Geometry from quantum mechanics : entanglement, energy conditions and the emergence of space

This thesis presents various examples of the application of quantum-mechanical methods to the understanding of the structure of space-time. It focuses on noncommutative geometry and the gauge/gravity duality as intermediaries between quantum mechanics and classical geometry. First, we numerically calculate entanglement entropy and mutual information for a massive free scalar field on commutative and noncommutative (fuzzy) spheres. To define a subregion with a well-defined boundary in the noncommutative geometry, we use the symbol map between elements of the noncommutative algebra and functions on the sphere. We show that the UV-divergent part of the entanglement entropy on a fuzzy sphere does not follow an area law. In agreement with holographic predictions, it is extensive for small (but fixed) regions. This is true even in the limit of small noncommutativity. Nonetheless, we find that mutual information (which is UV-finite) is the same in both theories. This suggests that nonlocality at short distances does not affect quantum correlations over large distances in a free field theory. Previous work has shown that a surface embedded in flat ℝ³ can be associated to any three Hermitian matrices. By constructing coherent states corresponding to points in the emergent geometry, we study this emergent surface when the matrices are large. We find that the original matrices determine not only shape of the emergent surface, but also a unique Poisson structure. We prove that commutators of matrix operators correspond to Poisson brackets. Through our construction, we can realize arbitrary noncommutative membranes. Finally, we use the gauge/gravity correspondence to translate the positivity of relative entropy on the boundary into constraints on allowable space-time metrics in the bulk. Using the Einstein equations, we interpret these constraints as energy conditions. For certain three-dimensional bulks, we obtain strict constraints coming from the positivity of relative entropy with a thermal reference state which turn out to be equivalent to a version of the weak energy condition near the boundary. In higher dimensions, we use the canonical energy formalism to obtain similar but weaker constraints.Science, Faculty ofGraduat