advanced_tools:group_theory:conformal_group
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advanced_tools:group_theory:conformal_group [2018/03/21 11:42] jakobadmin |
advanced_tools:group_theory:conformal_group [2018/05/27 13:52] 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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| + | ---- | ||
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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> | ||
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| + | <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 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 | |
| - | "//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 | + | |
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| - | + | ||
| - | 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 | + | |
| $$ g_{\mu\nu}\to \Omega(x)g_{\mu\nu} $$ | $$ g_{\mu\nu}\to \Omega(x)g_{\mu\nu} $$ | ||
| or equivalently | or equivalently | ||
| $$ d\tau \to \Omega(x) d\tau , $$ | $$ d\tau \to \Omega(x) d\tau , $$ | ||
| where $\Omega(x) = \mathrm{e}^{i \omega(x)}$ is a scalar factor. | 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> | ||
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advanced_tools/group_theory/conformal_group.txt · Last modified: 2018/05/27 13:52 by jakobadmin
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