Mathematical Problems in Quantum Field Theory and Infinite-Dimensional Analysis

(1)详细分析了R^3非简连通域上的(非交换)规范理论中的正则交换关系(CCR)的表示。这些表示是通过位置和物理动量算子的集合来实现的。阐明了由物理动量算符生成的强连续 1 参数酉群的性质（交换关系、不可约性等）以及与阿哈罗诺夫-玻姆效应、量子群表示和量子晶格规范理论的联系。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 空间和量化辐射场的 Fock 空间上发现了 CCR 的新表示形式。原创性发现。(2)利用单粒子基希尔伯特空间上狄拉克算子的强反交换性，刻画了玻色子-费米子福克空间上两个狄拉克算子强反交换的充要条件。(3)结合量子场模型中嵌入特征值的微扰问题，构造了一类具有无限自由度(或由无限维希尔伯特空间索引)的CCR表示。(4)导出了薛定谔算子基态能量的新估计。

(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.