Infrared, Semiclassical, and Microscopic Regimes

“Quantum gravity” can mean several inequivalent calculations. The decisive question is which degrees of freedom are quantized, which are retained as classical backgrounds, and which feedback terms are kept. This page compares the approximations by their state space, equations, and control parameters rather than by an informal distance scale.

Required background. Einstein Equations supplies the classical geometric response to stress-energy; Quantum Mechanics supplies quantum states, expectation values, and fluctuations.

Helpful background. Flat-Spacetime Quantum Field Theory supplies Fock-space and loop expansions; Quantum Information supplies the distinction between a state, a reduced state, a channel, and a measured correlation.

Degrees of freedom define the regime

Let $g_{\mu\nu}$ denote the spacetime metric and $\phi$ a matter field. The comparison axes are the status of geometry, the status of matter, and the feedback retained. The following hierarchy records what is dynamical and what is quantum.

Regime	Geometry	Matter	Feedback retained
Classical GR	Classical dynamical $g_{\mu\nu}$	Classical fields or fluids	Classical $T_{\mu\nu}$ sources the metric
QFT on curved spacetime	Prescribed classical $g_{\mu\nu}$	Quantum field $\hat\phi$	None; particle production does not change the chosen metric
Semiclassical gravity	Classical dynamical $g_{\mu\nu}$	Quantum state of $\hat\phi$	Renormalized expectation value $\langle\hat T_{\mu\nu}\rangle$
Stochastic gravity	Classical mean metric plus stochastic perturbations	Quantum matter	Stress-tensor noise kernel drives metric fluctuations
Gravitational EFT	Classical background plus quantized $h_{\mu\nu}$	Quantum matter and gravitons	Perturbative loops and higher-dimension operators
Microscopic quantum gravity	Candidate quantum geometry or pregeometry	Model-dependent	Must recover the preceding regimes in a controlled limit

The table is not a chronology. Several descriptions can overlap for the same physical system. A coherent graviton state, for example, may be described classically at leading order and quantum mechanically when fluctuations or scattering amplitudes are probed.

Fixed background and semiclassical backreaction

In QFT on curved spacetime, the metric is input data. Matter obeys an operator equation such as

$$(\Box_g-m^2-\xi R)\hat\phi=0,$$

and its state determines correlation functions, but there is no equation updating $g_{\mu\nu}$. This is the appropriate regime for field propagation and particle creation when backreaction is negligible. Those subjects are treated fully in the curved-spacetime QFT volume rather than duplicated here.

At leading curvature order, semiclassical gravity adds the mean response,

$$G_{\mu\nu}[g]+\Lambda g_{\mu\nu} =8\pi G\,\langle\Psi\lvert\hat T_{\mu\nu}^{\mathrm{ren}}[g]\rvert\Psi\rangle.$$

Renormalization prescription, quantum state, and boundary conditions are part of the equation. The full renormalized equation also contains the variations of local curvature-squared terms with measured or matched coefficients; the display suppresses them only in a regime where they are perturbative. Small relevant curvature components compared with the operator suppression scales help control a derivative expansion but are not sufficient: large stress-tensor fluctuations or an unstable metric response can make a mean source unreliable. The smeared fluctuation diagnostic must be tied to a specified measurement and response; dividing by a vanishing mean is not a validity test.

Quantized weak geometry as an effective theory

Gravitational EFT chooses a smooth background $\bar g_{\mu\nu}$ and expands

$$g_{\mu\nu}=\bar g_{\mu\nu}+\kappa h_{\mu\nu}, \qquad \kappa^2=32\pi G.$$

The fluctuation $h_{\mu\nu}$ is quantized after gauge fixing. For a characteristic invariant energy $E$, each additional graviton loop is suppressed by $\kappa^2E^2/(16\pi^2)\sim GE^2$, up to convention-dependent numerical factors, while higher-curvature operators carry their own Wilson coefficients. This is already quantum gravity: it predicts low-energy graviton scattering and universal long-distance corrections without knowing the ultraviolet completion. Donoghue 2023, §§ 2–6 develops this separation between calculable infrared effects and unknown short-distance matching.

An independent check is the classical limit. Holding a macroscopic source fixed while taking the relevant commutators or loop factors to zero must recover the classical Einstein solution. A second check is gauge independence: an on-shell amplitude or relational observable may not depend on the gauge parameter used for $h_{\mu\nu}$, even though intermediate propagators do.

Microscopic descriptions and correspondence

A microscopic proposal changes the basic Hilbert space, operator algebra, path integral, causal structure, or notion of geometry. Calling it “nonperturbative” does not by itself establish that it contains general relativity. It must construct physical states and dynamics, identify observables, and exhibit a limit in which local QFT, semiclassical backreaction, and Einstein gravity are recovered with controlled corrections.

The boundaries matter. Established semiclassical thermodynamics belongs to the semiclassical-gravity treatment; generic flat-spacetime field theory belongs to its preparation route; classical alternative gravities and string or holographic realizations have separate canonical homes. This chapter compares regimes common to candidate programmes without importing their microscopic conclusions.

Common category errors

Particle creation is not metric quantization. A quantum field can produce particles on a completely prescribed classical geometry. The calculation becomes semiclassical only when $\langle\hat T_{\mu\nu}\rangle$ is allowed to update that geometry.

A graviton is not a complete ultraviolet theory. Quantizing $h_{\mu\nu}$ gives a controlled low-energy theory. It does not settle what variables describe arbitrarily high curvature, but ultraviolet ignorance does not erase its infrared predictions.

Exercises

A calculation quantizes a scalar field on a fixed Schwarzschild metric and obtains an outgoing flux. The metric is never updated. Another calculation solves the Einstein equation with the renormalized expectation value of that flux. Classify both calculations and state one missing check in the second.

Solution

The first calculation is QFT on curved spacetime: matter is quantum, geometry is prescribed, and there is no feedback. The second is semiclassical gravity because a classical metric responds to $\langle\hat T_{\mu\nu}^{\mathrm{ren}}\rangle$. It must still specify the quantum state and renormalization prescription and check that stress-tensor fluctuations, curvature corrections, and accumulated errors remain small enough for a mean-field equation to be reliable.

References

• John F. Donoghue, “Quantum General Relativity and Effective Field Theory,” arXiv:2211.09902v2 (2023), Open manuscript.