Wheeler–DeWitt Quantization and the Problem of Time

Canonical quantization promotes the Hamiltonian constraint to an operator equation, schematically $\widehat{\mathcal H}\Psi=0$. This does not by itself define a quantum theory: factor ordering, regulator, solution space, physical inner product, observables, and a choice of relational time remain.

Required background. Canonical Constraints, Dirac Observables, and Constraint Algebras supplies the classical system; Self-Adjointness, Extensions, and Unitary Evolution supplies operator domains.

Helpful background. Bilinear and Hermitian Forms, Adjoints, and Isometries supplies inner products; Physical Gauge Hilbert Spaces and Constraint Enforcement supplies constrained Hilbert spaces.

Functional equation and missing data

In metric variables the formal equation is

$$\left[ -16\pi G\,G_{abcd}(h) \frac{\delta^2}{\delta h_{ab}\delta h_{cd}} -\frac{\sqrt h}{16\pi G}\left({}^{(3)}R-2\Lambda\right) +\widehat H_{\rm matter} \right]\Psi[h,\varphi]=0.$$

Products of functional derivatives at one point require regularization, and the superspace metric $G_{abcd}$ creates factor-ordering ambiguities. The kinematical measure is not automatically the physical inner product. DeWitt’s original formulation already emphasizes these structural problems DeWitt 1967.

First application: parametrized nonrelativistic particle

Promote Newtonian time to a coordinate with constraint

$$C=p_t+\frac{p_x^2}{2m}=0.$$

Quantization on $L^2(dt\,dx)$ gives

$$\widehat C\Psi=0 \quad\Longleftrightarrow\quad i\partial_t\Psi(t,x) =-\frac1{2m}\partial_x^2\Psi(t,x).$$

Group averaging projects kinematical states:

$$\eta[\psi]=\int_{-\infty}^{\infty}d\alpha\, e^{i\alpha\widehat C}\psi, \qquad \langle\eta[\psi_1],\eta[\psi_2]\rangle_{\rm phys} =\langle\psi_1,\delta(\widehat C)\psi_2\rangle_{\rm kin}.$$

Conditioning on the clock $t=\tau$ recovers unitary Schrödinger evolution and the usual $L^2(dx)$ inner product. The model shows that a “frozen” constraint can encode evolution, but only because a globally good clock and self-adjoint Hamiltonian were supplied.

Semiclassical time and multiple choice

In a Born–Oppenheimer ansatz $\Psi[h,\varphi]=e^{iM_{\rm Pl}^2S_0[h]}\chi[h,\varphi]$, the leading phase solves a gravitational Hamilton–Jacobi equation and the next order can yield a Schrödinger equation for $\chi$ along that semiclassical trajectory. Backreaction, branch interference, and turning points limit this time.

Adversarial control: change clock or ordering

Use $x$ rather than $t$ as an internal clock. It is not monotone on reflected trajectories and can require a different positive-frequency split. In genuinely constrained systems, inequivalent clocks can produce inequivalent quantum theories. Change the ordering of $G_{abcd}\pi^{ab}\pi^{cd}$ and the kernel and conserved current change as well.

Thus formal solutions of the Wheeler–DeWitt equation have no probabilities until a domain and physical inner product are given. Clock-conditioned evolution is a construction to test, not a unique resolution of time.

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, Bryce S. “Quantum Theory of Gravity. I. The Canonical Theory.” Physical Review 160, 1113–1148 (1967). DOI.