Lattice Gauge Theories as Problems of Constructive Quantum Field Theory

作为构造量子场论问题的格子规范理论

• 批准号: 10640161
• 负责人: YAMAMOTO Kazuhiro
• 金额: $ 0.83万
• 依托单位: Nagoya Institute of Technology
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640161/
• 关键词: Renormalization transform; Lattice model; Partition function; Abelian lattice gauge theory; Quantum field theory; 格子ゲージ場理論; 構成的場の理論

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.

这个计划的目的是给一个数学刚性理论的模型出现在量子场论。具体目标是：量子电动力学中的阿贝尔Higgs-Kibble模型在三维有限格点中的紫外稳定性，格点空间连续极限的存在性及其所满足的物理公理。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学中使用的重整化变换的严格定义。利用Dirac δ -函数的一个性质，即δ-函数的Haar测度的形式不变性证明了Faddeev-Popov过程的合理性，给出了δ-函数的形式定义.但我们定义了一个重正化变换作为测度，并证明了所有必要的性质。其次，我们证明了三维阿贝尔Higgs-Kibble模型的紫外稳定性。现在我们正在准备这个结果。为了验证这个一致估计，我们需要在新的图类型中出现新的Ward-Takahasi恒等式。

项目成果

Y. Nishimura and Z. Yosimura: "The quasi KO -types of weighted mod 4 len spaces"Osaka Journal of Mathematics. 35. 895-914 (1998)

Y. Nishimura 和 Z. Yosimura：“加权 mod 4 len 空间的准 KO 型”大阪数学杂志。

T. Adachi: "Distribution of length spectrum of circle on a complex hyperbolic space."Nagoya Mathematical Journal. Vol.153. 119-140 (1999)

T. Adachi：“复双曲空间上圆的长度谱的分布。”名古屋数学杂志。

T.Adachi: "Distribution of length spectrum of circle on a complex hyperbolic space"Nagoya Mathematical Journal. 153. 119-140 (1999)

T.Adachi：“复双曲空间上圆的长度谱的分布”名古屋数学杂志。

Y. Nishimura and Z. Yosimura: "The quasi KO -types of weighted mod 4 len spaces"Osaka Journal of Mathematics. Vol.35. 895-914 (1998)

Y. Nishimura 和 Z. Yosimura：“加权 mod 4 len 空间的准 KO 型”大阪数学杂志。

Y.Nishimura and Z.Yoshimura: "The quasi KO_*-types of weighted mod 4 len spaces"Osaka Journal of Mathematics. 35. 895-914 (1998)

Y.Nishimura 和 Z.Yoshimura：“加权 mod 4 len 空间的准 KO_* 型”大阪数学杂志。