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advanced_tools:group_theory:central_extension [2017/12/17 12:11]
jakobadmin [Student]
advanced_tools:group_theory:central_extension [2017/12/17 12:26]
jakobadmin [Why is it interesting?]
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 For example, the projective representations of  $SO(3,1)$ correspond to regular representations of $SL(2,​\mathbb{C})$. ​ For example, the projective representations of  $SO(3,1)$ correspond to regular representations of $SL(2,​\mathbb{C})$. ​
 +
 +<​blockquote>"​Central extensions play an important role in quantum mechanics: one of the earlier encounters is by means of Wigner’s theorem which states that a symmetry of a quantum mechanical system determines a (anti-) unitary transformation of the Hilbert space, which is unique up to a phase factor $e^{iϑ}$. As an immediate consequence of this phase factor, one deduces that given a quantum mechanical symmetry group $G$ there exists an extension $G_0$ of $G$ by $U(1)$ (the phase factors) which acts as a group of unitary transformations on the Hilbert space. **In most cases physicists have been succesful in hiding these central extensions by using larger symmetry groups**"​ <​cite>​http://​math.univ-lille1.fr/​~gmt/​PaperFolder/​CentralExtensions.pdf</​cite></​blockquote>​
  
 <tabbox Layman> ​ <tabbox Layman> ​
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 <tabbox Examples> ​ <tabbox Examples> ​
  
---> Example1# 
  
 +--> Galilean group -> Bargmann group#
 +
 +The classical Galilean group needs to be extended by the introduction of a central charge, called //mass//, and this yields the Bargmann group. (This is shown very nicely in QUANTIZATION ON A LIE GROUP: HIGHER-ORDER POLARIZATIONS by V. Aldaya, J. Guerrero and G. Marmo).
    
 <-- <--
  
---> ​Example2:#+--> ​SO(3) -> SU(2)# 
 + 
 +The standard spatial rotation group $SO(3)$ needs to be extended by $\mathbb{Z}_2$,​ which yields $SU(2)$, because otherwise we are not able to describe spin $\frac{1}{2}$ particles. 
 +  
 +<-- 
 + 
 + 
 +--> Mickelsson-Faddeev algebra#
  
 +The algebra of fermionic non-Abelian charge densitites needs to be extended to the Mickelsson-Faddeev algebra (See [[http://​physics.stackexchange.com/​a/​76653/​37286|this answer]])
    
 <-- <--
advanced_tools/group_theory/central_extension.txt · Last modified: 2017/12/17 12:26 by jakobadmin