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}

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