On the application of quantum perturbation theory to gravitational interactions
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Part 1: The formalism preliminary to a quantum perturbation treatment of the interaction of wave fields with gravitation is here developed. Since spinor fields are of importance, a resumé is given of Pauli’s treatment of spinors in general coordinates, involving the introduction of generalized Dirac operators. The essential points of the Einstein-Mie theory are outlined, and spin angular momentum is discussed from the general coordinate viewpoint with the aid of the generalized orthogonal group. The commutation law for covariant differentiation is obtained for arbitrary fields. The symmetric stress tensor can be constructed either from the canonical energy-momentum tensor together with the spin angular momentum tensor or directly through variation of the metric tensor. The concepts of energy, momentum and spin angular momentum, and hence the Hamiltonian formalism itself, can be introduced for the gravitational field only with respect to a "background space" which has a flat metric of no physical geometrical significance. In the background space only Lorentz transformations have immediate invariant significance, and general coordinate transformations appear as "gauge transformations" in the gravitational and accompanying matter fields. Only the total integrated energy, momentum and spin angular momentum quantities are invariant under these “gauge transformations.”Part 2: The Hamiltonian formulation of the “linearized” gravitational field equations is introduced. The longitudinal field components may be eliminated with the help of two auxiliary vector fields. The interaction Hamiltonian density for the interacting gravitational and scalar meson fields is constructed. When the longitudinal gravitational field is eliminated, a “Newtonian” term is introduced into the interaction. G-gauge transformations are discussed from the point of view of the interaction representation. Perturbation approximation methods are applied in the calculation of the mesic stress induced in the vacuum by an impressed gravitational field. The induced stress is found to be logarithmically divergent and structure dependent, and hence not interpretable as a stress renormalization. The gravitational self-mass of the scalar meson is next calculated, and is found to be ambiguous, depending, to second order, on what gravitational tensor is chosen with which to make power expansions. By appropriate choice of this tensor the self-mass can, however, be made to assume the unique finite value [mathematical equation]. The self-energy operator, itself, remains quadratically divergent. The same calculations are also carried out for the interacting gravitational and electromagnetic fields. In order that the supplementary Lorentz condition always be maintained, it is shown that every G-gauge transformation must be accompanied by a transformation in the electromagnetic gauge. The vacuum induced electromagnetic stress is found to be also logarithmically divergent and structure dependent. Owing to the tracelessness of the stress tensor, on the other hand, the self-energy operator is in this case unambiguous and is found, moreover, to be identically zero.