Mathematics – Logic
Scientific paper
Jan 1994
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1994phdt........25m&link_type=abstract
PhD Dissertation, California Univ. Santa Barbara, CA United States
Mathematics
Logic
1
Astronomical Models, Cosmology, Euclidean Geometry, Quantum Theory, Relativity, Universe, Degrees Of Freedom, Gravitation, Invariance, Isotropy, Wave Functions
Scientific paper
To describe the evolution of the universe, from the earliest times to the present, a theory of the initial conditions of the universe is required. When the universe is small, quantum effects must predominate. However, there is no well-defined quantum theory of cosmology. We propose a general method of fixing the remaining degrees of freedom in the Hartle-Hawking path integral formulation of quantum cosmology, and present important new results in the field of lattice gravity. Euclidean path integral calculations generally require a reduction of the infinite degrees of freedom to a finite number. This is called reducing to minisuperspace. Regge calculus, the natural lattice version of general relativity, is one way to achieve such a reduction. We calculate when Regge's piecewise flat simplicial geometries have analogs of the continuum's diffeomorphism invariance and contracted Bianchi identities. The geometries must be near-flat and the specific conditions that must hold on individual simplices are derived. We next apply Regge methods to construct a simple cosmological model and quantize it using Euclidean path integral techniques. The initial conditions are fixed by our proposal for the integration contour over geometries and matter fields. It is a higher dimensional analog of a steepest descents contour, and passes through the classical geometry with the smallest real part of the action. This actually leads to a family of contours that all yield the same semi-classical wave-function of the universe. Our homogeneous isotropic model with matter minimally coupled to gravity has only a small number of degrees of freedom. We find that the universe exhibits Euclidean behavior early and evolves later to a Lorentzian or classical era, in agreement with the calculations of the continuum model. This dissertation is the first time that matter has been included in a Regge quantization. Finally, an anisotropic model is examined. It predicts that the universe is anisotropic when small, and that it will in general evolve into an isotropic and classical state at late times, as we indeed observe today.
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