Quantum Theory on Manifolds and its Application to Gauge Theory

流形的量子理论及其在规范理论中的应用

基本信息

批准号：
07804015
负责人：
TSUTSUI Izumi
金额：
$ 1.02万
依托单位：
University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1995
资助国家：
日本
起止时间：
1995 至 1996
项目状态：
已结题

项目摘要

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.

我将齐次空间 G/H 上的量化视为在具有非平凡拓扑的流形上进行量化的第一步。我表明，通过将系统视为约束系统并使用适用于此类系统的狄拉克程序，可以从 G 群的量子理论中重现空间 G/H 上已知的不等价量子化。然后我证明了空间 G/H 上出现的感应规范电势是规范联系，并且它与贝里相位标准设置中出现的规范电势本质上相同。接下来，作为另一类具有非平凡流形的模型，我考虑了通过 WZNW 模型的哈密顿约简得到的 SL (n) Toda 晶格模型。我对以这种方式获得的 n=2,3,4 的所有可能类型的相空间进行了分类，特别是对于最简单的情况 n=2，明确地构建了量子理论，发现该理论的特征在于角度参数rheta。与上述研究完全独立，我还研究了 G/H 量化的路径积分方法。我发现，通过推广多连通空间的方法，如果我们添加一个权重系数由子群 H 的不可约表示给出，这恰好导致具有上述约束的系统。这个结果对规范理论的影响也在这个路径积分框架中得到了检验。