QFT in Curved Spacetime and Quantum Gravity

General relativity and quantum field theory are the two pillars of modern physics, each experimentally triumphant in its domain — yet they are mutually incompatible at a fundamental level. GR treats spacetime as a smooth, classical, dynamical geometry; QFT is formulated on a fixed, non-dynamical background and quantizes everything on it. Their union is the central unsolved problem of theoretical physics. This page maps the terrain: the successful halfway house of quantum fields on curved backgrounds, why naive quantization of gravity fails, the conceptual obstructions (the problem of time), and the leading programs.

We use $c=1$ and keep $\hslash$, $G$ explicit; the Planck scale sets the stakes.

The intermediate regime: QFT in curved spacetime

Before quantizing gravity itself, one can quantize matter fields on a fixed but curved background $g_{\mu \nu }$ — a controlled approximation valid when spacetime curvature is far below Planckian but strong enough that flat-space QFT fails. This semiclassical framework is well-defined and yields robust predictions:

• Hawking radiation. A black hole radiates thermally at $T_H=\hslash \kappa /2\pi$ — the flagship result, mixing the horizon of GR with the quantum vacuum.
• The Unruh effect. A uniformly accelerated observer in the ordinary Minkowski vacuum detects a thermal bath at temperature $T=\hslash a/2\pi$ (proportional to proper acceleration $a$). The very notion of "particle" is observer-dependent: the inertial vacuum is a thermal state to the accelerated detector. This is the flat-space shadow of Hawking's result (via the equivalence principle) and shows that particle number is not a covariant concept once gravity/acceleration enters.
• Cosmological particle creation. An expanding FLRW universe amplifies vacuum fluctuations into real quanta; in the inflationary era this seeds the primordial density perturbations imprinted on the CMB — arguably an observed quantum-gravitational (or at least quantum-field-in-curved-spacetime) effect.

The semiclassical back-reaction is captured by the semiclassical Einstein equation $G_{\mu \nu }=8\pi G⟨{\hat{T}}_{\mu \nu }⟩$, sourcing classical geometry with the expectation value of the quantum stress tensor (requiring subtle renormalization, and generically violating the energy conditions). This regime is as far as we can go with confidence — and its very successes (thermal horizons, the information paradox) are what demand a full quantum theory of gravity.

Why quantizing gravity is hard

Treating the metric perturbation $h_{\mu \nu }$ (linearized gravity) as just another quantum field — a massless spin-2 graviton on flat space — works at low energies but fails as a fundamental theory:

• Non-renormalizability. Newton's constant is dimensionful: $G\sim 1/M_{\text{Pl}}^2$ with the Planck mass $M_{\text{Pl}}=\sqrt{\hslash /G}\approx 1.2\times 10^{19}$ GeV. A coupling with negative mass dimension makes perturbative quantum gravity non-renormalizable: loop divergences proliferate uncontrollably, requiring infinitely many counterterms (explicit two-loop divergences were computed by Goroff–Sagnocchi). The theory loses predictivity at the Planck scale ${\ell }_{\text{Pl}}=\sqrt{G\hslash }\approx 1.6\times 10^{-35}$ m, where quantum-gravitational effects become order one. As an effective field theory, low-energy quantum gravity is fine (one can compute quantum corrections to the Newtonian potential); as a fundamental theory it is incomplete.
• Gravity is geometry, not a field on a stage. In QFT the spacetime background is fixed and provides the causal structure, the notion of time, and the vacuum. In GR the metric is the dynamical variable — so quantizing it means quantizing causal structure and time themselves. There is no fixed arena in which to define states, particles, or the S-matrix. This background independence is the conceptual heart of the difficulty.
• Singularities. Classical GR predicts its own breakdown at black-hole and Big Bang singularities, exactly where curvature reaches the Planck scale — precisely the regime a quantum theory must resolve.

The problem of time

Background independence has an acute consequence in the canonical (ADM) formulation. There, time evolution is generated by the Hamiltonian constraint, which on quantization becomes the Wheeler–DeWitt equation

$$\hat{\mathcal{H}}\Psi [\gamma ]=0.$$

The "wavefunction of the universe" is annihilated by the Hamiltonian — it does not evolve in any external time, because in a background-independent theory there is no external time; time is internal to the geometry. Reconciling this frozen formalism with the manifest experience of temporal evolution is the problem of time — as much philosophical as technical, and treated on the philosophy side under The Problem of Time in Quantum Gravity. It is a sharp illustration that quantum gravity is not merely a harder computation but a conceptual reworking of dynamics.

The main programs

No approach is complete or experimentally confirmed. The leading candidates take opposite stances on background independence:

• String theory. Replaces point particles with one-dimensional strings; the graviton emerges automatically as a string vibrational mode, and the ultraviolet divergences are tamed by the string length. It is perturbatively finite and unifies gravity with the other forces, at the cost of extra dimensions, supersymmetry, and a vast landscape of vacua. Its greatest success bearing on gravity is AdS/CFT (Maldacena): a holographic duality equating quantum gravity in an anti-de Sitter bulk with an ordinary conformal field theory on the boundary — a concrete, background-dependent but non-perturbative definition of quantum gravity, and the arena where the black-hole information paradox has seen real progress (the Page curve from replica wormholes).
• Loop quantum gravity. Quantizes GR non-perturbatively and background-independently, using Ashtekar variables. Geometry is quantized: area and volume operators have discrete spectra, so space is built from a spin network, and spacetime from a spin foam. It predicts a granular structure at ${\ell }_{\text{Pl}}$ and suggests singularity resolution (a "big bounce" replacing the Big Bang), but recovering smooth classical spacetime and the low-energy limit remains difficult.
• Other approaches. Causal dynamical triangulations, asymptotic safety (a nontrivial UV fixed point rescuing renormalizability), causal set theory (discrete causal order as fundamental), and emergent/entropic-gravity proposals (gravity as thermodynamics of underlying degrees of freedom, à la Jacobson's derivation of the Einstein equation from horizon entropy).

The scale of the problem — and prospects

Direct experimental access is daunting: the Planck energy $\sim 10^{19}$ GeV is fifteen orders of magnitude beyond the LHC, and ${\ell }_{\text{Pl}}$ is far below any probe. But indirect windows are opening: the CMB may carry imprints of quantum-gravitational or trans-Planckian physics from inflation; searches for Lorentz-invariance violation in high-energy astrophysical photons constrain discrete-spacetime models; gravitational-wave observations probe strong-field GR and could reveal horizon-scale quantum structure (echoes); and analogue-gravity experiments realize Hawking/Unruh physics in the lab. Quantum gravity remains open, but it is no longer wholly beyond empirical reach.

Summary

• QFT in curved spacetime (fixed background) is well-defined and predicts Hawking radiation, the Unruh effect (particle number is observer-dependent), and inflationary particle creation; back-reaction via $G_{\mu \nu }=8\pi G⟨{\hat{T}}_{\mu \nu }⟩$.
• Full quantum gravity is obstructed by non-renormalizability ($G\sim 1/M_{\text{Pl}}^2$; predictive only as an effective theory below the Planck scale), by background independence (the metric is dynamical, so causal structure and time are quantized), and by singularities.
• The problem of time: the Wheeler–DeWitt equation $\hat{\mathcal{H}}\Psi =0$ has no external time — a conceptual crux, echoed in the philosophy of time.
• Leading programs: string theory (graviton as a string mode; AdS/CFT holography) and loop quantum gravity (discrete, background-independent geometry), plus asymptotic safety, causal sets, and emergent gravity — none yet confirmed.

Next

Experimental Status collects the confirmations of classical general relativity across scales — the theory that any quantum completion must reproduce in its low-energy limit.