Polarization of Photon: Exploring Lorentz Condition

[carllacan]

My Quantum Field Theory notes, after explaining the Lorentz condition, say this:

The Lorentz condition still allows a residual gauge invariance with transformations satisfying $$\square \Lambda = 0 $$, so we can impose yet another constraint on the potentials. Since there are 4 potentials and we can impose two arbitrary constraints we have two degrees of freedom, and therefore the photon has two physical polarizations.

I have some questions about this.
1) What exactly does the polarization of a photon mean?
2) Why do the degrees of freedom of the potentials determine the polarizations of the photon?
3) If instead of the Lorentz condition we used another condition that didn't left any invariance, would it affect the polarizations the photon would have in our theory? Does such a condition even exists, or can it be proved that any condition would leave some residual invariance?

Thank you for your time.

 

[dextercioby]

It's Lorenz, not Lorentz, suggest your professor that his notes need rewriting. The 2nd question is answered by Weinberg in his QFT book, 1st volume. The 3rd question has my answer only using BRST anti-bracket/anti-field formalism which allows for a full gauge-fixing. The 1st question is ill-posed, photons (photonic quantum states, to be precise) use helicity as a Casimir of the Poincare group. Polarization of light is a classical concept. The polarization of a photon is then a misnomer.