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.

The directions of the complex numbers and constraint have no simple relation to real directions in space. Just as every point in space can have a temperature, a single quantity, there are some qualities that a point in space can have that require a more complex description than a single number. A quantum wave function (a Pauli 2-spinor) is like this, where every point in space can be represented by two complex numbers under the above constraint. These directions are represented in an abstract four-dimensional space.

For a 2D real rotation on a unit vector, there is a similar constraint. The squares of the lengths of the projections of the vector onto the and axes must sum to 1. The 2D complex rotation has the same constraint except the projections are now the lengths of the two underlying complex components. If either of these lengths change, the other must also to keep the constraint valid. These two complex numbers can also change independently via a change of phase, which does not change their length. This phase change is what gives 2D complex rotation more degrees of freedom than a 2D real rotation. It can exhibit behavior similar to a 2D real rotation, but it can also change its underlying components in more ways than a real rotation.

References

B. A. Schumm, Deep Down Things: The Breathtaking Beauty of Particle Physics, Baltimore, MD: Johns Hopkins University Press, 2004.

G. Arfken, Mathematical Methods for Physicists, 3rd ed., Orlando, FL: Academic Press, 1985.