The final approved version of my PhD thesis for James Cook University.AbstractQuantum field theory is the mathematical language of nature at the deepest levels currently known. Viewed in this light, much of the last century of theoretical physics can be seen as quantum field theory calculations performed in a variety of approximations. However, despite having a very successful standard model of physics and a phenomenally useful perturbation theory for extracting predictions from it, persistent

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... roblems remain. Many of these are due to the inability to calculate in practice what is calculable in principle. The difficulty of these problems comes from the importance of a large number of degrees of freedom, or a strong coupling between degrees of freedom, or both. Only in the case of very simple systems or systems close to thermal equilibrium can their properties be determined with confidence. The remaining cases require a resummation, or re-organisation, of perturbation theory. However, ad hoc resummations are problematic because perturbation series are asymptotic in nature and the re-organisation of the terms of an asymptotic series is mathematically questionable. Systematic resummation schemes are required to guarantee consistency with the original non-perturbative theory. n-particle irreducible effective actions (nPIEAs; n=1,2,3,...) are a proven method of resummation which can be thought of as generalisations of mean field theory which are (a) elegant, (b) general, (c) in principle exact, and (d) have been promoted for their applicability to non-equilibrium situations. These properties make the nPIEAs compelling candidates for filling the gap between theory and experiment anywhere the problem is the inability to compute the behaviour of strongly interacting degrees of freedom or degrees of freedom which are out of thermal equilibrium. Unfortunately, nPIEAs are known to violate the symmetries of a field theory when they are truncated to a form that can be solved in practice. This c [...]