Degrees of freedom of the electromagnetic field in a vacuum
The electric and magnetic fields have in total 6 real components per spatial point, so naively we would have 6 degrees of freedom. However, they’re not independent, as Maxwell’s equations in the classical form are redundant: we have 8 equations for 6 components, so they’re overdetermined.
A first step in getting rid of the redundancy is to use two of Maxwell’s equations to notice that E and B admit a scalar and a vector potential, define V and A and rewrite Maxwell’s equations in terms of V and A. Now we have 4 components (1 for V and 3 for A) and 4 equations.
However now these underdetermine the system, because of gauge invariance. We can fix a gauge to introduce an extra constraint that the potentials must obey to reduce the number of degrees of freedom (Coulomb gauge, Lorenz gauge etc…). So now we have 3 degrees of freedom between V and A.
However I know that a classical EM wave in a vacuum only has 2 degrees of freedom, not 3. How do we cut the last one? I don’t think we can impose another gauge fixing condition.