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 advanced_tools:group_theory:central_extension [2017/12/17 11:12]jakobadmin advanced_tools:group_theory:central_extension [2017/12/17 10:26] (current) Both sides previous revision Previous revision 2017/12/17 11:26 jakobadmin [Why is it interesting?] 2017/12/17 11:12 jakobadmin 2017/12/17 11:11 jakobadmin [Student] 2017/12/17 11:08 jakobadmin [Why is it interesting?] 2017/12/17 11:07 jakobadmin [Student] 2017/12/17 11:07 jakobadmin created Next revision Previous revision 2017/12/17 11:26 jakobadmin [Why is it interesting?] 2017/12/17 11:12 jakobadmin 2017/12/17 11:11 jakobadmin [Student] 2017/12/17 11:08 jakobadmin [Why is it interesting?] 2017/12/17 11:07 jakobadmin [Student] 2017/12/17 11:07 jakobadmin created Line 8: Line 8: 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 ​  ​