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。我证明了已知的在G/H空间上允许的不等价量子化可以从关于群G的量子理论中复制出来，方法是将该系统视为约束系统，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G/H空间上的诱导规范势是正则连接，并且它本质上与在Berry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，我考虑了由WZNW模型的哈密顿约化得到的SL(N)Toda格模型。我对n=2，3，4的所有可能的相空间类型进行了分类，特别是对于最简单的情况n=2，明确地构造了量子理论，其中发现该理论是由一个角度参数来表征的。我还独立于上面的研究，研究了在G/H上量子化的路径积分方法。我发现，通过推广多重连通空间的方法，如果我们增加一个由H子群的不可约表示所给出的权重因子，就有可能恢复不等价的量子化，这恰好导致了具有上述约束的系统。这一结果对规范理论的意义也在这个路径积分框架中得到了检验。

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.