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The Wigner method

Why the spin half animations work.

The final 3 mathematica notebooks in the spin-half section are based on a method for removing jump points from the parametric ball first given by Wigner(1959). Each point on the parametric ball represents a rotation with the angle of rotation being given by the distance from the origin to that point and the axis of rotation being in the direction of the point. So for example the point {0,0,Pi} represents a rotation by pi radians in the z-axis and the point {0,0,-Pi} represents a rotation by Pi radians in the negative z-direction. However rotations of Pi radians in one direction and Pi radians in the opposite direction are are the same thing hence points {0,0,Pi} and {0,0,-Pi} are considered to be the same point in the parametric ball. The same is true for all rotations by Pi regard less of the axis of rotation. So any rotation can be represented by a point in a ball of radius Pi where each point on the surface of that ball is regarded as being the same as it's antipodal (opposite) point.

Now supose we want to represent a series of rotations in the in the z-axis of angle increasing from 0 to 4Pi this would be done using the path in fig. a) below. Notice that it is drawn in 3 stages: Stage 1 being a rotation from 0 to Pi, stage 2 from -Pi to Pi  and stage 3 from -Pi back to 0 again. Inbetween each of these stages is what is known as a jump point (given by the small red and blue dots) where the path jumps from a point on the surface of the parametric ball to its antipode since these represent the same rotation. If we supose that each point on the path shown by fig. a) defines a rotation to be applied to points on a string  of distance from the origin increasing with distance allong the path, then this path represents the string being wound around the z-axis. Then if, as time passes, the path in fig. a) is then continuously deformed through b), c), d) ,e) and finally f) then the string gets a whole series of rotations applied to it in a continuos manner untill it is in the state represented by fig. f) where no rotations at all are applied. Notice that the two end points of the path remain at the center of the parametric ball hence the string untangles it's self without the end points moving! Furthermore these rotations could be applied to any number of different strings and, since it would be the same rotation applied to all of them, they would all untangle themselves without crossing.

 

Jump point removal

 

Another thing to notice is that during the deformation of the path the jump points are removed. This occurs between figs. d) and e) and is achieved by collapsing the red jump point in to the blue one. Ofcourse it takes two jump points to colapse one in to the other which means that it is imposible to remove the single jump point that would, for example, be created by an initial rotation going from 0 to 2Pi. In other words this means you must go through 2 (or any even number) of whole rotations before the strings can be detangled.