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advanced_tools:group_theory:conformal_group [2018/03/21 11:35]
jakobadmin [Researcher]
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> ​
 +  * 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.
 +
 +<​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</​blockquote>​
 +
 +
 +
 +
 +
 +----
  
 \begin{eqnarray} \begin{eqnarray}
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 The light cone The light cone
 $ds^2 =0$ is left invariant by all these transformations. ​ $ds^2 =0$ is left invariant by all these transformations. ​
 +
 +
 +-----
 +
 +  * [[http://​galileo.phys.virginia.edu/​~pf7a/​msmCFT.pdf|A conformal field theory primer by Paul Fendley]]
 +  * [[https://​arxiv.org/​abs/​1506.01399|Living Without Supersymmetry -- the Conformal Alternative and a Dynamical Higgs Boson by Philip D. Mannheim]]
 +  * http://​physics.stackexchange.com/​questions/​63595/​what-exactly-is-meant-by-the-conformal-group-of-minkowski-space
 +  * [[https://​www.staff.science.uu.nl/​~henri105/​Seminars/​AQFTtalk2.pdf|Very quick introduction to the conformal group and cft]]
    
 <tabbox Researcher> ​ <tabbox Researcher> ​
 +**Properties**
  
   * "the conformal group is non-compact and semisimple"​ [[https://​arxiv.org/​pdf/​hep-th/​0107008.pdf|Source]]   * "the conformal group is non-compact and semisimple"​ [[https://​arxiv.org/​pdf/​hep-th/​0107008.pdf|Source]]
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   * A detailed discussion of the geometrical interpretation of the conformal group can be found in [[http://​journals.aps.org/​rmp/​abstract/​10.1103/​RevModPhys.34.442|Fulton et al.]]"   * A detailed discussion of the geometrical interpretation of the conformal group can be found in [[http://​journals.aps.org/​rmp/​abstract/​10.1103/​RevModPhys.34.442|Fulton et al.]]"
  
 +
 +----
 +
 +**Lie Algebra**
 +
 +Take note that $\mathfrak{su}(4)$ and $\mathfrak{su}(2,​2)$ are both real forms of the same complex lie algebra $\mathfrak{sl}(4,​\mathbb{C})$. [[http://​math.stackexchange.com/​questions/​1925808/​representations-of-real-forms-of-simple-lie-algebra|Source]]
 +
 +See page 8 in http://​dias.kb.dk/​downloads/​dias:​67?​locale=da for a good summary of the generators and the commutation relations.
 +
 +----
 +
 +**Representations**
 +
 +In physical models, we need unitary representations to make sense of a probabilistic interpretation in a quantum field theory. ​
 +The conformal group is non-compact and therefore all unitary representations are infinite dimensional (except for the trivial representation).
 +
 +This can be understood, because the conformal algebra is equivalent to the $SO(2,4)$, the algebra, which describes rotations and boosts in a six-dimensional space with two spacelike directions. Therefore, as for the [[the_standard_model:​poincare_group|Poincare group]] it is always possible to boost the momentum label to higher and higher values.
 +
 +A classification of the representations of the conformal group in a similar spirit as the classification of the representations of the Poincare group (= making use of the "​little group" approach) can be found in section 3.4.1 in http://​homepages.uc.edu/​~argyrepc/​cu661-gr-SUSY/​susy2001.pdf.
 +
 +As for the Poincare group, we focus on the maximal compact part of the group. For the conformal group this maximal compact part is
 +$$ SO(2) \times SO(4) \subset SO(4,2).$$
 +In addition, we have the isomorphisms $SO(4) \simeq SU(2) \times SU(2)$ and $SO(2) \simeq U(1) $. Therefore, we can label the irreducible representation of the conformal group through "​left"​ and "​right"​ spins $(j_L,​j_R)$,​ which correspond to Casimir operators of the two $SU(2)$ factors, and one number, which labels the $U(1)$ representations.
 +
 +"//The non-compact generators act as raising and lowering operators, taking us between different states in a given representation."//​ from p. 165 in  http://​homepages.uc.edu/​~argyrepc/​cu661-gr-SUSY/​susy2001.pdf
 +
 +All unitary irreducible representations of the conformal algebra were classified as follows
 +^ Name      ^ $j_L,​j_R$ ​    ​^ ​ $d$        ^
 +| identity ​   | $j_L=j_R=0 $ | $d=0    $      |
 +| free chiral ​   | $j_Lj_R=0 $   | $d=j_L+j_R+1$|
 +| chiral ​   |  $j_Lj_R=0 $    | $d>​j_L+j_R+1$|
 +| free general ​   |  $j_Lj_R\neq 0 $    |$ d=j_L+j_R+2$|
 +| chiral ​   |  $j_Lj_R\neq 0 $     | $d>​j_L+j_R+2$|
 +Source: Mack, Commun. Math. Phys. 55 (1977) 1.
 +
 +See also:​[[http://​www.sciencedirect.com/​science/​article/​pii/​0022123682900040|Irreducible unitary representations of $SU(2, 2)$ by A.W Knapp, B Speh]] and section 4.3 in http://​dias.kb.dk/​downloads/​dias:​67?​locale=da
  
 ---- ----
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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>​
  
  
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