(Received June 19, 1970)
It is attempted to derive the anomalous commutators of the axial-vector current and the gravitational field variables by the reduction formula and the Bjorken-Johnson-Low limiting procedure for the axial-vector current-two graviton matrix element. For this purpose, the additional Lagrangian density for the gravitational field is introduced so that the gravitational interaction reduces to a renormalizable one. In spite of such a technique, the derived commutators are not consistent except for several ones and show that the Bjorken-Johnson-Low behavior does not hold in a strict sense. With the aid of the commutators which seem to be rigorous, the other commutators are determined so as to be consistent with each other, with the divergence equation and with the quantization procedure of the gravitational field. In the second place, a model is presented to confirm the appropriateness of the anomalous commutators. By setting out the field-current identity and the canonical commutators, the divergence equation of the axial-vector current is derived. The additional gravitational Lagrangian density plays an important role also in this model. Finally, the relation between the two methods is discussed.
URL : http://ptp.ipap.jp/link?PTP/44/1353/
DOI : 10.1143/PTP.44.1353