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advanced_tools:group_theory:conformal_group [2018/03/21 11:42]
jakobadmin [Student]
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> ​
   ​   ​
 <tabbox History> ​ <tabbox History> ​
 +<​blockquote>​
 +While the theory of representations of finite dimensional groups such as the ones Weyl studied in 1925-6 was a well-developed part of mathematics by the 1960s, little was known about the representations of **infinite dimensional groups such as the group of conformal transformations in two dimensions**. Without some restrictive condition on the groups and the representations to be considered, the general problem of understanding representations of infinite dimensional groups appears to be completely intractable. The mathematicians Victor Kac and Robert Moody in 1967 introduced some new algebraic structures that allowed the construction of a class of infinite dimensional groups, now known as **Kac-Moody groups**.
  
 +<​cite>​Not Even Wrong by P. Woit</​cite>​
 +</​blockquote>​
 </​tabbox>​ </​tabbox>​
  
  
advanced_tools/group_theory/conformal_group.1521628931.txt.gz · Last modified: 2018/03/21 10:42 (external edit)