Math 2.0

Relativity and gravitational theory Definition

Wheeler–DeWitt equation

Builds on ADM formalismHamiltonian constraintCanonical quantizationRiemannian manifold

Definition

Let Σ\Sigma be a closed three-manifold and let Ψ\Psi be a functional of Riemannian metrics hijh_{ij} on Σ\Sigma. Let RR be the scalar curvature of hh, Λ\Lambda the cosmological constant, and Gijkl=12h(hikhjl+hilhjk−hijhkl)G_{ijkl} = \frac{1}{2\sqrt{h}} \left( h_{ik} h_{jl} + h_{il} h_{jk} - h_{ij} h_{kl} \right) the DeWitt supermetric, where h=det⁡hijh = \det h_{ij}. The Wheeler–DeWitt equation (in units c=1c = 1) is (−16πGℏ2 Gijklδ2δhij δhkl−h16πG(R−2Λ))Ψ[h]=0\left( -16 \pi G \hbar^2 \, G_{ijkl} \frac{\delta^2}{\delta h_{ij} \, \delta h_{kl}} - \frac{\sqrt{h}}{16 \pi G} (R - 2\Lambda) \right) \Psi[h] = 0 at every point of Σ\Sigma. It is obtained from the Hamiltonian constraint 16πG Gijklπijπkl−h16πG(R−2Λ)=016 \pi G \, G_{ijkl} \pi^{ij} \pi^{kl} - \frac{\sqrt{h}}{16\pi G}(R - 2\Lambda) = 0 of the ADM formalism by the canonical quantization rule πij↦−iℏ δ/δhij\pi^{ij} \mapsto -i\hbar \, \delta / \delta h_{ij}.

Remarks

The wave function must also satisfy the momentum constraint ∇j(δΨ/δhij)=0\nabla_j \left( \delta \Psi / \delta h_{ij} \right) = 0, which says that Ψ\Psi is invariant under diffeomorphisms of Σ\Sigma. The product of two functional derivatives at the same point is ill defined, and the ordering of GijklG_{ijkl} and the derivatives is ambiguous, so the equation is formal. Example (minisuperspace): restricting to round three-spheres of radius aa reduces the equation to an ordinary differential equation in aa, which is the setting of the Hartle–Hawking state and other proposals. Non-example: the time-dependent equation iℏ ∂tΨ=H^Ψi \hbar \, \partial_t \Psi = \hat H \Psi is not of this form; here H^Ψ=0\hat H \Psi = 0 and no external time appears.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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ADM formalism, concept, no evidence levelADM formalismconceptCanonical quantization, concept, no evidence levelCanonical quantizationconceptHamiltonian constraint, concept, no evidence levelHamiltonian constraintconceptRiemannian metric, concept, no evidence levelRiemannian metricconceptSmooth manifold, concept, no evidence levelSmooth manifoldconceptRiemannian manifold, definition, no evidence levelRiemannian manifolddefinitionWheeler–DeWitt equation, definition, no evidence levelWheeler–DeWitt equati…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. ADM formalism concept
  2. Canonical quantization concept
  3. Hamiltonian constraint concept
  4. Riemannian metric concept
  5. Smooth manifold concept
  6. Riemannian manifold definition
  7. Wheeler–DeWitt equation definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Wheeler–DeWitt_equation
About this page
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Kinddefinition
WritingW0 stub
MSC83C45, 81S08
DefinesWheeler–DeWitt equation
UsesADM formalism, Hamiltonian constraint, Canonical quantization, Riemannian manifold
Changed3 November 2026

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