Chapter 7: Unitary Groups

### The Geometry of $SU(3)$

The unitary group in three complex dimensions is defined by $$\SU(3) = \{ M\in\CC^{3\times 3} : M^\dagger M = I, \det M = 1 \}$$ A $3\times3$ matrix has 9 complex components, the constraint $M^\dagger M=I$ imposes 9 real conditions, and the determinant restriction adds just one more (since the other conditions already imply that $|\det M|=1$). Thus, we expect there to be 8 independent “rotations” in $\SU(3)$. A set of generators is given by \begin{align} L_1 &= L_1(\alpha) = \begin{pmatrix} \cos\left(\frac\alpha2\right) & -i\sin\left(\frac\alpha2\right) & 0 \\ -i\sin\left(\frac\alpha2\right) & \cos\left(\frac\alpha2\right) & 0 \\ 0 & 0 & 1 \end{pmatrix}\\ L_2 &= L_2(\alpha) = \begin{pmatrix} \cos\left(\frac\alpha2\right) & -\sin\left(\frac\alpha2\right) & 0 \\ \sin\left(\frac\alpha2\right) & \cos\left(\frac\alpha2\right) & 0 \\ 0 & 0 & 1 \end{pmatrix}\\ L_3 &= L_3(\alpha) = \begin{pmatrix} e^{-i\alpha/2} & 0 & 0 \\ 0 & e^{i\alpha/2} & 0 \\ 0 & 0 & 1 \end{pmatrix}\\ L_4 &= L_4(\alpha) = \begin{pmatrix} \cos\left(\frac\alpha2\right) & 0 & -i\sin\left(\frac\alpha2\right) \\ 0 & 1 & 0 \\ -i\sin\left(\frac\alpha2\right) & 0 & \cos\left(\frac\alpha2\right) \end{pmatrix}\\ L_5 &= L_5(\alpha) = \begin{pmatrix} \cos\left(\frac\alpha2\right) & 0 & \sin\left(\frac\alpha2\right) \\ 0 & 1 & 0 \\ -\sin\left(\frac\alpha2\right) & 0 & \cos\left(\frac\alpha2\right) \end{pmatrix}\\ L_6 &= L_6(\alpha) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos\left(\frac\alpha2\right) & -i\sin\left(\frac\alpha2\right) \\ 0 & -i\sin\left(\frac\alpha2\right) & \cos\left(\frac\alpha2\right) \end{pmatrix}\\ L_7 &= L_7(\alpha) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos\left(\frac\alpha2\right) & -\sin\left(\frac\alpha2\right) \\ 0 & \sin\left(\frac\alpha2\right) & \cos\left(\frac\alpha2\right) \end{pmatrix}\\ L_8 &= L_8(\alpha) = \begin{pmatrix} e^{-i\alpha/2\sqrt3} & 0 & 0 \\ 0 & e^{i\alpha/2\sqrt3} & 0 \\ 0 & 0 & e^{-i\alpha/\sqrt3} \end{pmatrix} \label{su3gen} \end{align} where the factors of $2$ and $\sqrt3$ are again conventional. These matrices are closely related to the Gell-Mann matrices $$\lambda_m = 2i\frac{\partial L_m}{\partial\alpha}$$ which are explicitly given by 1) \begin{align} \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 &= \frac1{\sqrt{3}}\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -2 \end{pmatrix} \end{align} The Gell-Mann matrices are again Hermitian ($\sigma_m^\dagger=\sigma$) and tracefree ($\tr\sigma_m=0$), but do not share the other properties we listed for the Pauli matrices in the previous section. We will have more to say about the Gell-Mann matrices later.

The group $\SU(3)$ is the smallest of the unitary groups to be unrelated to the orthogonal groups; it's something new.

1) Our definition of $\lambda_5$ differs by an overall minus sign from the standard definition, in order to correct a minor but annoying lack of cyclic symmetry in the original definition.