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advanced_tools:spinors [2017/12/17 12:44]
jakobadmin [Why is it interesting?]
advanced_tools:spinors [2022/09/07 21:52] (current)
147.92.69.196 [FAQ]
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 ====== Spinors ====== ====== Spinors ======
  
-<tabbox Why is it interesting?> ​ 
  
-Spinors are the appropriate mathematical objects to describe particles with [[basic_notions:​spin|spin]] 1/2, like, for example, electrons. 
  
-<​blockquote>​ +<​tabbox ​Intuitive>  
-"One could say that a spinor is the most basic sort of mathematical object that can be Lorentz-transformed."​ +A spinor is a mathematical object similar to a [[basic_tools:​vector_calculus|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 [[advanced_tools:​internal_symmetry|internal space]]. ​
- +
-<​cite>​[[https://​arxiv.org/​abs/​1312.3824|An introduction to spinors]] by Andrew M. Steane</​cite>​ +
-</​blockquote>​ +
- +
-<​tabbox ​Layman>  +
-A spinor is a mathematical object similar to a [[basic_tools:​vector_calculus|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 [[advanced_tools:​internal_symmetries|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°. ​ 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°. ​
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   * https://​en.wikipedia.org/​wiki/​Plate_trick   * https://​en.wikipedia.org/​wiki/​Plate_trick
   ​   ​
-<​tabbox ​Student>  +<​tabbox ​Concrete>  
-Spinors arise as mathematical objects when one studies ​the [[advanced_tools:group_theory:​representation_theory|representations]] of the Lorentz group+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-[[advanced_notions:chirality|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 $e^-_R$). 
  
-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**. 
  
 ---- ----
  
   * A nice introduction is [[https://​arxiv.org/​abs/​1312.3824|An introduction to spinors]] by Andrew M. Steane   * A nice introduction is [[https://​arxiv.org/​abs/​1312.3824|An introduction to spinors]] by Andrew M. Steane
 +  * [[http://​www.weylmann.com/​spinor.pdf|A Child’s Guide to Spinors]] by William O. Straub
   * See also http://​www-personal.umich.edu/​~williams/​notes/​spinor.pdf   * See also http://​www-personal.umich.edu/​~williams/​notes/​spinor.pdf
   * https://​users.physics.ox.ac.uk/​~Steane/​teaching/​rel_C_spinors.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   * https://​physics.stackexchange.com/​questions/​74682/​introduction-to-spinors-in-physics-and-their-relation-to-representations/​112041#​112041
  
 +----
  
-**Things to take note of:**+**Things to take care 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>​ <​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. Reference 3 is  Winter, Rolf G., Quantum Physics, Wadsworth (1979), Chap. 9.
 +
 +
 +
  
    
-<​tabbox ​Researcher+<​tabbox ​Abstract 
 +Spinors arise as mathematical objects when we study 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**. 
 + 
 + 
 + 
 +<​blockquote>"​spinor representations are the square root of a principle [[advanced_tools:​fiber_bundles|fiber bundle]]”  
 + 
 +https://​particlephd.wordpress.com/​2008/​11/​28/​why-spinors/</​blockquote>​ 
 + 
 + 
 +----
  
   * [[https://​www.youtube.com/​watch?​v=SBdW978Ii_E|Sir Michael Atiyah, What is a Spinor ?]]   * [[https://​www.youtube.com/​watch?​v=SBdW978Ii_E|Sir Michael Atiyah, What is a Spinor ?]]
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 +<tabbox Why is it interesting?> ​
  
 +Spinors are the appropriate mathematical objects to describe particles with [[basic_notions:​spin|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>​ 
-<tabbox Examples+</blockquote>
  
---Example1#+<​blockquote>​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. <​cite>​Sir Michael Atiyah</​cite></​blockquote>
  
-  
-<-- 
  
---> Example2:#​ +  ​
- +
-  +
-<--+
  
 <tabbox FAQ> ​ <tabbox FAQ> ​
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 <​cite>​Student Friendly Quantum Field Theory by Klauber</​cite></​blockquote>​ <​cite>​Student Friendly Quantum Field Theory by Klauber</​cite></​blockquote>​
 +
 + * [[https://​geocalc.clas.asu.edu/​pdf/​SPINORPM.pdf| Spinor Particle Mechanics]] by David Hestenes
    
 <-- <--
   ​   ​
-<tabbox History> ​ 
  
 </​tabbox>​ </​tabbox>​
  
  
advanced_tools/spinors.1513511096.txt.gz · Last modified: 2017/12/17 11:44 (external edit)