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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.
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.
