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Gell-Mann Matrix


The eight Gell-Mann matrices lambda_i, i=1,...,8, are an example of the set of generators of the Lie algebra associated with the special unitary group SU(3). Explicitly, these matrices have the form

lambda_1=[0 1 0; 1 0 0; 0 0 0]
(1)
lambda_2=[0 -i 0; i 0 0; 0 0 0]
(2)
lambda_3=[1 0 0; 0 -1 0; 0 0 0]
(3)
lambda_4=[0 0 1; 0 0 0; 1 0 0]
(4)
lambda_5=[0 0 -i; 0 0 0; i 0 0]
(5)
lambda_6=[0 0 0; 0 0 1; 0 1 0]
(6)
lambda_7=[0 0 0; 0 0 -i; 0 i 0]
(7)
lambda_8=1/(sqrt(3))[1 0 0; 0 1 0; 0 0 -2].
(8)

Note that the eight Gell-Mann matrices are traceless and Hermitian and satisfy the relation Tr(lambda_ilambda_j)=2delta_(ij) where delta_(ij) denotes the Kronecker delta. Because of their properties, one can view the Gell-Mann matrices as a three-dimensional generalization of the 2×2 Pauli matrices, which (with slight modification) generate the Lie algebra associated to SU(2).

These matrices are particularly important in both mathematics and physics. For example, these matrices (and their generalizations) are important in Lie theory. In addition, they also play an important role in physics where they can be thought to model the eight gluons that mediate the strong force quantum chromodynamics, an analogue of the Pauli matrices well-adapted to applications in the realm of quantum mechanics.


See also

Antisymmetric Matrix, Diagonal Matrix, Generalized Gell-Mann Matrix, Hermitian Matrix, Kronecker Delta, Lie Algebra, Lie Group, Matrix Trace, Pauli Matrices, Special Unitary Group, Symmetric Matrix

This entry contributed by Christopher Stover

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References

Gell-Mann, M. "Symmetries of Baryons and Mesons." Phys. Rev. 125, 1067-1084, 1962.Haywood, S. "Lecture 4: SU(3)." http://hepwww.rl.ac.uk/Haywood/Group_Theory_Lectures/Lecture_4.pdf.

Cite this as:

Stover, Christopher. "Gell-Mann Matrix." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Gell-MannMatrix.html

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