Peskin and Schroeder Equation 3.23

Question

I've been trying (for a while) to prove that $S^{\mu\nu}:=\frac{i}{4}\left[\gamma^\mu,\,\gamma^\nu\right]$ is a representation of the Lorentz Lie algebra, that is, to prove that it satisfies the commutation relations of equation (3.17): $$ \left[J^{\mu\nu},\,J^{\rho\sigma}\right]=i\left(\eta^{\nu\rho}J^{\mu\sigma}-\eta^{\mu\rho}J^{\nu\sigma}-\eta^{\nu\sigma}J^{\mu\rho}+\eta^{\mu\sigma}J^{\nu\rho}\right)$$ where $\gamma$ are the gamma matrices defined by the relation $\left\{\gamma^\mu,\,\gamma^\nu\right\}=2\eta^{\mu\nu}\mathbb{1}_{n\times n}$. Peskin says "by repeated use of the defining relation of the gamma matrices it is easy to verify that these matrices satisfy the commutation relations." But I failed at that:

Answer 1

I struggled with this one as well and once I found I have written it in LaTeX which I will copy here below. Do note that I am using slightly different conventions than P&S, however it should still work out the same. \begin{equation} \begin{aligned} S^{\mu \nu} & = - \frac{i}{4}[\gamma^\mu,\gamma^\nu] \\& = - \frac{i}{4}(\gamma^\mu \gamma^\nu - \gamma^\nu \gamma^\mu) \\& = - \frac{i}{4}(\gamma^\mu \gamma^\nu - \left\{ \gamma^\mu, \gamma^\nu \right\} + \gamma^\mu \gamma^\nu) \\& = - \frac{i}{4}(2 \gamma^\mu \gamma^\nu - 2 g^{\mu \nu}) \\& = - \frac{i}{2}(\gamma^\mu \gamma^\nu - g^{\mu \nu}) \end{aligned} \end{equation} Subsequently, it will be convenient to first look at: \begin{equation} \begin{aligned} & [ S^{\mu \nu}, \gamma^\rho] = -\frac{i}{2}[(\gamma^\mu \gamma^\nu - g^{\mu \nu}),\gamma^\rho ] \\& =- \frac{i}{2} [\gamma^\mu \gamma^\nu,\gamma^\rho ] \\& = - \frac{i}{2}(\gamma^\mu \gamma^\nu \gamma^\rho - \gamma^\rho \gamma^\mu \gamma^\nu) \\& = - \frac{i}{2}(\gamma^\mu \gamma^\nu \gamma^\rho - \gamma^\rho \gamma^\mu \gamma^\nu - (\gamma^\mu \gamma^\rho \gamma^\nu - \gamma^\mu \gamma^\rho \gamma^\nu) - (\gamma^\mu \gamma^\rho \gamma^\nu - \gamma^\mu \gamma^\rho \gamma^\nu)) \\& = -\frac{i}{2}( \gamma^\mu \left\{ \gamma^\nu, \gamma^\rho \right\} - \left\{ \gamma^\rho, \gamma^\mu \right\} \gamma^\nu) \\& = - i( \gamma^\mu g^{\nu \rho} - g^{\rho \mu} \gamma^\nu) \end{aligned} \end{equation} and now: \begin{equation} \begin{aligned} & [S^{\mu \nu},S^{\rho \lambda}] = \left[S^{\mu \nu},-\frac{i}{2}(\gamma^\rho \gamma^\lambda - g^{\rho \lambda})\right] \\& = -\frac{i}{2} [S^{\mu \nu},\gamma^\rho \gamma^\lambda] \\& = - \frac{i}{2}(S^{\mu \nu} \gamma^\rho \gamma^\lambda - \gamma^\rho \gamma^\lambda S^{\mu \nu}) \\& = - \frac{i}{2}(S^{\mu \nu} \gamma^\rho \gamma^\lambda - \gamma^\rho S^{\mu \nu} \gamma^\lambda - \gamma^\rho \gamma^\lambda S^{\mu \nu}+\gamma^\rho S^{\mu \nu} \gamma^\lambda) \\& = - \frac{i}{2}\left( [S^{\mu \nu}, \gamma^\rho] \gamma^\lambda + \gamma^\rho [S^{\mu \nu},\gamma^\lambda] \right) \\& = -\frac{1}{2}\left( ( \gamma^\mu g^{\nu \rho} - g^{\rho \mu} \gamma^\nu) \gamma^\lambda + \gamma^\rho( \gamma^\mu g^{\nu \lambda} - g^{\lambda \mu} \gamma^\nu) \right) \\& = -\frac{1}{2}\left( g^{\nu \rho} \gamma^\mu \gamma^\lambda - g^{\rho \mu} \gamma^\nu \gamma^\lambda + g^{\nu \lambda} \gamma^\rho \gamma^\mu - g^{\lambda \mu} \gamma^\rho \gamma^\nu \right) \end{aligned} \end{equation} Now, we can write the first equation as: \begin{equation} \gamma^\mu \gamma^\nu = 2i S^{\mu \nu} + g^{\mu \nu} \end{equation} in order to write: \begin{equation} \begin{aligned} & [S^{\mu \nu},S^{\rho \lambda}] = -\frac{1}{2}\bigl( g^{\nu \rho} (2i S^{\mu \lambda} + g^{\mu \lambda}) - g^{\rho \mu} (2i S^{\nu \lambda} + g^{\nu \lambda}) \\& + g^{\nu \lambda} (2i S^{\rho \mu} + g^{\rho \mu}) - g^{\lambda \mu} (2i S^{\rho \nu} + g^{\rho \nu}) \bigr) \\& =- i g^{\nu \rho} S^{\mu \lambda} + i g^{\rho \mu} S^{\nu \lambda} - i g^{\nu \lambda} S^{\rho \mu} + i g^{\lambda \mu} S^{\rho \nu} \end{aligned} \end{equation} I hope/think this is the correct solution but I do not know if it is the quickest method.

Answer 2

It is proven in Tong's QFT script http://www.damtp.cam.ac.uk/user/dt281/qft.html section 4.1. in a quite nice fashion.