# lorentz invariance

by RedX
Tags: invariance, lorentz
 P: 969 In electrodynamics, the Coulomb gauge is specified by $$\nabla \cdot A=0$$, 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 $$\phi$$ (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?
P: 1,865
 Quote by RedX 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?
Well, the Schrödinger 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.
P: 1,128
 Quote by RedX In electrodynamics, the Coulomb gauge is specified by $$\nabla \cdot A=0$$, 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?
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.

 Quote by RedX 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.
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.

 Quote by RedX 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?
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.

 Quote by RedX Does the scalar potential and charge density really matter in field theory, or is just the 3-vector potential and 3-current important?
They absolutely matter. In Coulomb gauge, you get an explicit Coulomb interaction among pieces of the the charge density at different places.

P: 969