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--- equations:proca_equation [2018/03/28 10:24]
jakobadmin
+++ equations:proca_equation [2023/04/02 03:12] (current)
edi [Concrete]
@@ Line 16: / Line 16: @@
 &=\partial_\sigma F^{\sigma  \rho}
 \end{align}
+The general solution for the Proca equation is
+\begin{align} A_\mu &=  \int \frac{d^3 k}{\sqrt{ (2\pi)^3 2 \omega_k}} \left( \epsilon_{r,\mu}(k) a_r(k) {\mathrm{e}}^{-ikx} + \epsilon_{r,\mu}(k) a_r^\dagger(k) {\mathrm{e}}^{ikx}  \right) \notag \\
+ \label{eq:aplusminus} &\equiv  A_\mu^+ + A_\mu^-   \end{align}
+where $\epsilon_{r,\mu}(k)$ are basis vectors called polarization vectors.
+----
+**Graphical Summary**
+The diagram below shows the Proca equation and its Lagrangian in various forms. For a more detailed explanation see [[https://esackinger.wordpress.com/blog/lie-groups-and-their-representations/#proca_maxwell|Fun with Symmetry]].
+{{:equations:proca_maxwell.jpg?nolink}}
 <tabbox Abstract>