 Gell-Mann matrices

The eight Gell-Mann matrices ${\lambda }_1,\dots ,{\lambda }_8$ form a conventional orthogonal basis of the real vector space of traceless Hermitian matrices of size three. In terms of matrix units $E_{ij}$, one standard ordering isDirect matrix multiplication gives $\mathrm{tr}({\lambda }_a{\lambda }_b)=2{\delta }_{ab}$. Thus a traceless Hermitian matrix $H$ has the expansion $H=\frac{1}{2}{\sum }_a\mathrm{tr}(H{\lambda }_a){\lambda }_a$. Multiplication by $i$ gives a real basis of the special unitary Lie algebra $\mathfrak{su}(3)$; physicists often use the Hermitian generators $T_a={\lambda }_a/2$ instead.

The first three matrices contain the Pauli matrices in the first two coordinates, so $[{\lambda }_1,{\lambda }_2]=2i{\lambda }_3$. This displays the special unitary Lie algebra $\mathfrak{su}(2)$ inside the three-state matrix algebra and provides concrete generators for calculations in the special unitary group $SU(3)$.

Gell-Mann matrices are a set of matrices that are used in quantum mechanics, particularly in the context of quantum chromodynamics (QCD) and the mathematical description of the behavior of particles such as quarks and gluons. They are a generalization of the Pauli matrices used for spin-1/2 particles and are essential for modeling the non-abelian gauge symmetry of the strong interaction.