Deformation quantization, superintegrability and Nambu mechanics
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Phase Space is the framework best suited for quantizing superintegrable systems—systems with more conserved quantities than degrees of freedom. In this quantization method, the symmetry algebras of the hamiltonian invariants are preserved most naturally. We illustrate the power and simplicity of the method through new applications to nonlinear σ-models, specifically for Chiral Models and de Sitter N-spheres, where the symmetric quantum hamiltonians amount to compact and elegant expressions, in accord with the Groenewold-van Hove theorem. Additional power and elegance is provided by the use of Nambu Brackets (linked to Dirac Brackets) involving the extra invariants of superintegrable models. The quantization of Nambu Brackets is then successfully compared to that of Moyal, validating Nambu’s original proposal, while invalidating other proposals.
Keywordsdeformation quantization superintegrability Nambu Brackets
PACS02.30.Ik 11.30.Rd 11.30.-j
- 3.G. Velo and J. Wess, Nuovo Cim. A1 (1971) 177; H. Lin, W. Lin and R. Sugano, Nucl. Phys. B16 (1970) 441; M. Lakshmanan and K. Eswaran, J. Phys. A8 (1975) 1658; also see A. Leznov, Nucl. Phys. B640 (2002) 469 [hep-th/0203225]; P. Higgs, J. Phys. A12 (1979) 309; H. Leemon, J. Phys. A12 (1979) 489.CrossRefADSGoogle Scholar
- 8.L. van Hove, Proc. R. Acad. Sci. Belgium 26 (1951) 1.Google Scholar