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 g_i, with i taking values from 1 to 8. They obey the commutation relations [g_i, g_j] = i f^ijk g_k where a sum over the index k is implied. The structure constant f^ijk is completely antisymmetric in the three indices and has values

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

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 θ_i g_i), where θ_i 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

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

They are traceless, Hermitian, and obey the extra relation Tr(λ_iλ_j) = 2δ_ij. 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.