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8.9: Representations of SU(3) and Multiplication Rules

• • Contributed by Niels Walet
• Professor (Physics) at University of Manchester

A very important group is SU(3), since it is related to the color carried by the quarks, the basic building blocks of QCD. The transformations within SU(3) are all those amongst a vector consisting of three complex objects that conserve the length of the vector. These are all three-by-three unitary matrices, which act on the complex vector $$\psi$$ by

\begin{aligned} \psi &\rightarrow &U \psi\nonumber\\ &= \left( \begin{array}{lll} U_{11} & U_{12} & U_{13}\\ U_{21} & U_{22} & U_{23}\\ U_{31} & U_{32} & U_{33} \end{array} \right) \left( \begin{array}{l} \psi_1\\ \psi_2 \\ \psi_3 \end{array}\right)\end{aligned}

The complex conjugate vector can be shown to transform as

$\psi^* \rightarrow \psi^* U^\dagger,$

with the inverse of the matrix. Clearly the fundamental representation of the group, where the matrices representing the transformation are just the matrix transformations, the vectors have length $$3$$. The representation is usually labelled by its number of basis elements as $${\mathbf{3}}$$. The one the transforms under the inverse matrices is usually denoted by $$\bar{\mathbf{3}}$$.

What happens if we combine two of these objects, $$\psi$$ and $$\chi^*$$? It is easy to see that the inner product of $$\psi$$ and $$\chi^*$$ is scalar, $\chi^* \cdot \psi \rightarrow \chi^* U^\dagger U \psi = \chi^* \cdot \psi,$ where we have used the unitary properties of the matrices the remaining $$8$$ components can all be shown to transform amongst themselves, and we write

${\mathbf{3}}\otimes \bar{\mathbf{3}} = {\mathbf{1}}\oplus{\mathbf{8}}.$

Of further interest is the product of three of these vectors,

${\mathbf{3}}\otimes {\mathbf{3}}\otimes {\mathbf{3}} = {\mathbf{1}}\oplus{\mathbf{8}}\oplus{\mathbf{8}}\oplus{\mathbf{10}}.$