Incompleteness of Representation Theory: Hidden symmetries and Quantum Non-Integrability

When regular regions are discovered in the parameter space of a system, where the parameter might be an external field applied to the Hydrogen atom, it usually indicates the existence of new (approximate) integrals of motion, and consequently quantum numbers [1]. In studies of many–body systems, the large number of degrees of freedom generally precludes one from using methods introduced in simpler one and two dimensional problems. However, one often starts with a group theoretical formalism, and while classical analyses stop being practical, group theory can readily identify exactly solvable limits of such systems, and correspondingly, quantum numbers. These exactly solvable limits are referred to as dynamical symmetries since they arise from the nature of the interactions. It has long been assumed that there is a precise relation between the exactly solvable limits or integrability of Hamiltonians based on some Lie algebra G [2], and the dynamical symmetries obtained from representation theory [3]. Indeed there is now a large literature on studies relating classical chaos to breaking of dynamical symmetries, its consequences in random matrix theory, as well as relations to exactly solvable systems [4]. It is interesting to consider now, whether group theory as it stands actually identifies all integrable limits, or whether there are hidden symmetries lingering in the parameter space of interactions. A common starting point of group theoretical analyses is the identification of a dynamical algebra G for a given quantum system. There is such an algebra when a Hamiltonian H can be expressed in terms of the generators of G. (For this study, we will focus on models based on real forms of simple and semi-simple classical Lie algebras, denoted by G.) The next step is to use the techniques of representation theory to identify all subalgebra embeddings, or group chains, consistent with the problem at hand. If there are n such group chains, one arrives at a decomposition of the form: G ⊃ G11 ⊃ G12 ⊃ · · · .. Gn1 ⊃ Gn2 ⊃ · · · . (1)