Quantum Cosmology. I

Abstract

The Hamiltonian methods of Arnowitt, Deser, and Misner can be applied to homogeneous cosmological models, and prove to be an efficient way both of constructing the Einstein equations and of studying their solutions. By using an appropriate form for the metric, one finds that the constraint equations for these models can be solved explicitly, and the resulting problem in Hamiltonian mechanics resembles that of a particle in a potential well. The most unusual feature of the Hamiltonian is that it is explicitly time dependent. There is an easy and attractive choice of factor orderings which allows one to pass on to a quantum theory (by imposing canonical commutation relations on the independent canonical variables) while maintaining the signature of the quantized metric. For the closed-space cosmological model (Bianchi type IX) which is studied in most detail, a classical (high-quantum-number) state remains classical as the wave function is followed back in time toward the initial singularity. There is no tendency for significant contributions from states of low quantum number to develop even when the radius of the universe is much less than ${\left(\frac{G\hslash }{c^3}\right)}^{\frac{1}{2}}={10}^{-33\mathrm{}}$ cm.