Wave function of the Universe

TLDR

The quantum state of a spatially closed universe is encoded in a wave function that is a functional of three‑manifold geometries and matter fields, and it satisfies the Wheeler–DeWitt equation. The authors propose that the ground‑state amplitude for a given three‑geometry is obtained by a path integral over all compact, positive‑definite four‑geometries bounded by that three‑geometry. By requiring a Hermitian Hamiltonian, boundary conditions for the Wheeler–DeWitt equation are fixed, and the authors compute ground and excited states in a minisuperspace model with a scale factor and conformally invariant scalar field, while the path integral also accommodates topology change and allows estimation of the probability of multiple spacelike components. They find that the ground state reduces to de Sitter space in the classical limit, whereas excited states describe universes that expand from zero volume, reach a maximum size, recollapse, but possess a small tunneling probability to a de Sitter‑type eternally expanding state.

Abstract

The quantum state of a spatially closed universe can be described by a wave function which is a functional on the geometries of compact three-manifolds and on the values of the matter fields on these manifolds. The wave function obeys the Wheeler-DeWitt second-order functional differential equation. We put forward a proposal for the wave function of the "ground state" or state of minimum excitation: the ground-state amplitude for a three-geometry is given by a path integral over all compact positive-definite four-geometries which have the three-geometry as a boundary. The requirement that the Hamiltonian be Hermitian then defines the boundary conditions for the Wheeler-DeWitt equation and the spectrum of possible excited states. To illustrate the above, we calculate the ground and excited states in a simple minisuperspace model in which the scale factor is the only gravitational degree of freedom, a conformally invariant scalar field is the only matter degree of freedom and $\ensuremath{\Lambda}>0$. The ground state corresponds to de Sitter space in the classical limit. There are excited states which represent universes which expand from zero volume, reach a maximum size, and then recollapse but which have a finite (though very small) probability of tunneling through a potential barrier to a de Sitter-type state of continual expansion. The path-integral approach allows us to handle situations in which the topology of the three-manifold changes. We estimate the probability that the ground state in our minisuperspace model contains more than one connected component of the spacelike surface.