$ m^2 A^\rho = \partial_\sigma F^{\sigma \rho}$ ====== Proca Equation ====== 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. \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{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}} The motto in this section is: //the higher the level of abstraction, the better//. 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. * $\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: $F^{\sigma \rho} \equiv \partial^\sigma A^\rho - \partial^\rho A^\sigma$.