Semiclassical WKB Time and Recovery of QFT

Quantum field theory on a classical spacetime can emerge from a timeless gravitational constraint when the geometry occupies a narrow WKB branch and changes much more slowly than the matter state. A Born–Oppenheimer expansion then turns the phase of the gravitational wave functional into a time derivative for light fields. The result is approximate, branch-dependent, and controlled only while backreaction, branch interference, and higher-order quantum-gravity terms remain small.

Required background. Variational and Symplectic Geometry supplies Hamilton–Jacobi theory, constrained phase space, and canonical flows.

Helpful background. Flat-Spacetime Quantum Field Theory supplies functional Schrödinger evolution; Einstein Equations supplies the classical background limit; Quantum Information supplies entanglement and reduced states; Formulations of General Relativity supplies canonical variables; and Quantum Gravity supplies expansion-regime conventions.

Heavy geometry and light matter

Write a regulated Wheeler–DeWitt equation schematically on superspace coordinates $q^A$ as

$$\left[-\frac{1}{2M}\nabla_G^2+M V(q)+\widehat H_{\mathrm{m}}(q,\varphi)\right]\Psi(q,\varphi)=0,$$

where $G^{AB}$ is the indefinite superspace metric, $M$ is proportional to $G^{-1}$, $V$ contains spatial curvature and the cosmological term, and $\varphi$ denotes light matter modes. The regulator, factor ordering, superspace measure, and boundary conditions are part of the definition. The Born–Oppenheimer hierarchy treats $M$ as large relative to the matter scales under study.

Use

$$\Psi(q,\varphi)=e^{iMS_0(q)}A(q)\chi(q,\varphi).$$

Terms of order $M$ give the gravitational Hamilton–Jacobi equation

$$\frac12G^{AB}\partial_A S_0\partial_B S_0+V=0.$$

Its integral curves are classical spacetime histories. This leading equation is the first independent check: a wrong sign in the DeWitt metric or gravitational potential fails to recover the classical constraint.

Time from a Hamilton–Jacobi branch

Along one Hamilton–Jacobi branch define

$$\frac{\partial}{\partial t_{\mathrm{WKB}}} =G^{AB}\partial_A S_0\nabla_B.$$

At the next order, derivatives of $A$ can be chosen to satisfy the gravitational transport equation. After removing the corresponding geometric phase, the remaining terms yield

$$i\frac{\partial\chi}{\partial t_{\mathrm{WKB}}} =\widehat H_{\mathrm{m}}(q(t),\varphi)\chi.$$

This is the functional Schrödinger equation for matter on the classical geometry selected by $S_0$. Kiefer and Sandhöfer show this hierarchy and its cosmological specialization in Kiefer and Sandhöfer 2022, §5, Eqs. (13)–(14). The time orientation changes under $S_0\mapsto-S_0$. In a reduced cosmological model these conjugate branches can describe expansion and contraction. In the full field theory, a lapse and foliation must also be specified: the shorthand time derivative represents evolution along that chosen classical history, not a preferred time shared by every geometry.

The derivation uses more than a slowly varying scale factor. The phase must vary rapidly compared with $A$ and $\chi$; the matter energy must not destroy the hierarchy; and the state must be concentrated around a family of classical geometries. Decoherence between WKB branches can justify treating one branch autonomously, but tracing out an environment introduces approximation and does not turn the branch into a fundamental external time.

Quantum-gravitational correction

The first omitted term can be identified without guessing a universal correction Hamiltonian. With the Laplace–Beltrami ordering displayed above and $A$ obeying the transport equation, direct substitution gives

$$i\partial_{t_{\mathrm{WKB}}}\chi =\widehat H_{\mathrm m}\chi -\frac{1}{2MA}\nabla_G^2(A\chi).$$

The final term is suppressed by $M^{-1}$ only if derivatives of $A\chi$ remain bounded on the scales of the expansion. It contains derivatives in geometry space and is not yet an operator acting on matter alone. To rewrite it as an effective matter correction, one must use the leading evolution equation, choose a branch and phase convention, and treat matter backreaction consistently. This explains both the origin and the limits of terms sometimes proportional to $H_{\mathrm m}^2$. Kiefer and Singh derive such corrections and discuss norm conservation in Kiefer and Singh 1991, pp. 1069–1072.

An apparent non-Hermitian term can sometimes be shifted by redefining the WKB prefactor or including backreaction consistently. It cannot simply be discarded: one must state the inner product and check probability conservation to the retained order. A dimension check also matters. Since $M$ carries inverse gravitational coupling, every correction must vanish in the controlled $G\to0$ or large-$M$ limit at fixed low-energy matter data.

Where emergent time fails

In a one-coordinate WKB problem, a turning point has $S_0'=0$, and the time vector degenerates. In several dimensions, a turning point of one chosen coordinate need not stop the entire Hamilton–Jacobi flow, but caustics or a failed clock coordinate can still invalidate the branch expansion. If expanding and contracting branches interfere, there is no single phase from which to define time. Strong matter backreaction invalidates the heavy–light separation, and broad geometrical wave packets do not define one classical background. The derivation therefore recovers quantum field theory only in a semiclassical region of state space.

Nor does this derivation reproduce the ultraviolet definition of flat-space QFT, semiclassical black-hole thermodynamics, or a string or holographic completion; those belong to their dedicated treatments. It explains the correspondence limit that a canonical quantum state must possess if conventional QFT on curved spacetime is to emerge.

Exercises

For one heavy coordinate $Q$, insert $\Psi=e^{iMS_0(Q)}A(Q)\chi(Q,x)$ into

$$\left[-\frac{1}{2M}\partial_Q^2+MV(Q)+\widehat H_x(Q)\right]\Psi=0.$$

Keep the leading two orders and identify the WKB time. Why does the construction fail where $S_0'(Q)=0$?

Solution

The order-$M$ terms give $(S_0')^2/2+V=0$. At order one, choose $A$ so that $(A^2S_0')'=0$; the remaining cross term is $-iS_0'\partial_Q\chi+\widehat H_x\chi=0$. Defining $\partial_{t_{\mathrm{WKB}}}=S_0'\partial_Q$ gives $i\partial_{t_{\mathrm{WKB}}}\chi=\widehat H_x\chi$, consistently with the superspace definition above on the selected branch. At $S_0'=0$, the map from $Q$ derivatives to time derivatives is singular, the WKB wavelength is no longer short, and neglected second derivatives of $A\chi$ compete with the retained terms. A connection formula or a different clock patch is then required.

References

• Kiefer, Claus, and Barbara Sandhöfer. “Quantum Cosmology.” Zeitschrift für Naturforschung A 77 (2022): 543–559. DOI. Open manuscript, v3.
• Kiefer, Claus, and T. P. Singh. “Quantum Gravitational Corrections to the Functional Schrödinger Equation.” Physical Review D 44 (1991): 1067–1076. DOI.