✎ Proca Equation [The Physics Travel Guide]

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<WRAP lag>$ m^2 A^\rho =    \partial_\sigma F^{\sigma  \rho}$</WRAP>

====== Proca Equation ======

<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}

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 \\
 \label{eq:aplusminus} &\equiv  A_\mu^+ + A_\mu^-   \end{align}
where $\epsilon_{r,\mu}(k)$ are basis vectors called polarization vectors.

<note tip>
The motto in this section is: //the higher the level of abstraction, the better//.
</note>

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>

* $\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>

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