The Wheeler–DeWitt Equation

The Wheeler–DeWitt equation is the quantum Hamiltonian constraint: a wave functional of spatial geometry and matter is physical only if the constraint operator annihilates it. That sentence is compact, but it does not by itself define an operator, its domain, a physical inner product, or evolution. The equation is therefore best read as a structured quantization problem rather than as a single settled differential equation.

Required background. Variational and Symplectic Geometry supplies constrained phase space, canonical generators, and the distinction between gauge directions and physical variations.

Helpful background. Flat-Spacetime Quantum Field Theory supplies functional Schrödinger quantization; Einstein Equations supplies the classical limit; Quantum Information supplies states and inner products; Formulations of General Relativity supplies ADM variables; and Quantum Gravity supplies the broader regime distinctions.

The two quantum constraint equations

In metric variables a kinematical state is a functional $\Psi[h_{ij},\phi]$, where $\phi$ denotes matter. Replacing $\pi^{ij}$ by $-i\delta/\delta h_{ij}$ is formal because functional derivatives at the same point must be regulated. With

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

one representative ordering of the Hamiltonian constraint is

$$\widehat{\mathcal H}_\perp\Psi= \left[-16\pi G\,G_{ijkl}\frac{\delta^2}{\delta h_{ij}\delta h_{kl}} -\frac{\sqrt h}{16\pi G}\left({}^{(3)}R-2\Lambda\right) +\widehat{\mathcal H}_{\perp}^{\mathrm m}\right]\Psi=0.$$

Physical states must also satisfy the momentum, or spatial-diffeomorphism, constraint

$$\widehat{\mathcal H}_i\Psi= \left[2i h_{ik}D_j\frac{\delta}{\delta h_{kj}}+\widehat{\mathcal H}^{\mathrm m}_i\right]\Psi=0.$$

Here the functional derivative transforms as the contravariant momentum density, and $D_j$ acts on that density. The factor $h_{ik}$ lowers the remaining spatial index, as in the classical constraint $-2h_{ik}D_j\pi^{kj}$. The Hamiltonian constraint governs normal deformations of a slice; the momentum constraint governs tangential ones. Solving only the first does not make a state invariant under spatial coordinate changes. DeWitt’s original superspace construction and the signature of $G_{ijkl}$ are given in DeWitt 1967, §§4–7.

Ordering, regularization, and operator domain

The displayed formula hides three independent choices. First, $G_{ijkl}(h)$ and the functional derivatives do not commute, so a Laplace–Beltrami ordering on superspace differs from placing the supermetric to the left. Second, the coincident product $\delta^2/\delta h\delta h$ needs a regulator and renormalization prescription. Third, an unbounded operator is not defined until its domain and boundary conditions are supplied. These choices can affect constraint closure and the candidate inner product.

A finite-dimensional analogy makes the point. On $q\gt0$, the classical expression $q p^2$ can become $-q\partial_q^2$, $-\partial_q q\partial_q$, or a one-parameter family related to a chosen measure. They share the same principal symbol but differ at $q=0$ and can have different self-adjoint extensions. In full gravity, the analogous “boundary” lies in the stratified space of three-geometries, where metrics with symmetries and degenerate limits require care.

Regularization must also preserve the relation between the Hamiltonian and momentum constraints. A regulator that produces an anomalous commutator may change the physical state space even if each separate equation has many formal solutions. These limitations are emphasized in the canonical and minisuperspace survey Kiefer and Sandhöfer 2022, §§2–4.

Why the equation looks timeless

There is no external parameter $t$ in $\widehat{\mathcal H}_\perp\Psi=0$ because changing the embedding of a closed spatial slice is gauge. That does not imply that all gauge-invariant correlations are constant. A matter variable or geometric degree of freedom may serve as an internal clock, or a WKB gravitational branch may define approximate time for lighter fields. Both moves add assumptions that are absent from the bare constraint.

The DeWitt supermetric has one negative direction per spatial point, associated locally with conformal rescaling. In a symmetry-reduced model this often makes the Wheeler–DeWitt equation hyperbolic in a scale variable. One can then define a Klein–Gordon-like conserved current, but it is generally indefinite. Choosing a positive-frequency sector may work in a regime with an approximate time-translation symmetry; it is not a universal physical-inner-product prescription.

A minisuperspace control problem

For a flat FLRW truncation with no cosmological constant and a free homogeneous minimally coupled scalar $\varphi$, set $\alpha=\ln(a/a_0)$ for a fixed reference scale $a_0$ and define the dimensionless scalar $\phi=\sqrt{4\pi G/3}\,\varphi$. On a fixed comoving cell, multiplying the classical constraint by a positive factor gives $-p_\alpha^2+p_\phi^2=0$. With the constant-coefficient factor ordering, its quantum equation can be multiplied by $-1$ and written

$$\left(-\partial_\alpha^2+\partial_\phi^2\right)\Psi(\alpha,\phi)=0.$$

The general solution is $F(\phi+\alpha)+G(\phi-\alpha)$. This example exposes the hyperbolic structure and allows $\phi$ to be used locally as a clock. It does not establish the full equation: minisuperspace removes inhomogeneous gravitational and matter modes before quantization, and its ordering and inner product remain model choices. String or holographic constructions and semiclassical thermodynamics are separate topics; the present scope is canonical general relativity and its controlled reductions.

An independent classical-limit check inserts $\Psi=A\exp(iS)$ and keeps the leading term in a rapidly varying phase. The result is the Hamilton–Jacobi constraint for $S[h]$; the next order transports $A$ and eventually yields approximate matter evolution. Failure to recover the Hamilton–Jacobi equation signals a sign, normalization, or ordering mistake in the principal part.

Exercises

For the minisuperspace equation above, verify that $\Psi_k=e^{ik(\phi-\alpha)}$ is a solution. Compute $J_A=(i/2)\Psi^*\overleftrightarrow{\partial_A}\Psi$ for coordinates $(\alpha,\phi)$ and the conserved current $J^A=\eta^{AB}J_B$, where $\eta^{AB}=\operatorname{diag}(-1,1)$. Explain why selecting $k\gt0$ is an extra choice rather than a consequence of the constraint.

Solution

Both second derivatives give $-k^2\Psi_k$, so $(-\partial_\alpha^2+\partial_\phi^2)\Psi_k=0$. The lower-index components are $J_\phi=-k\lvert\Psi_k\rvert^2$ and $J_\alpha=k\lvert\Psi_k\rvert^2$. Raising with the superspace metric gives $J^\alpha=J^\phi=-k\lvert\Psi_k\rvert^2$. It is $\partial_AJ^A=0$ that follows from the wave equation for general solutions. Complex conjugation reverses the current and gives an equally valid solution in the opposite frequency sector. In the global convention $e^{-i\omega\phi}$, the displayed mode has $\omega=-k$, so $k\gt0$ is not positive frequency with respect to increasing $\phi$. The constraint alone supplies a conserved indefinite flux, not a preferred positive norm. Declaring $\phi$ to be a clock and choosing one sector requires a branch and orientation choice.

References

• DeWitt, Bryce S. “Quantum Theory of Gravity. I. The Canonical Theory.” Physical Review 160 (1967): 1113–1148. DOI.
• Kiefer, Claus, and Barbara Sandhöfer. “Quantum Cosmology.” Zeitschrift für Naturforschung A 77 (2022): 543–559. DOI. Open manuscript, v3.