Been reading some more about physics, due to perennially trying to figure out what the heck crazy awesome stuff John Baez is actually talking about. Surprising revelations:
1. The Dirac ladder operators used in finding solutions for the quantum harmonic oscillator have something to do with differentiation and "multiply by x" familiar from generating-functions/species theory a la Joyal/McBride-Lawvere-Blass type-derivative nonsense.
2. (I'm not sure if I've got this right but as far as I understand) one can explain the existence of current-induced magnetic fields just by invoking Lorentz invariance: the electric field around a moving current in a wire is zero, because there are just as many electrons as protons, so the net electric charge density is zero. But if you are a little test charge out in space near that wire, moving parallel to it, then the protons and the electrons in the wire get Lorentz contracted "by different amounts" (which is to say that you're using the same coefficient, but the resulting inter-ion distance is different) and so from your point of view there is a nonzero charge distribution now, and so you must feel an electric attraction or repulsion. This is exactly what someone at rest with respect to the wire would call the effect on you of the magnetic field.