I think a big hole in my thinking would be addressed if I had a physical, geometric intuition about the electromagnetic tensor. We have covered it briefly in a chapter of the course, but only as a piece of mathematics, an abstract operator.
However, I saw in a Wikipedia article here an interesting reference to Haskell's work where he claims that Maxwell's equations fail in accelerating frames. This was quite news to me as I had hitherto believed that Maxwell's equations were classically 'correct' in the sense of correctly predicting the 'large-scale' phenomena, and that QED was really an "implementation" of Maxwell at the quantum level, maybe providing predictive corrections at that scale.
I am also not completely comfortable that there is no mathematical distinction between a vector field which is constant and stationary over all x, y, z, t and a vector field which has a constant value over all x, y, z, t but which is moving with respect to the coordinate axes at some constant velocity. And what if we want a 'field which is constant in value over all x, y, z, t' to accelerate wrt some axis?
Perhaps the deep truth here is that unlike particles, fields can't move at all - only disturbances in fields can propagate.