In electrodynamics, the Coulomb gauge is specified by [tex]\nabla \cdot A=0 [/tex], i.e., the 3-divergence of the 3-vector potential is zero. This condition is not Lorentz invariant, so my first question is how can something that is not Lorentz invariant be allowed in the laws of physics? My second question concerns the photon polarization vector of a photon of 3-momentum k. Is this polarization vector a 3-vector or a 4-vector? If it's a 4-vector, what is the time component of the vector? The only condition seems to be that the 3-momentum k is perpendicular to the space-components of the polarization vector. My last question is this. Suppose your photon has 3-momentum k entirely in the z-direction, and in your frame of reference the 4-vector polarization e=(0,1,0,0), i.e., entirely in the x-direction. If you Lorentz boost your frame in the x-direction, then this 4-vector will receive some time component, say e'=(sqrt(2),sqrt(3),0,0). So when calculating a scattering amplitude, how do we know what the time component of our photon polarization vector is? In field theory, if the photon polarization vector has a non-zero time component, then the time component of the source, J^{0}, plays an important role. However, J^{0} is associated with the scalar potential [tex]\phi [/tex] (they are conjugate variables). Does the scalar potential and charge density really matter in field theory, or is just the 3-vector potential and 3-current important?
Well, the SchrÃ¶dinger equation isn't Lorenz-invariant either, but we certainly use it a lot! It's allowed because if the relative velocities of the interacting particles is small, the speed of light is "infinite" to a good approximation. The corrections for a retarded potential (AKA the Breit interaction, in an atomic system) are typically fairly small.
The physics is gauge invariant (that is, independent of the choice of gauge condition), so it's OK to choose a non-Lorentz-invariant gauge condition. The polarization is a 4-vector, and its dot product with the 4-momentum must be zero. In Coulomb gauge, the space components are orthogonal as well. So, in Coulomb gauge (but not in other gauges, in general) the time component of the polarization 4-vector is zero. If we start in a non-Lorentz-invariant gauge, then boosting takes us out of that gauge. So if you're going to specify Coulomb gauge (in which time components of polarization vectors are zero), then you're not allowed to boost. They absolutely matter. In Coulomb gauge, you get an explicit Coulomb interaction among pieces of the the charge density at different places.
thanks all, that made sense. If you experimentally prepare a photon, don't you always have to prepare it in the Coulomb gauge? That probably didn't make sense, since gauge is not physical. But what I mean is if you know a photon has a certain wavelength and direction and polarization, then where's the time component?