A Groenewold-Van Hove Theorem for 𝑆²

We prove that there does not exist a nontrivial quantization of the Poisson algebra of the symplectic manifold S 2 S^2 which is irreducible on the su(2) subalgebra generated by the components { S 1 , S 2 , S 3 } \{S_1,S_2,S_3\} of the spin vector. In fact there does not exist such a quantization of the Poisson subalgebra P P consisting of polynomials in { S 1 , S 2 , S 3 } \{S_1,S_2,S_3\} . Furthermore, we show that the maximal Poisson subalgebra of P P containing { 1 , S 1 , S 2 , S 3 } \{1,S_1,S_2,S_3\} that can be so quantized is just that generated by { 1 , S 1 , S 2 , S 3 } \{1,S_1,S_2,S_3\} .