Manifestly-covariant canonical formalisms of non-abelian gauge theory

This paper introduces the manifestly-covariant canonical formalism of the non-abelian gauge theory. There are three essential ingredients in this formalism, which are the B-field formalism, invariance under Becchi-Rouet-Stora (BRS) transformation, and new hermiticity assignment for the Faddeev-Popov (FP) ghosts. It has been customary that the FP ghosts are regarded as hermitian conjugates each other. Then, the total Lagrangian becomes non-hermitian. The non-hermiticity of the Lagrangian causes serious troubles in the operator formalism. Kugo and Ojima (KO) have cleaned up all those troubles by introducing some new hermiticity assignment for the FP ghosts. They assume that both FP ghosts are hermitian. Then, the total Lagrangian becomes hermitian. The BRS transformation can be defined in abstract way. The abstract BRS transformation is the local mapping of the polynomial algebra satisfying four rules, one of which holds only when the KO hermiticity assignment for the FP ghosts is adopted. Another rule is valid only when the B-field formalism is employed. The unitarity of the physical S-matrix is proved. Extending the KO formalism for the non-abelian gauge theory to quantum gravity, the manifestly covariant quantum field theory of general relativity in the framework of the canonical operator formalism can be formulated. Another formalism of the manifestly covariant quantum field theory of gravity is proposed. Satisfactory operator formalism of quantum gravity can be obtained. (Kato, T.)