UniversalWaveFunction

The Hartle-Hawking state is a proposal by James Hartle and Stephen Hawking to resolve the issue of initial conditions in cosmology. It suggests that the universe's initial state is determined by the requirement that it have no boundary or edge in imaginary time.

The Hartle-Hawking wave function $\Psi$ is given by a path integral over all compact Euclidean metrics $g_{\mu\nu}$ and matter fields $\phi$, which can be formally expressed as:

$$ \Psi[g_{\mu\nu},\phi] = \int_{\mathcal{M}} e^{-I[g_{\mu\nu},\phi]} Dg_{\mu\nu} D\phi$$

Here, $I[g_{\mu\nu},\phi]$ is the Euclidean action for the gravitational field $g_{\mu\nu}$ and the matter fields $\phi$. The path integral is over all metrics $g_{\mu\nu}$ and matter fields $\phi$ on a compact manifold $\mathcal{M}$, which is understood to be regular everywhere and has no boundary.

The no-boundary proposal posits that the wave function of the universe $\Psi$ is a superposition of all spatial geometries that are smoothly connected to a given 3-geometry. The weight of each geometry in the superposition is given by $e^{-I}$, where $I$ is the Euclidean action for the gravitational field and matter fields.

The no-boundary wave function can also be expressed as a sum over all possible 4-geometries that are compact and have no boundary, where the Euclideanized Einstein-Hilbert action is used for the gravitational part.

The Hartle-Hawking wave function has profound implications for cosmology and quantum gravity, though it still poses several unresolved challenges and questions. It has been considered a significant step towards understanding the nature of the initial conditions of the universe and the unification of quantum mechanics and general relativity.

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 Λ>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.