[Math] Trace of product of three Pauli matrices

Question

Consider the four $2\times 2$ matrices $\{\sigma_\mu\}$, with $\mu = 0,1,2,3$, which are defined as follows
$$
\sigma_0 =\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right)
$$
$$
\sigma_1 =\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right)
$$
$$
\sigma_2 =\left( \begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right)
$$
$$
\sigma_3 =\left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right)
$$
i.e. the identity matrix and the three Pauli matrices. For the trace of the product of any two matrices $\sigma_\mu$ one has the identity $\text{tr}(\sigma_\mu \sigma_\nu)= 2 \delta_{\mu \nu}$. I was wondering if a similar identity can be derived for the product of three sigma matrices,
$$
\text{tr}(\sigma_\mu \sigma_\nu \sigma_\lambda)= \;?
$$

Answer 1

You can check that $\sigma_1\sigma_2\sigma_3 = i I$ so $\mathrm{tr}(\sigma_1\sigma_2\sigma_3) = 2i$. By cyclic invariance of the trace, we also have $$\mathrm{tr}(\sigma_1\sigma_2\sigma_3) = \mathrm{tr}(\sigma_2\sigma_3\sigma_1) = \mathrm{tr}(\sigma_3\sigma_1\sigma_2) = 2i$$ Since $\sigma_r \sigma_s = - \sigma_s \sigma_r$ for distinct $r,s \in \{1,2,3\}$, we also get $$\mathrm{tr}(\sigma_2\sigma_1\sigma_3) = \mathrm{tr}(\sigma_3\sigma_2\sigma_1) = \mathrm{tr}(\sigma_1\sigma_3\sigma_2) = -2i$$ On the other hand, if two of $i,j,k \in \{1,2,3\}$ are equal then, from the the commutation relation mentioned above and the fact that $\sigma_r^2 = I$ for $r \in \{1,2,3\}$, you can conclude that $\sigma_i \sigma_j \sigma_k \in \{ \pm \sigma_1 ,\pm \sigma_2 , \pm \sigma_3\}$ and, in particular, $\mathrm{tr}(\sigma_i \sigma_j \sigma_k) = 0$.

This can be summarized by saying that, for $i,j,k \in \{1,2,3\}$, $$\mathrm{tr}(\sigma_i \sigma_j \sigma_k) = \varepsilon_{ijk} 2i$$ where $\varepsilon_{ijk}$ is the Levi-Cevita symbol given by $$\varepsilon_{ijk} = \begin{cases} 1 && \text{ if } ijk \text{ is an even permutation of } 123 \\ -1 && \text{ if } ijk \text{ is an odd permutation of } 123 \\ 0 && \text{ if } ijk \text{ is not a permutation of } 123 \\ \end{cases}$$