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In this thesis we explore the infrared problem perturbatively in massless field theory. We review the current conventional methods and theorems that are applied in the calculation of QCD jet observables and then discuss the formulation of an alternative approach called the Asymptotic Interaction Picture (AIP). The AIP is based on a unitary transformation such that long-ranged interactions are present in the asymptotic Lagrangian and thus the states associated with this picture are no longer free Fock states but are asymptotic states containing soft and collinear interactions. Under the guidance of the AIP we are led to modifying conventional perturbation theory, cutting up amplitudes in a manner that allows for the construction of infrared finite amplitudes that are in correspondence with the asymptotic states of the AIP. We apply this formalism to several NLO corrections to QCD observables and construct dressed states who's amplitudes are finite in all regions of phase space. Using these amplitudes we compute several observables and show agreement with the conventional calculations in infrared safe regions. Higher-order calculations are then investigated in ϕ (^3)theory and the infrared pole structure is shown to behave as expected such that NNLO corrections to dressed states are obtained. Finally we present part of the NNLO correction to the dressed two-parton amplitude in QCD and show that, with several provisos, this approach may potentially be applied to the precision calculations of observables at the International Linear Collider (ILC). We therefore give a possible alternative to current subtraction methods at NNLO when no initial state radiation is present.

We review quantum field theories on noncommutative Minkowski spaces (NCQFTs), concentrating on the mixing between ultra-violet and infra-red degrees of freedom in such theories. We use background field perturbation theory at the one-loop level to calculate the three and four point functions in supersymmetric NCQFT. We use the results of this calculation to show that the infra-red logarithmic divergences that arise as a result of the UV/IR mixing can be reproduced by an explicitly gauge-invariant low-energy effective action expressed in terms of Wilson lines. We present a noncommutative gauge theory that has the ordinary Standard Model as its low-energy limit. The model is based on the gauge group U (4) x U (3) x U (2) and satisfies all the key constraints that are imposed by noncommutativity: the UV/IR mixing effects, restrictions on representations and charges of matter fields and the cancellation of noncommutative gauge anomalies. At energies well below the noncommutative mass scale our model flows to the commutative Standard model plus additional free U {1) degrees of freedom which decouple due to the UV/IR mixing. Our model also predicts the values of the hypercharges of the Standard Model fields. We find that in order to accomodate the matter content of the Standard Model it is necessary to introduce extra, as yet undetected, matter fields.

Scattering amplitudes are fundamental and remarkably rich observables in quantum field theory. The basic observation that makes scattering amplitudes fascinating is that in theories with massless particles of spin s ≥ 1, they are much simpler than might be expected from traditional Feynman diagram techniques for computing them. This simple observation might ultimately have profound consequences for our view of quantum field theory. The broad aim of this thesis is to understand and exploit the hidden simplicity and structure in scattering amplitudes. The quantum field theory with the simplest scattering amplitudes in four dimensions is planar N = 4 supersymmetric Yang-Mills theory. This theory has provided considerable inspiration in developing new computational techniques and has provided many important theoretical insights. In this theory, there is a remarkable correspondence between scattering amplitudes and null polygonal Wilson loops, observables which on first inspection look very different. In this thesis, we will provide new insights into this correspondence using methods from twistor theory and exploit the symmetries of the problem to find new ways of computing scattering amplitudes.