Utente:Grasso Luigi/sandbox4/contrazione gruppo
In theoretical physics, Eugene Wigner and Erdal İnönü have discussed[1] the possibility to obtain from a given Lie group a different (non-isomorphic) Lie group by a group contraction with respect to a continuous subgroup of it. That amounts to a limiting operation on a parameter of the Lie algebra, altering the structure constants of this Lie algebra in a nontrivial singular manner, under suitable circumstances.[2][3]
For example, the Lie algebra of the 3D rotation group Template:Math, Template:Math, etc., may be rewritten by a change of variables Template:Math, Template:Math, Template:Math, as
The contraction limit Template:Math trivializes the first commutator and thus yields the non-isomorphic algebra of the plane Euclidean group, Template:Math. (This is isomorphic to the cylindrical group, describing motions of a point on the surface of a cylinder. It is the little group, or stabilizer subgroup, of null four-vectors in Minkowski space.) Specifically, the translation generators Template:Math, now generate the Abelian normal subgroup of Template:Math (cf. Group extension), the parabolic Lorentz transformations.
Similar limits, of considerable application in physics (cf. correspondence principles), contract
- the de Sitter group Template:Math to the Poincaré group Template:Math, as the de Sitter radius diverges: Template:Math; or
- the super-anti-de Sitter algebra to the super-Poincaré algebra as the AdS radius diverges Template:Math; or
- the Poincaré group to the Galilei group, as the speed of light diverges: c → ∞;[4] or
- the Moyal bracket Lie algebra (equivalent to quantum commutators) to the Poisson bracket Lie algebra, in the classical limit as the Planck constant vanishes: ħ → 0.
Notes
[modifica | modifica wikitesto]References
[modifica | modifica wikitesto]- On contractions of semisimple Lie groups (PDF), in Transactions of the American Mathematical Society, vol. 289, n. 1, 1985, pp. 185–202, DOI:10.2307/1999695.
- Robert Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications, Dover Publications, 2006, ISBN 0486445291.
- On the Contraction of Groups and Their Representations, in Proc. Natl. Acad. Sci., vol. 39, n. 6, 1953, pp. 510–24, DOI:10.1073/pnas.39.6.510.
- Contraction of Lie Groups, in Journal of Mathematical Physics, vol. 2, n. 1, 1961, pp. 1–21, DOI:10.1063/1.1724208.
- A class of operator algebras which are determined by groups, in Duke Mathematical Journal, vol. 18, 1951, p. 221, DOI:10.1215/S0012-7094-51-01817-0.