Representation of Spinors by Two-Dimensional Complex Rotations
Representation of Spinors by Two-Dimensional Complex Rotations
This Demonstration shows rotation in two complex dimensions and how this can represent a spinor. The matrix on the left shows the action of a spinor transform of the given angle on a unit vector. Remember that this is a unit vector in 2D complex space, so it has two complex components that can be represented by a two-row vector. The two circles on the left are graphical representations of these complex components. The circle at the far right shows the length of each complex component as one of the projections of a unit vector in the 2D real plane. This circle represents the constraints on the values of the complex components. Each component may have a different value, but the sum of the squares of their two lengths must equal 1.
You can change the overall angle of the system, which changes the lengths of both components but preserves the constraint. You can also change the length of each component independently, but because of the constraint, the other will also change. Finally, you can change the phase of each component, which does not change the length of either.
To see the spinorial qualities of the system, set the angle and phases to zero. Note the blue complex number and constraint are pointing to the right. Now slowly increase the angle and watch the interplay between the complex numbers. When you get to 360°, note that the blue complex number and constraint are now pointing left. The system is in the opposite state. You must move the angle to 720° to return it to its original state.