First I will play the old gauge symmetry game repeated everywhere for EM. I will then teach that old dog a new trick that ends up with the same bone. I will then struggle a little over the lingo to describe what I am interested in, the center of the quaternion group. I will then show how a new new trick, the new trick shifted to the right just enough to work, will lead to a gauge symmetry proof.

Let's review gauge symmetry in EM. Start with an arbitrary scalar function f. Take the conjugate of the 4-derivative operator and apply it to f:

Fire a machine gun at a wall with two slits. Look at the pattern bullets make that pass through. The pattern is the sum of two Gaussian curves. Classical physics makes sense.

Repeat the exercise with electrons or photons. Where there is constructive interference, the signal is strong. Where there is destructive interference, the signal can drop to zero. Waves behave that way all the time. Yet quantum interference works even if particles are fired at extremely low rates, such as once a day.