gauge symmetry

Quaternions do not commute in general, but they will commute if two quaternions point in the same direction. The common way to represent the group U(1) is with a normalized complex number. The same thing can be done with a quaternion. This will commute with a unitary quaternion if they both use the same quaternion pointing in the same direction. Electroweak symmetry uses all the degrees freedom available in a quaternion.

This groups is the completely uniform unit quaternion sphere, starting from t=-1, expanding to its maximal size at t=0, then contracting to t=+1. For an observer is now at the center of their private Universe - (0, 0, 0, 0) - when they see an event, no matter what the cause, the event can be scaled to fit on this sphere. The norm of any event in the unit sphere is exactly 1, even if with rulers and atomic clocks a big or small sized measurement could be made.

<p>Because the symmetries U(1), SU(2) and U(1)xSU(2) are formally subgroups of SU(3), there is no need for a larger group to unify these groups. A rather large effort is still required to connect to all we know of the standard model.

If one takes the vector part of a quaternion and takes the exponential, the norm is always equal to 1. The animation starts out at 8 points, those for exp(0, +/-1, +/-1, +/-1). These points grow into each other until they form a sphere. That sphere then shrinks to the point (1, 0, 0, 0), the furthest into the future the exponential can reach.

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.