SU(2) Traces

Dru B. Renner
www.DruBryantRenner.org

19 April 2003

The Pauli matrices satisfy the following.

$$\mathrm{tr}(\sigma^i)=0 ~~~~ \sigma^i\sigma^j=\delta^{ij}+i\epsilon^{ijk}\sigma^k$$

These properties can be used to simplify traces of products of Pauli matrices. The first few traces are as follows.

$$\begin{eqnarray*} \mathrm{tr}(\sigma^i)&=&0\\ \mathrm{tr}(\sigma^i\sigma^j)&=&2... ...a^{ij}\delta^{kl}-\delta^{ik}\delta^{jl}+\delta^{il}\delta^{jk}) \end{eqnarray*}$$

These results can be derived as follows.

$$\begin{eqnarray*} \mathrm{tr}(\sigma^i\sigma^j)=\mathrm{tr}(\delta^{ij}+i\epsilon^{ija}\sigma^a)=2\delta^{ij} \end{eqnarray*}$$

$$\mathrm{tr}(\sigma^i\sigma^j\sigma^k)=\mathrm{tr}((\delta^{i... ...\sigma^a\sigma^k)=i\epsilon^{ija}2\delta^{ak}=2i\epsilon^{ijk}$$

$$\begin{eqnarray*} &&\mathrm{tr}(\sigma^i\sigma^j\sigma^k\sigma^l)=\mathrm{tr}((\... ...a^{ij}\delta^{kl}-\delta^{ik}\delta^{jl}+\delta^{il}\delta^{jk}) \end{eqnarray*}$$