Quantum Theory on Manifolds and its Application to Gauge Theory

我认为在齐次空间G/H上的量子化是在具有非平凡拓扑的流形上量子化的第一步。我证明了已知在空间G/H上允许的不等价量子化可以从群G上的量子理论中再现，方法是把系统看作一个约束系统，并使用适用于这种系统的狄拉克程序。然后，我证明了出现在空间G/H上的诱导规范势是正则联络，并且它本质上与在贝里相的标准设置中出现的规范势相同。其次，作为具有非平凡流形的另一类模型，我考虑了从WZNW模型通过Hamilton约化得到的SL（n）户田晶格模型。我分类所有可能类型的相位空间获得这种方式为n= 2，3，4，特别是对于最简单的情况n=2，明确构建量子理论，发现该理论的特点是一个角度参数rheta。相当独立的上述研究路线，我还研究了路径积分的方法来量化G/H。我发现，通过将该方法推广到多连通空间，如果我们添加由子群H的不可约表示给出的权重因子，就有可能恢复不等价的量化，并且这精确地导致具有上述约束的系统。这一结果的规范理论的含义也检查在这个路径积分框架。

I considered quantization on a homogeneous space G/H as a first step toward quantizing on a manifold having a non-trivial topology. I showed that the inequivalent quantizations known to be allowed on the space G/H can be reproduced from the quantum theory on the group G by regarding the system as a constraint system and using Dirac's procedure applicable to such systems. I then showed that the induced gauge potential that appears on the space G/H is the canonical connection, and that it is essentially identical to the gauge potential which arises in the standard setting of Berry's phase. Next, as another class of models possessing a non-trivial manifold, I considered SL (n) Toda lattice models obtained by Hamiltonian reduction from the WZNW model. I classified all possible types of phases spaces obtained this way for n=2,3,4, and, in particular for the simplest case n=2, constructed the quantum theory explicitly where it is found that the theory is characterized by an angle parameter rheta.Quite independently of the above line of research, I also studied the path-integral approach to quantizing on G/H.I found that, by generalizing the approach for multiply-connected spaces, it is possible to recover the inequivalent quantizations if we add a weight factor given by irreducible representations of the subgroup H,and that this leads precisely to the system with the constraints mentioned above. Implication of this result to gauge theories is also examined in this path-integral framework.