# Wave function of the Universe

## 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 ..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 tomore »

- Authors:

- Publication Date:

- Research Org.:
- Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637 and Institute for Theoretical Physics, University of California, Santa Barbara, California 93106

- OSTI Identifier:
- 5310722

- Resource Type:
- Journal Article

- Journal Name:
- Phys. Rev. D; (United States)

- Additional Journal Information:
- Journal Volume: 28:12

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; UNIVERSE; WAVE FUNCTIONS; BOUNDARY CONDITIONS; DIFFERENTIAL EQUATIONS; FEYNMAN PATH INTEGRAL; HAMILTONIANS; PROBABILITY; PROPAGATOR; QUANTUM MECHANICS; SCALAR FIELDS; EQUATIONS; FUNCTIONS; INTEGRALS; MATHEMATICAL OPERATORS; MECHANICS; QUANTUM OPERATORS; 640106* - Astrophysics & Cosmology- Cosmology; 645400 - High Energy Physics- Field Theory

### Citation Formats

```
Hartle, J.B., and Hawking, S.W..
```*Wave function of the Universe*. United States: N. p., 1983.
Web. doi:10.1103/PhysRevD.28.2960.

```
Hartle, J.B., & Hawking, S.W..
```*Wave function of the Universe*. United States. doi:10.1103/PhysRevD.28.2960.

```
Hartle, J.B., and Hawking, S.W.. Thu .
"Wave function of the Universe". United States. doi:10.1103/PhysRevD.28.2960.
```

```
@article{osti_5310722,
```

title = {Wave function of the Universe},

author = {Hartle, J.B. and Hawking, S.W.},

abstractNote = {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 ..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.},

doi = {10.1103/PhysRevD.28.2960},

journal = {Phys. Rev. D; (United States)},

number = ,

volume = 28:12,

place = {United States},

year = {1983},

month = {12}

}