QFT in Curved Spacetime

How does a quantum field behave when spacetime itself expands, curves, or contains a horizon? This volume develops QFT on a classical gravitational background, then examines how quantum matter can change that background and how low-energy gravitational corrections can be calculated.

A geometry alone does not fix a quantum prediction. One must also choose the field, the state, and the observable. This distinction connects particle production, detector response, Hawking radiation, renormalized stress tensors, and cosmological correlations. It also makes clear when a fixed-background calculation needs backreaction or a different approximation.

Where to begin

• For a first course in curved-spacetime QFT, begin with fields and local algebras, then Chapters 2 and 4 for states and detectors. Chapters 7–9 connect local observables to the response of the geometry.
• For black holes and horizons, follow particles and detectors into Chapter 6, using Chapters 1–2 for the field and state construction. Add Chapters 7, 9, and 11 when the question involves flux, backreaction, or entropy.
• For cosmology, begin with quantum fields in cosmology, after the state and mode methods in Chapters 2 and 4. Choose Chapters 15–18 according to whether the target is a transition history, an inflationary correlator, infrared evolution, or analytic reconstruction.

Helpful background. Manifolds and tensors supply the geometric language; distributional kernels and operator algebras supply the field-theory tools. Review causality, in-out versus in-in expectation values, and gravitational EFT when those questions enter your route.

Chapters

1. Fields and Local Algebras on Curved Backgrounds — Construct fields from geometric, causal, and boundary data before selecting a quantum state.
2. States, Hadamard Structure, and Microlocal Control — Identify admissible short-distance behavior and distinguish it from a prescription for a preferred state.
3. Interacting Fields and Local Covariant Renormalization — Build interacting observables with causal factorization and local curvature counterterms.
4. Particles, Detectors, and Nonadiabatic Production — Compare particle definitions, localized detector responses, and production by time-dependent backgrounds.
5. Curved Channels, Communication, and Entanglement Harvesting — Specify a sender–field–receiver experiment and separate signaling from extracting existing correlations.
6. Horizons and Hawking Radiation — Connect horizon geometry, state regularity, and scattering to the radiation measured far away.
7. Local Observables, Stress Tensors, and Anomalies — Renormalize composite observables while tracking conservation and finite curvature ambiguities.
8. Matter Effective Actions in Curved Spacetime — Extract local and nonlocal gravitational response from quantum matter.
9. Mean Semiclassical Backreaction — Solve for a classical geometry sourced by quantum expectation values and test its consistency.
10. Stochastic Gravity and Metric Fluctuations — Determine which metric correlations are captured by stress fluctuations in a controlled approximation.
11. Generalized Entropy and Quantum Extremal Surfaces — Combine matter entropy with gravitational terms and examine surface variations and their scope.
12. Energy Conditions, Inequalities, and Focusing — Apply quantum energy bounds with their geometric, state, and smearing hypotheses.
13. Gravity as Effective Field Theory — Calculate low-energy effects of quantized fields and gravity with explicit power counting.
14. Quantum Fields in Cosmology — Follow normalized modes through expansion, state selection, and observable subtraction.
15. Gravitational Vacuum Decay and Cosmological Transitions — Connect gravitational decay calculations to expansion, completion, dilution, and relic signals.
16. Inflationary Correlators and Cosmological Perturbation Theory — Calculate cosmological correlations with gauge constraints, initial states, and the in-in contour.
17. de Sitter Infrared Physics and Stochastic Inflation — Compare infrared approximations while controlling late-time limits and the observables being matched.
18. Cosmological Analyticity and Bootstrap — Use symmetry and singularities to constrain wavefunction coefficients and correlators without identifying them.
19. Validity, Breakdown, and the Need for Quantum Gravity — Decide which approximation controls a requested observable and what evidence marks its failure.

How the ideas connect

Take a free scalar field on a globally hyperbolic spacetime without boundary. Its wave operator fixes the causal propagator and commutator. For a centered Gaussian state, a positive two-point bisolution with the required commutator supplies the state data. Requiring Hadamard short-distance behavior permits the standard local subtraction used for composite observables; it does not select a unique vacuum. The construction is developed in Hollands and Wald 2015, §§ 2.1 and 3.1.

Now ask two different questions about that same state. A detector’s transition probability depends on its trajectory, coupling, and switching. A renormalized stress tensor describes local energy and momentum, with specified finite curvature terms. Neither is determined by choosing a particle basis alone. The field-construction map and explanation show how geometry, boundary conditions, the algebra, and state enter separately.

Next allow matter to influence the geometry. Mean semiclassical gravity uses a stress expectation value as a source; stochastic gravity also treats selected effects of stress fluctuations. Gravitational EFT can include quantized metric fluctuations with a low-energy expansion. These are distinct approximations, not successive labels for the same determinant. The extension to quantized gravitational perturbations is discussed in Hollands and Wald 2015, § 4.2.

This sequence explains the reading order: establish propagation, choose a state, construct an observable, and then assess the approximation. Near a horizon, a thermal detector response and an outgoing energy flux are different observables. In cosmology, state construction, particle production, and subtraction are different operations. At each step, the relevant errors depend on energy, curvature, coupling, duration, and the quantity being measured.

Compare fixed-background, semiclassical, stochastic, and EFT descriptions

Regime and claim-domain table

The table classifies calculations by what is held fixed, what fluctuates, and what controls the result.

Regime	Prescribed data	Quantum or fluctuating data	Typical output	Required controls	Stop condition
Fixed-background QFT	$(M,g)$ and boundary/initial structure	Matter fields	Algebraic relations, detector response, $\langle T_{\mu\nu}\rangle_{\rm ren}$, particle-production or horizon flux	State admissibility, renormalization, coupling and derivative expansion, negligible backreaction for the chosen observable	The computed stress or duration changes the background at the target precision
Mean semiclassical gravity	Initial geometry and quantum state data	Matter expectation value sources a classical metric	A self-consistent solution of the semiclassical Einstein equation	Causal in-in response, constraints, finite curvature terms, stability, loop or large-$N$ ordering	Stress fluctuations or metric quantum correlations affect the observable at the claimed accuracy
Stochastic or large-$N$ gravity	Mean background and controlled state	Stress fluctuations and induced metric correlations	Noise kernels, Einstein–Langevin solutions, symmetrized metric correlators	Smearing, gauge-invariant observables, dissipation, cumulant truncation, large-$N$ matching	Higher cumulants, intrinsic quantum geometry, or nonperturbative fluctuations are unsuppressed
Low-energy gravitational EFT	Light fields, symmetries, matching conditions, cutoff	Matter and, when declared, metric and ghost fluctuations	Long-distance amplitudes, relational observables, nonanalytic corrections, background response	Operator basis, field redefinitions, gauge independence, loop content, $E/\Lambda$, curvature and derivative power counting	Omitted operators or new degrees of freedom are not smaller than the error target
Quantum-gravity-required	No classical geometry is assumed adequate for the requested question	Quantum geometry or UV-complete variables	Program-dependent observables	A definition and evidence standard supplied by the quantum-gravity framework	The chosen quantum-gravity framework must define the observable and its dynamics

A low curvature alone does not guarantee validity: local energies, long evolution, enhanced fluctuations, or omitted operators can invalidate the approximation for a particular observable.

Where to go next

• For proofs of existence, Hadamard properties, and local covariance, continue to Mathematical QFT.
• For boundary dualities, black-hole information, and proposed microscopic gravitational theories, use Holography and Quantum Gravity.
• For the information-theoretic definitions behind entropy and channels, use Quantum Information in QFT. For transport and open-system approximations, use Thermal and Nonequilibrium QFT.

References

• Hollands, Stefan, and Robert M. Wald. 2015. “Quantum Fields in Curved Spacetime.” Physics Reports 574: 1–35. DOI. Open text, arXiv:1401.2026v2.