User Tools

Site Tools


advanced_tools:spinors

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
Next revision
Previous revision
advanced_tools:spinors [2018/03/30 14:49]
jakobadmin [Researcher]
advanced_tools:spinors [2022/09/07 21:52] (current)
147.92.69.196 [FAQ]
Line 1: Line 1:
 ====== 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>​ 
-"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>​ 
  
-<​tabbox ​Layman>  +<​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°. ​
Line 24: Line 16:
   * https://​en.wikipedia.org/​wiki/​Plate_trick   * https://​en.wikipedia.org/​wiki/​Plate_trick
   ​   ​
-<​tabbox ​Student+<​tabbox ​Concrete
 A Dirac spinor field $\Psi$ and its conjugate $\overline\Psi$ A Dirac spinor field $\Psi$ and its conjugate $\overline\Psi$
 are equivalent to two left-handed Weyl spinors $\chi$ and $\tilde\chi$ are equivalent to two left-handed Weyl spinors $\chi$ and $\tilde\chi$
 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 $e^-_R$).
  
  
Line 38: Line 30:
  
   * 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>​
Line 53: Line 47:
  
    
-<​tabbox ​Researcher+<​tabbox ​Abstract
 Spinors arise as mathematical objects when we study the [[advanced_tools:​group_theory:​representation_theory|representations]] of the Lorentz group. ​ Spinors arise as mathematical objects when we study the [[advanced_tools:​group_theory:​representation_theory|representations]] of the Lorentz group. ​
  
Line 72: Line 66:
  
  
 +<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> ​
Line 95: Line 90:
  
 <​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.1522414191.txt.gz · Last modified: 2018/03/30 12:49 (external edit)