A Groenewold-Van Hove Theorem for S^2

TL;DR

This paper proves the impossibility of certain irreducible quantizations of the Poisson algebra on the sphere S^2, extending Groenewold-Van Hove type no-go theorems to this setting.

Contribution

It establishes a no-go theorem for irreducible quantizations of the Poisson algebra on S^2, identifying the maximal quantizable subalgebra.

Findings

No nontrivial irreducible quantization of the full Poisson algebra on S^2

No such quantization exists for the polynomial subalgebra in {S_1,S_2,S_3}

The maximal quantizable subalgebra is generated by {1,S_1,S_2,S_3}

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 subalgebra generated by the components {S_1,S_2,S_3} of the spin vector. We also show that there does not exist such a quantization of the Poisson subalgebra P consisting of polynomials in {S_1,S_2,S_3}. Furthermore, we show that the maximal Poisson subalgebra of 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}.