Analysis of Operator Formalism of Quantum Field Theory

在量子场论的算符形式中，重新考虑了基于BRS形式的Kato-Ogawa弦理论。给出了各种规范中二维量子引力算符形式的精确解中普遍存在的场方程反常现象与BRS Noether电荷的零能级击穿之间的关系。场方程的异常比BRS Noether电荷的异常更基本。我们可以显式构造一个无反常的BRS电荷，即BRS对称性本身就是无反常的。讨论了微扰法或路径积分法与算符形式主义方法的根本区别。在前一种方法中，T^*-积(协方差T-积)存在一些缺点。结果表明，在洛伦兹时空理论中，用路径积分方法得到的Faddeev-Popov鬼数电流反常现象是不存在的。TH…的精确解在光锥规中建立了更多的二维BF理论和Yang-Mills理论。精确解的结构与共形规范中二维量子引力的结构非常相似。场方程和BRS Noether电荷中存在反常，但始终存在无反常的BRS电荷，即BRS对称性本身是无反常的。基于这一事实，对基于BRS Noether电荷反常推导弦理论临界维度的推理提出了批评，并通过Cuntz代数阐明了费米子汽车代数的结构。明确地构造了Cuntz代数的Car代数和仿射代数的嵌入。通过将Cuntz代数推广到伪Cuntz代数，构造了二维量子引力中Faddeev-Popov鬼零模算符的非平凡表示。这一结果作为题为《弦理论中的伪Cuntz代数和递归FP Ghost系统》的论文提交给了一份期刊。通过Cuntz代数，证明了汽车代数存在无穷多个自同构，这些自同构用推广了著名的Bogoliubov变换的非线性变换来表示。这一结果被作为题为《汽车费米子代数的非线性变换群》的论文提交给期刊。较少

In the operator formalism of quantum field theory, the Kato-Ogawa string theory based on the BRS formalism is reconsidered. The relation between the breakdown of nilpotency of the BRS Noether charge and the field-equation anomaly, which is a common phenomenon in the exact solutions to the operator formalism of two-dimensional quantum gravity in various gauges, is shown. The field-equation anomaly is more fundamental than the anomaly of the BRS Noether charge. We can explicitly construct an anomaly-free BRS charge, that is, the BRS symmetry itself is anomaly free. The fundamental difference between the perturbative or path-integral approach and the operator-formalism approach is discussed. In the former approach, there are some disadvantages come from the T^*-product (covariantized T-product). It is shown that the Faddeev-Popov ghost number current anomaly, which is derived in the path-integral approach, does not exist in the theories on the Lorenzian spacetime.The exact solutions to th … More e two-dimensional BF and Yang-Mills theories in the light-cone gauge are constructed. The structure of the exact solutions is very similar to that of the two-dimenaional quantum gravity in the conformal gauge.There exist anomalies in a field equation and a BRS Noether charge, while there always exist a anomaly-free BRS charge, that is, the BRS symmetry itself is anomaly free. Based on this fact, a criticism is made on the reasoning of deriving the critical dimension of string theory on the basis of the anomaly of the BRS Noether charge.The structure of the CAR algebra of fermions is clarified through the Cuntz algebra. Embeddings of the CAR algebra and the paraferion algebra into the Cuntz algebra are explicitly constructed. By extending the Cuntz algebra to the pseudo Cuntz algebra, nontrivial representations of the zero-mode operators of the Faddeev-Popov ghost in the two-dimensional quantum gravity are constructed. This result is submitted to a journal as the paper entitled "Pseudo Cuntz Algebra and Recursive FP Ghost System in String Theory." Through the Cuntz algebra, it is also shown that there exist infinite number of automorphisms of the CAR algebra, which are expressed in terms of nonlinear transformations generalizing the well-known Bogoliubov transformations. This result is submitted to a jouranl as the paper entitled "Nonlinear Transformation Group of CAR Fermion Algebra." Less