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* Page found: Weitere Eigenschaften der Dirac-Gleichung (eq math.2682.3)

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\begin{align}

& \mathfrak{i} {{\Psi }^{+}}\dot{\Psi }\quad ={{\Psi }^{+}}\left( \underline{\alpha }.\hat{\underline{p}}+\beta m \right)\Psi  \\

& -\mathfrak{i} {{{\dot{\Psi }}}^{+}}\Psi \quad ={{\left( \underline{p}\Psi  \right)}^{+}}\underline{\alpha }\Psi +m{{\Psi }^{+}}\beta \Psi  \\

& ------------------------ \\

& \mathfrak{i} {{\partial }_{t}}\underbrace{\left( {{\Psi }^{+}}\Psi  \right)}_{:=\rho }\quad ={{\Psi }^{+}}\underline{\alpha }\left( \underline{p}\Psi  \right)-{{\left( \underline{p}\Psi  \right)}^{+}}\underline{\alpha }\Psi  \\

& \quad =-\mathfrak{i} \sum\limits_{k}{{{\Psi }^{+}}{{\alpha }_{k}}\left( {{\partial }_{k}}\Psi  \right)-{{\left( {{\partial }_{k}}\Psi  \right)}^{+}}{{\alpha }_{k}}\Psi } \\

& \quad =-\mathfrak{i} \sum\limits_{k}{{{\partial }_{k}}\underbrace{\left( {{\Psi }^{+}}{{\alpha }_{k}}\Psi  \right)}_{:={{j}_{k}}}} \\

\end{align}

TeX (checked): 

{\begin{aligned}&{\mathfrak {i}}{{\Psi }^{+}}{\dot {\Psi }}\quad ={{\Psi }^{+}}\left({\underline {\alpha }}.{\hat {\underline {p}}}+\beta m\right)\Psi \\&-{\mathfrak {i}}{{\dot {\Psi }}^{+}}\Psi \quad ={{\left({\underline {p}}\Psi \right)}^{+}}{\underline {\alpha }}\Psi +m{{\Psi }^{+}}\beta \Psi \\&------------------------\\&{\mathfrak {i}}{{\partial }_{t}}\underbrace {\left({{\Psi }^{+}}\Psi \right)} _{:=\rho }\quad ={{\Psi }^{+}}{\underline {\alpha }}\left({\underline {p}}\Psi \right)-{{\left({\underline {p}}\Psi \right)}^{+}}{\underline {\alpha }}\Psi \\&\quad =-{\mathfrak {i}}\sum \limits _{k}{{{\Psi }^{+}}{{\alpha }_{k}}\left({{\partial }_{k}}\Psi \right)-{{\left({{\partial }_{k}}\Psi \right)}^{+}}{{\alpha }_{k}}\Psi }\\&\quad =-{\mathfrak {i}}\sum \limits _{k}{{{\partial }_{k}}\underbrace {\left({{\Psi }^{+}}{{\alpha }_{k}}\Psi \right)} _{:={{j}_{k}}}}\\\end{aligned}}

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𝔦 Ψ + Ψ ˙   = Ψ + ( α ¯ . p ¯ ^ + β m ) Ψ - 𝔦 Ψ ˙ + Ψ   = ( p ¯ Ψ ) + α ¯ Ψ + m Ψ + β Ψ - ⁣ - ⁣ - ⁣ - ⁣ - ⁣ - ⁣ - ⁣ - ⁣ - ⁣ - ⁣ - ⁣ - ⁣ - ⁣ - ⁣ - ⁣ - ⁣ - ⁣ - ⁣ - ⁣ - ⁣ - ⁣ - ⁣ - ⁣ - 𝔦 ∂ t ( Ψ + ⁢ Ψ ) ⏟ := ρ   = Ψ + α ¯ ( p ¯ Ψ ) - ( p ¯ Ψ ) + α ¯ Ψ = - 𝔦 ⁢ ∑ k Ψ + ⁢ α k ⁢ ( ∂ k ⁡ Ψ ) - ( ∂ k ⁡ Ψ ) + ⁢ α k ⁢ Ψ = - 𝔦 ⁢ ∑ k ∂ k ⁡ ( Ψ + ⁢ α k ⁢ Ψ ) ⏟ := j k absent fragments i superscript Ψ ˙ Ψ   superscript Ψ fragments ( ¯ 𝛼 . ^ ¯ 𝑝 β m ) Ψ absent fragments i superscript ˙ Ψ Ψ   superscript fragments ( ¯ 𝑝 Ψ ) ¯ 𝛼 Ψ m superscript Ψ β Ψ absent absent fragments i subscript 𝑡 subscript ⏟ superscript Ψ Ψ assign absent 𝜌   superscript Ψ ¯ 𝛼 fragments ( ¯ 𝑝 Ψ ) superscript fragments ( ¯ 𝑝 Ψ ) ¯ 𝛼 Ψ absent absent 𝔦 subscript 𝑘 superscript Ψ subscript 𝛼 𝑘 subscript 𝑘 Ψ superscript subscript 𝑘 Ψ subscript 𝛼 𝑘 Ψ absent absent 𝔦 subscript 𝑘 subscript 𝑘 subscript ⏟ superscript Ψ subscript 𝛼 𝑘 Ψ assign absent subscript 𝑗 𝑘 {\displaystyle{\displaystyle\begin{aligned} \par&\displaystyle\mathfrak{i}{{% \Psi}^{+}}\dot{\Psi}\quad={{\Psi}^{+}}\left(\underline{\alpha}.\hat{\underline% {p}}+\beta m\right)\Psi\\ \par&\displaystyle-\mathfrak{i}{{{\dot{\Psi}}}^{+}}\Psi\quad={{\left(% \underline{p}\Psi\right)}^{+}}\underline{\alpha}\Psi+m{{\Psi}^{+}}\beta\Psi\\ \par&\displaystyle------------------------\\ \par&\displaystyle\mathfrak{i}{{\partial}_{t}}\underbrace{\left({{\Psi}^{+}}% \Psi\right)}_{:=\rho}\quad={{\Psi}^{+}}\underline{\alpha}\left(\underline{p}% \Psi\right)-{{\left(\underline{p}\Psi\right)}^{+}}\underline{\alpha}\Psi\\ \par&\displaystyle\quad=-\mathfrak{i}\sum\limits_{k}{{{\Psi}^{+}}{{\alpha}_{k}% }\left({{\partial}_{k}}\Psi\right)-{{\left({{\partial}_{k}}\Psi\right)}^{+}}{{% \alpha}_{k}}\Psi}\\ \par&\displaystyle\quad=-\mathfrak{i}\sum\limits_{k}{{{\partial}_{k}}% \underbrace{\left({{\Psi}^{+}}{{\alpha}_{k}}\Psi\right)}_{:={{j}_{k}}}}\\ \par\end{aligned}}}

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iΨ+Ψ˙=Ψ+(α_.p_^+βm)Ψ−iΨ˙+Ψ=(p_Ψ)+α_Ψ+mΨ+βΨ−−−−−−−−−−−−−−−−−−−−−−−−i∂t(Ψ+Ψ)⏟:=ρ=Ψ+α_(p_Ψ)−(p_Ψ)+α_Ψ=−i∑kΨ+αk(∂kΨ)−(∂kΨ)+αkΨ=−i∑k∂k(Ψ+αkΨ)⏟:=jk

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