Quantum-Cosmology Observables and the Problem of Time

A quantum-cosmology prediction requires solutions of all constraints, a positive physical inner product, relational observables, a clock and its validity range, and records or decoherence that support an operational interpretation. The formal Wheeler–DeWitt wavefunction $\Psi$ is not itself a probability density on configuration space.

Required background. Wheeler–DeWitt Quantization and the Problem of Time supplies constraint quantization. Relational and Gauge-Invariant Gravitational Observables supplies the observable standard.

Helpful background. Constraint Solving and the Observable Dictionary and Relativistic Protocols, Clocks, Encoding, and Channel Tomography supply gauge and protocol checks.

Relational time in a constrained system

For a deparametrizable constraint

$$C=p_T+H(q,p)=0,$$

quantization gives

$$i\frac{\partial}{\partial T}\Psi(q,T)=\widehat H\Psi(q,T).$$

If $\widehat H$ is self-adjoint, the physical inner product

$$\langle\Psi_1|\Psi_2\rangle_{\mathrm{phys}} =\int dq\,\Psi_1^*(q,T_0)\Psi_2(q,T_0)$$

is independent of $T_0$. The conditional probability for $q\in\Delta$ when the clock reads $\tau$ is the Born probability on that slice. This construction works only while $T$ is monotonic and its quantum fluctuations permit a clock interpretation.

Application: a scalar clock and scale factor

In a flat massless-scalar minisuperspace, a common form is

$$\partial_\phi^2\Psi(a,\phi)=-\Theta\Psi(a,\phi),$$

with positive self-adjoint $\Theta$ after domain and factor-ordering choices. Select the positive-frequency sector

$$-i\partial_\phi\Psi=\sqrt{\Theta}\Psi$$

and normalize with its conserved physical inner product. For projector $P_\Delta$ onto $a\in\Delta$,

$$\Pr(a\in\Delta\mid\phi=\phi_0) =\frac{\langle\Psi(\phi_0)|P_\Delta|\Psi(\phi_0)\rangle_{\mathrm{phys}}} {\langle\Psi|\Psi\rangle_{\mathrm{phys}}}.$$

The clock, frequency sector, operator domain, and measure are all part of the result. Page and Wootters give a general conditional-time framework Page and Wootters 1983.

Clock and observable stress tests

Repeat with another monotonic clock. Predictions for genuine Dirac observables should be related, while approximate conditional observables can differ. Test turning points where $\dot T=0$, broad clock states, and multiple-choice quantizations. A conserved indefinite current cannot be used as probability until a positive-frequency or group-averaged sector is justified.

Scope

Relational evolution can solve the frozen-formalism problem in deparametrizable models, not every global problem of time. The next page derives what is lost before quantization in Minisuperspace Reductions and Approximation Control.

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

• Page, D. N., and W. K. Wootters. “Evolution without Evolution: Dynamics Described by Stationary Observables.” Physical Review D 27 (1983): 2885–2892. DOI.
• Rovelli, C. “Time in Quantum Gravity: An Hypothesis.” Physical Review D 43 (1991): 442–456. DOI.