Mathematical analysis on quantum fields

2014 Fiscal Year Final Research Report

• Project/Area Number: 24540154
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Hokkaido University
• Principal Investigator: ARAI Asao 北海道大学, 理学(系)研究科(研究院), 教授 (80134807)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Keywords: 漸近的摂動論 / 基底状態 / 量子場 / 無限次元ディラック作用素 / 汎関数積分 / フォック空間 / ハミルトニアン / スペクトル理論

Outline of Final Research Achievements

(1) A new asymptotic perturbation theory is constructed for a class of linear operators on a Hilbert space and applied to a model of mass-less quantum fields.
As a result, it is shown that the ground state energy of the model has an asymptotic expansion in the coupling constant up to any finite order. (2) A spectral analysis is made on a class of infinite dimensional Dirac operators Q on the abstract boson-fermion Fock space. In particular, it is shown that Q has non-zero eigenvalues in a strong coupling region independently of whether or not the unperturbed part of Q has a non-zero eigenvalue. (3) Functional integral representations are derived for a general class of boson-fermion systems with application to statistical mechanics.

Free Research Field

数学