MATHEMATICAL FOUNDATIONS OF RENORMALIZATION GROUP AND THEIR APPLICATIONS TO MATHEMATICAL SCIENCES

1.伊藤和田村(金泽大学)用重整化群(块自旋变换)方法研究了经典的O(N)对称自旋模型。在第一阶段，他们论证了泛函行列式det^&lt；-N/2&gt；(1+2iGψ/√&lt；N&gt；)关于ψ的可积性，其中ψ是引入傅里叶变换的轴向场。利用聚合物(团簇)膨胀技术，他们证明了反临界温度β_c在二维上服从约束β_c&gt；N log N，这意味着存在强偏差。(β_c-N表示大于或等于3的维度。)在本模型中，β_c=∽。为了建立这一猜想，Ito递归地将BST应用到模型中，将行列式分解为来自不同距离尺度的波动的多个行列式的乘积。他证明了递归关系的主要部分是相当简单的，并再现了威尔逊-戴森类型的递阶逼近的流。他不是…进一步将本方法(引入ψ场)应用于原理上具有相同结构的格点规范理论。Teramoto和Ito研究了湍流的性质，其中包括关于能量耗散和偏离能量的柯尔莫戈洛夫定律。他们试图从纳维-斯托克斯方程推导出偏差，但今年未能得到具体结果。岛田考虑了具有δ函数的3D薛定谔算子，该算子类似于半径为0的球的球面上的势支承。通过预解收敛，讨论了作为→0的哈密顿的收敛。广岛研究了被认为是经典量子电动力学的Pauli-Fierz模型。广岛证明了哈密顿量的基态属于光子数算符的范围，并且如果引入电子自旋，基态是双重简并的。广岛和伊藤研究了Pauli-Fierz模型。尽管如果没有引入动量截断，QED被认为是微不足道的，但Pauli-Fierz模型可能并非如此。他们将重整化群类型自变量应用于Pauli-Fierz模型(这个想法最初是由J.Froelich(ETH)提出的)。但他们的分析仍有待观察。较少

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