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advanced_tools:spinors [2018/03/30 14:50]
jakobadmin
advanced_tools:spinors [2022/09/07 21:52] (current)
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 <tabbox Intuitive> ​ <tabbox Intuitive> ​
-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 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]]. ​
  
 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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 and their right-handed conjugates $\chi^\dagger$ and $\tilde\chi^\dagger$;​ 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 $\chi$ and $\chi^\dagger$ describe the left-[[advanced_notions:​chirality|chiral]] fermion and the right-chiral
-antifermion (\eg\ $e^-_L$ and $e^+_R$),+antifermion (e.g. \ $e^-_L$ and $e^+_R$),
 while $\tilde\chi$ and $\tilde\chi^\dagger$ describe while $\tilde\chi$ and $\tilde\chi^\dagger$ describe
-the left-chiral antifermion and the right-chiral fermion (\eg\ $e^+_L$ and $e^-_R$).+the left-chiral antifermion and the right-chiral fermion (e.g.  ​$e^+_L$ and $e^-_R$).
  
  
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   * 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>​
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 <​cite>​[[https://​arxiv.org/​abs/​1312.3824|An introduction to spinors]] by Andrew M. Steane</​cite>​ <​cite>​[[https://​arxiv.org/​abs/​1312.3824|An introduction to spinors]] by Andrew M. Steane</​cite>​
 </​blockquote>​ </​blockquote>​
 +
 +<​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>​
  
  
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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
    
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advanced_tools/spinors.1522414216.txt.gz · Last modified: 2018/03/30 12:50 (external edit)