equations:proca_equation
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equations:proca_equation [2018/03/13 11:27] jakobadmin [Why is it interesting?] |
equations:proca_equation [2025/03/04 00:47] (current) edi [Concrete] |
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| + | <WRAP lag>$ m^2 A^\rho = \partial_\sigma F^{\sigma \rho}$</WRAP> | ||
| + | |||
| ====== Proca Equation ====== | ====== Proca Equation ====== | ||
| - | <note tip> $$ m^2 A^\rho = \partial_\sigma ( \partial^\sigma A^\rho - \partial^\rho A^\sigma) \quad \text { or } \quad m^2 A^\rho = \partial_\sigma F^{\sigma \rho} $$ | ||
| - | -->Definitions# | ||
| - | * $\partial_{\sigma} $ denotes the partial derivative, | + | <tabbox Intuitive> |
| - | * $m$ denotes the mass of the particle, | + | |
| - | * $A^\rho$ is either the wave function of the spin $1$ particle if we use the Proca equation in a particle theory, or describes the spin $1$ field if we work in a field theory. | + | |
| - | * $F^{\sigma \rho}$ is the electromagnetic field tensor: $F^{\sigma \rho} \equiv \partial^\sigma A^\rho - \partial^\rho A^\sigma$. | + | |
| - | <-- | + | <note tip> |
| + | Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party. | ||
| + | </note> | ||
| + | |||
| + | <tabbox Concrete> | ||
| + | \begin{align}m^2 A^\rho &= \partial_\sigma ( \partial^\sigma A^\rho - \partial^\rho A^\sigma) \\ | ||
| + | &=\partial_\sigma F^{\sigma \rho} | ||
| + | \end{align} | ||
| - | </note> | ||
| - | <tabbox Why is it interesting?> | + | The general solution for the Proca equation is |
| - | The Proca equation is a generalization of the [[equations:maxwell_equations|Maxwell equation]] for massive spin $1$ particles. Formulated differently, the Maxwell equation is only a special case of the Proca equation for massless particles/fields. | + | \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. | ||
| - | The Proca equation is important because it correctly describes massive spin $1$ particles/fields. | + | ---- |
| - | <tabbox Layman> | + | **Graphical Summary** |
| - | <note tip> | + | The diagram below shows the Proca equation and its Lagrangian in various forms. For a more detailed explanation see [[https://esackinger.wordpress.com/appendices/#field_equations|Fun with Symmetry]]. |
| - | Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party. | + | |
| - | </note> | + | |
| - | + | ||
| - | <tabbox Student> | + | |
| - | <note tip> | + | {{:equations:proca_maxwell.jpg?nolink}} |
| - | In this section things should be explained by analogy and with pictures and, if necessary, some formulas. | + | <tabbox Abstract> |
| - | </note> | + | |
| - | + | ||
| - | <tabbox Researcher> | + | |
| <note tip> | <note tip> | ||
| Line 39: | 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 Examples> | + | <tabbox Definitions> |
| - | --> Example1# | ||
| - | |||
| - | <-- | ||
| - | --> Example2:# | + | * $\partial_{\sigma} $ denotes the partial derivative, |
| + | * $m$ denotes the mass of the particle, | ||
| + | * $A^\rho$ is either the wave function of the spin $1$ particle if we use the Proca equation in a particle theory, or describes the spin $1$ field if we work in a field theory. | ||
| + | * $F^{\sigma \rho}$ is the electromagnetic field tensor: $F^{\sigma \rho} \equiv \partial^\sigma A^\rho - \partial^\rho A^\sigma$. | ||
| - | |||
| - | <-- | ||
| - | <tabbox FAQ> | ||
| - | | ||
| - | <tabbox History> | ||
| </tabbox> | </tabbox> | ||
equations/proca_equation.1520936859.txt.gz · Last modified: 2018/03/13 10:27 (external edit)
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