Contraction of broken symmetries via Kac-Moody formalism

Abstract

I investigate contractions via Kac-Moody formalism. In particular, I show how the symmetry algebra of the standard 2-D Kepler system, which was identified by Daboul and Slodowy as an infinite-dimensional Kac-Moody loop algebra, and was denoted by ${\mathbb{H}}_2$, gets reduced by the symmetry breaking term, defined by the Hamiltonian \[ H(β)= \frac 1 {2m} (p_1^2+p_2^2)- \frac αr - βr^{-1/2} \cos ((ϕ-γ)/2). \] For this $H(\beta )$ I define two symmetry loop algebras ${\mathfrak{L}}_i(\beta ),i=1,2$, by choosing the `basic generators' differently. These ${\mathfrak{L}}_i(\beta )$ can be mapped isomorphically onto subalgebras of ${\mathbb{H}}_2$, of codimension 2 or 3, revealing the reduction of symmetry. Both factor algebras ${\mathfrak{L}}_i(\beta )/I_i(E,\beta )$, relative to the corresponding energy-dependent ideals $I_i(E,\beta )$, are isomorphic to $\mathfrak{s}o(3)$ and $\mathfrak{s}o(2,1)$ for $E<0$ and $E>0$, respectively, just as for the pure Kepler case. However, they yield two different non-standard contractions as $E\to 0$, namely to the Heisenberg-Weyl algebra ${\mathfrak{h}}_3={\mathfrak{w}}_1$ or to an abelian Lie algebra, instead of the Euclidean algebra $\mathfrak{e}(2)$ for the pure Kepler case. The above example suggests a general procedure for defining generalized contractions, and also illustrates the {\em `deformation contraction hysteresis'}, where contraction which involve two contraction parameters can yield different contracted algebras, if the limits are carried out in different order.

21 pages, 1 figure

References in corpus (2)

• Completeness of superintegrability in two-dimensional constant curvature spaces
• Exact Solvability of Superintegrable Systems