Editing 20: Sir Roger Penrose - Plotting the Twist of Einstein’s Legacy

Spinor-valued functions or sections of spinor bundles are the primary objects of particle physics, however they can be realized in non-quantum ways as a model for the electromagnetic field per Penrose's books. When Atiyah stated that the geometrical significance of spinors is not fully understood, it is at this level rather than the well-understood representation algebra level. Analyzing the analogue of the vectors in this scenario, they represent tangents through curves at a particular point in space/on a manifold. The quadratic form then becomes a field of its own, operating on the tangent spaces of each point of the manifold independently but in a smoothly varying way. Arc length of curves can be calculated by integrating the norm of tangent vectors to the curve along it. In general relativity, this is the metric that is a dynamical variable as the solution to Einstein's equations. In order to follow the curvature of the manifold, vector fields are only locally vector valued functions, requiring transitions between regions as in the tribar example previously. Similarly, because the spin and orthogonal groups are so strongly coupled, we can apply the same transitions to spinor-valued functions. The physical reason to do this is that coordinate changes which affect vectors, thus also affect spinor fields in the same way.