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, J0, plays an important role. However, J0 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?