An alternative dynamical description of Quantum Systems
For the analysis of finite-dimensional classical completely integrable Hamiltonian systems the representation by action-angle-variables is an essential tool. This representation can be carried over to the infinite-dimensional situation by use of mastersymmetries, thus leading to a suitable Viasoro algebra in the vector fields. For the quantum case, however, such a structure cannot exist in the corresponding operator algebra, due to a classical theorem of Kaplansky, although the concept of mastersymmetries can be used to give a formal description of the infinitedimensional symmetry group of those “nonlinear” Quantum systems which are accessible by the Quantum Inverse Scattering Transform. In order to make that precise and to transfer classical notions and methods to the quantum case an alternative dynamical concept for quantum systems is proposed. We give two examples, in the discrete case we consider spin chains, like the Heisenberg anisotropic spin chain, and in the continuous case, where additional difficulties arise, we consider the quantization of the KdV.
KeywordsQuantum System Poisson Bracket Spin Chain Hamiltonian Structure Recursion Operator
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- I. Ya. Dorfman and A. S. Fokas: Hamiltonian Theory over noncommutative Rings and Integrability in Multidimensions, INS #181, Clarkson University, Potsdam New-York, p.1–20, 1991Google Scholar
- B. Fuchssteiner: Mastersymmetries for completely integrable systems in Statistical Mechanics, in: Springer Lecture Notes in Physics 216 (L. Garrido ed.) Berlin-Heidelberg-New York, p. 305–315, 1985Google Scholar
- B. Fuchssteiner and U. Falck: Computer algorithms for the detection of completely integrable quantum spin chains, in: Symmetries and nonlinear phenomena (D. Levi and P. Winternitz ed.), Singapore, World Sc. Publishers, p.22–50, 1988Google Scholar
- B. Fuchssteiner: Hamiltonian structure and Integrability, in: Nonlinear Systems in the Applied Sciences, Math. in Sc. and Eng. Vol. 185 Academic Press, C. Rogers and W. F. Ames eds., p.211–256, 1991Google Scholar
- I. Kaplansky: Functional Analysis, in Surveys in Applied Mathematics, Wiley, New York, 4, p.1–34, 1958Google Scholar
- F. Magri and C. Morosi: Old and New Results on Recursion Operators: An alebraic approach to KP equation, in: Topics in Soliton Theory and Exactly solvable Nonlinear equations (eds: M. Ablowitz, B. Fuchssteiner, M. Kruskal) World Scientific Publ., Singapore, 1987 p.78–96Google Scholar
- M. Mathieu: Is there an unbounded Kleinecke-Shirokov theorem, Sem. Ber. Funkt. anal. Universität Tübingen, SS 1990, p.137–143, 1956Google Scholar
- W. Oevel: A Geometrical approach to Integrable Systems admitting time dependent Invariants, in: Topics in Soliton Theory and Exactly solvable Nonlinear equations ( eds: M. Ablowitz, B. Fuchssteiner, M. Kruskal) World Scientific Publ., Singapore, p.108–124, 1987Google Scholar