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advanced_tools:group_theory:conformal_group [2018/03/21 11:42]
jakobadmin [History]
advanced_tools:group_theory:conformal_group [2018/05/27 13:52] (current)
jakobadmin [Why is it interesting?]
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 <tabbox Why is it interesting?> ​ <tabbox Why is it interesting?> ​
 +The maximal spacetime symmetry group of massless particles is the conformal group.
 +
 +----
  
  
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 <​cite>​page 621 Einstein Gravity in a Nutshell - A. Zee</​cite>​ <​cite>​page 621 Einstein Gravity in a Nutshell - A. Zee</​cite>​
 </​blockquote>​ </​blockquote>​
 +
 +<​blockquote>"​The simplest example of conformal matter is a perfect fluid of radiation. In the context of cosmology, this is extremely well motivated since the early Universe was, we believe, radiation dominated."​ <​cite>​https://​arxiv.org/​pdf/​1612.02792.pdf</​cite></​blockquote>​
  
 <tabbox Layman> ​ <tabbox Layman> ​
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 <tabbox Student> ​ <tabbox Student> ​
-<WRAP tip>The conformal group is $SO(4,2)$ or its double cover $SU(2,2)$ [[http://​math.ucr.edu/​home/​baez/​symmetries.html|Source]]</WRAP>+  * The conformal group is $SO(4,2)$ or its double cover $SU(2,2)$ [[http://​math.ucr.edu/​home/​baez/​symmetries.html|Source]] 
 +  * For a nice discussion, see [[http://​aip.scitation.org/​doi/​pdf/​10.1063/​1.1665843|On Representations of the Conformal Group Which When Restricted to Its Poincare or Weyl Subgroups Remain Irreducible]] by J. Mickelsson 
 +  * For the definition of the group and the algebra, see [[https://​books.google.de/​books?​id=H90XDQAAQBAJ&​lpg=PA188&​ots=5JxDa7kfpc&​dq=%22su(2%2C2)%22%20Lie%20algebra&​hl=de&​pg=PA188#​v=onepage&​q&​f=false|this chapter]].  
 +  * The conformal group is a subgroup of the diffeomorphism group. Under a conformal transformation,​ the metric changes as 
 +$$ g_{\mu\nu}\to \Omega(x)g_{\mu\nu} $$ 
 +or equivalently 
 +$$ d\tau \to \Omega(x) ​ d\tau , $$ 
 +where $\Omega(x) = \mathrm{e}^{i \omega(x)}$ is a scalar factor.
  
-For a nice discussionsee [[http://aip.scitation.org/doi/pdf/10.1063/1.1665843|On Representations of the Conformal Group Which When Restricted to Its Poincare or Weyl Subgroups Remain Irreducible]] by J. Mickelsson+<​blockquote>"//​the conformal algebra is equivalent to SO(24), the algebra of rotations and boosts in a six dimensional 
 +space with two time-like directions.//" ​http://homepages.uc.edu/~argyrepc/cu661-gr-SUSY/susy2001.pdf</blockquote>​
  
-"//the conformal algebra is equivalent to SO(2, 4), the algebra of rotations and boosts in a six dimensional 
-space with two time-like directions.//"​ http://​homepages.uc.edu/​~argyrepc/​cu661-gr-SUSY/​susy2001.pdf 
  
  
-For the definition of the group and the algebra, see [[https://​books.google.de/​books?​id=H90XDQAAQBAJ&​lpg=PA188&​ots=5JxDa7kfpc&​dq=%22su(2%2C2)%22%20Lie%20algebra&​hl=de&​pg=PA188#​v=onepage&​q&​f=false|this chapter]]. ​ 
- 
-The conformal group is a subgroup of the diffeomorphism group. Under a conformal transformation,​ the metric changes as 
-$$ g_{\mu\nu}\to \Omega(x)g_{\mu\nu} $$ 
-or equivalently 
-$$ d\tau \to \Omega(x) ​ d\tau , $$ 
-where $\Omega(x) = \mathrm{e}^{i \omega(x)}$ is a scalar factor. 
  
  
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   ​   ​
-<tabbox Examples> ​ 
- 
---> Example1# 
- 
-  
-<-- 
- 
---> Example2:# 
- 
-  
-<-- 
  
 <tabbox FAQ> ​ <tabbox FAQ> ​
advanced_tools/group_theory/conformal_group.1521628950.txt.gz · Last modified: 2018/03/21 10:42 (external edit)