The Group U(1), Electromagnetism's Gauge Symmetry

Posted by doug
The group for the unit circle in a complex plane, U(1), can be at an arbitrary angle in 3D space. The transverse waves of EM have this symmetry.
If one picks a quaternion at random, normalize it, then take n powers of that number, one ends up with this animation. It looks like tilted circles in the complex planes. The motion is fasted at the creation and annihilation. The velocity of the dots is slowest when the two have their largest separation. This group is symmetric in both time and space reflection. Recall that time reflection requires recall, memories of the path taken, while space reflection involves mirrors. <p>On pleasing aspect of this animation is that it starts to make sense of a transverse wave. Mapping that wave to electric or magnetic fields will require considerably more work.
q_graph -dir vp -out group_u1 -loop 0 -box 1.1 -command "g_u1 1 2 3 4 1000"
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\frac{A}{|A|} \in U(1)