advanced_tools:group_theory:group_contraction
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advanced_tools:group_theory:group_contraction [2017/12/17 12:15] jakobadmin [Student] |
advanced_tools:group_theory:group_contraction [2018/10/11 16:23] (current) jakobadmin [Student] |
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| - | >"//The mechanism which is at work, according to well established results of QFT, goes under the general name | + | <blockquote>"//The mechanism which is at work, according to well established results of QFT, goes under the general name |
| of spontaneous breakdown of symmetry and involves the physical phenomena of the Bose | of spontaneous breakdown of symmetry and involves the physical phenomena of the Bose | ||
| - | condensation and the mathematical structure of the (Ïnonü–Wigner) group contraction//" from Group Contraction in Quantum Field Theory by Giuseppe Vitiello | + | condensation and the mathematical structure of the (Ïnonü–Wigner) group contraction//" <cite>from Group Contraction in Quantum Field Theory by Giuseppe Vitiello</cite></blockquote> |
| <tabbox Layman> | <tabbox Layman> | ||
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| Both concepts are mutually the opposite. However while one can always deform to a group where we contracted from, the opposite procedure is not always possible. | Both concepts are mutually the opposite. However while one can always deform to a group where we contracted from, the opposite procedure is not always possible. | ||
| + | |||
| + | To **deform** a Lie algebra, we redefine the Lie brackets as a power series in some parameter $t$ | ||
| + | $$ | ||
| + | f_t(a,b)=[a,b]+tF_1(a,b)+t^2 F_2(a,b)+\ldots,\quad a,b\in\frak{g}\,, | ||
| + | $$ | ||
| + | and demand that the series converges in some neighbourhood of the origin. | ||
| + | |||
| + | |||
| + | ---- | ||
| + | |||
| <blockquote> | <blockquote> | ||
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| * A nice discussion can be found here: {{ :advanced_tools:group_theory:20100209000459_wigner-inonu_contraction.pdf |}} | * A nice discussion can be found here: {{ :advanced_tools:group_theory:20100209000459_wigner-inonu_contraction.pdf |}} | ||
| + | * See also [[advanced_tools:group_theory:https://aip.scitation.org/doi/10.1063/1.1705338|Deformation and Contraction of Lie Algebras]] by Levy-Nahas | ||
advanced_tools/group_theory/group_contraction.1513509329.txt.gz · Last modified: 2017/12/17 11:15 (external edit)
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