MATHEMATICAL STUDY OF RENORMALIZATION GROUP AND APPLICATIONS

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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.