Perturbative Aspects of Non-local and Non-commutative Quantum Field Theories

The understanding of fundamental interactions is based on the concept of quantum field theory in combination with gauge fields. It represents a unification of the theory of special relativity and quantum mechanics. Elementary particle processes are described with amazing precision. But phenomena related to gravity are not included in a satisfactory way. This would require the unification of quantum theory and the general theory of relativity. Thereby, one faces the following problem: In quantum mechanics, particles can in principal be localised within arbitrarily small spaces, provided that one spends sufficiently high energies. Taking the general theory of relativity into account, particles could create black hole horizons for the corresponding localisation. This would contradict our knowledge and experiences. Thus, one tries for example to exclude particle states with arbitrary localisation from quantum theory. This can be realised by replacing the continuous space-time coordinates by non-commuting, Hermitian operators, which leads to non-commutative geometry. The corresponding noncommutative quantum field theories are discussed in this thesis. One possible description applies the so-called Weyl quantisation. This means that in noncommutative quantum field theory, the local field products of the corresponding local theory are replaced by non-local Moyal-Weyl products, which are characterised by a deformation parameter of the dimension of an area. Thus, one introduces non-localities, which cause serious problems to renormalisation and unitarity, for example. The latter property is violated as soon as there are non-localities in time. Unitarity can now be preserved by strictly constructing the theory on the basis of time-ordered perturbation theory. Within this approach, the validity of the usual Feynman rules is therefore extended to a large class of non-local interactions to be defined in this thesis. The connection to the usual Feynman rules of local quantum field theory becomes clear in contrast to existing results. In coordinate space, the propagator is replaced by the contractor, which embeds a more general type of time-ordering. The prescription for vertices becomes also more complicated. The diagrammatic techniques in momentum space are considerably simpler. It is remarkable that the functions associated with vertices do not depend on the full off-shell momenta, but simply on their space components. Properties like unitarity, or the behaviour under Lorentz transformations, time reversal, and parity are investigated on a very general basis. It turns out that especially the requirement for Lorentz invariance represents a serious problem, whose solution seems to be nearly impossible within the framework of the investigations carried out here. The problem of renormalisation was basically studied in two ways: First, a theory free of ultraviolet divergences was constructed from smeared field operators. Secondly, the influence of an additional, harmonic oscillator term in the action of a non-commutative quantum field theory was investigated. Thus, one drops translation invariance. Simple calculations show that the UV/IR mixing problem, which is a serious obstacle to the renormalisation program, seems to be cured, which is in agreement with existing results.