Problems of Quantization on Closed Manifolds and the Gauge Structure

迄今为止，对闭流形上的量化问题的研究都是在狄拉克方法的基础上进行的，狄拉克方法是很久以前通过推广常用的正则量化方法而形成的，但最近landman - linden和Ohnuki-Kitakado独立地提出了更基本的方法来解决这个问题。作为这些研究的结果，揭示了通过这种量子化推导的理论可以具有如此显著的性质，以至于它自动配备了某种类型的规范势。利用群论的诱导表示技术，我们成功地构造了S^D （D=1,2, -）上规范势的显式形式。在此基础上，我们明确了规范电位的数学性质，可以表述为：1。在S^D （D=1,2, -）上出现的量子化规范势在这个球上满足Yang-Mills方程。2. 对于D=1和D=2n (n=1,2, -)，规范p…更多的势在拓扑上是非平凡的。它们与D=1时的Aharonov-Bohm规范势、D=2时的磁单极规范势、D=4时的瞬子解以及D=6,8,10， -时的Fujii广义瞬子构型相同。特别地，对于D=4p (p=1,2, -)，我们可以定义规范域之间的对偶（或反对偶）关系。3. 另一方面，当D=2n+1 （n=1,2, -）时，规范势在拓扑上都是平凡的。有趣的是，已知的S^D上的Yang-Mills方程的拓扑非平凡解完全被我们用代数方法得到的规范势所耗尽。结果在几个国际会议上发表，研究量子力学基本问题的人似乎对我们的方法很感兴趣。与此相关的是，从物理和数学的角度来考察规范势的上述性质在多大程度上是正确的，这是非常重要的。最近，我们推导出了量子化在格拉斯曼流形上运动的粒子U (n+m) /U (n) *U (m)所引起的规范势的所有可能形式。对其数学性质的研究正在进行中。详细内容将于近期与相关主题一起发布。少

Quantization problem on a closed manifold have been studied so far on the basis of Dirac's method, which was formulated long time ago by generalizing the usual method of canonical quantization Recently, however, more fundamental approach to this problem was independently achieved by Landsman-Linden and by Ohnuki-Kitakado. As a result of these investigations, it was revealed that the theory derived through this quantization can have a so remarkable property that it is automatically equipped with a certain type of gauge potential. Applying the induced representation technique of group theory to perform this quantization we succeeded in constructing the explicit form of the gauge potential on S^D (D=1,2, -). Based on this result we have made clear those mathematical properties of the gauge potentials, which may be stated as follows :1. The gauge potential emerging in quantization on S^D (D=1,2, -) satisfies the Yang-Mills equation on this sphere. 2. For D=1 and D=2n (n=1,2, -) the gauge p … More otentials become topologically non-trivial. They are shown to be the same as the Aharonov-Bohm gauge potential for D=1, the magnetic monopole gauge potential for D=2, the instanton solution for D=4, and Fujii's generalized instanton configuration for D=6,8,10, -. Especially, for D=4p (p=1,2, -) we can define a duality (or anti-duality) relation among the gauge fields. 3. On the other hand, for D=2n+1 (n=1,2, -) the gauge potentials are found to be all topologically trivial.It is interesting to note that the topologically non-trivial solutions to the Yang-Mills equation on S^D, which have been known already, are completely exhausted by our gauge potentials obtained by an algebraic method. The results were presented in several international conferences and people working on basic problems of quantum mechanics seemed to have much interest in our approach. Related to this it would be quite important from physical and mathematical view to examine in what extent the above properties of gauge potentials hold true. Very recently we have derived out all possible forms of the gauge potential induced in quantizing a particle moving on the Grassmann manifold U (n+m) /U (n) *U (m). A study of its mathematical properties is in progress. The details will be published in near future together with related topics. Less