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 advanced_tools:group_theory:central_extension [2017/12/17 11:26] advanced_tools:group_theory:central_extension [2017/12/17 12:26] (current)jakobadmin [Why is it interesting?] Both sides previous revision Previous revision 2017/12/17 12:26 jakobadmin [Why is it interesting?] 2017/12/17 12:12 jakobadmin 2017/12/17 12:11 jakobadmin [Student] 2017/12/17 12:08 jakobadmin [Why is it interesting?] 2017/12/17 12:07 jakobadmin [Student] 2017/12/17 12:07 jakobadmin created Previous revision 2017/12/17 12:26 jakobadmin [Why is it interesting?] 2017/12/17 12:12 jakobadmin 2017/12/17 12:11 jakobadmin [Student] 2017/12/17 12:08 jakobadmin [Why is it interesting?] 2017/12/17 12:07 jakobadmin [Student] 2017/12/17 12:07 jakobadmin created Line 1: Line 1: + ====== Central Extension ====== + +  ​ + + Central extensions are a standard trick to convert projective representations of some group into true representations of another group. + + This is necessary, because when we only consider the "​naive"​ normal representations of a group like the Lorentz group, we miss an important representation (the spin $\frac{1}{2}$) representation). Thus, we can either use a less restrictive definition of a representation,​ i.e. use projective representations instead of true representations,​ or we could simply work with true representations of the central extension of the given group. ​ + + 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​ + +  ​ + + + Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party. + ​ + ​ +  ​ + + + The central extension $\hat G$ of a given group $G$ by an abelian group $A$ is defined as a group such that $A$ is a subgroup of the center of $\hat G$ and that the quotient $\hat G/A = G$. ​ + + * See page 178 in Moonshine beyond the Monster by Terry Gannon + + + + +  ​ + + + The motto in this section is: //the higher the level of abstraction,​ the better//. + ​ + + ​ +  ​ + + + --> 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). + + <-- + + --> 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]]) + + <-- + +  ​ + ​ +  ​ + + ​ + 