This shows you the differences between two versions of the page.
Both sides previous revision Previous revision Next revision | Previous revision Last revision Both sides next revision | ||
advanced_tools:group_theory:group_contraction [2017/12/17 12:15] jakobadmin [Student] |
advanced_tools:group_theory:group_contraction [2017/12/17 12:16] jakobadmin [Why is it interesting?] |
||
---|---|---|---|
Line 34: | Line 34: | ||
- | >"//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> | ||
Line 50: | Line 50: | ||
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> |