Divergence of Axial-Vector Current in the Gravitational Field. II

Abstract

1191 To clarify the similarity and dissimilarity between the electromagnetic and gravitational fields, the vacuum expectation value of the divergence of the axial-vector current is evaluated in the external gravitational field. Within the framework of the perturbation approach in the Minkowski space, the requirement of gauge invariance yields the finite extra term in addition to the expected term. The additional term comes from the diagrams of the triangle closed loop and the two-sided closed loop which is linked to the former to maintain gauge invariance. Within the covariant formalism in the Riemannian space-time, it is also examined whether the additional term can be inferred directly from the field equations by giving a proper definition of the operator of the axial-vector current. To maintain gauge invariance explicitly, the spinor fields in the current are separated infinitesimally and are parallel-transported back. A difficulty lies in defining the correct current which gives the same additional term as given by the perturbation treatment in the Minkowski space. The origin of the difficulty is discussed in connection with the non-local nature of the gravity.