A Groenewold-Van Hove Theorem for S2
Mark J. Gotay, Hendrik Grundling, C. A. Hurst
Abstract
We prove that there does not exist a nontrivial quantization of the Poisson algebra of the symplectic manifold S2 which is irreducible on the subalgebra generated by the components S1,S2,S3 of the spin vector. We also show that there does not exist such a quantization of the Poisson subalgebra P consisting of polynomials in S1,S2,S3. Furthermore, we show that the maximal Poisson subalgebra of P containing 1,S1,S2,S3 that can be so quantized is just that generated by 1,S1,S2,S3.
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