Why does the EM field have 6 degrees of freedom but the 4-potential only 3?

[kye]

What does it mean that the electromagnetic field strength includes six degrees of freedom (three for the electric field strength, three for the magnetic field strength), whereas the four-potential includes only three degrees of freedom (two for the photon, one for the Higgs boson)? What is the four-potential? Why isn't it adjusted to be able to encompass 6 degrees of freedom too?

 

[thegreenlaser]

You don't really have 6 degrees of freedom... You can't just arbitrarily pick all the electric and magnetic field components and expect them to satisfy Maxwell's equations. As a simple example, suppose $E_x(x,y,z,t) = cos(\omega t + k z)$ and set everything else to zero for all space and time ($E_y = E_z = B_x = B_y = B_z = 0$). Those particular electric and magnetic fields are not allowed by Maxwell's equations. Specifically, if you plug them into Faraday's law
$$\vec{\nabla}\times\vec{E} = -\frac{\partial \vec{B}}{\partial t}$$
you'll find that the left hand side is non-zero while the right hand side is zero, which is obviously a contradiction. There's no charge density/current density combination I can pick which will give me those particular fields.

So you don't really have 6 degrees of freedom because not all combinations of $(E_x, E_y, E_z, B_x, B_y, B_z)$ are allowed. They need to satisfy Maxwell's equations.