In this thesis we investigate the role played by fundamental properties of QED in determining the non-perturbative fermion-boson vertex. These key features are gauge invariance and multiplicative renormalisability. We use the Schwinger-Dyson equations as the non- perturbative tool to study the general structure of the fermion-boson vertex in QED. These equations, being an infinite set, have to be truncated if they are to be solved. Such a truncation is made possible by choosing a suitable non-perturbative ansatz for the fermion-boson vertex. This choice must satisfy these key properties of gauge invariance and multiplicative renormalisability. In this thesis we develop the constraints, in the case of massless unquenched QED, that have to be fulfilled to ensure that both the fermion and photon propagators are multiplicatively renormalisable-at least as far as leading and subleading logarithms are concerned. To this end, the Schwinger-Dyson equations are solved perturbatively for the fermion and photon wave-function renormalisations. We then deduce the conditions imposed by multiplicative renormalisability for these renormalisation functions. As a last step we compare the two results coming from the solution of the Schwinger-Dyson equations and multiplicative renormalisability in order to derive the necessary constraints on the vertex function. These constitute the main results of this part of the thesis. In the weak coupling limit the solution of the Schwinger-Dyson equations must agree with perturbation theory. Consequently, we can find additional constraints on the 3- point vertex by perturbative calculation. Hence, the one loop vertex in QED is then calculated in arbitrary covariant gauges as an analytic function of its momenta. The vertex is decomposed into a longitudinal part, that is fully responsible for ensuring the Ward and Ward-Takahashi identities are satisfied, and a transverse part. The transverse part is decomposed into 8 independent components each being separately free of kinematic singularities in any covariant gauge in a basis that modifies that proposed by Ball and Chiu. Analytic expressions for all 11 components of the O(a) vertex are given explicitly in terms of elementary functions and one Spence function. These results greatly simplify in particular kinematic regimes. These are the new results of the second part of this thesis.

This thesis considers the problem of determining Casimir-Lifshitz forces in inhomogeneous media. The ground-state energy of the electromagnetic field in a piston-geometry is discussed. When the cavity is empty, the Casimir pressure on the piston is finite and independent of the small-scale physics of the media that compose the mirrors. However, it is demonstrated that, when the cavity is filled with an inhomogeneous dielectric medium, the Casimir energy is cut-off dependent. The local behavior of the stress tensor commonly used in calculations of Casimir forces is also determined. It is shown that the usual expression for the stress tensor is not finite anywhere within such a medium, whatever the temporal dispersion or index profile, and that this divergence is unlikely to be removed by modifying the regularisation. These findings suggest that the value of the Casimir pressure may be inextricably dependent on the detailed behavior of the mirror and the medium at large wave vectors. This thesis also examines two exceptions to this rule: first, the case of an idealised metamaterial is considered which, when introduced into a cavity, reduces the magnitude of the Casimir force. It is shown that, although the medium is inhomogeneous, it does not contribute additional scattering events but simply modifies the effective length of the cavity, so the predicted force is finite and can be stated exactly. Secondly, a geometric argument is presented for determining a Casimir stress in a spherical mirror filled with the inhomogeneous medium of Maxwell's fish-eye. This solution questions the idea that the Casimir force of a spherical mirror is repulsive, but prompts additional questions concerning regularisation and the role of non-local effects in determining Casimir forces.