Prompt:
$$ \left[\frac{\partial}{\partial t}\,, \vec{\nabla}\right]\varphi=\frac{\partial}{\partial t}\left(\vec{\nabla}\varphi\right)-\vec{\nabla}\left(\frac{\partial \varphi}{\partial t}\right)\,\qquad\qquad\qquad\qquad \qquad(A1) $$ $$ \left[\frac{\partial}{\partial t}\,, \vec{\nabla}\cdot\right]\vec{F}=\frac{\partial}{\partial t}\left(\vec{\nabla}\cdot\vec{F}\right)-\vec{\nabla}\cdot\left(\frac{\partial \vec{F}}{\partial t}\right)\,\qquad \qquad\qquad\qquad (A2) $$ $$ \left[\frac{\partial}{\partial t}\,, \vec{\nabla}\times\right]\vec{F}=\frac{\partial}{\partial t}\left(\vec{\nabla}\times\vec{F}\right)-\vec{\nabla}\times\left(\frac{\partial \vec{F}}{\partial t}\right)\,\qquad\qquad\qquad\qquad (A3) $$ $$ \left[\frac{\partial}{\partial t}\,\,,\vec{\nabla}\times\right](\vec{F}\times\vec{G})=\vec{F}\times\left(\frac{\partial}{\partial t}(\vec{\nabla}\times\vec{G})-\vec{\nabla}\times\frac{\partial\vec{G}}{\partial t}\right)+\left(\vec{\nabla}\times\frac{\partial\vec{F}}{\partial t}-\frac{\partial}{\partial t}(\vec{\nabla}\times\vec{F})\right)\times\vec{G}\qquad (A4) $$ $$ \left[\frac{\partial}{\partial t}\,\,,\vec{\nabla}\right](\vec{F}\cdot\vec{G})=\vec{F}\left(\frac{\partial}{\partial t}(\vec{\nabla}\cdot\vec{G})-\vec{\nabla}\cdot\frac{\partial\vec{G}}{\partial t}\right)+\left(\vec{\nabla}\cdot\frac{\partial\vec{F}}{\partial t}-\frac{\partial}{\partial t}(\vec{\nabla}\cdot\vec{F})\right)\vec{G}\qquad\qquad\qquad\qquad (A5) $$ $$ \left[\frac{\partial^2}{\partial t^2}\,, \vec{\nabla}\right]\vec{\varphi}= \left[\frac{\partial}{\partial t}\,, \vec{\nabla}\right]\frac{\partial \varphi}{\partial t}\,, \left[\frac{\partial^2}{\partial t^2}\,, \vec{\nabla}\cdot\right]\vec{F}= \left[\frac{\partial}{\partial t}\,, \vec{\nabla\cdot}\right]\frac{\partial \vec{F}}{\partial t}\,, \left[\frac{\partial^2}{\partial t^2}\,, \vec{\nabla}\times\right]\vec{F}= \left[\frac{\partial}{\partial t}\,, \vec{\nabla\times}\right]\frac{\partial \vec{F}}{\partial t}\,(A6) $$ \end{document} If we now consider the interaction of a fermion particle with electromagnetic field ($A_\mu$), the spinor derivative will become \begin{equation} P'_\mu=p_\mu-eA'_\mu\,,\qquad A'_\mu=A_\mu-\frac{m_0c}{e}\,\gamma_\mu\,. \end{equation}
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