equations:proca_equation
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| The general solution for the Proca equation is | The general solution for the Proca equation is | ||
| - | \begin{equation} m^2 A^\rho = \frac{1}{2} \partial_\sigma ( \partial^\sigma A^\rho - \partial^\rho A^\sigma) \end{equation} | ||
| - | is, analogous to the spin $0$ field solution, of the form | ||
| \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 \\ | \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} | \label{eq:aplusminus} &\equiv A_\mu^+ + A_\mu^- \end{align} | ||
| where $\epsilon_{r,\mu}(k)$ are basis vectors called polarization vectors. | 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]]. | ||
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| + | {{:equations:proca_maxwell.jpg?nolink}} | ||
| <tabbox Abstract> | <tabbox Abstract> | ||
equations/proca_equation.1522669981.txt.gz ยท Last modified: 2018/04/02 11:53 (external edit)
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