User Tools

Site Tools


advanced_tools:group_theory:central_extension

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
Previous revision
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?]
Line 1: Line 1:
 +====== Central Extension ======
 +
 +<tabbox Why is it interesting?> ​
 +
 +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</​cite></​blockquote>​
 +
 +<tabbox Layman> ​
 +
 +<note tip>
 +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.
 +</​note>​
 +  ​
 +<tabbox Student> ​
 +
 +
 +<WRAP tip> 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 $\hat G/A = G$. </​WRAP>​
 +
 +  * See page 178 in Moonshine beyond the Monster by Terry Gannon
 +
 +
 +
 + 
 +<tabbox Researcher> ​
 +
 +<note tip>
 +The motto in this section is: //the higher the level of abstraction,​ the better//.
 +</​note>​
 +
 +  ​
 +<tabbox Examples> ​
 +
 +
 +--> 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]])
 + 
 +<--
 +
 +<tabbox FAQ> ​
 +  ​
 +<tabbox History> ​
 +
 +</​tabbox>​
 +