Noncommutative integrability from classical to quantum mechanics. (English) Zbl 1278.81106
Summary: We propose in this work a definition of integrable quantum system, which is based upon the correspondence with the concept of noncommutative integrability for classical mechanical systems. We then determine sufficient conditions under which, given an integrable classical system, it is possible to construct an integrable quantum system by means of a quantization procedure based on the symmetrized product of operators. As a first example of application of such an approach, we will consider the possible cases of noncommutative integrability for systems with rotational symmetry in an \(n\)-dimensional Euclidean configuration space.
MSC:
| 81R12 | Groups and algebras in quantum theory and relations with integrable systems |
| 70H06 | Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics |
| 81S05 | Commutation relations and statistics as related to quantum mechanics (general) |
| 20G45 | Applications of linear algebraic groups to the sciences |
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