advanced_tools:group_theory:central_extension
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advanced_tools:group_theory:central_extension [2017/12/17 12:12] jakobadmin |
advanced_tools:group_theory:central_extension [2017/12/17 12:26] jakobadmin [Why is it interesting?] |
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| 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 $SL(2,\mathbb{C})$. | ||
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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> | ||
| <tabbox Layman> | <tabbox Layman> | ||
advanced_tools/group_theory/central_extension.txt · Last modified: 2017/12/17 12:26 by jakobadmin
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