2002 Fiscal Year Final Research Report Summary
MATHEMATICAL FOUNDATIONS OF RENORMALIZATION GROUP AND THEIR APPLICATIONS TO MATHEMATICAL SCIENCES
| Project/Area Number |
13640227
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | SETSUNAN UNIVERSITY |
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Principal Investigator |
ITO Keiichi SETSUNAN UNIVERSITY, MATHEMATICS DEPARTMENT, PROFESSOR, 工学部, 教授 (50268489)
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| Co-Investigator(Kenkyū-buntansha) |
SHIMADA Shin-ichi SETSUNAN UNIVERSITY, MATHEMATICS DEPARTMENT, ASSOCIATE PROFESSOR, 工学部, 助教授 (40196481)
HIROSHIMA Fumio SETSUNAN UNIVERSITY, MATHEMATICS DEPARTMENT, ASSOCIATE PROFESSOR, 工学部, 助教授 (00330358)
ONO Hiroaki SETSUNAN UNIVERSITY, PHYSICS DEPARTMENT, PROFESSOR, 工学部, 教授 (50100780)
WATARAI Seizo SETSUNAN UNIVERSITY, MATHEMATICS DEPARTMENT, ASSOCIATE PROFESSOR, 工学部, 助教授 (20131500)
TERAMOTO Yoshiaki SETSUNAN UNIVERSITY, MATHEMATICS DEPARTMENT, ASSOCIATE PROFESSOR, 工学部, 助教授 (40237011)
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| Project Period (FY) |
2001 – 2002
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| Keywords | Renormalization Group / Recursion Formula / Scaling / O(N) Spin Model / point interaction / resolvent / Pauli-Fierz Model / Ground State |
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| Research Abstract |
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
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