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advanced_tools:group_theory:group_contraction [2017/12/17 12:15]
jakobadmin [Student]
advanced_tools:group_theory:group_contraction [2018/10/11 16:23] (current)
jakobadmin [Student]
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->"//​The mechanism which is at work, according to well established results of QFT, goes under the general name+<​blockquote>"//​The mechanism which is at work, according to well established results of QFT, goes under the general name
 of spontaneous breakdown of symmetry and involves the physical phenomena of the Bose  of spontaneous breakdown of symmetry and involves the physical phenomena of the Bose 
-condensation and the mathematical structure of the (Ïnonü–Wigner) group contraction//"​ from Group Contraction in Quantum Field Theory by Giuseppe Vitiello+condensation and the mathematical structure of the (Ïnonü–Wigner) group contraction//" ​<​cite>​from Group Contraction in Quantum Field Theory by Giuseppe Vitiello</​cite></​blockquote>​
  
 <tabbox Layman> ​ <tabbox Layman> ​
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 <tabbox Student> ​ <tabbox Student> ​
  
-**Deformation:​** Continuously modify the structure constants! +  * **Deformation:​** Continuously modify the structure constants! 
-**Contraction:​** Generators are multiplied with contraction parameters that are then sent to zero or infinity.+  ​* ​**Contraction:​** Generators are multiplied with contraction parameters that are then sent to zero or infinity.
  
 Both concepts are mutually the opposite. However while one can always deform to a group where we contracted from, the opposite procedure is not always possible. Both concepts are mutually the opposite. However while one can always deform to a group where we contracted from, the opposite procedure is not always possible.
 +
 +To **deform** a Lie algebra, we redefine the Lie brackets as a power series in some parameter $t$
 +$$
 +f_t(a,​b)=[a,​b]+tF_1(a,​b)+t^2 F_2(a,​b)+\ldots,​\quad a,​b\in\frak{g}\,,​
 +$$
 +and demand that the series converges in some neighbourhood of the origin.
 +
 +
 +----
 +
  
 <​blockquote>​ <​blockquote>​
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   * A nice discussion can be found here: {{ :​advanced_tools:​group_theory:​20100209000459_wigner-inonu_contraction.pdf |}}   * A nice discussion can be found here: {{ :​advanced_tools:​group_theory:​20100209000459_wigner-inonu_contraction.pdf |}}
 +  * See also [[advanced_tools:​group_theory:​https://​aip.scitation.org/​doi/​10.1063/​1.1705338|Deformation and Contraction of Lie Algebras]] by Levy-Nahas
  
  
advanced_tools/group_theory/group_contraction.1513509321.txt.gz · Last modified: 2017/12/17 11:15 (external edit)