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Definitionen:

  • Feldstärketensor:
    \[F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}\]
  • Viererstrom:
    \[j^{\mu}(x)=\left(\rho,\vec{j}\right)^{\mu}\]
  • Vierervektor:
    \[x=\left(x^{\mu}\right),\,\mu\in\left\{ 0,1,2,3\right\}\]

 

Lagrangedichte und Herleitung der Bewegungsgleichungen:

Lagrangedichte:

\[\mathfrak{L}_{em}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-j_{\mu}A^{\mu}\]

\[\Rightarrow\]

\[\partial_{\mu}F^{\mu\nu}=j^{\nu}\]

 

Beweis:

Euler-Lagrange-Gleichungen:

\[\partial_{\nu}\frac{\partial\mathfrak{L}}{\partial(\partial_{\nu}A^{\mu})}-\frac{\partial\mathfrak{L}}{\partial A^{\mu}}=0\]

 

\[\small\frac{\partial\mathfrak{L}_{em}}{\partial A^{\mu}}=-j_{\mu}\]

 

Umformung, damit abgeleitet werden kann:

\[\small-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}=-\frac{1}{4}(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})(\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu})=-\frac{1}{4}(\partial_{\mu}A_{\nu}\partial^{\mu}A^{\nu}-\partial_{\mu}A_{\nu}\partial^{\nu}A^{\mu}-\partial_{\nu}A_{\mu}\partial^{\mu}A^{\nu}+\partial_{\nu}A_{\mu}\partial^{\nu}A^{\mu})=\]

\[\small-\frac{1}{2}(\partial_{\mu}A_{\nu}\partial^{\mu}A^{\nu}-\partial_{\nu}A_{\mu}\partial^{\mu}A^{\nu})=-\frac{1}{2}(\partial_{\mu}A^{\tau}g_{\tau\nu}g^{\mu\sigma}\partial_{\sigma}A^{\nu}-\partial_{\mu}A^{\tau}g_{\tau\nu}g^{\sigma\nu}\partial_{\sigma}A^{\mu})\]

 

Ableiten:

\[\Rightarrow\small\frac{\partial}{\partial(\partial_{\mu}A^{\tau})}\left[-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\right]=-\frac{1}{2}(g_{\nu\tau}g^{\mu\sigma}\partial_{\sigma}A^{\nu}+\partial_{\mu}A^{\tau}g_{\nu\tau}g^{\mu\sigma}\delta_{\sigma}^{\mu}\delta_{\tau}^{\nu}-g_{\tau\nu}g^{\sigma\nu}\partial_{\sigma}A^{\mu}-\partial_{\mu}A^{\tau}g_{\tau\nu}g^{\sigma\nu}\delta_{\sigma}^{\mu}\delta_{\tau}^{\mu})=\]

\[\small-\frac{1}{2}(\partial^{\mu}A_{\tau}+\partial^{\sigma}A_{\nu}\delta_{\sigma}^{\mu}\delta_{\tau}^{\nu}-g_{\tau\nu}\partial^{\nu}A^{\mu}-\partial_{\mu}A_{\nu}g^{\nu\sigma}\delta_{\sigma}^{\mu}\delta_{\tau}^{\mu})=-\frac{1}{2}(\partial^{\mu}A_{\tau}+\partial^{\mu}A_{\tau}-\partial_{\tau}A^{\mu}-\partial_{\tau}A^{\mu})=\]

\[\small-\partial^{\mu}A_{\tau}+\partial_{\tau}A^{\mu}=-F_{\tau}^{\mu}\]

 

Daraus folgen die Maxwellgleichungen:

\[\Rightarrow\partial_{\nu}\frac{\partial}{\partial(\partial_{\mu}A^{\tau})}\left[-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\right]=-\partial_{\nu}F_{\mu}^{\nu}\]

\[\Rightarrow-\partial_{\nu}F_{\mu}^{\nu}=-j_{\mu}\]

 

Kontinuitätsgleichung:

\[\partial_{\nu}j^{\nu}=0\]

Beweis:

\[\partial_{\nu}j^{\nu}=\partial_{\nu}\partial_{\mu}F^{\mu\nu}=-\partial_{\nu}\partial_{\mu}F^{\nu\mu}=-\partial_{\mu}\partial_{\nu}F^{\mu\nu}=-\partial_{\nu}\partial_{\mu}F^{\mu\nu}=0\]

 

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