Gauge fields and quantum theory

Abstract

This thesis investigates the problems within quantum mechanics for the Bohm model caused by Lorentz invariance and the existence of photons. A model describing the electromagnetic interactions of fermions is produced which does not use photons and avoids these problems. It is then shown how these techniques can be extended to linearised gravitational interactions. Finally semi-classical gravity and the possibility of gravitationally induced collapse are considered. In the first part of the thesis two modifications to the Bohm model are proposed. One takes account of Lorentz invariance, and the other is capable of describing photons. The main part of the thesis is devoted to describing interactions in a way which does not need extra gauge particles, and so is in the same spirit as the Bohm model. Electromagnetic interactions are formed using a 4-potential operator which is calculated directly, without imposing commutation relations on the 4-potential. This leads to an expression for the 4-potential in terms of the Dirac field, and results in there being no photon states. There are various ways of constructing the theory and the scattering matrix of standard QED is compared to the scattering matrix of the version which appears to be most similar. Considering only the matrix elements between fermion states, they are found to be in agreement at the order e(^2), but disagree at the order e(^4). It follows that this model, which otherwise appears to be a self consistent theory of QED, cannot agree with experiment. The same techniques can be used to quantise General Relativity when it is linearised about the Minkowski metric. The metric operator is calculated in terms of the Dirac field. The interaction is similar to that of electrodynamics, being of order 4 in the Dirac field. Finally issues relating to gravitational collapse are discussed.