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We describe the modern approach to quantum cosmology, as initiated by Hartle and Hawking, Linde, Vilenkin and others. The primary aim is to explain how one determines the consequences for the late universe of a given quantum theory of cosmological initial or boundary conditions. An extensive list of references is included, together with a guide to the literature. 1 2 ,φ = 0, on the k = 0 curve, because at this point the model undergoes inflation. This point is an attractor for all the expanding k = 0 and k = −1 solutions. The k = +1 solutions, however, with which one is primarily concerned in quantum cosmology, do not all end up on the attractor: if they start out away from the k = 0 curve with |φ| large they recollapse before getting anywhere near the attractor.Inflation occurs, therefore, only for the subset of k = +1 solutions with reasonably small † In the case k = 0, one can eliminateα using the constraint, and the phase portrait becomes two-dimensional. This has been constructed for various inflationary potentials by Belinsky et al.(1985) and Piran and Williams (1985).

In the no-boundary proposal for the initial conditions of a closed cosmology, the wave function of the universe is the integral of expi-action) over a contour of four-geometries and matter-field configurations on compact manifolds having only that boundary necessary to specify the arguments of the wave function. There is no satisfactory covariant Hamiltonian quantum mechanics of closed cosmologies from which the contour may be derived, as there would be for defining the ground states of asymptotically flat spacetimes. No compelling prescription, such as the conformal rotation for asymptotically flat spacetimes, has been advanced. In this paper it is argued that the contour of integration can be constrained by simple physical considerations: (1) the integral defining the wave function should converge; (2) the wave function should satisfy the constraints implementing diffeomorphism invariance; (3) classical spacetime when the universe is large should be a prediction;(4) the correct field theory in curved spacetime should be reproduced in this spacetime; (5) to the extent that wormholes make the cosmological constant dependent on initial conditions the wave function should predict its vanishing. We argue that the convergence criterion is readily satisfied by choosing a suitable complex contour. The constraints will be satisfied if the end points of the contour are suitably restricted. For classical spacetime to be a prediction, the contour must be dominated by one or more saddle points at which the four-metric is complex. We discuss the conditions under which such complex solutions to the Einstein equations arise and their interpretation. Because the action is double valued in the space of complex metrics, every solution of the Einstein equations corresponds to two saddle points: one with R e ( d F ) > 0, the other with Reid? < 0.They differ only in the sign of their action. We find that criteria (4) and (5) imply that the contour should not be dominated by a saddle point with ~e i d z ) i 0 . This restriction may be difficult to satisfy in the path-integral forms of the "tunneling" boundary condition proposals of Linde, Vilenkin, and others. Although all of these physical considerations constrain the contour and largely determine the semiclassical predictions of the wave function, there is still remaining freedom. Until fixed by more fundamental considerations, the remaining freedom in the contour means that there are many corresponding no-boundary proposals.

The destruction of quantum interference, decoherence, and the destruction of entanglement both appear to occur under the same circumstances. To address the connection between these two phenomena, we consider the evolution of arbitrary initial states of a two-particle system under open system dynamics described by a class of master equations which produce decoherence of each particle. We show that all initial states become separable after a finite time, and we produce the explicit form of the separated state. The result extends and amplifies an earlier result of Diósi. We illustrate the general result by considering the case in which the initial state is an EPR state (in which both the positions and momenta of a particle pair are perfectly correlated). This example clearly illustrates how the spreading out in phase space produced by the environment leads to certain disentanglement conditions becoming satisfied.