Dirac operators and field equations of the gravitational field and matter fields

Abstract

Dirac operators on curved space-times are introduced with the help of a new point-view that observers have to be included in the formulation of natural laws. The class of Dirac operators are Lorentz invariant in the sense that the transformation rule is specified under diffeomorphisms of the space-time which preserve the time orientation and the gravitational field. Moreover these Dirac operators, like the original Dirac's operator with the special relativity, satisfy the Hamiltonian relation required by the general theory of relativity up to a correction due to the setup of a reference frame. In order to generalise (or discover) Dirac operators, which have been an important building block in the quantum field theories (QFTs), to curved space-times, we bring observers (which are elements of the principal orthonormal bundle over the space-time with its structure group being the proper Lorentz group) into the description of Fermion fields in the presence of a gravitational field. As a consequence field equations combining the gravitational field and matter fields may be formulated. This work suggests that observational effects (due to the setup of references) in the presence of a strong gravitational field, i.e. matter, may be unavoidable.