Mathematics
Scientific paper
Feb 1991
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1991phdt........22k&link_type=abstract
Ph.D. Thesis Pennsylvania State Univ., University Park.
Mathematics
3
Astronomical Models, Born-Oppenheimer Approximation, Cosmology, Eigenvalues, Eigenvectors, Scalars, Scaling Laws, Wave Packets, Wentzel-Kramer-Brillouin Method, Absolute Zero, Boundary Value Problems, Cauchy Problem, Differential Equations, Dimensionless Numbers, Euclidean Geometry, Matrices (Mathematics)
Scientific paper
The Wheeler-DeWitt equations for the Friedmann-Robertson-Walker cosmology conformally and minimally coupled to scalar fields with power-law potentials are expanded in the eigenstates of the scalar field parts. The gravitational parts become a diagonal matrix-valued differential equation for a conformal scalar field, and a coupled matrix-valued differential equation for a minimally coupled scalar field. The coupling matrix for the minimal scalar field is found to be a dimensionless quantity by the Symanzik scaling law, invalidating the Born-Oppenheimer approximation. There also occur returning points depending on the energy eigenvalues of the scalar field, contrary to the WKB approximation obtained by neglecting the kinetic energy term for the scalar field. The Cauchy initial value problem is defined with respect to the intrinsic timelike coordinate, and the wavefunctions incorporating initial data are constructed using the product integral formulation. The packetlike wavefunctions around classical turning points are shown possible in the product integral formulation, and the returning wavepackets near the returning point of the classical Friedmann-Robertson-Walker cosmology are constructed. The wavefunctions to the Wheeler-DeWitt equation minimally coupled to the scalar field are constructed by two different methods, the master equation and the enlarged matrix equation. The spectrum for the wavefunctions regular at the infinite size of universe is found, and these are interpreted as the Hawking-Page spectrum of wormholes connecting two asymptotically Euclidean regions. The quantum Friedmann-Robertson-Walker cosmology is extended to the minimal scalar field with the inflationary potential having a first order phase transition. The Wheeler-DeWitt equation is expanded in the eigenstates of the scalar field, and the gravitational part becomes a coupled matrix-valued differential equation. When the symmetry of the scalar field is broken at a certain stage of evolution of the universe, the relative energy difference of a local minimum to the global minimum (absolute zero) behaves as an effective cosmological constant responsible for inflation for the modes whose eigenvalues are almost determined by the local minimum. As the universe evolves further, the effective cosmological constant vanishes due to disappearance of the local minimum. This quantum inflationary cosmological model avoids not only ad hoc potentials having very weak-coupling but also the a priori cosmological constant.
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