量化障碍
Obstructions to Quantization
摘要：
量子化并不是一个直截了当的命题，正如格罗内沃尔德和范霍夫在50多年前发现量子化的“障碍”所表明的那样。他们的“不通过定理”断言，不可能以物理上有意义的方式一致地证明相空间R^{2n}上可观察到的每个经典多项式。对于S^2和T*S^1也发现了类似的障碍，这支持了一个共同的信念，即不通过定理应该具有某种普遍性。令人惊讶的是，事实并非如此--已经证明，对T^2或T*R_+进行量子化没有任何障碍。在本文中，我们努力描绘的情况下，这种障碍会出现，并了解产生它们的机制。我们的目标是猜想-并在某些情况下证明-广义Groenewold-Van Hove定理，并确定最大的李子代数的观测，可以一贯量化。这需要研究泊松代数的结构及其表示。为了达到这些目的，我们包括了预量子化（在扩展的意义上）和量子化理论的阐述，这里用“可观测量的基本代数”来表述。然后，我们详细回顾了R^{2n}，S^2，T*S^1，T^2和T*R_+的已知结果，以及最近的理论工作。
英文摘要：
Quantization is not a straightforward proposition, as shown by Groenewold's and Van Hove's discovery, more than fifty years ago, of an "obstruction" to quantization. Their "no-go theorems" assert that it is impossible to consistently quantize every classical polynomial observable on the phase space R^{2n} in a physically meaningful way. Similar obstructions have been found for S^2 and T*S^1, buttressing the common belief that no-go theorems should hold in some generality. Surprisingly, this is not so--it has been proven that there are no obstructions to quantizing either T^2 or T*R_+. In this paper we work towards delineating the circumstances under which such obstructions will appear, and understanding the mechanisms which produce them. Our objectives are to conjecture--and in some cases prove--generalized Groenewold-Van Hove theorems, and to determine the maximal Lie subalgebras of observables which can be consistently quantized. This requires a study of the structure of Poisson algebras and their representations. To these ends we include an exposition of both prequantization (in an extended sense) and quantization theory, here formulated in terms of "basic algebras of observables." We then review in detail the known results for R^{2n}, S^2, T*S^1, T^2, and T*R_+, as well as recent theoretical work.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
