Bihamiltonian quantum systems and Weyl quantization.
By means of quantization in the sense of Weyl, the authors construct a number of non-standard representation of the canonical commutation relations arising from the one and two-dimensional (isotropic) harmonic oscillator. Part of their construction results in a family of non-canonical pairs of operators, obtained through a nonlinear transformation from the standard CCR representation (which the reviewer takes to be isomorphic to the Schrödinger representation), indexed by a collection of functions. These functions follow from considering the most general symplectic structure compatible with the oscillator vector field (for one degree of freedom). For each such non-canonical pair, the authors then construct two canonical pairs, canonical meaning satisfying the usual commutation relations. In the last section, they turn to the construction of representation of the group \(SU(2)\) obtained from pairs of canonical pairs, so to speak, the Jordan-Schwinger construction. Armed with these representations of the CCR, they substitute them into the Jordan-Schwinger construction to obtain what one might call non-standard representation of \(SU(2)\) (bearing in mind that the representation theory of \(SU(2)\) is known). This paper is interesting not only for the results obtained and the description of the geometric aspects of Weyl quantization, but for the questions it raises. For example, in view of the fact that the new CCR representations are nonlinearly related to the standard one, how does the Stone-von Neumann uniqueness theorem come into play? Are there function forms for (at least some of) the new pairs? How do the representations of \(SU(2)\) obtained in this way fit into the decomposition of the regular representation, and could this be connected to the symplectic structure aspect of the construction?
scientific article; zbMATH DE number 1799980