Final PhD Oral Examination (Thesis Title: “Geometry from Quantum Mechanics: Entanglement, Energy Conditions and the Emergence of Space”)

Event Date and Time: 
Mon, 2016-07-18 12:30 - 14:30
Room 203, Graduate Student Centre
Local Contact: 
Physics and Astronomy, UBC
Intended Audience: 

This thesis presents various examples of the application of quantummechanical
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 UVfinite)
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 associated
to any three Hermitian matrices. We study this emergent surface when the matrices
are large, by constructing coherent states corresponding to points in the emergent
geometry. We find 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
or 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

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