Indefinite-Metric Quantum Field Theory of General Relativity. XV: Tensorlike Four-Dimensional Commutation Relation

Abstract

In the manifestly covariant canonical formalism of quantum gravity, it is known that the equal·time commutator between a tensor field and the B field bp is consistent with the rules of tensor analysis. Another tensorlike commutation relation is shown to exist for the equal·time commutator between a tensor and 6 p, but at the same time its limitation is clarified. The quantum-gravity extension of the invariant D function is defined and proved to be affine· invariant. The four·dimensional commutation relation between a tensor and bp is investigated, and it is shown that the commutator consists of a completely tensorlike, manifestly affine· covariant part and a remainder, which is clearly distinguishable from the former. In the manifestly covariant canonical formalism of quantum gravity, one of the most remarkable features is the existence of the tensorlike commutation relation.!) That is, it has been verified by a large number of examples that the equal-time commutation relation between a local field (j)Pl·~·;'.':))s(x) which is a tensor at the classical level (for brevity, we call it a tensor hereafter) and the B field bp(x') can be written as