A Groenewold-Van Hove Theorem for 𝑆²

Gotay, Mark; Grundling, Hendrik; Hurst, C.

doi:10.1090/S0002-9947-96-01559-0

A Groenewold-Van Hove Theorem for $S^2$
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by Mark J. Gotay, Hendrik Grundling and C. A. Hurst PDF

Trans. Amer. Math. Soc. 348 (1996), 1579-1597 Request permission

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 $\mathcal {P}$ consisting of polynomials in $\{S_1,S_2,S_3\}$. Furthermore, we show that the maximal Poisson subalgebra of $\mathcal {P}$ 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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