Wheeler–DeWitt Cosmology: Boundary Conditions, Inner Products, and Probabilities

The Wheeler–DeWitt equation is a constraint, not an external-time Schrödinger equation. Boundary conditions select solutions, while factor ordering, self-adjoint domain, physical inner product, and a clock determine whether those solutions support probabilities. A conserved superspace flux is not positive on the full solution space.

Required background. Minisuperspace Reductions and Approximation Control supplies the reduced model. Wheeler–DeWitt Quantization and the Problem of Time supplies the general constraint.

Helpful background. Bilinear and Hermitian Forms, Adjoints, and Isometries and Self-Adjointness, Extensions, and Unitary Evolution supply mathematical controls.

WKB branches and superspace current

Let $\alpha=\log a$ and use a massless scalar clock. In a simple region,

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

For $U=0$,

$$\Psi_{\omega,\pm} =e^{-i\omega\phi\pm i\omega\alpha}.$$

The Klein–Gordon-like current

$$j^A=\frac{i}{2}\Psi^* \overleftrightarrow{\partial^A}\Psi$$

is conserved, but its flux changes sign between positive- and negative-frequency branches.

Application: compare three inner products

Flux through a constant-$\phi$ surface distinguishes expanding and contracting WKB branches but is indefinite on their unrestricted superposition. Selecting $\omega>0$ gives a positive-frequency inner product

$$(\Psi_1,\Psi_2)_{\mathrm{KG}} =i\int d\alpha\, \Psi_1^*\overleftrightarrow{\partial_\phi}\Psi_2.$$

Group averaging the constraint can induce the same sector with a measure determined by the spectral resolution. Alternatively, deparametrize using $\phi$ and use the Schrödinger inner product at $\phi_0$. These agree only when the operator is self-adjoint, frequency selection is stable, and $\phi$ is a good clock.

At $a=0$ or another endpoint, boundary conditions select a self-adjoint extension. DeWitt’s condition $\Psi(0)=0$ is one proposal; it does not by itself prove zero probability of a physical singularity DeWitt 1967.

Boundary and measure stress tests

Change factor ordering, which changes both differential operator and natural measure. Classify self-adjoint extensions and propagate each. Mix positive and negative frequencies or move through a clock turning point; positivity can fail. A normalized answer must state which branch and measure were selected.

Scope

The equation organizes constrained wavefunctions and semiclassical branches, but does not choose a unique cosmological state. Competing path-integral boundary prescriptions are treated on No-Boundary and Tunneling Wavefunction Proposals.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• DeWitt, B. S. “Quantum Theory of Gravity. I. The Canonical Theory.” Physical Review 160 (1967): 1113–1148. DOI.
• Higuchi, A. “Quantum Linearization Instabilities of de Sitter Spacetime. I.” Classical and Quantum Gravity 8 (1991): 1961–1981. DOI.