I'm having problems seeing the global SU(2) invariance of the QED Lagrangian. My specific problem is seeing why \begin{equation} e^{-i a_i \sigma_i} \gamma_\mu e^{i a_i \sigma_i} = \gamma_\mu \end{equation}

In every book i looked it up, it was written this is trivial and i couldn't find a computation, so i guess i'm missing something trivial. Any tip or reading recommendation where the global SU(2) invariance is shown explicitly would be much appreciated.


1 Answer 1


The $\gamma$ matrix acts on spinor indices, while the $SU(2)$ transformation most likely transforms two spinors into each other, i.e. it acts on a different space.

To be correct, if $a, b$ are $SU(2)$ indices, and $\alpha, \beta$ are spinor indices, the full structure would look like $$ \left[ \big(e^{-i a_i \sigma_i}\big)_{ab} \delta_{\alpha \beta}\right] \left[ \delta_{bc} \gamma_{\mu}^{\beta \delta} \right] \left[ \big(e^{i a_i \sigma_i}\big)_{cd} \delta_{\delta \epsilon}\right] = \left[ \delta_{ad} \gamma_{\mu}^{\alpha \epsilon} \right]$$ where i did not pay special attention to index placement. In order to make the structure more clear, I put square brackets around any object.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.