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advanced_tools:group_theory:group_contraction [2017/12/17 12:14] jakobadmin |
advanced_tools:group_theory:group_contraction [2017/12/17 12:16] jakobadmin [Why is it interesting?] |
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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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<tabbox Student> | <tabbox Student> | ||
- | A nice discussion can be found here: {{ :advanced_tools:group_theory:20100209000459_wigner-inonu_contraction.pdf |}} | + | * **Deformation:** Continuously modify the structure constants! |
+ | * **Contraction:** Generators are multiplied with contraction parameters that are then sent to zero or infinity. | ||
+ | |||
+ | 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> | ||
+ | "There exists a plethora of definitions for both contractions and deformations. [...] [W]e discuss and compare the mutually opposite procedures of deformations and contractions of Lie algebras." | ||
+ | |||
+ | <cite>[[http://www.emis.de/journals/SIGMA/2006/Paper048/|On Deformations and Contractions of Lie Algebras]] by A. Fialowski and M. de Montigny</cite> | ||
+ | </blockquote> | ||
+ | |||
+ | |||
+ | * A nice discussion can be found here: {{ :advanced_tools:group_theory:20100209000459_wigner-inonu_contraction.pdf |}} | ||
+ | |||
+ | |||
<tabbox Researcher> | <tabbox Researcher> |