MATHEMATICAL STUDY OF RENORMALIZATION GROUP AND APPLICATIONS

1.Ito和Tamura（金泽大学）研究了二维的经典O(N)对称自旋模型，该模型被认为没有任何相变。通过傅里叶变换，它被转换成格林函数G^<(ψ)>(x,y)=(-Δ+m^2+2iψ/√<N>)^<-1>(x,y)描述的系统，其中{ψ(x)；x∈Z^2}为复随机势。然后他们推测，由于安德森局域化，G^<(ψ)> (x,y)是短程的，因此热力学量可能是收敛的（无相变）。伊藤和田村（金泽大学）也研究了粒子的准静力学和准玻色子的玻色-爱因斯坦凝聚的起源。点过程测量的正性意味着粒子必须服从准色或准费米统计。他们证明了对称群的表示理论产生了由点过程导出的对粒子的配分函数。Hirosima和Ito研究了相对论性Pauli-Fierz模型的重整化能力。虽然QED被认为在去掉紫外线截断后是平凡的，但Pauli-Fierz模型可能定义了一个非平凡的模型。他们明确地计算了四阶质量位移，并表明存在Λ^2阶的散度。这种散度太过强烈，无法被传统的重整化群论证所控制。他们推测，Dirac方程型的a Pauli-Fierz模型可能有更轻微的散度（阶为（log Λ）^P），并且可能是可重整的。

1.Ito and Tamura (Kanazawa Univ.) studied classical O(N) symmetric spin model in two dimensions which is believed to be free from any phase transitions. By Fourier transformation, this is transformed into a system which is described by the Green's function G^<(ψ)>(x,y)=(-Δ+m^2+2iψ/√<N>)^<-1>(x,y), where {ψ(x);x ∈Z^2} are complex random potentials. Then they conjecture that G^<(ψ)> (x,y) is short range thanks to the Anderson localization, and thus thermodynamic quantities may be convergent (no phase transition).2.Ito and Tamura (Kanazawa Univ.) also studied the origin of para-statics of particles and Bose-Einstein condensation of para-bosons. The positivity of the measure of point processes implies that partcles must obey para-bose or para-fermi statistics. They showed that the representation theory of the symmetric group yields the partition functions of para particles which are derived from the point processes.3.Hirosima and Ito investigated the renormalizability of the relativistic Pauli-Fierz Model. Though QED is believed to be trivial after removing ultra-violet cut-offs, the Pauli-Fierz model may define a non-trivial model. They explicitly calculated the fourth order mass-shift and showed that there exists a divergence of order Λ^2. This divergence is too strong to be controled by the conventional renormalization group argument. They conjectured that the a Pauli-Fierz model of Dirac equation type may have more mild divergences (of order (log Λ)^P) and may be renormalizable.