Showing that a given matrix provides the desired Lorentz transformation [closed]

I am working on problem 5.12 in Halzen and Martin’s Quarks and Leptons. We are asked to show that the matrix

$$S = 1 - \frac{i}{4} \sigma_{\mu \nu} \varepsilon^{\mu \nu}$$

represents the infinitesimal Lorentz transformation

$$\Lambda^\mu {}_\nu = \delta^\mu {}_\nu + \varepsilon^\mu {}_\nu ,$$

in the sense that

$$S^{-1} \gamma^\mu S = \Lambda^\mu {}_\nu \gamma^\nu .$$

I have already proven that $S^{-1} = \gamma^0 S^\dagger \gamma^0$, so I can begin with the left-hand side of this third equation:

\begin{align} S^{-1} \gamma^\mu S &= \gamma^0 S^\dagger \gamma^0 \gamma^\mu S \\ &= \gamma^0 \left( 1 + \frac{i}{4} \sigma_{\mu \nu} \varepsilon^{\mu \nu} \right) \gamma^0 \gamma^\mu \left( 1 - \frac{i}{4} \sigma_{\mu \nu} \varepsilon^{\mu \nu} \right) \\ &= \gamma^0 \gamma^0 \gamma^\mu - \gamma^0 \gamma^0 \gamma^\mu \frac{i}{4} \sigma_{\mu \nu} \varepsilon^{\mu \nu} + \gamma^0 \frac{i}{4} \sigma_{\mu \nu} \varepsilon^{\mu \nu} \gamma^0 \gamma^\mu - \gamma^0 \frac{i}{4} \sigma_{\rho \tau} \varepsilon^{\rho \tau} \gamma^0 \gamma^\mu \frac{i}{4} \sigma_{\mu \nu} \varepsilon^{\mu \nu} \\ &= \gamma^\mu + \frac{1}{16} \gamma^0 \sigma_{\rho \tau} \varepsilon^{\rho \tau} \gamma^0 \gamma^\mu \sigma_{\mu \nu} \varepsilon^{\mu \nu} \\ &= \gamma^\mu + \frac{1}{16} \left( \sigma_{\rho \tau} \varepsilon^{\rho \tau} \sigma_{\mu \nu} \varepsilon^{\mu \nu} \right) \gamma^\mu . \end{align}

Remember, we want this to be equal to

$$\Lambda^\mu {}_\nu \gamma^\nu = \left( \delta^\mu {}_\nu + \varepsilon^\mu {}_\nu \right) \gamma^\nu = \gamma^\mu + \varepsilon^\mu {}_\nu \gamma^\nu .$$

The first term is there already, but I have no idea how that second term is going to work out to $\varepsilon^\mu {}_\nu \gamma^\nu$. Can anyone give me a hint, or tell me what I’ve done wrong?

-

closed as off-topic by Danu, Sebastian Riese, Daniel Griscom, ACuriousMind, Kyle KanosJan 3 at 22:18

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" – Danu, Sebastian Riese, Daniel Griscom, ACuriousMind, Kyle Kanos
If this question can be reworded to fit the rules in the help center, please edit the question.

To order $\epsilon$, isn't $S^{-1}$ just $1+\frac{i}{4}\sigma_{\mu\nu}\epsilon^{\mu\nu}$ ? Then try expanding out $S^{-1}\gamma^{\mu}S$ keeping terms of order $\epsilon$ and using the Clifford algebra relations when you have to commute $\gamma$s through $\sigma_{\mu\nu}$ terms...