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Contribution to the quantum theory of gravitation Freeman, Michael James

Abstract

A quantum theory of gravitation is constructed, by considering the gravitational field in the linear approximation to be a rank II tensor field, which has imposed upon it the auxiliary conditions of symmetry, transversality, and tracelessness. Extensive use is made of the close analogy between the electromagnetic field as a special case of a vector field, and the gravitational field as a special case of a tensor field. This analogy includes the necessity of introducing an indefinite metric in order to make the auxiliary conditions compatible with the commutation relations. A complete theory of gravitation must take into account the gravitating nature of gravitation and hence must be a nonlinear theory. A method proposed by Gupta of iterating the linear field equations for this purpose is investigated, and it is shown that this method fails, because the Lagrangian for the second order equations does not exist. An alternative method of iteration is proposed which avoids this problem, and which yields a functional equation for the Lagrangian of the full nonlinear theory. Finally, the problem of photon-photon scattering due to the gravitational interaction is investigated. This is done by constructing an interaction Hamiltonian by using the principle of the compensating field and then applying the standard methods of quantum electrodynamics. It is found that for sufficiently high frequencies this process dominates the purely electrodynamic scattering of photons by photons.

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