  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.
