Lattice Gauge Theories as Problems of Constructive Quantum Field Theory

本程序的目的是为量子场论中出现的模型提供一个数学上严格的理论。具体地说，下面两个主题是具体的目标：量子电动力学中的三维有限格子上的阿贝尔Higgs-Kibble模型的紫外线稳定性，以及它的格子空间的连续极限的存在性和连续有限空间所满足的所需的物理公理。但我们可以得到以下有趣的结果。首先，我们可以给出理论物理中使用的重整化变换的严格定义。它是利用狄拉克的δ函数及其性质之一，即Faddeev-Popov过程由δ函数的Haar测度的形式不变性来证明的。但我们定义了一个重整化变换作为度量，并证明了所有必要的性质。其次，证明了三维阿贝尔Higgs-Kibble模型的紫外稳定性。现在我们正在准备这个结果。为了验证这一一致估计，我们需要在新型图中出现新的Ward-Takahasi恒等式。

The Aim of this program is to give a mathematically rigid theory for models appeared in quantum field theory. In particular the following two themes are concrete targets ; Ultraviolet stability for Abelian Higgs-Kibble model in a three dimensional finite lattice, which is a model in quantum electrodynamics, and existence of its continuous limite of the lattice space and the required physical axioms satisfied by the continuous limited space.For two years research we can not get the completely results for the above problems. But we can got the following interesting results. First we can give a rigorous definition of the renormalization transform used in theoretical physics. That is formally defined by making use of Dirac's δ -function and one of it's properties, that is, Faddeev-Popov procedure is justified by the formal invariance of Haar measure for δ-function. But we defined a renormalization transform as the measure and prove the all required properties. Secondly, we can prove the ultra-violet stability for three dimensional Abelian Higgs-Kibble model. Now we are preparing this result. In order to verify this uniform estimate we need new Ward-Takahasi identity appearing in new type graphs.