Relativity and gravitational theory Definition

Hartle–Hawking state

Builds on Euclidean quantum gravity Wheeler–DeWitt equation Riemannian manifold

Definition

Let $\Sigma$ be a closed three-manifold with a Riemannian metric $h$ and matter field configuration $\phi_0$ on $\Sigma$ . The Hartle–Hawking state is the wave function $\Psi_{\mathrm{HH}}[h, \phi_0] = \int \mathcal{D}[g] \, \mathcal{D}[\phi] \; e^{-I[g, \phi]/\hbar},$ where $I$ is the Euclidean action of gravity and matter as in Euclidean quantum gravity. The formal integral runs over compact four-manifolds $M$ with $\partial M = \Sigma$ and over Riemannian metrics $g$ on $M$ inducing $h$ . It also runs over matter fields $\phi$ on $M$ restricting to $\phi_0$ .

Remarks

The integral has no other boundary, hence the name no-boundary proposal. Formally $\Psi_{\mathrm{HH}}$ satisfies the Wheeler–DeWitt equation, because the integrand is invariant under diffeomorphisms and the integral over the lapse imposes the Hamiltonian constraint. Example: take the minisuperspace model with a cosmological constant $\Lambda > 0$ and $\Sigma$ a round three-sphere of radius $a$ . For small $a$ the dominant saddle is a portion of a round four-sphere of radius $\sqrt{3/\Lambda}$ capping off $\Sigma$ . For $a > \sqrt{3/\Lambda}$ it continues as half a four-sphere glued to Lorentzian de Sitter space, and the wave function oscillates. Non-example: a path integral over cylinders $[0,1] \times \Sigma$ with a fixed initial metric on $\set{0} \times \Sigma$ is not the Hartle–Hawking state, since it has a second boundary. Vilenkin's tunnelling wave function is a different proposal with different boundary conditions on the same equation.

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Tag	`MRY9JZTK9V`
URI	`m2://MRY9JZTK9V`
Kind	definition
Writing	W0 stub
MSC	83C45, 83F05
Defines	Hartle–Hawking state
Uses	Euclidean quantum gravity, Wheeler–DeWitt equation, Riemannian manifold
Changed	3 November 2026

Hartle–Hawking state

Definition

Remarks

Reading path

Further reading

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