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Consider a real field $V^{\mu}(x)$ defined on a 4-dimensional Minkowski space. Acted by a transformation $\Lambda = \Lambda^{\mu}{}_{\nu} $ it transforms like $$V^{\mu}(x) \to V^{'\mu}(x) = \Lambda^{\ …

Thanks to @G. Smith and @mike stone, I've come to a solution. Expanding in Taylor Series, $$ S(\delta_x) = e^{\frac{\delta_x}{2} \begin{pmatrix} 0 & \sigma_x \\ \sigma_x & 0 \end{pmatrix}} = \sum_{n …

Thanks to @Charlie and @Cosmas Zachos I was able to find the correct answer. It simply suffices to develop the sum $$\frac{\omega^{\alpha \beta}}{2}\left(J_{\alpha \beta} \right)^{\mu}{}_{\nu} = -\del …

Let $\psi_\vec{0}^+$ be a Dirac wavefunction describing a motionless particle, $$\psi_\vec{0}^+(x) = \sqrt{2m} \begin{pmatrix} \chi \\ 0 \end{pmatrix} e^{ip \cdot x}$$ where $p = (m, \vec{0})$. Acte …

Let $\Lambda^{\alpha}{}_{\beta}$ denote a generic Lorentz transformation. Then, an infinitesimal transformation can be written like $$\Lambda^{\mu}{}_{\nu} = \delta^{\mu}{}_{\nu} + \omega^{\mu}{}_{\n …