The Crystal Girl

On this page, I want to discuss some thoughts in regard to the classification of spinors in quantum field theory/the standard model. The initial impotus behind this was the question "Why U(1)xSU(2)xSU(3)?" And I wasn't really able to find any answers online about this. A lot of Grand Unified Theories (GUTs) have suggested that "U(1)xSU(2)xSU(3)" because it is a subgroup of some larger group. But I think this approach does not answer the essential question of *why* these symmetry groups should exist in the first place.

One might argue that questions of *why* aren't science. Science concerns itself with explaining observation through models. And U(1)xSU(2)xSU(3) *is* the model that effectively explains the observation. Further asking for an explanation of U(1)xSU(2)xSU(3) is "removed" from the initial observations.

Still, I press on as I hopelessly ambitious angel. My opinion became that U(1)xSU(2)xSU(3) is a classifying space for spinors. This came about from a mathematical perspective: If spinors are just sections of vector bundles, what additional information is there to distinguish spinors of different mass, charge, etc. Is it that the sections are different or is it that they belong to a different space altogether? A priori, if one had a gauge field G on a manifold, it's action on a vector bundle should be the same regardless of the section that's chosen (I think). This is what led me to the conclusion that it must be classifying the vector bundles themselves.

Based on this ad-hoc conclusion, I began doing research into the classification of vector bundles, and the first result I found was that the Grassmannian would classify these bundles. However based on a topological analysis of the fundamental groups, I found that Gr(2, 4) is of completely different topology to U(1)x SU(2)x SU(3). I think maybe I was misguided and Gr(2, inf) is the correct classifying space, but I think the issue probably gets worse. Gr(2, inf) is a huge space, so it is hard to believe that it classifies spinors.

A further thought supporting this is the null-vector construction of spinors. Dirac spinors are, roughly speaking sections of the exterior algebra bundle of a maximally isotropic subspace of the tangent space. This introduces a set of restrictions onto spinors that aren't satisfied by a vector bundle in general.