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Spinors are the appropriate mathematical objects to describe particles with spin 1/2, like, for example, electrons.
Things to take note of:
Representing the u, as vectors is a heuristic oversimplification though, and in fact is not really correct, as operations like spinor addition work a little differently than vector addition. (See Winter 3.) However, temporarily visualizing them as such can aid in our understanding of how they and spin behave, relative to the at-rest coordinate system, for varying particle velocities.page 99 in Student Friendly Quantum Field Theory, by R. Klauber
Reference 3 is Winter, Rolf G., Quantum Physics, Wadsworth (1979), Chap. 9.
FAQ:
[V]ia the Pauli exclusion principle, fermions cannot occupy the same state within the same macro system. So, whereas photons (bosons) can occupy the same state and a lot of them can therefore reinforce one another to produce a macroscopic electromagnetic field, spinors (fermions) cannot do so. In other words, we have no classical macroscopic spinor fields to sense, interact with, and study experimentally. And thus, we have no classical theory of spinors.
Student Friendly Quantum Field Theory by Klauber