Mathematical Problems in Quantum Field Theory and Infinite-Dimensional Analysis

量子场论和无限维分析中的数学问题

• 批准号: 08454021
• 负责人: ARAI Asao
• 金额: $ 5.44万
• 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: 1996
• 资助国家: 日本
• 起止时间: 1996 至 1997
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-08454021/
• 关键词: canonical commutation relations (CCR); gauge theory; qauntum field; Dirac operator; supersymmetric qauntum field theory; Fock space; Schrodinger operator; 強反可換性; 一般化されたスピン-ボソンモデル; 埋蔵固有値; 非可逆的ボゴリューボフ変換; 基底状態

(1) Representations of canonical commutation relations (CCR) appearing in a (non-commutative) gauge theory on a non-simply connected region of R^3 have been analyzed indetail. These representations are realized by the set of the position and the physical momentum operators. Properties of the strongly continuous 1-parameter unitary groups generated by the physical momentum operators (commutation relations, irreducibility etc.) were clarified as well as connections to the Aharonov-Bohm effect, representations of quantum groups and quantum lattice gauge theory. Moreover, the coupling of the quantum system to a quantized radiation field was considered. As a result, new classes of representations of CCR were discovered on the tensor product Hibert space of L^2 (R^3) and the Fock space of the quantized radiation field. These are original discoveries.(2) A necessary and sufficient condition for two Dirac operators on the boson-fermion Fock space to strongly anticommute was characterized in terms of the strong anticommutativity of Dirac operators on the one-particle base Hilbert space.(3) A class of representations of CCR with infinite degrees of freedom (or indexed by an infinite dimensional Hilbert space) was constructed in connection with perturbation problem of embedded eigenvalues in quantum field models.(4) A new estimate for the groundstate energy of the Schrodinger operator was derived.

(1)详细分析了非对易规范理论中的正则对易关系(CCR)在R^3的非单连通区域上的表示。这些表示是通过位置和物理动量算符的集合来实现的。由物理动量算子生成的强连续单参数酉群的性质(对易关系、不可约性等)以及与Aharonov-Bohm效应、量子群的表示和量子格点规范理论的联系。此外，还考虑了量子系统与量子化辐射场的耦合。结果，在L^2(R^3)的张量积Hibert空间和量子化辐射场的Fock空间上发现了CCR的新的表示类。(2)利用单粒子基Hilbert空间上Dirac算符的强反对易性刻画了玻色子-费米子Fock空间上的两个Dirac算符强反对易的充要条件。(3)针对量子场模型中嵌入本征值的微扰问题，构造了一类具有无限自由度(或由无限维希尔伯特空间标引的)CCR的表示。(4)给出了薛定谔算符基态能量的新估计。

项目成果

Asao, Arai: "On the exrstence and unaqueness of a generalized spn-boson model" Journal of Functional Anolysts. 151. 455-503 (1997)

Asao，Arai：“论广义 spn-玻色子模型的存在性和唯一性”《功能分析学杂志》。

新井朝雄: "場の量子論の数学的方法入門" 大阪大学 Osaka Mathematical Publications, 226 (1997)

新井朝雄：《量子场论的数学方法导论》大阪数学刊物，226（1997）

新井 朝雄: "ヒルベルト空間と量子力学" 共立出版社, 276 (1997)

Asao Arai：《希尔伯特空间与量子力学》Kyoritsu Shuppansha，276（1997）

Asao Arai: "Strong anticommutativity of Dirac operators on boson-fermion Fock spaces and representations of a supersymmetry algebra." Mathematrsche Nachrochten. (in press). (1998)

Asao Arai：“狄拉克算子在玻色子-费米子福克空间和超对称代数表示上的强反交换性。”

Asao Arai: "On the exostence and undqueness of graund states of a generalized spmboson model" Journal of Functional Analysis. 151. 455-503 (1997)

Asao Arai：“论广义 spmboson 模型的大态的存在性和不正当性”《泛函分析杂志》。