advanced_tools:group_theory:central_extension
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advanced_tools:group_theory:central_extension [2017/12/17 12:07] jakobadmin created |
advanced_tools:group_theory:central_extension [2017/12/17 12:26] (current) jakobadmin [Why is it interesting?] |
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| <tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||
| - | Central extensions are a standard trick to convert [[group_theory:notions:projective_representation|projective representations]] of some group into true representations of another group. | + | 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. | 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 $SL(2,\mathbb{C})$. | 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> | <tabbox Layman> | ||
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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 [[group_theory:notions:quotient_group|quotient]] $\hat G/A = G$. </WRAP> | + | <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 | * See page 178 in Moonshine beyond the Monster by Terry Gannon | ||
| - | **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. (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 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 Examples> | <tabbox Examples> | ||
| - | --> Example1# | ||
| + | --> 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). | ||
| <-- | <-- | ||
| - | --> Example2:# | + | --> 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]]) | ||
| <-- | <-- | ||
advanced_tools/group_theory/central_extension.1513508853.txt.gz · Last modified: 2017/12/17 11:07 (external edit)
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