advanced_tools:group_theory:central_extension

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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?] |
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+ | ====== Central Extension ====== | ||

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+ | <tabbox Why is it interesting?> | ||

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+ | Central extensions are a standard trick to convert projective representations of some group into true representations of another group. | ||

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+ | 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. | ||

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+ | For example, the projective representations of $SO(3,1)$ correspond to regular representations of $SL(2,\mathbb{C})$. | ||

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+ | <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> | ||

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+ | <tabbox Layman> | ||

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+ | <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> | ||

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+ | <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> | ||

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+ | * See page 178 in Moonshine beyond the Monster by Terry Gannon | ||

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+ | <tabbox Researcher> | ||

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+ | <note tip> | ||

+ | The motto in this section is: //the higher the level of abstraction, the better//. | ||

+ | </note> | ||

+ | |||

+ | | ||

+ | <tabbox Examples> | ||

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+ | --> Galilean group -> Bargmann group# | ||

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+ | 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). | ||

+ | |||

+ | <-- | ||

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+ | --> SO(3) -> SU(2)# | ||

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+ | 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. | ||

+ | |||

+ | <-- | ||

+ | |||

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+ | --> Mickelsson-Faddeev algebra# | ||

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+ | 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]]) | ||

+ | |||

+ | <-- | ||

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+ | <tabbox FAQ> | ||

+ | | ||

+ | <tabbox History> | ||

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+ | </tabbox> | ||

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