# Spin-One Klein Gordon Equation

Summary:
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
Homework Helper
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
Demystifier
Gold Member
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
Homework Helper
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##.

Demystifier