Curved-Spacetime QFT and Semiclassical Validity

Quantum field theory on a curved metric and semiclassical gravity are controlled by several independent hierarchies. Low curvature alone is not a validity test: the state may contain short proper wavelengths, the mean stress may backreact strongly, or connected stress fluctuations may produce a large metric response. The practical question is whether a specified observable is insensitive to every first omitted effect at the resolution and duration being claimed.

Helpful background. Differential equations for relativity supplies the response and stability language used below. Quantum Fields in Curved Spacetime supplies admissible states and renormalized observables, while Semiclassical Gravity supplies the coupled mean and fluctuation hierarchies.

Two independent axes of control

The fixed-background axis asks whether the quantum field problem is defined on the prescribed geometry. A globally hyperbolic field problem, an admissible Hadamard or microlocal state, and locally covariant renormalized observables are parts of that specification. Local covariance organizes comparisons between spacetimes; it is a framework, not a higher order in one approximation series. The relevant Library routes are Hadamard structure and quantum states, locally covariant QFT, and the renormalized stress–energy tensor.

The response axis asks how much of the metric dynamics is retained. A test-field calculation neglects the field’s gravitational response. Semiclassical gravity instead solves a renormalized mean equation schematically of the form

$$G_{\mu\nu}+\Lambda g_{\mu\nu} +\alpha H^{(1)}_{\mu\nu}+\beta H^{(2)}_{\mu\nu} =8\pi G\,\langle T_{\mu\nu}\rangle_{\mathrm{ren}},$$

with the finite local curvature couplings fixed as part of the theory. Stochastic gravity also retains the connected stress fluctuations and their retarded propagation into metric correlations. These mean, fluctuation, and response tests probe different omissions, as emphasized by Hu and Verdaguer 2008, §§2–3 and §§4.1–5.2.

A validity vector rather than one threshold

Let $L_R$ be the shortest relevant curvature or background-variation scale, $\Lambda_{\mathrm{EFT}}$ the proper cutoff, and $\omega_{\mathrm{prop}}$ a proper frequency present in the state or observable. A minimal local hierarchy contains

$$\epsilon_R=(\Lambda_{\mathrm{EFT}}L_R)^{-2}\ll1, \qquad \epsilon_\omega=\frac{\omega_{\mathrm{prop}}}{\Lambda_{\mathrm{EFT}}}\ll1, \qquad \epsilon_{\mathrm{ad}}= \left\lvert\frac{\dot\omega_{\mathrm{prop}}} {\omega_{\mathrm{prop}}^2}\right\rvert\ll1$$

when an adiabatic expansion is actually used. None implies either of the others. The full check is better recorded as a vector:

Question	Controlled quantity	Independent check or failure signal
Is fixed-background QFT adequate?	Curvature and proper frequencies below the cutoff for the chosen state and smearing scale	Change resolution or compare with the first omitted local operator
Is mean backreaction perturbative?	The metric change induced by $8\pi G\langle T_{\mu\nu}\rangle_{\rm ren}$ in a declared gauge-invariant observable	Solve or linearize the coupled mean equation rather than compare one stress component with one curvature component
Are fluctuations negligible?	A smeared connected stress correlator after retarded propagation into the observable	Vary the sampling scale and compare rms response with the stated tolerance
Is a large-$N$ estimate controlled?	Mean and connected correlators with the species correlations stated	Recover $N^{-1/2}$ rms suppression only for independent or sufficiently weakly correlated species
Is the prediction uniform in time?	Accumulated phase, response, and secular terms over the claimed interval	Shorten the interval, resum, or compare with direct evolution

Pointwise stress variance is generally too singular and too resolution dependent to serve as this test. The sampling function, state, boundary conditions, observable, and retarded response belong in the same comparison. Likewise, a small mean source does not entail a small connected fluctuation. The large-$N$ and stochastic hierarchy, including this distinction, is developed in Hu and Verdaguer 2008, §§3.1–3.3.1.

Causal response and the stopping rule

Linear response supplies a sharper mean-field test: perturb the state and metric, use the retarded stress-tensor response, enforce the Ward identities, and inspect gauge-invariant solutions within the low-energy domain. Growing solutions or cutoff-sensitive modes invalidate the claimed mean solution; the flat-space result does not automatically transfer to another geometry or state. This criterion and its Minkowski example are given in Anderson, Molina-París, and Mottola 2003, §§II–V.

The approximation stops when any required component of the validity vector is order unity, when a causal response is unstable below the cutoff, or when the state and renormalization data do not define the observable being claimed. Passing a curvature test cannot compensate for failing a frequency, fluctuation, response, or secular test. The full derivation and a six-step workflow live in Semiclassical Regime, Expansion, and Validity; this card is the compact decision surface.

References

• Anderson, Paul R., Carmen Molina-París, and Emil Mottola. 2003. “Linear Response, Validity of Semi-Classical Gravity, and the Stability of Flat Space.” Physical Review D 67, 024026. DOI. Open preprint.
• Hu, Bei-Lok, and Enric Verdaguer. 2008. “Stochastic Gravity: Theory and Applications.” Living Reviews in Relativity 11, 3. DOI. Open preprint.