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advanced_notions:chirality [2017/10/23 10:55]
jakobadmin [Student]
advanced_notions:chirality [2018/03/30 13:19] (current)
jakobadmin
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 ====== Chirality ====== ====== Chirality ======
 +//see also [[basic_notions:​spin]] and [[advanced_notions:​helicity|]]//​
 +
 +<tabbox Intuitive> ​
 +
 +<​blockquote>​One of the things we observe in everyday life is that things have a distinct left and right. The simplest case is just the hands of a human: Obviously, the left hand and the right hand are different from each other. That is a very general thing in nature that things can be 'like a left hand' or 'like a right hand'. Of course, they do not need to be so. A ball has obviously no distinct left or right. But things can have. This fact is known in science as chirality, originating from a Greek word for hand.
 +<​cite>​http://​axelmaas.blogspot.de/​2011/​11/​chiral-or-why-left-and-right-is-not.html</​cite></​blockquote>​
  
-<tabbox Why is it interesting?> ​ 
-Chirality is one of the fundamental labels we use to identify [[advanced_notions:​elementary_particles|elementary particles]]. (Other labels are the mass or the electric charge.) ​ 
-<tabbox Layman> ​ 
 <​blockquote>​Positive and negative chirality fermions are often described as being right-handed or left-handed,​ respectively;​if one shines a beam of positive chirality fermions (particles described math-matically as sections of S+) into a block of matter, it will begin to spin in a right-handed sense."​ <​cite>​[[http://​www.mathunion.org/​ICM/​ICM1986.1/​Main/​icm1986.1.0267.0306.ocr.pdf|from Geometry and Physics by E. Witten]]</​cite></​blockquote> ​ <​blockquote>​Positive and negative chirality fermions are often described as being right-handed or left-handed,​ respectively;​if one shines a beam of positive chirality fermions (particles described math-matically as sections of S+) into a block of matter, it will begin to spin in a right-handed sense."​ <​cite>​[[http://​www.mathunion.org/​ICM/​ICM1986.1/​Main/​icm1986.1.0267.0306.ocr.pdf|from Geometry and Physics by E. Witten]]</​cite></​blockquote> ​
  
  
  
-<​tabbox ​Student+<​tabbox ​Concrete 
 + 
 +For a nice discussion see http://​www.quantumfieldtheory.info/​Chirality_vs_Helicity_chart.pdf and http://​www.quantumfieldtheory.info/​ChiralityandHelicityindepth.pdf
  
 Chirality arises as a quantum number related to the Lorentz group. Form the [[http://​notes.jakobschwichtenberg.com/​doku.php?​id=the_standard_model:​poincare_group#​representations_of_the_lorentz_group|representation theory of the Lorentz group]], we know that the corresponding Lie algebra, can be interpreted as two copies of the $SU(2)$ Lie algebra $\mathfrak{su}(2)$. Therefore, we labelled each representation by two numbers: $j_L$ and $j_R$ which indicate which $\mathfrak{su}(2)$ representations are used to construct the Lorentz algebra representations. For example, the label $(\frac{1}{2},​0)$ means that we used to fundamental representation for one $\mathfrak{su}(2)$ and the trivial, one-dimensional representation for the other $\mathfrak{su}(2)$. ​ Chirality arises as a quantum number related to the Lorentz group. Form the [[http://​notes.jakobschwichtenberg.com/​doku.php?​id=the_standard_model:​poincare_group#​representations_of_the_lorentz_group|representation theory of the Lorentz group]], we know that the corresponding Lie algebra, can be interpreted as two copies of the $SU(2)$ Lie algebra $\mathfrak{su}(2)$. Therefore, we labelled each representation by two numbers: $j_L$ and $j_R$ which indicate which $\mathfrak{su}(2)$ representations are used to construct the Lorentz algebra representations. For example, the label $(\frac{1}{2},​0)$ means that we used to fundamental representation for one $\mathfrak{su}(2)$ and the trivial, one-dimensional representation for the other $\mathfrak{su}(2)$. ​
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 <​cite>​Quantum Field Theory and Standard Model by M. Schwartz</​cite></​blockquote>​ <​cite>​Quantum Field Theory and Standard Model by M. Schwartz</​cite></​blockquote>​
    
-<​tabbox ​Researcher+<​tabbox ​Abstract
  
 <note tip> <note tip>
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 </​note>​ </​note>​
  
---> Does the opposite chirality only emerge dynamically?#​ 
  
-<​blockquote>"//​because fundamentally all fermion particles are left-handed and all fermion antiparticles are right-handed,​ with the opposite handedness emerging dynamically for massive fermions. Such dynamical emergence of handed-ness is described by L. B. Okun, in his book Leptons and Quarks (North-Holland (2nd printing 1984) page 11) where he said:  “… a particle with spin in the direction opposite to that of its momentum …[is]… said to possess left-handed helicity, or left-handed polarization. A particle is said to possess right-handed helicity, or polarization,​ if its spin is directed along its momentum. The concept of helicity is not Lorentz invariant if the particle mass is non-zero. The helicity of such a particle depends oupon the motion of the observer’s frame of reference. For example, it will change sign if we try to catch up with the particle at a speed above its velocity. Overtaking a particle is the more difficult, the higher its velocity, so that helicity becomes a better quantum number as velocity increases. It is an exact quantum number for massless particles … The above space-time structure … means … that at …[ v approaching the speed of light ]… particles have only left-handed helicity, and antparticles only right-handed helicity.//"​ [[http://​arxiv.org/​pdf/​1504.03695.pdf|On the chirality of the SM and the fermion content of GUTs  by Renato M. Fonseca]]</​blockquote>​ 
  
-  +<tabbox Why is it interesting?>​  
-<--+Chirality is one of the fundamental labels we use to identify [[advanced_notions:​elementary_particles|elementary particles]]. (Other labels are the mass or the electric charge.) ​
  
---> Common Question 2# 
  
-  +<​tabbox ​FAQ
-<-- +
-   +
-<​tabbox ​Examples+
  
---> ​Example1#+--> ​Does the opposite chirality only emerge dynamically?​#
  
-  +<blockquote>"//​because fundamentally all fermion particles are left-handed and all fermion antiparticles are right-handed, with the opposite handedness emerging dynamically for massive fermions. Such dynamical emergence of handed-ness is described by L. B. Okun, in his book Leptons and Quarks (North-Holland (2nd printing 1984) page 11) where he said ​“… a particle with spin in the direction opposite to that of its momentum …[is]… said to possess left-handed helicity, or left-handed polarization. A particle is said to possess right-handed helicity, or polarization,​ if its spin is directed along its momentum. The concept of helicity is not Lorentz invariant if the particle mass is non-zero. The helicity of such a particle depends oupon the motion of the observer’s frame of reference. For example, it will change sign if we try to catch up with the particle at a speed above its velocity. Overtaking a particle is the more difficult, the higher its velocity, so that helicity becomes a better quantum number as velocity increases. It is an exact quantum number for massless particles … The above space-time structure … means … that at …[ v approaching the speed of light ]… particles have only left-handed helicity, and antparticles only right-handed helicity.//"​ [[http://​arxiv.org/​pdf/​1504.03695.pdf|On the chirality of the SM and the fermion content of GUTs  by Renato M. Fonseca]]</​blockquote>​
-<-- +
- +
---> Example2:#+
  
    
 <-- <--
   ​   ​
-<tabbox History> ​ 
  
 </​tabbox>​ </​tabbox>​
  
  
advanced_notions/chirality.1508748916.txt.gz · Last modified: 2017/12/04 08:01 (external edit) · Currently locked by: 162.158.167.47,35.170.66.78