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A spinor is a mathematical object similar to a vector. However, while a vector points in some spatial direction, like, for example, in the direction of the north pole, a spinor points in a direction in an internal space.
A curious property of a spinor is that if you rotate it by 360° it isn't the same but get's a minus sign. Only after a rotation by 720° a spinor is again the same. In contrast a vector is completely unchanged if you rotate it by 360°.
This crazy property can be illustrated as shown, for example, here:
A Dirac spinor field $\Psi$ and its conjugate $\overline\Psi$ are equivalent to two left-handed Weyl spinors $\chi$ and $\tilde\chi$ and their right-handed conjugates $\chi^\dagger$ and $\tilde\chi^\dagger$; $\chi$ and $\chi^\dagger$ describe the left-chiral fermion and the right-chiral antifermion (e.g. \ $e^-_L$ and $e^+_R$), while $\tilde\chi$ and $\tilde\chi^\dagger$ describe the left-chiral antifermion and the right-chiral fermion (e.g. $e^+_L$ and $e^-_R$).
Things to take care 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.
Spinors arise as mathematical objects when we study the representations of the Lorentz group.
The objects that transform under the $(\frac{1}{2},0)$ or $(0,\frac{1}{2})$ representation of the Lorentz group are called Weyl spinors, objects transform under the (reducible) $(\frac{1}{2},0) \oplus (0,\frac{1}{2})$ representation are called Dirac spinors.
"spinor representations are the square root of a principle fiber bundle”
Spinors are the appropriate mathematical objects to describe particles with spin 1/2, like, for example, electrons.
"One could say that a spinor is the most basic sort of mathematical object that can be Lorentz-transformed."
An introduction to spinors by Andrew M. Steane
No one fully understands spinors. Their algebra is formally understood, but their geometrical significance is mysterious. In some sense they describe the ‘‘square root’’ of geometry and, just as understanding the concept of $\sqrt{-1}$ took centuries, the same might be true of spinors. Sir Michael Atiyah
[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
* Spinor Particle Mechanics by David Hestenes