OPERATOR-ANALYTICAL STUDY OF SINGULARITIES OF HAMILTONIANS IN QUANTUM PHYSICS

量子物理中哈密尔顿奇点的算子分析研究

基本信息

批准号：
13640215
负责人：
HIROKAWA Masao
金额：
$ 1.86万
依托单位：
OKAYAMA UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640215/
关键词：
nonrelativistic QED infrared catastrophe ultraviolet catastrophe Pauli-Fierz model Nelson model パウリ・フイエルツ模型場の量子論 Nelson Model spin-boson model Wigner-Weisskoph model Pauli-Fierz model

项目摘要

We studied Nelson's model derived from the Pauli-Fierz model through several physical approximations. The Pauli-Fierz model describes an electron coupled with the quantized radiation field in nonrelativistic quantum electrodynamics, when we regard the electron as a nonrelativistic particle. We proved that the Nelson model has infrared catastrophe when its Hamiltonian has the Coulomb potential appearing in the structure of atoms. Developing the proof and using the Carleman operator, we clarified and characterized a mathematical mechanism which causes infrared safe or infrared divergence. The Carleman operator is derived from the so-called pull-through formula, and we gave the exact operator-theoretical proof for the formula, which is the fast to succeed in it. By this proof, we can investigate mathematical properties of the domain of the Carleman operator and pull-through formula, which resulted in our results. Because Nelson's model has infrared catastrophe by our results, we find another representation in which the model has a ground state. This representation describes the actual physical phenomenon. So, we removed both, infrared and ultraviolet cutoffs, and proved the Nelson model without both cutoffs has a ground state in the representation.We studied the norm resolvent convergence for the Hamiltonian describing relativistic particle coupled with the Aharonov-Bohm field in 2-dim. space. We investigated which self-adjoint extension has the most suitable representation to the actual physics among several self-adjoint extensions corresponding to the boundary conditions around singularities.

我们通过几个物理近似研究了纳尔逊模型从Pauli-Fierz模型中得出的模型。 Pauli-Fierz模型描述了一种电子，当我们将电子视为非同性粒子时，在非递归量子电动力学中的量化辐射场结合。我们证明，当尼尔森模型的哈密顿量在原子结构中出现库仑电位时，尼尔森模型具有红外灾难。开发证明并使用卡尔曼操作员，我们澄清并表征了一种数学机制，该机制导致红外安全或红外发散。卡尔曼操作员源自所谓的拉装公式，我们给出了该公式的确切操作者理论证明，这是成功的速度。通过此证明，我们可以研究Carleman操作员和Pull-Trough Formula域的数学特性，这导致了我们的结果。由于尼尔森的模型通过我们的结果有红外灾难，因此我们找到了模型具有基态的另一种表示形式。该表示描述了实际的物理现象。因此，我们删除了紫外线和紫外线的截止，并证明了没有两个截止的尼尔森模型在表示中具有基态。我们研究了汉密尔顿的标准分辨率收敛，描述了相对论粒子，并在2二维中与aharonov-bohm场相连。空间。我们调查了哪些自偶会扩展具有与几个自偶会扩展的实际物理形式，这些伸展与奇点周围的边界条件相对应。