Add a new page:
This is an old revision of the document!
$ m^2 A^\rho = \partial_\sigma F^{\sigma \rho}$
\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.
The Proca equation is a generalization of the 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.
The Proca equation is important because it correctly describes massive spin $1$ particles/fields.