Recovering SU(2) gauge symmetry in cosmological implementations of loop quantum gravity

This thesis investigates the formal aspects of applying the quantisation techniques developed in Loop Quantum Gravity within a cosmological context. Its principal contribution is the proposal of a novel symmetry-reduction procedure that defines the cosmological sector of the Ashtekar-Barbero-Immirzi formulation of General Relativity for arbitrary cosmological models. Crucially, this method preserves the local SU(2) gauge symmetry, thereby enabling a quantisation scheme closely aligned with Loop Quantum Gravity, in which spin-network states on suitably symmetric graphs arise naturally. Loop Quantum Gravity is regarded as one of the most promising candidates for a theory of quantum gravity, primarily owing to its prediction that geometric quantum operators associated with area and volume have discrete spectra. The application of the loop quantisation scheme to cosmology leads to Loop Quantum Cosmology. Although Loop Quantum Cosmology has achieved significant success in predicting novel and intriguing phenomena, its reliability has been questioned, particularly concerning its consistency with the full Loop Quantum Gravity theory. The original contributions of this thesis begin with an extension of the Loop Quantum Cosmology framework to more general cosmological models not thoroughly examined in the literature: the non-diagonal models, which are relevant to the Belinski-Khalatnikov-Lifshitz conjecture. We show that these models can be quantised through a minor modification of the standard approach, introducing angular variables that describe the directions of cosmic expansion. In this framework, the fate of the internal SU(2) symmetry is analysed, revealing that the minisuperspace approximation effectively Abelianises the theory: the Gauss constraint reduces to three independent Abelian constraints, equivalent even at the quantum level. To restore the local SU(2) gauge symmetry, we need to move beyond the minisuperspace approach. To achieve this, we develop an appropriate geometric framework by employing principal bundle theory and emphasising the role of spin structures in the description of both the Ashtekar variables and the associated constraints. We apply the framework to cosmology, defining a precise notion of homogeneity for the Ashtekar connection that is consistent with Wang’s theorem. This enables us to construct the classical cosmological sector while preserving the local SU(2) gauge symmetry. Furthermore, we explore the properties of the moduli space of homogeneous Ashtekar connections, uncovering significant topological features that help address typical issues concerning the quantisation of gauge field theories. Finally, we propose a novel interpretation of a class of theories often claimed to incorporate quantum-gravitational effects in an effective manner: the Generalised Uncertainty Principle theories. We introduce a semiclassical framework in which such models are described as symplectic manifolds endowed with a non-standard symplectic structure, conjecturing a link between modified symplectic geometry and the quantum-deformed Poisson algebra. Applying this formalism to constrained theories, with particular attention to cosmological scenarios, we demonstrate that the deformation is consistent with symplectic reduction and thus compatible with the imposition of constraints.