equations:proca_equation
Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision | ||
|
equations:proca_equation [2018/03/26 16:41] jakobadmin |
equations:proca_equation [2023/04/02 03:12] (current) edi [Concrete] |
||
|---|---|---|---|
| Line 1: | Line 1: | ||
| - | ====== Proca Equation: $\quad m^2 A^\rho = \partial_\sigma F^{\sigma \rho}$ ====== | + | <WRAP lag>$ m^2 A^\rho = \partial_\sigma F^{\sigma \rho}$</WRAP> |
| + | ====== Proca Equation ====== | ||
| - | <tabbox Why is it interesting?> | ||
| - | The Proca equation is a generalization of the [[equations:maxwell_equations|Maxwell equation]] for [[basic_notions:mass|massive]] [[basic_notions:spin|spin]] $1$ particles. Formulated differently, the Maxwell equation is only a special case of the Proca equation for massless particles/fields. | ||
| - | The Proca equation is important because it correctly describes massive spin $1$ particles/fields. | + | <tabbox Intuitive> |
| - | + | ||
| - | <tabbox Layman> | + | |
| <note tip> | <note tip> | ||
| Line 14: | Line 11: | ||
| </note> | </note> | ||
| | | ||
| - | <tabbox Student> | + | <tabbox Concrete> |
| \begin{align}m^2 A^\rho &= \partial_\sigma ( \partial^\sigma A^\rho - \partial^\rho A^\sigma) \\ | \begin{align}m^2 A^\rho &= \partial_\sigma ( \partial^\sigma A^\rho - \partial^\rho A^\sigma) \\ | ||
| &=\partial_\sigma F^{\sigma \rho} | &=\partial_\sigma F^{\sigma \rho} | ||
| \end{align} | \end{align} | ||
| - | + | ||
| - | <tabbox Researcher> | + | |
| + | 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> | ||
| <note tip> | <note tip> | ||
| Line 26: | Line 37: | ||
| </note> | </note> | ||
| + | |||
| + | <tabbox Why is it interesting?> | ||
| + | |||
| + | The Proca equation is a generalization of the [[equations:maxwell_equations|Maxwell equation]] for [[basic_notions:mass|massive]] [[basic_notions:spin|spin]] $1$ particles. Formulated differently, the Maxwell equation is only a special case of the Proca equation for massless particles/fields. | ||
| + | |||
| + | The Proca equation is important because it correctly describes massive spin $1$ particles/fields. | ||
| | | ||
| <tabbox Definitions> | <tabbox Definitions> | ||
equations/proca_equation.1522075319.txt.gz · Last modified: 2018/03/26 14:42 (external edit)
Page Tools
Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Share Alike 4.0 International
