A Spin-One Klein Gordon Equation

[Jogging-Joe]

TL;DR

The spin zero Klein Gordon equation is commonly discussed. How about the spin one Klein Gordon Equation?

What is the spin one Klein Gordon Equation? What is the formula for the conserved current, i.e. the electric current density four-vector?

 

[Gaussian97]

I don't know what do you understand by the spin-one KG equation...
But the KG equation is just $(\square - m^2)\phi=0$.
So I would say that the KG equation for spin-one is just $$(\square - m^2)A^\mu=0$$

 

[Demystifier]

I don't know what do you understand by the spin-one KG equation...
But the KG equation is just $(\square - m^2)\phi=0$.
So I would say that the KG equation for spin-one is just $$(\square - m^2)A^\mu=0$$

Yes, and it's called Proca equation.

 

[Gaussian97]

Well, technically the Proca equation is $(\square - m^2)A^\mu = \partial^\mu \partial^\nu A_\nu$.
Klein-Gordon equation should (as far as I know), be fulfilled by any field with definite mass, since it's the quantum version of $p^2 + m^2 = 0$.
To obtain the Proca equation you need the extra condition $\partial_\nu A^\nu=0$.