The Yang-Mills field

These are two-component spinors rotationg by 4π. In the complex plane, where each axis is a complex dimension, we can describe a spinor as a point on the circle with radius one. Now, this is the most interesting topic in quantum physics. Because rotating spinors by 2π we get its negative! In order to get the original spinor we need to keep rotating by another 2π. This also helps you understand the difference between the elements of a Lie group and the elements of its Lie algebra. These 2-spinors take values in the Lie group SU(2). For rotating them as quantum particles, which is the same as changing their angular momentum, we take the Lie group elements, then convert them to their Lie algebra elements, and finally exponentiate them and apply them to the spinors in the form of matrix-vector multiplication. We do this operation in a continuous and pointwise manner for each spinor. It's a linear operation. And we have control over the rotation degree by exponentiating our Pauli matrix as the operator and supplying the exponential map by an argument. Then of course the left-invariance of the Lie group SU(2) helps with composing rotations. It's surprising that after exponentiating our operator for 2π, π half by π half to be precise, we get the negative of the identity operator, not the original identity! This fact sparked the matter and anti-matter research. Finally, the failure of applying the isomorphism between unit quaternions and SU(2) to our operator gives us a hint at the difference between a Lie algebra and its corresponding Lie group. We are able to go from Lie algebra elements to their Lie group elements by the exponential map. But the rules are different for the two and so we should take note of the difference despite the similar appearence in the matrix form.

On the bottom right, you can see the Yang-Mills field which is the pullback of the spinors using a connection one-form on the Hopf bundle. The Yang-Mills field describes every force carrier other than gravity, while spinors describe matter fields in the Standard Model of particle physics. However, we define spinors with associated vector bundles of the principal bundle and use the structure group to make spinor bundles. So, it's a work in progress until we get a complete picture of reality.

The Hopf Fibration Explained
   • The Hopf Fibration Explained