The Gell-Mann matrices, developed by Murray Gell-Mann, are a set of eight linearly independent 3×3 traceless Hermitian matrices used in the study of the strong interaction in particle physics.
They span the Lie algebra of the SU(3) group in the defining representation.
These matrices are traceless, Hermitian, and obey the extra trace orthonormality relation, so they can generate unitary matrix group elements of SU(3) through exponentiation.
[1] These properties were chosen by Gell-Mann because they then naturally generalize the Pauli matrices for SU(2) to SU(3), which formed the basis for Gell-Mann's quark model.
In mathematics, orthonormality typically implies a norm which has a value of unity (1).
Thus, the trace of the pairwise product results in the ortho-normalization condition where
In this three-dimensional matrix representation, the Cartan subalgebra is the set of linear combinations (with real coefficients) of the two matrices
There are three significant SU(2) subalgebras: where the x and y are linear combinations of
are completely antisymmetric in the three indices, generalizing the antisymmetry of the Levi-Civita symbol
For the present order of Gell-Mann matrices they take the values In general, they evaluate to zero, unless they contain an odd count of indices from the set {2,5,7}, corresponding to the antisymmetric (imaginary) λs.
Using these commutation relations, the product of Gell-Mann matrices can be written as where I is the identity matrix.
Namely, using the dot to sum over the eight matrices and using Greek indices for their row/column indices, the following identities hold, and One may prefer the recast version, resulting from a linear combination of the above, A particular choice of matrices is called a group representation, because any element of SU(3) can be written in the form
are real numbers and a sum over the index j is implied.
Given one representation, an equivalent one may be obtained by an arbitrary unitary similarity transformation, since that leaves the commutator unchanged.
The matrices can be realized as a 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
[1] The squared sum of the Gell-Mann matrices gives the quadratic Casimir operator, a group invariant, where
These matrices serve to study the internal (color) rotations of the gluon fields associated with the coloured quarks of quantum chromodynamics (cf.
A gauge colour rotation is a spacetime-dependent SU(3) group element