We discuss a certain generalization of the Hilbert space of states in noncommutative quantum mechanics that, as we show, introduces magnetic monopoles into the theory. Such generalization arises very naturally in the considered model, but can be easily reproduced in ordinary quantum mechanics as well. This approach offers a different viewpoint on the Dirac quantization condition and other important relations for magnetic monopoles. We focus mostly on the kinematic structure of the theory, but investigate also a dynamical problem (with the Coulomb potential).

Many-body quantum mechanics is the fundamental theory behind many areas of modern science, such as condensed-matter physics, nuclear physics, and quantum chemistry. It is also notoriously hard: the classical picture of particles with well-defined positions and velocities is replaced by an intricate interference pattern between all their possible trajectories, captured by the quantum wave function. The exponentially large information content of wave functions makes direct simulation of large, strongly interacting quantum systems impossible, and necessitates strategies to manage the complexity in an analytically or computationally tractable manner. The bulk of this thesis explores two such strategies in the context of quantum spin liquids. In these materials, competition between incompatible interactions results in robust, massive entanglement, down to zero temperature. Such ground states give rise to a range of exotic behaviour, such as topological order and fractionalised excitations: understanding these is a central challenge in the physics of strongly correlated materials. Several quantum-spin-liquid phases are underpinned by strict local conservation laws, which give rise to lattice gauge theories with exotic quasiparticle excitations, such as emergent photons or magnetic monopoles. We developed a systematic approach, based on a large-S bosonisation formalism, to extract gauge-theoretic descriptions from such constrained Hamiltonians automatically, and thus make them amena