Gell-Mann matrices

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The Gell-Mann matrices, named for Murray Gell-Mann, are one possible representation of the infinitesimal generators of the special unitary group called SU(3).

This group has eight generators, which we can write as gi, with i taking values from 1 to 8. They obey the commutation relations [gi,gj] = ifijkgk where a sum over the index k is implied. The structure constant fijk is completely antisymmetric in the three indices and has values

f123 = 1, f147 = f165 = f246 = f257 = f345 = f376 = 1/2, f458 = f678 = √3/2.

Any set of Hermitian matrices which obey these relations are allowed. A particular choice of matrices is called a group representation, because any element of SU(3) can be written in the form exp(i θi gi), where θi are real numbers and a sum over the index i is implied. Given one representation, another may be obtained by an arbitrary unitary transformation, since that leaves the commutator unchanged.

An important representation involves 3×3 matrices, because the group elements then act on complex vectors with 3 entries, i.e., on the fundamental representation of the group. A particular choice of this representation is

\lambda_1 = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \lambda_2 = \begin{pmatrix} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \lambda_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \end{pmatrix}
\lambda_4 = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix} \lambda_5 = \begin{pmatrix} 0 & 0 & -i \\ 0 & 0 & 0 \\ i & 0 & 0 \end{pmatrix} \lambda_6 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}
\lambda_7 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \end{pmatrix} \lambda_8 = \frac{1}{\sqrt{3}} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -2 \end{pmatrix}

Where gi = λi /2. They are traceless, Hermitian, and obey the extra relation Tr(λiλj) = 2δij. These properties were chosen by Gell-Mann because they then generalize the Pauli matrices.

In this representation it is clear that the Cartan subalgebra is given by the set of two matrices λ3 and λ8, which commute with each other. There are 3 independent SU(2) subgroups: {λ1, λ2, x}, {λ4, λ5, y}, and {λ6, λ7, z}, where the x, y, z must consist of linear combinations of λ3 and λ8.

These matrices form a useful representation for computations in the quark model, and, to a lesser extent, in quantum chromodynamics.

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