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The SU(3) group

Main article: Special unitary group

The special unitary group SU is the group of unitary matrices whose determinant is equal to 1. This set is closed under matrix multiplication. All transformations characterized by the special unitary group leave norms unchanged. The SU(3) symmetry appears in quantum chromodynamics, and, as already indicated in the light quark flavour symmetry dubbed the Eightfold Way (physics). The quarks possess colour quantum numbers and form the fundamental (triplet) representation of an SU(3) group.

The group SU(3) is a subgroup of group U(3), the group of all 3×3 unitary matrices. The unitarity condition imposes nine constraint relations on the total 18 degrees of freedom of a 3×3 complex matrix. Thus, the dimension of the U(3) group is 9. Furthermore, multiplying a U by a phase, e^iφ leaves the norm invariant. Thus U(3) can be decomposed into a direct product of U(1)⊗SU(3). Because of this additional constraint, SU(3) has dimension 8.

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Generators of the Lie algebra

Every unitary matrix U can be written in the form

$U=e^{iH}\,$

where H is hermitian. The elements of SU(3) can be expressed as

$U=e^{i\sum {a_{k}\lambda _{k}}}$

where $\lambda _{k}$ are the 8 linearly independent matrices forming the basis of the Lie algebra of SU(3), in the tripet representation. The unit determinant condition requires the $\lambda _{k}$ matrices to be traceless, since

$\det(e^{A})=e^{\operatorname {tr} (A)}~.$

An explicit basis in the fundamental, 3, representation can be constructed in analogy to the Pauli matrix algebra of the spin operators. It consists of the Gell-Mann matrices,

${\begin{array}{ccc}\lambda _{1}={\begin{pmatrix}0&1&0\\1&0&0\\0&0&0\end{pmatrix}}&\lambda _{2}={\begin{pmatrix}0&-i&0\\i&0&0\\0&0&0\end{pmatrix}}&\lambda _{3}={\begin{pmatrix}1&0&0\\0&-1&0\\0&0&0\end{pmatrix}}\\\\\lambda _{4}={\begin{pmatrix}0&0&1\\0&0&0\\1&0&0\end{pmatrix}}&\lambda _{5}={\begin{pmatrix}0&0&-i\\0&0&0\\i&0&0\end{pmatrix}}\\\\\lambda _{6}={\begin{pmatrix}0&0&0\\0&0&1\\0&1&0\end{pmatrix}}&\lambda _{7}={\begin{pmatrix}0&0&0\\0&0&-i\\0&i&0\end{pmatrix}}&\lambda _{8}={\frac {1}{\sqrt {3}}}{\begin{pmatrix}1&0&0\\0&1&0\\0&0&-2\end{pmatrix}}.\end{array}}$

These are the generators of the SU(3) group in the triplet representation, and they are normalized as

$\operatorname {tr} (\lambda _{j}\lambda _{k})=2\delta _{jk}.$

The Lie algebra structure constants of the group are given by the commutators of $\lambda _{k}$

$[\lambda _{j},\lambda _{k}]=2if_{jkl}\lambda _{l}~,$

where $f_{jkl}$ are the structure constants completely antisymmetric and are analogous to the Levi-Civita symbol $\epsilon _{jkl}$ of SU(2).

In general, they vanish, unless they contain an odd number of indices from the set {2,5,7}, corresponding to the antisymmetric λs. Note $f_{ljk}={\frac {-i}{2}}\mathrm {tr} ([\lambda _{l},\lambda _{j}]\lambda _{k})$.

Moreover,

$\{\lambda _{j},\lambda _{k}\}={\frac {4}{3}}\delta _{jk}+2d_{jkl}\lambda _{l}$

where $d_{jkl}$ are the completely symmetric coefficient constants. They vanish if the number of indices from the set {2,5,7} is odd.