arXiv 1404.70052014-04hep-th

2+1维Yang-Mills真空波泛函

The Yang-Mills Vacuum Wave Functional in 2+1 Dimensions

Sebastian Krug

备注：: PhD thesis, 229 pages, added a reference

摘要：

本文研究了薛定谔表象中2+1维的Yang-Mills理论。薛定谔图像是有趣的，因为它非常适合于探索非微扰区域中真空态的性质。然而，在这个问题上没有做太多的分析工作，甚至薛定谔表象中的微扰理论也没有很好地发展，特别是在规范理论的情况下。在Hatfield的一篇论文[Phys.Lett.B147，435（1984）]中，SU（2）理论的真空波泛函被计算为O（e）。在非微扰状态下，Karabali等人在一系列论文中开发了最复杂的分析方法（参见[Nucl.Phys.B 824，387（2010）]和其中的参考文献）。本论文的目的是通过对这两种方法的改进，计算一般规范群SU $(N_c)$到O $(e^2)$的真空波泛函，使薛定谔表象中的微扰理论有更坚实的基础。这是很重要的，因为它为我们提供了一个工具，用于测试非微扰的方法，这应该重现微扰的结果在一个适当的限制。此外，正则化和重整化在薛定谔图像中也没有得到很好的理解。Karabali等人提出的正则化方法在用上述两种不同的方法计算真空波泛函时会导致相互矛盾的结果。我们的目标是澄清如何实现正则化，并开发一种新的正则化方法，使这两个表达式一致，提供了一个强有力的检查所采用的正则化。我们认为，这种正规化过程是不特定的情况下，这里研究。它应该以同样的方式应用于薛定谔图像中任何维度的任何量子场论。

英文摘要：

We investigate Yang-Mills theory in 2+1 dimensions in the Schroedinger representation. The Schroedinger picture is interesting because it is well suited to explore properties of the vacuum state in the non-perturbative regime. Yet, not much analytical work has been done on this subject, and even the topic of perturbation theory in the Schroedinger representation is not well developed, especially in the case of gauge theories. In a paper by Hatfield [Phys.Lett.B 147, 435 (1984)] the vacuum wave functional for SU(2) theory was computed to O(e). In the non-perturbative regime, the most sophisticated analytical approach has been developed by Karabali et al. in a series of papers (see [Nucl.Phys.B 824, 387 (2010)] and references therein). This thesis aims to put perturbation theory in the Schroedinger representation on more solid ground by computing the vacuum wave functional for a general gauge group SU$(N_c)$ up to O$(e^2)$, utilizing modifications of these two methods. This is important since it provides us with a tool for testing non-perturbative approaches, which should reproduce the perturbative result in an appropriate limit. Furthermore, regularization and renormalization are also not well understood in the Schroedinger picture. The regularization method proposed by Karabali et al. leads to conflicting results when applied to the computation of the vacuum wave functional with the two different methods mentioned above. We aim to clarify how regularization should be implemented and develop a new regularization approach, which brings these two expressions into agreement, providing a strong check of the regularization employed. We argue that this regularization procedure is not specific to the cases studied here. It should be applied in the same way to any quantum field theory in any dimension in the Schroedinger picture.

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英文摘要：

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