BREAKING A QUANTUM SYMMETRY ON THE TABLETOP. A recurrent theme in art and science, the concept of symmetry has become a powerful scientific tool for the analysis of physical systems. However, under special circumstances, a "quantum anomaly" occurs: the laws of quantum physics break a system's apparent symmetry. After a long search, a research group (Horacio Camblong, University of San Francisco, [email protected], and collaborators at Universidad Nacional de La Plata, Argentina) has found a relatively simple example of a quantum anomaly: the interaction of a polar molecule with an electron. A polar molecule, despite being neutral, has a permanent separation of electric charge--a dipole. This dipole produces an electric field, which can capture electrons if it is strong enough. Can such an arrangement exist as a stable ion, with its "extra" electron? The researchers formulated the answer to this question in the language of symmetry. In physics, symmetry means that a system, such as the molecule-electron arrangement, behaves the same after you perform a change to it, such as stretching the molecule to larger scales and making appropriate adjustments to other variables in the system. At first glance, the electron-molecule interaction exhibits a remarkable scale invariance: the system "looks" the same when viewed from different scales in space and time--at least in a classical physics description which treats the molecule as a dipole and the electron as a point of charge. But this tidy picture breaks down with a proper treatment of the system, as prescribed by quantum field theory. A quantum field theory treatment requires the process of renormalization, which removes certain mathematical infinities and inconsistencies from the quantum approach. This process also makes the molecule's energy levels discrete or quantized rather than continuous. Examining the system this way, the researchers found that the scale invariance broke down. In fact, a large body of existing evidence, both experimental and numerical, supports their conclusion. While all other known quantum anomalies occur at high energies (an example is chiral symmetry in nuclear physics), the work suggests that quantum symmetry breaking can occur at much lower energies, in the domain of interacting electrons and molecules. (Camblong et al., Physical Review Letters, 26 November 2001.)

Authors: Klishevich, Sergey; Plyushchay, Mikhail
The nonlinear supersymmetry of one-dimensional systems is investigated in the context of the quantum anomaly problem. Any classical supersymmetric system characterized by the nonlinear in the Hamiltonian superalgebra is symplectomorphic to a supersymmetric canonical system with the holomorphic form of the supercharges. Depending on the behaviour of the superpotential, the canonical supersymmetric systems are separated into the three classes. In one of them the parameter specifying the supersymmetry order is subject to some sort of classical quantization, whereas the supersymmetry of another extreme class has a rather fictive nature since its fermion degrees of freedom are decoupled completely by a canonical transformation. The nonlinear supersymmetry with polynomial in momentum supercharges is analysed, and the most general one-parametric Calogero-like solution with the second order supercharges is found. Quantization of the systems of the canonical form reveals the two anomaly-free classes, one of which gives rise naturally to the quasi-exactly solvable systems. The quantum anomaly problem for the Calogero-like models is ''cured'' by the specific superpotential-dependent term of order hbar2. The nonlinear supersymmetry admits the generalization to the case of two-dimensional systems.