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 has eight generators, which we can write as "g_i", with "i" taking values from 1 to 8. They obey the commutation relations $[g\_i,\; g\_j]\; =\; if^\{ijk\}\; g\_k$ where a sum over the index "k" is implied. The structure constant $f^\{ijk\}$ is completely antisymmetric in the three indices and has values: $f^\{123\}\; =$ 1, $f^\{147\}\; =\; f^\{165\}\; =\; f^\{246\}\; =\; f^\{257\}\; =\; f^\{345\}\; =\; f^\{376\}\; =$ 1/2, $f^\{458\}\; =\; f^\{678\}\; =$ √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 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:Where "g_i = λ_i /2". They are traceless, Hermitian, and obey the extra relation Tr("λ_iλ_j") = 2"δ_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 "λ"₃ and "λ"₈, which commute with each other. There are 3 independent SU(2) subgroups: {"λ"₁, "λ"₂, "x"}, {"λ"₄, "λ"₅, y}, and {"λ"₆, "λ"₇, "z"}, where the "x", "y", "z" must consist of linear combinations of "λ"₃ and "λ"₈.

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

ee also

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

References and external links

* "Lie algebras in particle physics", by Howard Georgi (ISBN 0-7382-0233-9)
* " [http://www.amazon.com/o/asin/B0006BYWZW The quark model] ", by J. J. J. Kokkedee