Non-perturbative Quantum Electrodynamics in low dimensions

Field theories provide the relevant modern framework to account for the quantum physics of the smallest constituents of matter and their interactions, save for the gravitational interaction. While the perturbative approach to the quantisation of these theories has proved to be so successful in describing all such physical phenomena of interest, important questions remain beyond its reach. Through the use and development of specific non-perturbative approaches and techniques, this thesis aims at describing low dimensional gauge models, namely massless Quantum Electrodynamics (QED) in two and in three space-time dimensions. Firstly, by emphasising the relevance of the topological sector of two-dimensional QED, we recover its exact solution given by a free massive pseudo-scalar boson, while the composition of this bound state in terms of fermion pairs as well as its quantum ground state are made explicit through the intricate role played by the axial quantum anomaly. Then, the vacuum state of three-dimensional QED is found to be approximated by a neutral pair condensate. Subsequently, the confinement of charged states is argued to be a consequence of this dynamical mechanism, as well as is the spontaneous breaking of parity. Finally, still within the context of three-dimensional QED the vacuum structure of a system of relativistic fermions, constrained in a plane and subjected to a constant magnetic field, is investigated to unravel an intricate pattern of degenerate quantum states in the Lowest Landau Level.