QFT in Curved Spacetime

Everything so far was formulated on flat Minkowski spacetime, where the Poincaré symmetry picks out a unique vacuum and an unambiguous notion of "particle." On a curved background — the regime of general relativity — those foundations weaken: the vacuum becomes observer-dependent and particles are no longer absolute. This semiclassical framework (quantum fields on a classical curved geometry) yields the Unruh effect and Hawking radiation, and marks the boundary where QFT hands off to quantum gravity.

Conventions: $\hslash =c=1$; metric signature mostly-minus, now a general $g_{\mu \nu }(x)$.

Fields on a curved background

The semiclassical program keeps gravity classical (a fixed metric $g_{\mu \nu }$) and quantizes matter fields on top. The scalar action is covariantized by the minimal-coupling rule — replace ${\eta }_{\mu \nu }\to g_{\mu \nu }$, ${\partial }_{\mu }\to {\nabla }_{\mu }$, and $d^{4}x\to d^{4}x\sqrt{-g}$:

$$S=\int d^{4}x\sqrt{-g}\left[\frac{1}{2}{g}^{\mu \nu }{\nabla }_{\mu }\varphi {\nabla }_{\nu }\varphi -\frac{1}{2}{m}^2{\varphi }^2-\frac{1}{2}\xi R{\varphi }^2\right],$$

with $R$ the Ricci scalar and $\xi$ a curvature coupling. The canonical quantization machinery still applies locally, but the global step — expanding in modes $a_{\mathbf{p}},a_{\mathbf{p}}^{†}$ — runs into trouble.

The ambiguity of the vacuum

In flat space the mode expansion splits fields into positive- and negative-frequency parts using the global time translation symmetry, defining a unique vacuum $a_{\mathbf{p}}|0⟩=0$ that all inertial observers agree on. A general curved spacetime has no timelike Killing vector, so there is no preferred notion of positive frequency and hence:

$$\boxed{\text{No unique vacuum, no observer-independent particle number.}}$$

Two observers with different notions of time define different creation/annihilation operators, related by a Bogoliubov transformation

$$a_i^{\prime }=\sum _j({\alpha }_{ij}{a}_j+{\beta }_{ij}^{\ast }{a}_j^{†}).$$

When the mixing coefficient ${\beta }_{ij}\ne 0$, one observer's vacuum contains particles in the other's counting: $⟨0|a^{\prime †}{a}^{\prime }|0⟩=\sum |\beta {|}^2\ne 0$. Particle number is not an invariant — it depends on the observer and the geometry.

The Unruh effect

The simplest and most striking instance occurs already in flat space, for a non-inertial observer. A uniformly accelerated (Rindler) observer with proper acceleration $a$ does not see the inertial Minkowski vacuum as empty — a Bogoliubov calculation shows they perceive a thermal bath at the Unruh temperature

$$\boxed{T_{\text{Unruh}}=\frac{a}{2\pi }.}$$

The inertial vacuum is a thermal state (with exactly the finite-temperature Bose–Einstein spectrum) from the accelerated frame. This makes vivid that "vacuum" and "particle" are frame-dependent, and it is the flat-space template for Hawking radiation.

Hawking radiation

Applying the same logic to the black-hole geometry gives Hawking's 1974 result: a black hole is not black — it radiates thermally. Modes that fall in versus escape to infinity are related by a Bogoliubov transformation across the horizon, and the escaping flux is a blackbody spectrum at the Hawking temperature

$$\boxed{T_{\text{Hawking}}=\frac{\kappa }{2\pi }=\frac{1}{8\pi GM},}$$

with $\kappa$ the surface gravity. The black hole thereby has a genuine thermodynamics (entropy $S=A/4G$, the Bekenstein–Hawking area law) and slowly evaporates. This result — quantum field theory on a curved classical background — is one of the deepest in theoretical physics, and it exposes a genuine puzzle:

The information paradox and the handoff to quantum gravity

If a black hole forms from a pure state and evaporates into exactly thermal (mixed) radiation, unitarity — the bedrock of quantum mechanics and the S-matrix — appears violated: information is lost. The black-hole information paradox signals that the semiclassical approximation (quantum field on classical geometry) must break down when the geometry itself is quantum. Its resolution requires a full theory of quantum gravity, where the metric is dynamical and quantized — the point at which QFT as developed in this section reaches its limit and hands off to string theory, loop quantum gravity, and holography. The AdS/CFT correspondence — a CFT dual to quantum gravity — has provided the strongest evidence that information is preserved, making this a rare case where holography constrains real physics.

Why gravity is different: non-renormalizability

Attempting to quantize gravity itself as a QFT (gravitons as spin-2 excitations of $g_{\mu \nu }$) fails by the power-counting criterion: Newton's constant $G$ has negative mass dimension ($[G]=-2$), so general relativity is non-renormalizable. By the EFT viewpoint this is not fatal — GR is a perfectly good effective field theory of gravity below the Planck scale $M_{\text{Planck}}\sim 10^{19}$ GeV, making reliable low-energy predictions (e.g. quantum corrections to the Newtonian potential). It simply must be completed by new physics in the deep UV — the domain of quantum gravity.

Summary

• On a curved background there is no unique vacuum and no observer-independent particle number; frames are related by Bogoliubov transformations.
• Unruh: an accelerated observer sees the inertial vacuum as thermal at $T=a/2\pi$.
• Hawking: black holes radiate thermally at $T=1/8\pi GM$ and have entropy $S=A/4G$; evaporation raises the information paradox.
• Gravity is a non-renormalizable EFT (valid below $M_{\text{Planck}}$); its UV completion is quantum gravity.

Where this leads

• Quantum gravity proper: GR frontiers — quantum gravity.
• The thermal analogy: finite-temperature QFT.
• Holographic duality: conformal field theory and AdS/CFT.
• Gravity as an EFT: effective field theory.

References

• Birrell & Davies, Quantum Fields in Curved Space.
• Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics.
• Hawking, Commun. Math. Phys. 43, 199 (1975).
• Unruh, Phys. Rev. D 14, 870 (1976).