Problems of Quantization on Closed Manifolds and the Gauge Structure

关于闭流形上的量子化问题，人们一直是在Dirac方法的基础上进行研究的，Dirac方法是很久以前提出的，最近推广了通常的正则量子化方法，然而，对这一问题更基本的方法是由Landsman-Linden和Ohnuki-Kitakado独立地实现的。这些研究的结果表明，通过这种量子化得到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。应用群论的诱导表示技术进行量子化，我们成功地构造了S^D(D=1，2，-)上规范势的显式形式。根据这一结果，我们明确了规范势的数学性质，它可以表述如下：1.在S^D(D=1，2，-)上量子化出现的规范势满足该球面上的杨-米尔斯方程。2.当D=1且D=2n(n=1，2，-)时，规范p…更多的奥特莱斯在拓扑上变得不是微不足道的。它们与D=1的Aharonov-Bohm规范势、D=2的磁单极规范势、D=4的瞬子解以及D=6、8、10、-时的Fujii广义瞬子组态相同。特别地，对于D=4p(p=1，2，-)，我们可以定义规范场之间的对偶(或反对偶)关系。3.另一方面，对于D=2n+1(n=1，2，-)，规范势都是拓扑平凡的。有趣的是，已知的S^D上的杨-Mills方程的拓扑非平凡解已经被我们用代数方法得到的规范势完全耗尽了。结果在几个国际会议上公布，从事量子力学基本问题研究的人们似乎对我们的方法很感兴趣。与此相关的是，从物理和数学角度考察规范势的上述性质在多大程度上成立将是相当重要的。最近，我们导出了量子化在Grassmann流形U(n+m)/U(N)*U(M)上运动的粒子所产生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。细节将在不久的将来与相关主题一起发布。较少

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