Andrade e Silva, RodrigoWe develop the non-perturbative reduced phase space quantization of causal diamondsin (2+1)-dimensional gravity with a nonpositive cosmological constant. The system is defined as the domain of dependence of a spacelike topological disk with fixed boundary metric. By solving the constraints in a constant-mean-curvature time gauge and removing all the spatial gauge redundancy, we find that the phase space is the cotangent bundle of Diff+(S1)/PSL(2,R). Classically, the states correspond to causal diamonds embedded in AdS3 (or Mink3 if Λ = 0), with fixed corner length, and whose Cauchy surfaces have the topology of a disc. Because the phase space does not have a natural linear structure, a generalization of the standard canonical (coordinate) quantization is required. As the configuration space is a homogeneous space for the Diff+(S1) group, we apply Isham’s group-theoretic quantization scheme. We propose a quantization based on (projective) unitary irreducible representations of the BMS3 group. We find a class of suitable quantum theories labelled by a choice of a coadjoint orbit of the Virasoro group and an irreducible unitary representation of the corresponding little group. The most natural choice, justified by a Casimir matching principle, corresponds to a Hilbert space realized by wavefunctions on Diff+(S1)/PSL(2,R) valued in some unitary irreducible representation of SL(2,R). A surprising result is that the twist of the diamond corner loop is quantized in terms of the ratio of the Planck length to the corner perimeter.enDissertationTheoretical physicsCanonical quantizationGeneral relativityGroup quantizationQuantum gravityReduced phase spaceRepresentation theory