Analysis of Operator Formalism of Quantum Field Theory

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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