Nonlinear supersymmetry,quantum anomaly and quasi-exactly solvable systems |
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Institution: | 1. Department of Clinical Research, Nagasaki Kawatana Medical Center,Nagasaki,Japan;2. Department of Neurology, Nagasaki Kawatana Medical Center,Nagasaki,Japan;3. Department of Neurology, Sendai Nishitaga National Hospital, Japan;4. Department of Neurology, Iwaki National Hospital, Japan;5. Department of Neurology, Nishi-Niigata Chuo National Hospital, Japan;6. Department of Neurology, Iwate Hospital, Japan;7. Department of Neurology, Higashisaitama National Hospital, Japan;8. Department of Neurology, Hyogo-Chuo National Hospital, Japan;9. Department of Neurology, Higashi Nagoya National Hospital, Japan;10. Department of Neurology, Sagamihara National Hospital, Japan;11. Department of Neurology, Asahikawa Medical Center, Japan;12. Department of Neurology, Shizuoka Institute of Epilepsy and Neurological Disorders, Japan;13. Department of Neurology, Miyazaki Higashi Hospital, Japan;14. Department of Neurology, Minami Kyushu Hospital, Japan |
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Abstract: | 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 ℏ2. The nonlinear supersymmetry admits the generalization to the case of two-dimensional systems. |
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