关于SU(3)子群的闭性及类泡利σ矩阵代数性质的技术问询

Hey there, let's break down these SU(3)-associated matrices and their Pauli-like algebraic properties clearly:

First, here are the three σ matrices in question:
$$\sigma_1 =
\begin{pmatrix}
0 & 1 & 0 \
1 & 0 & 0 \
0 & 0 & 0
\end{pmatrix}, \quad \sigma_2 =
\begin{pmatrix}
0 & 0 & 1 \
0 & 0 & 0 \
1 & 0 & 0
\end{pmatrix},
\quad
\sigma_3 =
\begin{pmatrix}
0 & 0 & 0 \
0 & 0 & -i \
0 & i & 0
\end{pmatrix}.
$$

These matrices exhibit key algebraic behaviors that mirror the Pauli matrices—here are the core properties you highlighted:
$$[\sigma_j,\sigma_k] = i \sum_{\ell=1}^3\varepsilon_{jk\ell}\sigma_{\ell}, \qquad \mathrm{Tr}(\sigma_j\sigma_k\sigma_{\ell} )=i \varepsilon_{jk\ell}, \qquad \forall j,k,\ell\in{1,2,3},$$
where $\varepsilon_{jk\ell}$ denotes the Levi-Civita symbol (the fully antisymmetric tensor with $\varepsilon_{123}=1$).

You started to mention the exponential form $e^{i\theta \sigma_1} = ...$ but the rest of the expression was cut off. For context, computing this exponential is straightforward using Taylor series: since $\sigma_1^2 = \begin{pmatrix}1&0&0\0&1&0\0&0&0\end{pmatrix}$ (a projection matrix), the series simplifies to:
$$e^{i\theta \sigma_1} = I\cos\theta + i\sigma_1\sin\theta$$
where $I$ is the 3x3 identity matrix. This follows the same pattern as Pauli matrix exponentials, which makes sense given their shared commutation relations.

If you're looking to verify the subgroup closure of the generated group under multiplication (a key SU(3) property), you can check that products of these matrices (and their exponentials) remain within the SU(3) group—i.e., they have determinant 1 and are unitary. For example, $\sigma_1\sigma_2$ is a unitary matrix with determinant 1, fitting SU(3) criteria.

备注：内容来源于stack exchange，提问作者Xin Fu