Bihamiltonian quantum systems and Weyl quantization.
type: Item
instance of: scholarly article
title: Bihamiltonian quantum systems and Weyl quantization.
zbMATH Open document ID: 1040.81048
DOI: 10.1016/S0034-4877(01)80074-4
author: Giuseppe Marmo
author: A. Simoni
author: F. Ventriglia
published in: Reports on Mathematical Physics
publication date: 2002-09-12T00:00:00Z
review text: 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?
reviewed by: Daniel A. Dubin
Mathematics Subject Classification ID: 81S10
Mathematics Subject Classification ID: 37N20
zbMATH DE Number: 1799980
zbMATH Keywords: quantization
zbMATH Keywords: CCR representations
MaRDI profile type: Publication
cites work: The inverse problem in the calculus of variations and the geometry of the tangent bundle
cites work: Do the Equations of Motion Determine the Quantum Mechanical Commutation Relations?
cites work: Alternative Hamiltonian description for quantum systems
cites work: Canonical quantization
cites work: Q4759160
cites work: A generalization of the Jordan-Schwinger map: The classical version and its \(q\) deformation
Wikidata QID: Q127247392
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description: scientific article; zbMATH DE number 1799980