Cosmological Wheeler–DeWitt Solutions

Cosmological Wheeler–DeWitt equations resemble wave equations on minisuperspace. Their oscillatory solutions can encode expanding and contracting semiclassical branches, while exponential solutions describe classically forbidden regions. This identification uses phase gradients, a declared internal clock, and a current; it is not read from the sign of the phase alone.

Helpful background. Flat-Spacetime Quantum Field Theory supplies wave equations and conserved currents; Einstein Equations supplies classical cosmological branches; Quantum Information supplies state interpretation; Quantum Gravity supplies evidential scope; Causality and Global Structure supplies singular-boundary interpretation; and Quantum Fields in Curved Spacetime supplies semiclassical matter evolution.

A massless-scalar model

For a flat homogeneous universe with a free scalar, set $\alpha=\ln(a/a_0)$ and $\varphi=\sqrt{4\pi G/3}\,\phi$, where $a_0$ is a fixed reference scale and $\phi$ is the canonically normalized scalar. With the flat ordering of the rescaled constraint used in Minisuperspace Quantization, the equation becomes

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

The general solution is

$$\Psi(\alpha,\varphi)=F(\alpha-\varphi)+G(\alpha+\varphi).$$

Plane waves may be written

$$\Psi_{k,\pm}= \exp\!\left[ik\alpha\mp i|k|\varphi\right].$$

Choosing $\varphi$ as time, the minus sign in $e^{-i\omega\varphi}$ defines positive frequency with $\omega=|k|$. The constraint then becomes a Schrödinger-like evolution only after restricting to that sector:

$$i\partial_\varphi\Psi= \sqrt{-\partial_\alpha^2}\,\Psi.$$

Here $\alpha\in\mathbb R$, and $\sqrt{-\partial_\alpha^2}$ is the positive self-adjoint Fourier multiplier $|k|$ on $L^2(\mathbb R,d\alpha)$, with domain $\{\psi:\int dk\,k^2|\widetilde\psi(k)|^2\lt\infty\}$. No boundary condition is inserted at finite $\alpha$; the zero-volume limit lies at $\alpha\to-\infty$. Plane waves are generalized eigenfunctions, so normalizable states are wave packets. Keeping both frequency signs returns the second-order constraint and its indefinite Klein–Gordon inner product.

Phase gradients and expanding branches

For a WKB state $\Psi=Ae^{iS}$, canonical momenta are $p_\alpha=\partial_\alpha S$ and $p_\varphi=\partial_\varphi S$. From the reduced classical action, the sign of $p_\alpha$ is opposite to the sign of $\dot\alpha$ for positive lapse. Consequently, in this convention $p_\alpha\lt0$ labels expansion and $p_\alpha\gt0$ contraction.

The plane wave $e^{ik\alpha-i|k|\varphi}$ has $p_\alpha=k$. Thus $k\lt0$ is the expanding positive-frequency branch and $k\gt0$ the contracting one. Changing the definition of $\alpha$, the lapse orientation, or the phase convention changes this label; the invariant procedure is to derive the momentum–velocity relation and then use the phase gradient.

For the minisuperspace metric $d s_{\mathrm{mini}}^2=d\varphi^2-d\alpha^2$, use the conserved current

$$J^A=\frac{i}{2}\Psi^* \overset{\leftrightarrow}{\partial^{A}}\Psi, \qquad \partial_AJ^A=0$$

This current checks the assignment: a positive-frequency plane wave has $J^\varphi=|k|\gt0$. The integrated Klein–Gordon form is positive on the positive-frequency sector, but its pointwise density need not stay positive for a superposition. A separate Schrödinger representation uses the conserved norm $\int d\alpha\,|\Psi|^2$; mapping between the two normalizations requires a frequency weight. A superposition of both frequency signs has an indefinite integrated Klein–Gordon norm, which is why the constraint alone does not define probabilities.

A turning-point model

Curvature, a cosmological constant, or a scalar potential produces a minisuperspace potential. After separation in the clock variable, a common form is

$$\left[-\frac{d^2}{d\alpha^2}+U(\alpha)\right]\psi_\omega(\alpha) =\omega^2\psi_\omega(\alpha).$$

Where $\omega^2\gt U$, WKB solutions oscillate:

$$\psi_\omega\approx \frac{1}{\sqrt{p(\alpha)}} \exp\!\left(\pm i\int^\alpha p(\alpha')\,d\alpha'\right), \qquad p=\sqrt{\omega^2-U}.$$

Where $\omega^2\lt U$, $p=i|p|$ and the branches grow or decay exponentially. A turning point requires Airy-function matching; selecting only a decaying Euclidean branch is a boundary condition, not a consequence of the differential equation. Kiefer and Sandhöfer survey these solution types and their interpretive limits Kiefer and Sandhöfer 2022, §§ 3–5.

Wave packets and semiclassical time

A normalizable packet in $k$ can be peaked on a classical relation $\alpha(\varphi)$. Its width grows according to the spectral dispersion; for the exactly linear $\omega=|k|$ branch there is no ordinary dispersion away from $k=0$, while a potential causes spreading and reflection. Interference between expanding and contracting packets can invalidate a single WKB time.

With additional matter modes $q$, a Born–Oppenheimer ansatz introduces a large gravitational mass parameter $M^2$ proportional to $G^{-1}$ in the chosen rescaling (its numerical normalization is convention dependent):

$$\Psi(\alpha,q)\approx A(\alpha)e^{iM^2S_0(\alpha)}\chi(\alpha,q)$$

lets the gravitational phase define a derivative along a classical trajectory. At the next order $\chi$ obeys an approximate Schrödinger equation in WKB time. The construction fails near turning points where $\partial_\alpha S_0=0$ or where branches overlap strongly.

Inner product and boundary flux

A physical solution space needs a domain, boundary condition, and inner product. DeWitt’s suggestion $\Psi\to0$ near a singular boundary can suppress some configurations but is neither uniquely required nor sufficient to make curvature observables finite. Reflecting, outgoing, tunneling, and no-boundary conditions assign different flux at minisuperspace boundaries. Probability claims must state which one is used.

An independent check integrates current conservation over a minisuperspace region: flux through its boundaries must balance. A mismatch indicates an incorrect WKB prefactor, factor ordering, or boundary condition.

The current test also exposes a common error: $|\Psi|^2$ is not generally the conserved density of a hyperbolic constraint. It can be used only after a clock choice and positive-frequency reduction justify a Schrödinger inner product. Near a turning point or branch-mixing region, that reduction must be rebuilt rather than extrapolated.

Exercise

For $\Psi=e^{-ik\alpha-i|k|\varphi}$ with $k\gt0$, determine the signs of $p_\alpha$ and $p_\varphi$, and classify the geometric branch using the convention above.

Solution

The phase is $S=-k\alpha-|k|\varphi$, so

$$p_\alpha=\partial_\alpha S=-k\lt0, \qquad p_\varphi=\partial_\varphi S=-|k|\lt0.$$

The factor $e^{-i|k|\varphi}$ is positive frequency in the stated clock convention. Because positive lapse gives $\dot\alpha$ opposite in sign to $p_\alpha$, $p_\alpha\lt0$ implies $\dot\alpha\gt0$: this is the expanding branch. The classification follows from the momentum–velocity relation, not simply from seeing a minus sign in the exponential.

References

• Kiefer, Claus, and Barbara Sandhöfer (2022), “Quantum Cosmology,” Zeitschrift für Naturforschung A 77, 543–559, arXiv:0804.0672v3.