Gell-Mann matrices: Difference between revisions

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

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

and

${\displaystyle {\frac {1}{\sqrt {3}}}{\begin{pmatrix}1&0&0\\0&1&0\\0&0&-2\end{pmatrix}}}$.

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

${\displaystyle [\lambda _{i},\lambda _{j}]=i{\mathcal {K}}_{ijk}\lambda _{k}}$

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

${\displaystyle {\mathcal {K}}_{123}=2;\;{\mathcal {K}}_{147}=1;\;{\mathcal {K}}_{156}=-1;\;{\mathcal {K}}_{246}=1;\;{\mathcal {K}}_{257}=1;\;{\mathcal {K}}_{345}=1;\;{\mathcal {K}}_{367}=-1;\;{\mathcal {K}}_{458}={\sqrt {6}};\;{\mathcal {K}}_{678}={\sqrt {6}}}$

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