# Physics Notes by Jakob Schwichtenberg

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group_theory:notions:central_extension

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 group_theory:notions:central_extension [2017/04/09 14:24]jakobadmin [Central Extension] group_theory:notions:central_extension [2017/12/06 09:33] (current) Both sides previous revision Previous revision 2017/04/10 18:43 jakobadmin [Important Examples] 2017/04/09 14:24 jakobadmin [Central Extension] 2017/04/09 14:17 jakobadmin [Important Examples] 2017/04/09 13:41 jakobadmin 2017/04/09 08:40 jakobadmin created Next revision Previous revision 2017/04/10 18:43 jakobadmin [Important Examples] 2017/04/09 14:24 jakobadmin [Central Extension] 2017/04/09 14:17 jakobadmin [Important Examples] 2017/04/09 13:41 jakobadmin 2017/04/09 08:40 jakobadmin created Line 24: Line 24: ===== Important Examples ===== ===== Important Examples ===== - * The classical Galilean group needs to be extended by the introduction of a central charge, called //mass//, and this yields the 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). * 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. * 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. * 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]]) * 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]])