Gell-Mann matrices

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 (a real Lie algebra in fact) has dimension eight and therefore it has some set with eight linearly independent generators, which can be written as \( g_i, \) with i taking values from 1 to 8. They obey the commutation relations

\( [g_i, g_j] = if^{ijk} g_k \, \)

where a sum over the index k is implied. The structure constants \( f^{ijk} \) are completely antisymmetric in the three indices and have values

\( f^{123} = 1 \ , \quad f^{147} = f^{165} = f^{246} = f^{257} = f^{345} = f^{376} = \frac{1}{2} \ , \quad f^{458} = f^{678} = \frac{\sqrt{3}}{2} \ . \)

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 \mathrm{exp}(i \theta_j g_j), where \theta_j are real numbers and a sum over the index j 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

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

and \( g_i = \lambda_i/2 \). These matrices are traceless, Hermitian, and obey the extra relation \( \mathrm{tr}(\lambda_i \lambda_j) = 2\delta_{ij} \). 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 the set of linear combinations (with real coefficients) of the two matrices \( \lambda_3 \) and \( \lambda_8 \) , which commute with each other. There are 3 independent SU(2) subgroups: \( \{\lambda_1, \lambda_2, x\}, \{\lambda_4, \lambda_5, y\}, \) and \( \{\lambda_6, \lambda_7, z\} \), where the x, y, z must consist of linear combinations of \( \lambda_3 \) and \( \lambda_8 \).

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

Generalizations of Pauli matrices
Unitary groups and group representations
Quark model, colour charge and quantum chromodynamics

References

Lie algebras in particle physics, by Howard Georgi (ISBN 0-7382-0233-9)
George Arfken and Hans Weber. Mathematical Methods for Physicists. Harcourt/Academic Press, 2000.

External links

The quark model, by J. J. J. Kokkedee