标题： 关于量化$T^*S^1$ 显示英文标题

标题： On Quantizing $T^*S^1$

摘要： 本文我们继续研究了Groenewold-Van Hove 对量子化带来的障碍。我们证明，在圆柱体 $T^*S^1.$ 上存在这样的障碍。更确切地说，我们证明了不存在对 $T^*S^1$ 的泊松代数的量子化，使其在一个自然定义的 $e(2) \times R$ 子代数上不可约。此外，我们确定了可以一致量子化的最大“多项式”子代数，并完全刻画了这些量子化。这个例子支持了Gotay等人在1996年提出的猜想之一，但部分否定了另一个。通过扩展到覆盖空间，我们还得到了关于 $R^2$ 的一个更强的否定结果，比Groenewold和Van Hove最初发现的那些更强。 显示英文摘要

摘要： 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.