Operator Formalism of Quantum Field Theory and Operator Algebra

利用康茨代数的限制，阐明了描述费米子系统的CAR代数结构。首先，明确地构造了孔兹代数的嵌入、自同构、自同态和各种表示。其次，通过将它们约束到Cuntz代数中的嵌入CAR代数中，得到CAR代数的相应结构；利用一类自同态，证明了作为费米子系统混合态的KMS态是由纯态的Fock表示得到的。这一结果与von Neumann代数分类中的Araki-Woods因子理论相一致。通过限制自同构，明确构造了费米子系统的各种非平凡时间演化，给出了可解费米子系统的新构造。对于规范理论中的不定度量费米子系统，伪孔兹代数起着类似的作用。因此，弦理论中更多的FP幽灵的表示问题也得到了澄清。结果表明，用Alvarez-Gaume和Witten的推理不能建立二维重力异常的存在性。它的缺点是T^*积和T积的混淆。与T-积相反，T^*积随时间微分交换，但有些场方程明显被破坏。这个问题与异常无关。Alvarez-Gaume和Witten得到的结果在数量上与T^*积中场方程的明显分解相同。关于共形规范二维量子引力的可解性，研究了无限维守恒量所对应的无限维对称性的构造。由于场方程作为共形规范条件的特性，很难将场的对称变换从一般守恒量定义为对称发生器。这个过程对应于诺特定理的反问题。少

The structure of the CAR algebra describing a fermion system is clarified through a new approach using restrictions of the Cuntz algebra. First, embeddings, automorphisms, endomorphisms, and various representations of Cuntz algebras are explicitly constructed. Second, by restricting them to the embedded CAR algebra in the Cuntz algebra, the corresponding structures of the CAR algebra are obtained. Using a certain class of endomorphorphisms, it is shown that the KMS states as the mixed states of a fermion system are obtained from the Fock representation as the pure state. This result corresponds to the theory of Araki-Woods factors in the classification of von Neumann algebras. By restricting automorphisms, various nontrivial time evolutions of a fermion system are explicitly constructed, which give new constructions of solvable fermion systems. As for the indefinite-metric fermion system in gauge theories, the pseudo Cuntz algebra plays the similar role as above. Therefore, the problem … More of representation of the FP ghosts in string theory is clarified likewise.It is shown that the existence of the gravitational anomaly in two dimensions cannot be established by the reasoning of Alvarez-Gaume and Witten. Its drawback is the confusion of the T^*-product quantities with the T-product ones. In contrast with the T-product, the T^*-product commutes with the time differentiation, however some field equations are apparently broken. This problem has nothing to do with anomaly. The result obtaied by Alvarez-Gaume and Witten is shown to be quantitatively the same as the apparent breakdown of the field equation in the T^*-product.As concerns the solvability of the conformal-gauge two-dimensional quantum gravity, constructions of the infinite-dimensional symmetry corresponding to the infinite-number of conserved quantities are studied. Owing to the speciality of the field equation as the conformal-gauge condition, it is difficult to define the symmetry transformation of fields from a generic conserved quantitity as a symmetry generator. This procedure corresponds to the inverse problem of the Noether theorem. Less