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. 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. Gell-Mann's generalization further
extends to general SU(n). For their connection to the
standard basis of Lie algebras, see the
Weyl–Cartan basis.
Trace orthonormality In mathematics, orthonormality typically implies a norm which has a value of unity (1). Gell-Mann matrices, however, are normalized to a value of 2. Thus, the
trace of the pairwise product results in the ortho-normalization condition :\operatorname{tr}(\lambda_i \lambda_j) = 2\delta_{ij}, where \delta_{ij} is the
Kronecker delta. This is so the embedded Pauli matrices corresponding to the three embedded subalgebras of
SU(2) are conventionally normalized. In this three-dimensional matrix representation, 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 three
significant SU(2) subalgebras: • \{\lambda_1, \lambda_2, \lambda_3\} • \{\lambda_4, \lambda_5, x\}, and • \{\lambda_6, \lambda_7, y\}, where the and are linear combinations of \lambda_3 and \lambda_8. The SU(2)
Casimirs of these subalgebras mutually commute. However, any unitary similarity transformation of these subalgebras will yield SU(2) subalgebras. There is an uncountable number of such transformations.
Commutation relations The 8 generators of SU(3) satisfy the
commutation and anti-commutation relations : \begin{align} \left[ \lambda_a, \lambda_b \right] &= 2 i \sum_c f_{abc} \lambda_c, \\ \{ \lambda_a, \lambda_b \} &= \frac{4}{3} \delta_{ab} I + 2 \sum_c d_{abc} \lambda_c, \end{align} with the
structure constants : \begin{align} f_{abc} &= -\frac{1}{4} i \operatorname{tr}([\lambda_a , \lambda_b] \lambda_c ), \\ d_{abc} &= \frac{1}{4} \operatorname{tr}(\{\lambda_a , \lambda_b\} \lambda_c ). \end{align} The
structure constants d_{abc} are completely symmetric in the three indices. The
structure constants f_{abc} are completely antisymmetric in the three indices, generalizing the antisymmetry of the
Levi-Civita symbol \epsilon_{jkl} of . For the present order of Gell-Mann matrices they take the 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} \ . 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) s. Using these commutation relations, the product of Gell-Mann matrices can be written as : \lambda_a \lambda_b = \frac{1}{2} ([\lambda_a,\lambda_b] + \{\lambda_a,\lambda_b\}) = \frac{2}{3} \delta_{ab} I + \sum_c \left(d_{abc} + i f_{abc}\right) \lambda_c , where is the 3×3 identity matrix.
Fierz completeness relations Since the eight matrices and the identity are a complete trace-orthogonal set spanning all 3×3 matrices, it is straightforward to find two Fierz
completeness relations, (Li & Cheng, 4.134), analogous to that
satisfied by the Pauli matrices. Namely, using the dot to sum over the eight matrices and using Greek indices for their row/column indices, the following identities hold, :\delta^\alpha _\beta \delta^\gamma _\delta = \frac{1}{3} \delta^\alpha_\delta \delta^\gamma _\beta +\frac{1}{2} \lambda^\alpha _\delta \cdot \lambda^\gamma _\beta and :\lambda^\alpha _\beta \cdot \lambda^\gamma _\delta = \frac{16}{9} \delta^\alpha_\delta \delta^\gamma _\beta -\frac{1}{3} \lambda^\alpha _\delta \cdot \lambda^\gamma _\beta ~. One may prefer the recast version, resulting from a linear combination of the above, :\lambda^\alpha _\beta \cdot \lambda^\gamma _\delta = 2 \delta^\alpha_\delta \delta^\gamma _\beta -\frac{2}{3} \delta^\alpha_\beta \delta^\gamma _\delta ~. ==Representation theory==