Contraction of broken symmetries via Kac-Moody formalism
- Centro de Ciencias Fisicas, Universidad Nacional Autonoma de Mexico, Apartado Postal 48-3, Cuernavaca, Morelos 62251 (Mexico)
I investigate contractions via Kac-Moody formalism. In particular, I show how the symmetry algebra of the standard two-dimensional Kepler system, which was identified by Daboul and Slodowy as an infinite-dimensional Kac-Moody loop algebra, and was denoted by H{sub 2}, gets reduced by the symmetry breaking term, defined by the Hamiltonian H({beta})=(1/2m)(p{sub 1}{sup 2}+p{sub 2}{sup 2})-{alpha}/r-{beta}r{sup -1/2} cos(({phi}-{gamma})/2). For this H({beta}) I define two symmetry loop algebras L{sub i}({beta}), i=1,2, by choosing the 'basic generators' differently. These L{sub i}({beta}) can be mapped isomorphically onto subalgebras of H{sub 2}, of codimension two or three, revealing the reduction of symmetry. Both factor algebras L{sub i}({beta})/I{sub i}(E,{beta}), relative to the corresponding energy-dependent ideals I{sub i}(E,{beta}), are isomorphic to so(3) and so(2,1) for E<0 and E>0, respectively, just as for the pure Kepler case. However, they yield two different nonstandard contractions as E{yields}0, namely to the Heisenberg-Weyl algebra h{sub 3}=w{sub 1} or to an Abelian Lie algebra, instead of the Euclidean algebra e(2) for the pure Kepler case. The above-noted example suggests a general procedure for defining generalized contractions, and also illustrates the 'deformation contraction hysteresis', where contraction which involves two contraction parameters can yield different contracted algebras, if the limits are carried out in different order.
- OSTI ID:
- 20860769
- Journal Information:
- Journal of Mathematical Physics, Vol. 47, Issue 8; Other Information: DOI: 10.1063/1.2234726; (c) 2006 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); ISSN 0022-2488
- Country of Publication:
- United States
- Language:
- English
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