Quantum Geometrodynamics Beyond Minisuperspace

Beyond minisuperspace, the Wheeler–DeWitt state is a functional of three-geometry and matter. The Born–Oppenheimer/WKB expansion of Kiefer and Singh 1991 recovers a Schrödinger equation with suppressed quantum-gravitational corrections in a specified semiclassical regime. Regularization, constraint closure, inner products, backreaction, and mode convergence remain essential before that controlled expansion can be promoted further.

Required background. Wheeler–DeWitt Cosmology: Boundary Conditions, Inner Products, and Probabilities supplies constraint interpretation. BKL, Mixmaster, and Inhomogeneous Singularities supplies the omitted target.

Helpful background. 2PI Effective Actions and Conserving Truncations and Convergence, Extrapolation, and Error Certification supply backreaction and truncation methods.

Functional constraint and WKB expansion

Schematically,

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

Use

$$\Psi[h,\varphi] =e^{iM_{\mathrm P}^2S_0[h]/\hbar}\, \psi[h,\varphi].$$

The leading $M_{\mathrm P}^2$ equation is gravitational Hamilton–Jacobi. Along one WKB branch it defines emergent time, and the next order gives

$$i\hbar\frac{\delta\psi}{\delta t_{\mathrm{WKB}}} =\widehat H_{\mathrm m}\psi +O(M_{\mathrm P}^{-2}).$$

Application: one inhomogeneous mode

For a perturbation mode $v_k$,

$$\widehat H_k=\frac12 \left[-\hbar^2\partial_{v_k}^2+\omega_k^2(t)v_k^2\right].$$

Solve its time-dependent Schrödinger equation on the WKB background, then retain the next Born–Oppenheimer term, which schematically contains $\widehat H_k^2/(M_{\mathrm P}^2H_{\mathrm{bg}})$. Compute its correction to the Gaussian width and power spectrum. Increase the number of modes and include their expectation stress in the background Hamilton–Jacobi equation.

Constraint and convergence tests

Regulate coincident functional derivatives and verify the Hamiltonian and momentum constraints close without an anomaly at the retained order. Change WKB branch, mode cutoff, and factor ordering. Monitor norm in the physical inner product and energy transferred to omitted modes. A finite-mode numerical solution is evidence only after convergence.

Scope

This expansion derives QFT on a semiclassical spacetime and controlled corrections where WKB time exists. It fails near branch interference, turning points, or large backreaction. Observable initial-state consequences are treated on Initial-State and Trans-Planckian Interfaces.

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

• Kiefer, C., and T. P. Singh. “Quantum Gravitational Corrections to the Functional Schrödinger Equation.” Physical Review D 44 (1991): 1067–1076. DOI.