A Groenewold-Van Hove theorem for \(S^ 2\). (English) Zbl 0856.58016
Summary: 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 \(\text{su}(2)\) 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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