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Lorentz invariance 
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#1
Jun510, 05:22 PM

P: 969

In electrodynamics, the Coulomb gauge is specified by [tex]\nabla \cdot A=0 [/tex], i.e., the 3divergence of the 3vector 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 3momentum k. Is this polarization vector a 3vector or a 4vector? If it's a 4vector, what is the time component of the vector? The only condition seems to be that the 3momentum k is perpendicular to the spacecomponents of the polarization vector. My last question is this. Suppose your photon has 3momentum k entirely in the zdirection, and in your frame of reference the 4vector polarization e=(0,1,0,0), i.e., entirely in the xdirection. If you Lorentz boost your frame in the xdirection, then this 4vector 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 nonzero 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 3vector potential and 3current important? 


#2
Jun510, 05:43 PM

Sci Advisor
P: 1,866

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. 


#3
Jun610, 06:40 PM

Sci Advisor
P: 1,232




#4
Jun610, 06:53 PM

P: 969

Lorentz invariance
thanks all, that made sense.
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? 


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