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

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(iθⁱg_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, ie, on the fundamental representation of the group. A particular choice of this representation is

<math>\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},</math>

<math>\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},</math>

<math>\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}.</math>

They are traceless, Hermitian, and obey the extra relation trλ_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: one includes λ₁ and λ₂, one, λ₄ and λ₅, and the third, λ₆ and λ₇. The third element of each of these subgroups must consist of linear combinations of λ₃ and λ₈.