✎ Spinors [The Physics Travel Guide]

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====== Spinors ======

Spinors are the appropriate mathematical objects to describe particles with spin 1/2, like, for example, electrons.

<blockquote>
"One could say that a spinor is the most basic sort of mathematical object that can be Lorentz-transformed."

<cite>[[https://arxiv.org/abs/1312.3824|An introduction to spinors]] by Andrew M. Steane</cite>
</blockquote>

<note tip>
Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.
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Spinors arise as mathematical objects when one studies the [[advanced_tools:group_theory:representation_theory|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**.

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* A nice introduction is [[https://arxiv.org/abs/1312.3824|An introduction to spinors]] by Andrew M. Steane
  * See also http://www-personal.umich.edu/~williams/notes/spinor.pdf
  * https://users.physics.ox.ac.uk/~Steane/teaching/rel_C_spinors.pdf
  * https://physics.stackexchange.com/questions/74682/introduction-to-spinors-in-physics-and-their-relation-to-representations/112041#112041

**Things to take note of:**

<blockquote>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.<cite>page 99 in [[ftp://srdconsulting.com/USB_BackUp/Data/Articles/QFT/StudentFriendlyQFT/SF_QFT_Chap04.pdf|Student Friendly Quantum Field Theory]], by R. Klauber</cite></blockquote>

Reference 3 is  Winter, Rolf G., Quantum Physics, Wadsworth (1979), Chap. 9.

* [[https://www.youtube.com/watch?v=SBdW978Ii_E|Sir Michael Atiyah, What is a Spinor ?]]
  * The standard reference for spinors is https://arxiv.org/abs/0812.1594

--> Example1#

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--> Example2:#

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--> Why is there no classical theory of spinors?#

<blockquote>[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.

<cite>Student Friendly Quantum Field Theory by Klauber</cite></blockquote>
 
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