Gell-Mann matrices

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Murray Gell-Mann

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). The Lie algebra of 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.

Defining relations[edit]

These Lie Algebra elements 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 qualifies. 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.

Particular representations[edit]

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 for SU(2). They also naturally extend to general SU(n), cf. Generalizations of 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, \lambda_3\}, \{\lambda_4, \lambda_5, x\}, and \{\lambda_6, \lambda_7, y\}, where the x and y are linear combinations of \lambda_3 and \lambda_8.

The squared sum of the Gell-Mann matrices gives the Casimir operator:

 C = \sum_{i=1}^8 \lambda_i \lambda_i = 16/3 .

These matrices are needed to study the internal rotations between the different coloured quarks in Quantum Chromodynamics.

See also[edit]

References[edit]