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Transactions of the American Mathematical Society

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A Groenewold-Van Hove Theorem for $S^2$


Authors: Mark J. Gotay, Hendrik Grundling and C. A. Hurst
Journal: Trans. Amer. Math. Soc. 348 (1996), 1579-1597
MSC (1991): Primary 81S99; Secondary 58F06
DOI: https://doi.org/10.1090/S0002-9947-96-01559-0
MathSciNet review: 1340175
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Abstract: We prove that there does not exist a nontrivial quantization of the Poisson algebra of the symplectic manifold $S^2$ which is irreducible on the su(2) subalgebra generated by the components $\{S_1,S_2,S_3\}$ of the spin vector. In fact there does not exist such a quantization of the Poisson subalgebra $\cal P$ consisting of polynomials in $\{S_1,S_2,S_3\}$. Furthermore, we show that the maximal Poisson subalgebra of $\cal P$ containing $\{1,S_1,S_2,S_3\}$ that can be so quantized is just that generated by $\{1,S_1,S_2,S_3\}$.


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Additional Information

Mark J. Gotay
Affiliation: Department of Mathematics, University of Hawaii, 2565 The Mall, Honolulu, Hawaii 96822
Email: gotay@math.hawaii.edu

Hendrik Grundling
Affiliation: Department of Pure Mathematics, University of New South Wales, P. O. Box 1, Kensington, NSW 2033 Australia
Email: hendrik@solution.maths.unsw.edu.au

C. A. Hurst
Affiliation: Department of Physics and Mathematical Physics, University of Adelaide, G. P. O. Box 498, Adelaide, SA 5001 Australia
Email: ahurst@physics.adelaide.edu.au

DOI: https://doi.org/10.1090/S0002-9947-96-01559-0
Received by editor(s): March 23, 1995
Article copyright: © Copyright 1996 American Mathematical Society