MATHEMATICAL FOUNDATIONS OF RENORMALIZATION GROUP AND THEIR APPLICATIONS TO MATHEMATICAL SCIENCES

重正化群的数学基础及其在数学科学中的应用

基本信息

批准号：
13640227
负责人：
ITO Keiichi
金额：
$ 1.54万
依托单位：
SETSUNAN UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640227/
关键词：
Renormalization Group Recursion Formula Scaling O(N) Spin Model point interaction resolvent Pauli-Fierz Model Ground State Renormalization Group Recursion Formula Ο(N) Spin Model Pauli-Fierz Model Auxiliary Field Block Spin Trans

项目摘要

1. Ito and Tamura (Kanazawa Univ.) studied classical O(N) symmetric spin model by renormalization group (block spin transformation) method. In the first stage, they argued the integrability of the functional determinent det ^<-N/2>(1 + 2iGψ/√<N>) with respect to ψ, where ψ is the axially field introduced for Fourier Transformation. Using the technique called polymer (cluster) expansion, they showed that the inverse critical temperature β_c obeys the bound β_c > N log N in two dimensions, which implies the existence of strong deviation. (β_c - N for the dimension more than or equal to 3.) It is believed that β_c = ∽ in the present model.To establish this conjecture, Ito recursively applies the BST to the model to decompose the determinant into product of many determinants which comes from fluctuations of various distance scales. He showed that the main part of the recursion relations is quite simple, and reproduces the flow of the hiererchical approximation of Wilson-Dyson type.He is no … More w applying the present method (introduction of the ψ field) to the lattice gauge theory which has the same structure in principle.2. Teramoto and Ito investigated properties of turbulence, among them, the Kolmogorov law about the dissipation of energy and deviation from it. They tried to derive the deviation from the Navier-Stokes equation but they could not obtain concrete results this year.3. Shimada considered a 3D Schroedinger operator with the δ function like potential support on the sphere of the ball of radius a > 0. He discussed the convergence of the Hamiltoan as a → 0 through the resolvent convergence.4. Hiroshima investigated the Pauli-Fierz Model which is regarded as a classical Quantum Electrodynamics (QED). Hiroshima showed that the ground states of the Hamiltonian belong to the domain of the photon number operator, and also that the ground states are doubly degenerated if the spin of the electron is introduced.5. Hiroshima and Ito investigated the Pauli-Fierz Model. Though QED is believed to be trivial if no momentum cutoff is introduced, the Pauli-Fierz model may not. They apply the renormalization group type argument to the Pauli-Fierz model (this idea is originally due to J. Froelich (ETH)). But their analysis remains to be seen. Less

1。Ito和Tamura（Kanazawa Univ。）Studiod Classic O（N）对称旋转模型（通过重新归一化组（块自旋变换）方法。在第一阶段，他们争论了相对于ψ的功能确定的det ^<-n/2>（1 +2igψ/√n>）的整合，其中ψ是引入轴向的轴向场，用于傅立叶变换。他们使用称为聚合物（群集）扩展的技术，表明临界温度β_C在二维中遵守结合的β_C> n log n，这意味着存在强差。（对于维度的β_C -N超过或等于3。）据信，在本模型中，β_C=∽为了建立这种猜想，ITO递归地将BST应用于模型，将确定的确定量的乘积分解为来自各种距离尺度波动的许多确定器的乘积。他表明，递归关系的主要部分非常简单，并重现了威尔逊 - 迪森类型的层近似的流动。 Teramoto和ITO调查了湍流的属性，其中包括科尔莫戈罗夫法律，涉及能量耗散和偏离它。他们试图从Navier-Stokes方程中获得偏离，但今年无法获得具体的结果3。 Shimada考虑了一个3D Schroedinger操作员，其δ函数就像Radius A> 0的球体上的潜在支撑一样。他通过分辨率收敛讨论了Hamiltoan的收敛为A→0.4。 Hiroshima研究了Pauli-Fierz模型，该模型被认为是经典的量子电动力学（QED）。广岛表明，哈密顿量的基态属于光子数算子的域，并且如果引入了电子的旋转，则基态也会两倍地变性。5。广岛和ITO调查了Pauli-Fierz模型。尽管QED据信如果没有引入动量截止，但Pauli-Fierz模型可能不会。他们将重新归一化组类型参数应用于Pauli-Fierz模型（此想法最初是由于J. Froelich（ETH）造成的）。但是他们的分析还有待观察。较少的