Local interactions of the quantised electromagnetic field

Photons, i.e. the basic energy quanta of monochromatic waves, are highly non-localised and occupy all available space in one dimension. This non-local property can complicate the modelling of the quantised electromagnetic field in the presence of optical elements that are local objects. Therefore, this thesis takes an alternative approach and shows that a local second quantisation of the electromagnetic (EM) field is possible but requires an extension of conventional quantum theory. For light propagating in one dimension, we obtain highly localised bosonic Fock operators, which we do by doubling the usual photon Hilbert space with some photonic modes evolving according to the standard Schrodinger equation and others evolving according to the complex conjugated Schrodinger equation. We also view the quantised EM field as a biorthogonal system. However, we view it as a biorthogonal system where the intersection of the Hilbert space and its dual Hilbert space is non-zero. To the best of our knowledge, this is the first time such a construction of the EM field has been made. These highly localised bosonic Fock operators provide natural building blocks of wave packets of light and enable us to construct locally acting interaction Hamiltonians for two-sided semi-transparent mirrors. Using these Hamiltonians, we produce appropriate classical dynamics of the electric field near a mirror. The question of how to model local transformations of the EM field is a hot topic, as physicists often measure interactions between the EM field and local optical devices in experiments. Therefore, we expect our results to find large appeal across both the quantum optics and non-Hermitian communities. We finish by discussing possible future avenues of research.