# Analog of Klein-Gordon equation from Proca action

What would be the general form of Lagrange Equation when instead of a scalar field we have a vector potential? has anyone derived the klein gordon equation for a corresponding vector potential Lagrangian?

Yes, starting from a Lagrangian density of the form $$\mathcal L = -1/4 F_{\mu\nu} F^{\mu\nu} + m^2 A_\mu A^\mu$$ one can find the so-called Proca equation from the Euler-Lagrange, $$\partial_\mu F^{\mu\nu} + m^2 A^\nu = 0$$ If the field is massless, one can choose the Lorenz gauge, $\partial_\mu A^\mu =0$. If the field is massive, $\partial_\mu A^\mu =0$ follows from applying $\partial_\nu$ from the left-hand side. In both cases, one finds that the Proca equation is equivalent to the Klein-Gordon equation, $$(\partial^2 + m^2 )A^\mu =0$$ If the field is massless, we find Maxwell's equations in covariant form.
• With a massive gauge field, there is no need for a gauge fixing choice. The equation $\partial_\mu F^{\mu\nu} + m^2 A^\nu = 0$ implies $\partial_\mu A^\mu = 0$ whenever $m^2 \neq 0$ (act on the first equation with $\partial_\nu$). Commented Dec 1, 2013 at 15:17