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Quantum Gravity: Exploring the Causal Dynamical Triangulation Approach

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Norman S. Israel

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2012

Quantum Gravity is an attempt to unify the current understanding of the microworld (governed by quantum mechanics) with the current understanding of the macroworld (governed by general relativity). This thesis provides a brief overview of the problem of quantum gravity, including conceptual as well as technical overviews of general relativity and quantum mechanics. It explores a conservative approach to quantizing gravity known as Causal Dynamical Triangulation (CDT). CDT constructs spacetime from triangular-like building blocks known as simplexes. Combining quantum mechanics and general relativity by a sum-over-geometries, CDT demonstrates that microcausality implies classical global spacetime. This thesis focuses on a bare 1+1 dimensional universe. It converts the Einstein-Hilbert action of general relativity (from which one can recover the Einstein Field Equations using the principle of least action) from continuous to discrete form using the Gauss-Bonnet theorem, which is rigorously developed. This thesis proves a version of the theorem on two dimensional Euclidean polyhedra and sketches a proof on Pseudo-Riemannian spaces. Finally, it describes the design, construction, and operation of a computer simulation of a bare 1+1 dimensional CDT universe. It uses the simulation to find a critical value for the cosmological constant and computes the corresponding fluctuations in the spatial size of the unvierse, which it compares with analytical results.