Chemistry - Gell-Mann matrices

The Gell-Mann matrices, named after Murray Gell-Mann, are the infinitesimal generators of su(3). They are

$\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix},\ \begin{pmatrix} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix},\ \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \end{pmatrix},\ \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix},$

$\begin{pmatrix} 0 & 0 & -i \\ 0 & 0 & 0 \\ i & 0 & 0 \end{pmatrix},\ \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix},\ \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \end{pmatrix},\mathrm{and}\ \frac{1}{\sqrt{3}}\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -2 \end{pmatrix}.$

Like the Pauli matrices (the generators of SU(2)) the Gell-Mann matrices are traceless and hermitian. The Gell-Mann matrices describe color charge in much the same way that the Pauli matrices describe spin and isospin.

Also like the Pauli matrices the Gell-Mann matrices satisfy certain important commutation relations. These are

$[\lambda_i, \lambda_j]=i\mathcal{K}_{ijk} \lambda_k$

where the we sum over k, and K is totally antisymmetric with

$\mathcal{K}_{123}=2;\; \mathcal{K}_{147}=1;\; \mathcal{K}_{156}=-1;\; \mathcal{K}_{246}=1;\; \mathcal{K}_{257}=1;\; \mathcal{K}_{345}=1;\; \mathcal{K}_{367}=-1;\;$

$\mathcal{K}_{458}=\sqrt{3};\; \mathcal{K}_{678}=\sqrt{3}.$

and those elements whose indices are not permutations of these equal to zero.

Categories: Group theory | Matrices