Mathematical Studies on Systems of Particles Interacting with Quantum Fields

粒子与量子场相互作用系统的数学研究

• 批准号: 11440036
• 负责人: ARAI Asao
• 金额: $ 2.62万
• 依托单位: HOKKAIDO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B).
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-11440036/
• 关键词: quantum field; Fock space; essential spectrum; Dirac particle; quantum electrodynamics; Nelson model; infrared divergence; ground state; 本質的スパクトル

Head investigator(1) Spectral analysis has been made for the Hamiltonian H of a quantum system of a Dirac particle-a relativistic charged particle with spin 1/2-interacting with the quantized radiation field, where the Dirac particle is in an external potential V.The following have been proved : (i) existence of a physically natural self-adjoint extension of H which is connected with the charge conjugation and parity invariance of the external potential V.(ii) For a reasonable class of V, H is essentially self-adjoint.(iii) In the case V=0, H has a direct integral decomposition H=∫^<【symmetry】>_<R^3> H (p) dp.(2) A general structure of the essential spectrum of a class of Hamiltonians which describe particle-field interactions has been clarified.(3) Detailed analysis has been made on the existence or the absence of ground states of the generalized spin boson model.(4) The massless Nelson model has been considered in a non-Fock representation and the existence of a ground state of it has been established.Investigators(a) Existene of a global solution of nonlinear elastic waves (Agemi)(b) Studies on the Rohlin property for automorphisms of general non-commutative shifts (Kishimoto)(c) Studies on asymptotic behaviors of partial autocorrelation functions of stationary processes (Inoue)

（1）对Dirac粒子（自旋为1/2的相对论性带电粒子）与量子化辐射场相互作用的量子系统的哈密顿量H进行了谱分析，其中Dirac粒子处于外部势V中。（i）H存在一个物理上自然的自伴扩张，它与外势V的电荷共轭和宇称不变性有关。(ii)对于V的合理类，H本质上是自伴的。(iii)在V=0的情况下，H有一个直接的积分分解H= H（p）dp。(2)阐明了一类描述粒子-场相互作用的哈密顿量的本质谱的一般结构。(3)对广义自旋玻色子模型基态的存在与否进行了详细的分析。(4)在非Fock表象中考虑了无质量的纳尔逊模型，并建立了该模型基态的存在性。研究者（a）非线性弹性波整体解的存在性（Agemi）（B）一般非对易位移自同构的Rohlin性质的研究（Kishimoto）（c）平稳过程偏自相关函数的渐近行为的研究（Inoue）

项目成果

Asao Arai: "Mathematical analysis of a model in relativistic quantum electrodynamics"RIMS Kokyuroku. (発表予定).

Asao Arai：“相对论量子电动力学模型的数学分析”RIMS Kokyuroku（待提交）。

Rentaro Agemi: "Global existence of nonlinear elastic waves"Inventiones Mathematicae. 142. 225-250 (2000)

上美伦太郎：“非线性弹性波的整体存在”数学发明。

Asao Arai: "Essential spectrum of a self-adjoint operator on an abstract Hilbert space of Fock type and applications to quantum field Hamiltonians"Journal of Mathematical Analysis and Applications. (発表予定).

Asao Arai：“福克型抽象希尔伯特空间上的自伴算子的本质谱及其在量子场哈密顿量中的应用”《数学分析与应用杂志》（即将出版）。

Asao Arai: "Ground states of a general class of quantum field Hamiltonians"Rev.Math.Phys.. 12. 1085-1135 (2000)

Asao Arai：“量子场哈密顿量的一般类的基态”Rev.Math.Phys.. 12. 1085-1135 (2000)

Rentaro Agemi: "Global existence of nonlinear elastic waves"Invent.Math.. 141. 225-250 (2000)

上美连太郎：《非线性弹性波的整体存在性》Invent.Math.. 141. 225-250 (2000)