Gell-Mann matrices
The Gell-Mann matrices are denoted by $ \lambda _ {1} \dots \lambda _ {8} $. They form a family of traceless Hermitian $ ( 3 \times 3 ) $- matrices, orthonormalized as follows: $ { \mathop{\rm Tr} } ( \lambda _ {j} \lambda _ {k} ) = 2 \delta _ {jk } $. When multiplied by the complex unit they form a basis in the Lie algebra $ \mathfrak s \mathfrak u ( 3 ) $, in analogy with the Pauli matrices and the Lie algebra $ \mathfrak s \mathfrak u ( 2 ) $. Their explicit form is [a1]:
$$ \lambda _ {1} = \left (
\begin{array}{ccc}
0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \\
\end{array}
\right ) , \lambda _ {2} = \left (
\begin{array}{ccc}
0 &- i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \\
\end{array}
\right ) , $$
$$ \lambda _ {3} = \left (
\begin{array}{ccc}
1 & 0 & 0 \\ 0 &- 1 & 0 \\ 0 & 0 & 0 \\
\end{array}
\right ) , \lambda _ {4} = \left (
\begin{array}{ccc}
0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \\
\end{array}
\right ) , $$
$$ \lambda _ {5} = \left (
\begin{array}{ccc}
0 & 0 &- i \\ 0 & 0 & 0 \\ i & 0 & 0 \\
\end{array}
\right ) , \lambda _ {6} = \left (
\begin{array}{ccc}
0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \\
\end{array}
\right ) , $$
$$ \lambda _ {7} = \left (
\begin{array}{ccc}
0 & 0 & 0 \\ 0 & 0 &- i \\ 0 & i & 0 \\
\end{array}
\right ) , \lambda _ {8} = { \frac{1}{\sqrt 3 }
} \left (
\begin{array}{ccc}
1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 &- 2 \\
\end{array}
\right ) . $$
References
| [a1] | M. Gell-Mann, Y. Ne'eman, "The eightfold way" , Benjamin (1964) |
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