity equations, linking Dirac matrices to SU(4) structure.

Findings

Explicit parametrization of GL(4,C) matrices using four complex vectors.

Solutions for unitarity conditions leading to specific subgroups.

Formulas relating Dirac basis to SU(4) generators enabling subgroup separation.

Abstract

Parametrization of $4\times 4$-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 $4\times 4$ 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 $\Lambda_{k}$, being of Gell-Mann type, substantially differs from the basis $\lambda_{i}$ 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 used $\{\Lambda_k\} = \{\alpha_i\oplus\beta_j\oplus(\alpha_iV\beta_j = {\boldsymbol K} \oplus {\boldsymbol L}\oplus{\boldsymbol M})\}$, which permit to factorize SU(4) transformations according to $S = e^{i\vec{a}\vec{\alpha}} e^{i\vec{b}\vec\beta}} e^{i{\boldsymbol k}{\boldsymbol K}} e^{i{\boldsymbol l}{\boldsymbol L}} e^{i\boldsymbol m}{\boldsymbol M}}$, 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.