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

• 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

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从 Pauli-Fierz 模型导出的纳尔逊模型。当我们将电子视为非相对论粒子时，泡利-菲尔兹模型描述了非相对论量子电动力学中与量子化辐射场耦合的电子。我们证明了当尼尔森模型的哈密顿量具有原子结构中出现的库仑势时，尼尔森模型就会产生红外灾变。通过证明并使用卡尔曼算子，我们阐明并描述了导致红外安全或红外发散的数学机制。卡尔曼算子是从所谓的拉通公式推导出来的，我们对该公式给出了精确的算子理论证明，这是该公式获得成功的最快方法。通过这个证明，我们可以研究卡尔曼算子域的数学性质和推导公式，从而得出我们的结果。由于根据我们的结果，尼尔森的模型存在红外灾难，因此我们找到了该模型具有基态的另一种表示形式。这种表示描述了实际的物理现象。因此，我们删除了红外和紫外截止，并证明没有两个截止的尼尔森模型在表示中具有基态。我们研究了描述相对论粒子与二维阿哈罗诺夫-玻姆场耦合的哈密顿量的范数求解收敛性。空间。我们研究了在与奇点周围的边界条件相对应的几个自伴随扩展中，哪种自伴扩展最适合实际物理的表示。