On Quantizing $T^*S^1$

In this paper we continue our study of Groenewold-Van Hove obstructions to quantization. We show that there exists such an obstruction to quantizing the cylinder $T^*S^1.$ More precisely, we prove that there is no quantization of the Poisson algebra of $T^*S^1$ which is irreducible on a naturally defined $e(2) \times R$ subalgebra. Furthermore, we determine the maximal ``polynomial'' subalgebras that can be consistently quantized, and completely characterize the quantizations thereof. This example provides support for one of the conjectures in Gotay et al 1996, but disproves part of another. Passing to coverings, we also derive a no-go result for $R^2$ which is comparatively stronger than those originally found by Groenewold and Van Hove.