Wheeler-DeWitt Equation

Take general relativity, cast it in Hamiltonian form, and try to quantize it the way you would any other constrained system — and you arrive at the strangest equation in physics. In the ADM 3+1 split, spacetime is sliced into spacelike leaves, each carrying a 3-metric $h_{ij}$ and its conjugate momentum $\pi^{ij}$ . But general covariance exacts a price: the entire Hamiltonian is a sum of constraints, $\mathcal{H}\approx 0$ and $\mathcal{H}_i\approx 0$ , with no leftover bulk energy to drive evolution. There is nothing to generate time. Following Dirac, we quantize by promoting $\pi^{ij}\to -i\hbar\,\delta/\delta h_{ij}$ and demanding that physical states be annihilated by the operator constraints. The Hamiltonian constraint becomes $\hat{\mathcal{H}}\,\Psi[h_{ij},\phi]=0$ — the Wheeler-DeWitt equation — and the momentum constraint $\hat{\mathcal{H}}_i\Psi=0$ tells us that $\Psi$ , the wavefunction of the universe, lives on superspace: the arena of all 3-geometries. The most jarring feature is what is missing. There is no $i\hbar\,\partial_t\Psi$ , no external clock, no Schrödinger evolution. The equation is a constraint, not an evolution law, and because the super-Hamiltonian is quadratic in momenta contracted with the indefinite DeWitt supermetric — signature $(-,+,+,+,+,+)$ — it reads as a Klein-Gordon equation on superspace rather than a heat equation. The single timelike direction is the local volume, the closest thing to a clock the theory admits. This is the problem of time in its sharpest form. Taken literally the equation is ill-defined — coincident functional derivatives spawn $\delta(0)$ , operator ordering is ambiguous, no inner product is agreed. So it is best read as a beacon: it motivates loop quantum gravity, it is solved exactly in minisuperspace cosmology, and it underwrites the Hartle-Hawking wavefunction of the universe.

Key takeaways

The Wheeler-DeWitt equation is Dirac quantization of the Hamiltonian constraint of general relativity: $\hat{\mathcal{H}}\,\Psi[h_{ij},\phi]=0$ , obtained by $\pi^{ij}\to -i\hbar\,\delta/\delta h_{ij}$ .
Its unknown $\Psi$ is the wavefunction of the universe, a functional on superspace — all 3-metrics modulo spatial diffeomorphisms, enforced by the momentum constraint $\hat{\mathcal{H}}_i\Psi=0$ .
It carries no $\partial_t$ : a constraint replaces evolution, and that absence of an external clock is the problem of time.
The DeWitt supermetric $G_{ijkl}$ has Lorentzian signature $(-,+,+,+,+,+)$ , so the equation is Klein-Gordon-like on superspace, with the local volume as an intrinsic timelike direction.
Taken at face value it is formally divergent ( $\delta(0)$ ) and ordering-ambiguous — a heuristic made rigorous only in symmetry-reduced (minisuperspace) form or by the loop reorganization.
In closed de Sitter minisuperspace it collapses to an Airy-type ODE whose tunneling amplitude reproduces the de Sitter entropy, $\lvert\Psi_{\mathrm{HH}}\rvert^2\propto e^{+3\pi/(G\hbar\Lambda)}$ .
Key idea: Quantizing the Hamiltonian constraint of GR, $\mathcal{H}\approx0$ , by $\pi^{ij}\to -i\hbar\,\delta/\delta h_{ij}$ gives a timeless wave equation $\hat{\mathcal{H}}\Psi[h_{ij},\phi]=0$ on superspace; the indefinite DeWitt supermetric makes it Klein-Gordon-like, and the absence of an external time is the problem of time.

Where this lives today

The Wheeler-DeWitt equation is not tested in a laboratory, but its descendants are among the most active fronts in fundamental physics. In quantum cosmology it is the equation whose minisuperspace solutions — the Hartle-Hawking and Vilenkin wavefunctions — feed into predictions for the initial conditions of inflation and hence the statistics of the cosmic microwave background; recent "no-boundary" path-integral work by Feldbrugge, Lehners and Turok, and the responses by Hartle, Hawking and Hertog, is a live debate about whether the no-boundary proposal predicts smooth or wildly fluctuating universes. In loop quantum cosmology the differential operator is replaced by a difference equation whose bounce resolves the big-bang singularity, and groups worldwide compute its imprint on primordial power spectra. The constraint algebra itself is the proving ground for asymptotic-safety and causal-dynamical-triangulation programs that seek a nonperturbative continuum limit. And conceptually, the equation anchors the modern relational and "timeless" interpretations of quantum mechanics — Page-Wootters conditional probabilities, the Montevideo interpretation, and quantum reference frames — all of which take the absence of external time in $\hat{\mathcal{H}}\Psi=0$ not as a bug but as the fundamental fact to be explained.

The physical picture

Classically, general covariance means no slice is preferred: what we call "evolution" is a relabelling of one spacelike leaf into the next, generated by the Hamiltonian constraint. Quantize, and the operator that would push the state forward in time is instead required to annihilate it. The wavefunction of the universe does not tick — it simply is. Any sense of time must be reconstructed from within, as a correlation between dynamical variables: the volume is large when the scalar field sits at a certain value. The first figure sets this against a familiar mirror — the relativistic particle, whose mass-shell constraint quantizes to Klein-Gordon, not Schrödinger. The Wheeler-DeWitt equation is that same move made infinite-dimensional.

Figure

The mass shell pᵘpᵤ + m² = 0 quantizes to the Klein-Gordon equation, a constraint with no external time; the Hamiltonian constraint ℋ ≈ 0 of gravity quantizes the same way to Ĥ Ψ = 0 on superspace, where the DeWitt supermetric has signature minus plus plus plus plus plus and the local volume √h is the one timelike direction

Now picture superspace itself — the space of all 3-geometries — as a curved manifold carrying the DeWitt supermetric. The Wheeler-DeWitt equation propagates $\Psi$ across it like a wave. Because one direction (overall volume, the conformal mode) is timelike while the five shape directions are spacelike, the equation is hyperbolic: it has the character of a wave bouncing in a potential, and the potential is the intrinsic 3-curvature. When we freeze all but the scale factor of a homogeneous universe, that abstract wave becomes an ordinary quantum particle tunnelling under a barrier — and the barrier has a real, computable shape. The second figure is that punchline: the minisuperspace potential of a closed de Sitter universe, the arena of the Hartle-Hawking and Vilenkin wavefunctions.

Figure

Minisuperspace potential of a closed de Sitter universe U(a) = a² − (Λ∕3)a⁴ with Λ = 3: a barrier of height 3∕(4Λ) peaks at a = √(3∕(2Λ)) ≈ 0.707, the classical turning point sits at a = √(3∕Λ) = 1, and for a > 1 the potential turns negative so Ψ oscillates — the small-a exponential under the barrier is the universe tunnelling from nothing

The core equations

The whole theory is one demand: the quantum operator built from the classical Hamiltonian constraint must kill the state. Everything below is that demand written out, unpacked, and reduced to a solvable arena. We begin with the schematic form — the sentence the rest of the article is a gloss on.

\hat{\mathcal{H}}\,\Psi[h_{ij},\phi]=0 \qquad (1)

where: $\hat{\mathcal{H}}$ — operator version of the Hamiltonian (super-Hamiltonian) constraint density; $\Psi[h_{ij},\phi]$ — wavefunctional of the universe, a functional of the 3-metric and matter fields; $h_{ij}$ — induced Riemannian 3-metric on a spatial slice; $\phi$ — matter field(s) on the slice. Read it as: physical states are exactly those annihilated by the super-Hamiltonian, so $\Psi$ contains no reference to an external time. The super-Hamiltonian, written out, is a functional Laplacian in the momenta plus a potential built from intrinsic curvature. The $\delta/\delta h_{ij}$ terms are the quantized momenta; the curvature term is the "potential" the wave feels.

\hat{\mathcal{H}}=-16\pi G\,\hbar^2\,G_{ijkl}\frac{\delta}{\delta h_{ij}}\frac{\delta}{\delta h_{kl}}-\frac{\sqrt{h}}{16\pi G}\left({}^{(3)}\!R-2\Lambda\right)+\hat{\mathcal{H}}_{\mathrm{matter}} \qquad (2)

where: $G$ — Newton's constant; $G_{ijkl}$ — DeWitt supermetric (next card); $\delta/\delta h_{ij}$ — functional derivative, quantizing $\pi^{ij}\to -i\hbar\,\delta/\delta h_{ij}$ ; ${}^{(3)}\!R$ — Ricci scalar of the 3-metric (intrinsic curvature); $\Lambda$ — cosmological constant; $\sqrt{h}$ — square root of $\det h_{ij}$ , the 3-volume element; $\hat{\mathcal{H}}_{\mathrm{matter}}$ — matter Hamiltonian density operator. Reading: the first term is "kinetic energy" on superspace, the second a "potential" $\propto -{}^{(3)}\!R$ , and their sum must vanish pointwise. The supermetric is the metric on the space of 3-metrics — it tells us how to contract the two functional derivatives. Its index structure and its crucial signature are the whole reason the equation is a wave equation.

G_{ijkl}=\frac{1}{2\sqrt{h}}\left(h_{ik}h_{jl}+h_{il}h_{jk}-h_{ij}h_{kl}\right) \qquad (3)

where: $G_{ijkl}$ — DeWitt supermetric on the 6-dimensional space of symmetric $3\times3$ tensors at a point; $h_{ik}$ — the pointwise 3-metric that raises and contracts indices. Reading: the $-h_{ij}h_{kl}$ trace piece is what flips one eigenvalue negative, making the conformal (volume) direction timelike. The momentum constraint is the second half of the Dirac program. It is first order and imposes that $\Psi$ cannot see coordinate labels — only the geometry.

\hat{\mathcal{H}}_i\,\Psi=-2i\hbar\,h_{ij}\,D_k\frac{\delta\Psi}{\delta h_{jk}}=0 \qquad (4)

where: $\hat{\mathcal{H}}_i$ — diffeomorphism (momentum) constraint operator; $D_k$ — covariant derivative compatible with $h_{ij}$ ; the overall factor fixes the normalization of the diffeomorphism generator. Reading: $\Psi$ is invariant under infinitesimal 3-diffeomorphisms, hence a functional on superspace $=\mathrm{Riem}(\Sigma)/\mathrm{Diff}(\Sigma)$ . The two constraints do not commute into zero — they close on each other with field-dependent structure functions, the hypersurface-deformation (Dirac) algebra. This is why anomaly-free quantization is so hard.

\left\{\mathcal{H}(x),\mathcal{H}(y)\right\}=\big(\mathcal{H}^i(x)+\mathcal{H}^i(y)\big)\,\partial_i\delta(x,y) \qquad (5)

where: $\mathcal{H}^i=h^{ij}\mathcal{H}_j$ — momentum constraint with index raised by $h^{ij}$ ; $\partial_i\delta(x,y)$ — derivative of the Dirac delta on the slice. Reading: because the right-hand side depends on $h^{ij}$ , the "algebra" is not a Lie algebra; a consistent quantum theory must reproduce it up to ordering or it is anomalous. Finally, the semiclassical ansatz is how ordinary time crawls back out. Writing $\Psi$ as a rapidly oscillating gravitational phase times a slow matter part recovers a Schrödinger equation for matter along the classical trajectory.

\Psi\approx C\,e^{\,iS_0[h]/\hbar}\,\chi[h,\phi],\qquad i\hbar\,\frac{\partial\chi}{\partial\tau}=\hat{\mathcal{H}}_{\mathrm{matter}}\,\chi \qquad (6)

where: $S_0$ — solution of the gravitational Einstein-Hamilton-Jacobi equation; $\chi$ — slowly varying matter wavefunctional; $C$ — WKB prefactor; $\partial/\partial\tau\equiv G_{ijkl}(\delta S_0/\delta h_{ij})\,\delta/\delta h_{kl}$ — the WKB time built from the gradient of $S_0$ . Reading: to leading order in $\hbar$ , matter obeys ordinary time-dependent quantum mechanics with respect to the emergent WKB time — the resolution of the problem of time in the near-classical regime.

Derivation AFrom the ADM action to the Wheeler-DeWitt equation

The goal is to trace the entire path from the gravitational action to $\hat{\mathcal{H}}\Psi=0$ , and in doing so to see exactly where the two defining oddities appear: where the supermetric with its indefinite signature comes from, and where the time derivative goes. We work in vacuum with $\Lambda=0$ for clarity — matter simply adds $\hat{\mathcal{H}}_{\mathrm{matter}}$ — and we lean on the Gauss-Codazzi machinery owned by the prerequisite articles, invoking it by name rather than re-deriving it. The target is the boxed vacuum equation $\big[-16\pi G\hbar^2 G_{ijkl}\,\delta^2/\delta h_{ij}\delta h_{kl}-\sqrt{h}\,{}^{(3)}\!R/16\pi G\big]\Psi=0$ .

Step 1. Perform the ADM split $\mathrm{d}s^2=-N^2\mathrm{d}t^2+h_{ij}(\mathrm{d}x^i+N^i\mathrm{d}t)(\mathrm{d}x^j+N^j\mathrm{d}t)$ , with lapse $N$ , shift $N^i$ , and extrinsic curvature $K_{ij}=\tfrac{1}{2N}(\dot h_{ij}-D_iN_j-D_jN_i)$ . Why: this separates the metric into the spatial data that will become dynamical ( $h_{ij}$ ) and the gauge data ( $N,N^i$ ) that will become Lagrange multipliers.

Step 2. Use Gauss-Codazzi-Mainardi to rewrite the 4-Ricci scalar and drop the boundary term, giving the ADM action $S=\tfrac{1}{16\pi G}\int\mathrm{d}t\,\mathrm{d}^3x\,N\sqrt{h}\,({}^{(3)}\!R+K_{ij}K^{ij}-K^2)$ . Why: this is the first appearance of the "kinetic" combination $K_{ij}K^{ij}-K^2$ , whose DeWitt structure will survive into the quantum theory as the supermetric.

Step 3. Compute the canonical momentum $\pi^{ij}=\partial(\sqrt{h}L)/\partial\dot h_{ij}=\tfrac{\sqrt{h}}{16\pi G}(K^{ij}-Kh^{ij})$ , and note that $\partial L/\partial\dot N=\partial L/\partial\dot N^i=0$ . Why: the lapse and shift have no conjugate momenta — this is the smoking gun of a constrained system, and it forces the Hamiltonian to be pure constraint.

Step 4. Legendre transform to $H=\int\mathrm{d}^3x\,(N\mathcal{H}+N^i\mathcal{H}_i)$ , a sum of constraints. Inverting the momentum relation gives the Hamiltonian constraint $\mathcal{H}=16\pi G\,G_{ijkl}\pi^{ij}\pi^{kl}-\tfrac{\sqrt{h}}{16\pi G}{}^{(3)}\!R\approx0$ . Why: here the supermetric $G_{ijkl}$ is born precisely as the object that turns $K_{ij}K^{ij}-K^2$ into a quadratic form in the momenta — and its trace term is what makes it indefinite.

Step 5. Quantize à la Dirac: $\hat\pi^{ij}=-i\hbar\,\delta/\delta h_{ij}$ , $\hat h_{ij}=h_{ij}\times$ , acting on wavefunctionals $\Psi[h_{ij}]$ , and impose $\hat{\mathcal{H}}\Psi=0$ . Why: the classical constraint $\approx 0$ becomes an operator that must annihilate physical states — and since $\mathcal{H}$ carried no $\dot h_{ij}$ that survived as an energy, nothing becomes $i\hbar\partial_t$ . Time has evaporated. *Verified: matches MTW §43 and DeWitt (1967); the supermetric and constraint structure agree with Wald Appendix E; the two functional derivatives at a coincident point are the origin of the $\delta(0)$ ambiguity flagged throughout.*

Go deeper · Derivation B · 1 minSignature of the DeWitt supermetric — why the equation is Klein-Gordon-like

Derivation BSignature of the DeWitt supermetric — why the equation is Klein-Gordon-like

The claim to establish is precise and checkable: the supermetric $G_{ijkl}$ , viewed as a metric on the six-dimensional vector space of symmetric $3\times3$ matrices $\delta h_{ij}$ at a point, has exactly one negative eigenvalue and five positive ones — Lorentzian signature $(-,+,+,+,+,+)$ — and the negative (timelike) direction is the overall volume, or conformal, mode. This single fact is what makes the Wheeler-DeWitt equation hyperbolic rather than elliptic, and hence a wave equation with an intrinsic clock rather than a diffusion equation. We prove it by splitting any perturbation into its trace and traceless parts.

Step 1. Contract the supermetric with a perturbation twice: $G_{ijkl}\,\delta h_{ij}\delta h_{kl}=\tfrac{1}{2\sqrt{h}}\big(2\,\delta h_{ij}\delta h^{ij}-(\delta h^i{}_i)^2\big)$ , where indices are raised with $h^{ij}$ . Why: this quadratic form is the "line element" on the space of 3-metrics; its sign on each direction is exactly the signature we want.

Step 2. Take a pure trace perturbation $\delta h_{ij}=\tfrac13 s\,h_{ij}$ (a local conformal rescaling, i.e. a change of volume), so $\delta h^i{}_i=s$ and $\delta h_{ij}\delta h^{ij}=\tfrac13 s^2$ . The form gives $\tfrac{1}{2\sqrt{h}}(\tfrac23 s^2-s^2)=-\tfrac{s^2}{6\sqrt{h}}<0$ . Why: the volume direction has negative norm — it is the one timelike direction, the intrinsic clock of quantum geometrodynamics.

Step 3. Take any traceless perturbation, $\delta h^i{}_i=0$ (a pure shape change). The form reduces to $\tfrac{1}{\sqrt{h}}\,\delta h_{ij}\delta h^{ij}>0$ . Since traceless symmetric $3\times3$ tensors form a five-dimensional space, these are the five spacelike directions. Why: shape deformations all have positive norm, completing the signature to $(-,+,+,+,+,+)$ .

Step 4. Conclude: the super-Hamiltonian $\propto G_{ijkl}\pi^{ij}\pi^{kl}+(\text{potential})$ is a wave operator with one timelike direction, so $\hat{\mathcal{H}}\Psi=0$ is a Klein-Gordon-type equation on superspace, not a Schrödinger equation. Why: this is the structural reason there is no positive-definite conserved probability and why the local volume can serve as time — the deep root of the problem of time. *Verified: one negative plus five positive eigenvalues; trace mode timelike; consistent with DeWitt (1967) and Wald Appendix E.*

Go deeper · Derivation C · 2 minMinisuperspace reduction — the de Sitter wavefunction

Derivation CMinisuperspace reduction — the de Sitter wavefunction

The full functional equation is intractable, so the goal here is to reduce it to something we can actually solve: freeze every degree of freedom except the scale factor $a$ of a closed, homogeneous, isotropic universe, and watch the Wheeler-DeWitt equation collapse to a one-dimensional ordinary differential equation of Airy type. We take $\mathrm{d}s^2=-N^2\mathrm{d}t^2+a(t)^2\mathrm{d}\Omega_3^2$ on $S^3$ in vacuum with $\Lambda>0$ , and carry the constants so that the tunnelling exponent comes out as a real number in the worked example. The target is the boxed ODE $\big[\mathrm{d}^2/\mathrm{d}a^2-(p/a)\mathrm{d}/\mathrm{d}a-\hbar^{-2}(a^2-\tfrac{\Lambda}{3}a^4)\big]\Psi(a)=0$ .

Step 1. Insert the ansatz into the ADM action and integrate over $S^3$ (volume $2\pi^2$ ), using ${}^{(3)}\!R=6/a^2$ for the unit 3-sphere. This gives $S=\tfrac{3\pi}{4G}\int\mathrm{d}t\,N\big(-\tfrac{a\dot a^2}{N^2}+a-\tfrac{\Lambda}{3}a^3\big)$ . Why: symmetry reduction turns a field theory into a mechanics problem — a single "particle" with coordinate $a$ — because all spatial gradients vanish by homogeneity.

Step 2. Read off the momentum $\pi_a=\partial L/\partial\dot a=-\tfrac{3\pi}{2G}\tfrac{a\dot a}{N}$ and vary $N$ to get the constraint $\mathcal{H}=-\tfrac{G}{3\pi}\tfrac{\pi_a^2}{a}-\tfrac{3\pi}{4G}\big(a-\tfrac{\Lambda}{3}a^3\big)=0$ . Why: this is the minisuperspace Hamiltonian constraint — a single particle of position-dependent mass $\propto a$ moving in a potential, with the constraint fixing its "energy" to zero.

Step 3. Quantize with $\pi_a\to-i\hbar\,a^{-p/2}\tfrac{\mathrm{d}}{\mathrm{d}a}a^{p/2}$ , where $p$ is a factor-ordering parameter, and impose $\hat{\mathcal{H}}\Psi=0$ . With $p=0$ and absorbing the constant $9\pi^2/4G^2$ into units, this is the boxed ODE with potential $U(a)=a^2-\tfrac{\Lambda}{3}a^4$ . Why: $p$ is an irreducible echo of the operator-ordering ambiguity that plagues the full theory — even here it cannot be uniquely fixed.

Step 4. Read the two regimes off $U$ : the turning point is at $a_*=\sqrt{3/\Lambda}$ (where $U=0$ ). For $a<a_*$ the potential is positive — classically forbidden — and $\Psi$ is exponential (tunnelling from nothing); for $a>a_*$ , $U<0$ and $\Psi$ oscillates (a classically expanding de Sitter universe). Why: this is the "creation of the universe from nothing" barrier — small universes are quantum-suppressed, large ones classical.

Step 5. Fix the state by a boundary condition. The Hartle-Hawking prescription selects the real solution that grows under the barrier; Vilenkin's tunnelling prescription selects the outgoing solution — physically inequivalent solutions of the same equation. Why: the Wheeler-DeWitt equation alone does not pick a state; a boundary condition on superspace does, which is where the physics of the no-boundary proposal enters. *Verified: reduced action and constraint reproduce Halliwell/Kiefer; ${}^{(3)}\!R=6/a^2$ and $S^3$ volume $2\pi^2$ standard; turning point $a_*=\sqrt{3/\Lambda}$ confirmed against the tunnelling calculation in the worked example.*

Limiting cases

The full functional equation is a formal object; it is understood only through controlled reductions, each of which trades rigour for tractability in a different way.

Case	Result	Meaning
Minisuperspace (FLRW + scalar)	$\big[\tfrac{\hbar^2}{2}\tfrac{1}{a^p}\partial_a(a^p\partial_a)-\tfrac{\hbar^2}{2a^3}\partial_\phi^2-U(a,\phi)\big]\Psi=0$	Freezing inhomogeneities reduces the PDE-on-superspace to a 2D wave-type equation; $p$ encodes ordering, $U$ the curvature-plus-matter potential.
WKB / semiclassical	$\Psi\sim e^{iS/\hbar}\Rightarrow G_{ijkl}\tfrac{\delta S}{\delta h_{ij}}\tfrac{\delta S}{\delta h_{kl}}+\sqrt{h}\,{}^{(3)}\!R=0$	Leading order recovers the Einstein-Hamilton-Jacobi equation; classical spacetime and a WKB notion of time emerge from the phase.
Loop quantum gravity	$\hat{\mathcal{H}}\,\Psi[A]=0$ on spin-network states	Same constraint in Ashtekar connection variables; Thiemann's regularization yields a formally UV-finite operator on a rigorous kinematical Hilbert space.
Loop quantum cosmology	difference equation in volume, $\hat{\mathcal{H}}\Psi_n=0$	The differential WDW operator becomes a discrete recursion; the big-bang singularity is replaced by a bounce.

Common trapreading Ĥ Ψ = 0 as ordinary energy conservation or expecting Schrödinger evolution

Do not read $\hat{\mathcal{H}}\Psi=0$ as "the energy is zero" in the everyday sense, and do not wait for an $i\hbar\,\partial_t\Psi=\hat H\Psi$ that never comes. There is no external $t$ in the fundamental equation: the constraint is the entire content, and "dynamics" lives in correlations among the arguments of $\Psi$ . Two further traps. First, the equation is formally divergent — two functional derivatives at the same spatial point produce an undefined $\delta(0)$ — and the ordering of $G_{ijkl}$ relative to the derivatives is ambiguous, so the full functional equation is a heuristic, rigorous only in minisuperspace or under the loop reorganization. Second, the timelike direction of the supermetric is the volume, not any matter clock; the "Klein-Gordon" analogy is geometric, and the associated conserved current is not a positive-definite probability, which is exactly why defining an inner product on solutions is an open problem.

Worked examples

1 · Tunnelling from nothing — the de Sitter no-boundary amplitude (fully worked)

For the closed de Sitter minisuperspace of derivation 3, with potential $U(a)=a^2-\tfrac{\Lambda}{3}a^4$ , locate the barrier and turning point, compute the WKB tunnelling exponent, and show the Hartle-Hawking probability equals the de Sitter horizon entropy.

Step 1. Locate the turning point where $U=0$ : $a^2=\tfrac{\Lambda}{3}a^4\Rightarrow a_*=\sqrt{3/\Lambda}$ . The barrier peak sits where $U'=2a-\tfrac{4\Lambda}{3}a^3=0$ , i.e. $a=\sqrt{3/(2\Lambda)}$ , with height $U_{\max}=\tfrac{3}{2\Lambda}-\tfrac{\Lambda}{3}\big(\tfrac{3}{2\Lambda}\big)^2=\mathbf{3/(4\Lambda)}$ . For the plotted case $\Lambda=3$ : $a_*=1$ , peak at $a=0.707$ of height $0.25$ — exactly the figure.

Step 2. The physical (constants-restored) equation is $\Psi''-\tfrac{9\pi^2}{4G^2\hbar^2}U(a)\,\Psi=0$ , so under the barrier the WKB solution is $\Psi\sim\exp\big(\pm\tfrac{3\pi}{2G\hbar}\!\int_0^{a_*}\!\sqrt{U}\,\mathrm{d}a\big)$ . Evaluate the integral with the substitution $u=1-\tfrac{\Lambda}{3}a^2$ : $\int_0^{a_*}\!a\sqrt{1-\tfrac{\Lambda}{3}a^2}\,\mathrm{d}a=\tfrac{3}{2\Lambda}\int_0^1\!\sqrt{u}\,\mathrm{d}u=\tfrac{3}{2\Lambda}\cdot\tfrac23=\mathbf{1/\Lambda}$ .

Step 3. Hence the tunnelling exponent is $B=\tfrac{3\pi}{2G\hbar}\cdot\tfrac{1}{\Lambda}=\dfrac{3\pi}{2G\hbar\Lambda}$ . The Hartle-Hawking state grows under the barrier, so its probability is $\lvert\Psi_{\mathrm{HH}}\rvert^2\propto e^{+2B}=\boxed{\exp\!\big(+3\pi/(G\hbar\Lambda)\big)}$ , while the Vilenkin tunnelling state carries the opposite sign, $e^{-2B}$ .

Step 4. Identify the exponent physically. The de Sitter horizon has Hubble rate $H^2=\Lambda/3$ , area $A=4\pi/H^2=12\pi/\Lambda$ , and Bekenstein-Hawking entropy $S_{\mathrm{dS}}=A/(4G\hbar)=3\pi/(G\hbar\Lambda)$ . Therefore $\lvert\Psi_{\mathrm{HH}}\rvert^2\propto e^{+S_{\mathrm{dS}}}$ : the no-boundary probability of creating a de Sitter universe is the exponential of its horizon entropy, and it is largest for small $\Lambda$ — the no-boundary proposal favours the emptiest, largest universes.

2 · The supermetric signature by hand

Verify the claim $(-,+,+,+,+,+)$ by evaluating the DeWitt line element on a concrete perturbation of the flat slice $h_{ij}=\delta_{ij}$ (so $\sqrt{h}=1$ ).

Step 1. Use $G_{ijkl}\delta h_{ij}\delta h_{kl}=\tfrac12\big(2\,\delta h_{ij}\delta h_{ij}-(\mathrm{tr}\,\delta h)^2\big)$ at $h_{ij}=\delta_{ij}$ . Take a pure dilation $\delta h_{ij}=s\,\delta_{ij}$ : then $\mathrm{tr}\,\delta h=3s$ and $\delta h_{ij}\delta h_{ij}=3s^2$ , giving $\tfrac12(6s^2-9s^2)=\mathbf{-\tfrac32 s^2<0}$ — the volume mode is timelike.

Step 2. Take a traceless shear $\delta h=\mathrm{diag}(s,-s,0)$ : then $\mathrm{tr}\,\delta h=0$ and $\delta h_{ij}\delta h_{ij}=2s^2$ , giving $\tfrac12(4s^2-0)=\mathbf{+2s^2>0}$ — a shape mode is spacelike. The five independent traceless symmetric matrices all give positive norm.

Step 3. Count: one negative (trace) plus five positive (traceless) directions on the six-dimensional space of symmetric $3\times3$ tensors. Signature $(-,+,+,+,+,+)$ confirmed — the equation is hyperbolic, and the local volume is the intrinsic time.

Physics of the portrait

The equation carries two names because it took two visions to make it. John Wheeler saw superspace and the timeless "wavefunction of the universe"; Bryce DeWitt did the hard canonical work that turned the vision into an operator equation.

Origins · 1 minBryce DeWitt and how this was discovered

Bryce DeWitt

1923-2004 · canonical quantization of gravity and the wavefunction of the universe

Bryce Seligman DeWitt spent a career insisting that gravity must be quantized like everything else, and paying the technical price to try. Trained at Harvard under Julian Schwinger, he wrote a 1949 thesis on the quantum theory of the gravitational field when almost no one thought the problem tractable. His landmark trilogy "Quantum Theory of Gravity" (Phys. Rev., 1967) laid out both the canonical theory — including the equation now bearing his name and John Wheeler's — and the covariant, background-field approach that seeded modern gauge-theory quantization and the DeWitt-Faddeev-Popov ghosts. The story of the equation is famously prickly: Wheeler pressed the idea of a wave equation on superspace during a 1965 airport layover in Raleigh-Durham, and DeWitt, who found Wheeler's "wavefunction of the universe" grandiose, called it simply "that damned equation." DeWitt also pioneered the effective action in curved spacetime, championed the Everett many-worlds interpretation when it was deeply unfashionable, and computed heat-kernel coefficients that still bear his name. A mountaineer and, late in life, a decorated soldier's sardonic wit intact, he embodied the stubborn conviction that the universe as a whole has a wavefunction — and that someone had better write down the equation it obeys.

History

1959

— Arnowitt, Deser and Misner cast general relativity in Hamiltonian form

1962

— Wheeler introduces superspace and the idea of quantum geometrodynamics

1965

— Wheeler and DeWitt sketch the wave equation on superspace during a Raleigh-Durham layover

1967

— DeWitt publishes "Quantum Theory of Gravity I", the canonical theory and the equation

1983

— Hartle and Hawking propose the no-boundary wavefunction of the universe

1986

— Ashtekar introduces new variables, opening the road to loop quantum gravity

Going further

Texts. Misner, Thorne & Wheeler, Gravitation (1973), §43 (geometrodynamics and superspace) · DeWitt, "Quantum Theory of Gravity. I. The Canonical Theory", Phys. Rev. 160, 1113 (1967) · Wald, General Relativity (1984), Ch. 10 and Appendix E · Hartle & Hawking, "Wave function of the universe", Phys. Rev. D 28, 2960 (1983) · Kuchař, "Time and interpretations of quantum gravity", Int. J. Mod. Phys. D 20 (2011 reprint) · Isham, "Canonical Quantum Gravity and the Problem of Time", gr-qc/9210011 · Kiefer, Quantum Gravity (3rd ed., 2012), Ch. 5.

Provenance. A verifier should: (1) re-derive the ADM Hamiltonian constraint $\mathcal{H}=16\pi G\,G_{ijkl}\pi^{ij}\pi^{kl}-\sqrt{h}\,{}^{(3)}\!R/16\pi G$ from MTW §21, §43 and Wald App. E; (2) confirm $G_{ijkl}=\tfrac{1}{2\sqrt h}(h_{ik}h_{jl}+h_{il}h_{jk}-h_{ij}h_{kl})$ and its $(-,+,+,+,+,+)$ signature via the trace/traceless split (worked example 2); (3) check the minisuperspace reduction using ${}^{(3)}\!R=6/a^2$ on the unit $S^3$ (volume $2\pi^2$ ), the potential $U(a)=a^2-\tfrac{\Lambda}{3}a^4$ , and the tunnelling integral $\int_0^{a_*}\sqrt{U}\,\mathrm{d}a=1/\Lambda$ giving $\lvert\Psi_{\mathrm{HH}}\rvert^2\propto e^{+3\pi/(G\hbar\Lambda)}=e^{+S_{\mathrm{dS}}}$ ; (4) verify the hypersurface-deformation algebra with field-dependent structure functions. All formulae are standard; the only numerical anchors ( $a_*=\sqrt{3/\Lambda}$ , barrier $3/4\Lambda$ , $S_{\mathrm{dS}}=3\pi/G\hbar\Lambda$ ) are re-derived in the examples.

Prerequisites. ADM 3+1 Decomposition of Spacetime · ADM Hamiltonian Formulation of General Relativity · Hamiltonian and Momentum Constraints · Gauss-Codazzi-Mainardi Equations. Leads to. Loop Quantum Gravity · Hartle-Hawking No-Boundary Proposal · Initial-Value (Cauchy) Problem in General Relativity.

Ask the tutor: Want me to walk through how the same $\mathcal{H}=0$ that started as ADM initial data becomes this timeless wave equation, or solve the de Sitter minisuperspace ODE and compare the Hartle-Hawking and Vilenkin boundary conditions line by line?

Why is there no time in the Wheeler-DeWitt equation

Because the Hamiltonian of general relativity is a sum of constraints that vanish on physical solutions. In an ordinary theory the Hamiltonian is a nonzero energy that generates evolution through $i\hbar\,\partial_t\Psi=\hat H\Psi$ . In GR, general covariance means that pushing data from one slice to the next is a gauge transformation, not a physical change, so the generator $\hat{\mathcal{H}}$ must annihilate physical states rather than evolve them. What remains, $\hat{\mathcal{H}}\Psi=0$ , has no external time parameter at all. Time is not fundamental here; it must be recovered internally, as a correlation — "the universe is this big when the clock-field reads that value" — which is precisely the content of the problem of time.

What is superspace, and how is it different from the superspace of supersymmetry

Wheeler's superspace is the space of all Riemannian 3-geometries: all spatial metrics $h_{ij}$ modulo diffeomorphisms, $\mathrm{Riem}(\Sigma)/\mathrm{Diff}(\Sigma)$ . Each point is an entire 3-geometry, and the wavefunction of the universe is a function on this infinite-dimensional manifold. It has nothing to do with the "superspace" of supersymmetry, which is ordinary spacetime augmented by anticommuting Grassmann coordinates. The names collide by historical accident. Wheeler's usage (1960s) predates the supersymmetric one and refers to the configuration space of geometrodynamics; when you freeze all but finitely many degrees of freedom you get minisuperspace, the tractable arena of quantum cosmology.

Why does the DeWitt supermetric have a minus sign, and what does the timelike direction mean

The supermetric inherits its indefinite signature from the gravitational kinetic term $K_{ij}K^{ij}-K^2$ : the $-K^2$ piece, coming from the trace of the extrinsic curvature, gives the volume mode a negative norm. Concretely, a perturbation that rescales the overall volume of a slice is "timelike" in superspace, while shape-changing (shear) perturbations are "spacelike" — five of them, since traceless symmetric $3\times3$ tensors form a five-dimensional space. The upshot is signature $(-,+,+,+,+,+)$ , which makes the Wheeler-DeWitt equation hyperbolic like a wave equation. Physically, the volume (or scale factor) is the closest thing the theory offers to a clock, which is why in cosmology the scale factor $a$ plays the role of intrinsic time.

Is the Wheeler-DeWitt equation actually well-defined

Not in its full functional form. It contains the product of two functional derivatives evaluated at the same spatial point, which formally produces an undefined $\delta(0)$ , and the ordering of the supermetric relative to those derivatives is ambiguous. Worse, there is no agreed inner product that would turn the space of solutions into a Hilbert space with a sensible probability interpretation — the natural Klein-Gordon-type current is not positive definite. These are not cosmetic issues; they are the reasons the equation is usually treated as a heuristic. It becomes genuinely well-posed only in symmetry-reduced minisuperspace (an ordinary ODE) or when reorganized in loop variables, where Thiemann's regularization yields a densely defined, UV-finite Hamiltonian constraint on a rigorous kinematical Hilbert space.

How does ordinary time-dependent quantum mechanics re-emerge if the fundamental equation is timeless

Through a Born-Oppenheimer / WKB expansion. Write $\Psi\approx C\,e^{iS_0[h]/\hbar}\chi[h,\phi]$ , with a rapidly oscillating gravitational phase $S_0$ (a solution of the Einstein-Hamilton-Jacobi equation, describing a classical spacetime) multiplying a slowly varying matter part $\chi$ . The gradient of $S_0$ defines a preferred flow through superspace — the classical trajectory — and the directional derivative along it is what we call WKB time $\tau$ . To leading order in $\hbar$ , $\chi$ then satisfies an ordinary time-dependent Schrödinger equation $i\hbar\,\partial_\tau\chi=\hat{\mathcal{H}}_{\mathrm{matter}}\chi$ . So ordinary quantum mechanics with a ticking clock is an approximation valid when the gravitational field is nearly classical — which is why we never notice the underlying timelessness.

How is the Wheeler-DeWitt equation related to loop quantum gravity and the Hartle-Hawking proposal

They are two ways of taking it seriously. Loop quantum gravity quantizes the very same Hamiltonian and momentum constraints, but in Ashtekar connection variables using holonomies and fluxes; this reorganization tames the ultraviolet problems and yields a discrete spectrum of geometry, with Thiemann's constraint operator playing the role of $\hat{\mathcal{H}}$ . The Hartle-Hawking no-boundary proposal instead supplies a boundary condition: the wavefunction is a Euclidean path integral over compact 4-geometries with a single boundary, $\Psi[h]=\int_{\partial g=h}\mathcal{D}g\,e^{-I_E[g]/\hbar}$ , which formally solves both constraints and singles out one solution. The equation says which states are allowed; loop gravity says how to make the operator rigorous; Hartle-Hawking says which allowed state is ours.