Intuitive

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{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 Fun with Symmetry.

Abstract

Why is it interesting?

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.

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: $F^{\sigma \rho} \equiv \partial^\sigma A^\rho - \partial^\rho A^\sigma$.