QFT in Curved Spacetime

Quantum field theory in curved spacetime becomes a definite physical theory only after the background and its causal or boundary data, the quantum field algebra, the state, the observable, the renormalization prescription, and the approximation layer have all been specified. This volume develops that specification from fixed-background fields through detectors, horizon radiation, local observables, semiclassical and stochastic response, gravitational effective field theory, and cosmology. Its recurring discipline is simple: a result is exported only with the hypotheses and error controls that make it true.

Helpful background. Smooth manifolds and tensor fields prepare the geometric language; distributional kernels on manifolds and operator algebras and positive functionals prepare the algebraic formulation. Relativistic causality, in-out versus in-in expectation values, and gravity as effective field theory are route-specific preparation rather than universal entrance requirements.

The data that make a curved-space question definite

A useful problem statement can be compressed into the tuple

$$\mathfrak P= \bigl(M,g;\mathcal B;\mathcal F,\mathcal A;\omega;\mathcal O; \mathcal R;\mathcal E;\mathcal V\bigr).$$

Here $(M,g)$ is the spacetime, $\mathcal B$ denotes causal, initial, and boundary data, $\mathcal F$ the field system, $\mathcal A$ its observable algebra, $\omega$ the state, $\mathcal O$ the requested observable, $\mathcal R$ the regulator and renormalization prescription, $\mathcal E$ the collection of expansion parameters, and $\mathcal V$ the validation test. Two calculations that differ in one of these entries need not answer the same question even when their equations look similar.

The algebra–state separation is particularly important. The causal propagator and commutator are fixed by the differential operator and boundary problem, whereas a positive two-point function selects a state. Hadamard behavior constrains the ultraviolet singularity of that state but does not choose a preferred vacuum. Hollands and Wald develop this local, covariant viewpoint and its consequences for interacting fields and renormalized composites in Hollands and Wald 2015, §§ 2–4.

The diagram shows the dependency that persists across the volume. Inspect the split after the local algebra: state selection changes expectation values, while the causal algebra remains fixed until the background or boundary problem changes.

The fixed-background construction separates geometric and algebraic data from quantum-state data. Topology and timelike boundaries can alter the solution space or admissible algebra before a state is chosen; relative Cauchy evolution tests how local observables respond to compactly supported background changes. The map is schematic and does not assert existence on a background that fails the stated hyperbolicity or boundary hypotheses.

Regime and claim-domain table

The same matter theory can appear in several approximation layers. What distinguishes them is not curvature alone but which variables are quantized, which correlators are retained, and which expansion controls the requested observable.

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	This volume stops and hands the question to the exact framework owner

This table classifies claims, not entire spacetimes. A low-curvature background can still probe trans-cutoff local energies, an excessively long secular interval, a large species enhancement, or an uncontrolled fluctuation observable. Conversely, a large coordinate effect need not signal quantum gravity when invariant curvatures and measured energies remain controlled.

Check your preparation

Use these diagnostics independently; there is no single readiness score.

• Geometry and propagation. Can you state why a normally hyperbolic operator on a globally hyperbolic spacetime has retarded and advanced Green operators, and identify what a timelike boundary adds to the problem? If not, repair the geometry and PDE route before Chapters 1–3.
• States and distributions. Can you distinguish a state-dependent Wightman distribution from the state-independent causal propagator, and explain why their coincident limits are not ordinary products? If not, begin with Chapters 1–2 and the linked distributional preparation.
• Renormalization and real time. Can you explain why a finite local curvature counterterm may change a renormalized stress tensor, and why an in-out action does not generally give a causal expectation-value equation? If not, use Chapters 3, 7, and 9 with the in-in preparation linked above.
• Gravity and EFT. Can you identify the expansion parameter of a proposed gravitational calculation and distinguish a matter determinant from a graviton loop? If not, enter through Chapters 8 and 13.
• Cosmology. Can you normalize a time-dependent mode by its Wronskian and keep state construction, particle number, and observable subtraction separate? If not, follow Chapters 2, 4, 7, and 14 before the inflationary and de Sitter routes.
• Claims and evidence. Can you state the observable, state, regulator, limit order, independent check, and falsifying condition for a result? If not, use Chapter 19 as a companion to whichever scientific route you choose.

The nineteen chapters

Each chapter is a direct route, not a compulsory week in a single linear course.

Chapter	Central question	Capability at exit
1. Fields and Local Algebras on Curved Backgrounds	Which geometric, causal, bundle, topological, and boundary data make a field theory well posed?	Specify a Green-hyperbolic field system, reduced solution space, local algebra, and failure domain
2. States, Hadamard Structure, and Microlocal Control	Which states are ultraviolet admissible, and which additional principle selects among them?	Test a two-point function for positivity and Hadamard behavior without manufacturing a unique vacuum
3. Interacting Fields and Local Covariant Renormalization	How are interacting observables constructed without importing a preferred vacuum or global S-matrix?	Use causal factorization, local time-ordered products, curvature counterterms, and Ward identities
4. Particles, Detectors, and Nonadiabatic Production	Which operational statement is made by a particle number, a localized detector, or a production calculation?	Separate bases, switching, response, Stokes transitions, energy accounting, and asymptotic limits
5. Curved Channels, Communication, and Entanglement Harvesting	How does a specified sender–field–receiver system define a channel on a curved background?	Distinguish causal exchange, correlation extraction, restricted access, redshift, and resource constraints
6. Horizons and Hawking Radiation	Which ingredients turn horizon kinematics into a state-dependent asymptotic flux?	Keep surface gravity, state regularity, KMS behavior, greybody scattering, and superradiance distinct
7. Local Observables, Stress Tensors, and Anomalies	How are local composite observables renormalized and compared across prescriptions?	Track state dependence, finite curvature terms, conservation, anomalies, boundaries, and numerical checks
8. Matter Effective Actions in Curved Spacetime	What do matter determinants say about local and nonlocal metric response?	Use heat kernels, thresholds, imaginary parts, and variation within their operator and expansion domains
9. Mean Semiclassical Backreaction	When does a renormalized mean stress tensor source a controlled classical geometry?	Pose coupled state–geometry initial data, propagate constraints, and test causal stability and order reduction
10. Stochastic Gravity and Metric Fluctuations	Which metric correlations are captured by stress fluctuations?	Construct smeared noise and response kernels and state the Gaussian or large-$N$ evidence ceiling
11. Generalized Entropy and Quantum Extremal Surfaces	How are matter entropy, gravitational counterterms, and surface variation combined?	Renormalize generalized entropy and distinguish stationarity, GSL theorems, and holographic prescriptions
12. Energy Conditions, Inequalities, and Focusing	What quantum replacements for classical energy conditions are actually proved?	Carry smearing, state, dimension, affine normalization, anomalies, and counterexamples with each claim
13. Gravity as Effective Field Theory	What does quantized gravity predict at energies below a declared cutoff?	Build a reduced operator basis, identify loop content, match observables, and estimate truncation error
14. Quantum Fields in Cosmology	How do expansion and initial-state data affect fields, particles, and local observables?	Normalize FLRW modes, compare state prescriptions, match eras, and control relic and backreaction claims
15. Gravitational Vacuum Decay and Cosmological Transitions	How are gravitational decay and supplied microphysics integrated into an expanding history?	Distinguish saddle regimes and compute completion, dilution, yield, and gravitational-wave transfer conditions
16. Inflationary Correlators and Cosmological Perturbation Theory	How do gauge-invariant modes and an in-in contour produce observable correlators?	Derive spectra and interactions while controlling constraints, state, loops, soft limits, and decoherence
17. de Sitter Infrared Physics and Stochastic Inflation	Which infrared claims survive a fixed state, observable, gauge, regulator, and order of limits?	Compare resummation, stochastic, and open-system descriptions within a stated matching domain
18. Cosmological Analyticity and Bootstrap	What can symmetry and singularity data reconstruct about cosmological observables?	Translate among wavefunction coefficients and correlators and retain cuts, contacts, subtractions, and state assumptions
19. Validity, Breakdown, and Quantum-Gravity Handoffs	Which approximation answers this observable to the requested accuracy, and where must it stop?	Produce an observable-specific error contract and an exact rigorous, Research, or quantum-gravity handoff

Routes through the volume

Graduate curved-QFT core. Chapters 1 and 2 establish fields and admissible states; Chapter 4 makes particle language operational; Chapters 7–9 develop renormalized observables and mean response. Use the relevant parts of Chapter 19 to state the final validity domain.

Particles and black holes. Follow Chapters 1 → 2 → 4 → 6 → 7 → 9 → 11. This route begins with local field theory, not horizon thermality, because a Hawking claim needs a state, a collapse or stationary setting, scattering data, and an observable flux. Endpoint and information questions then pass to Holography and Quantum Gravity.

Local covariance and rigorous structure. Follow Chapters 1 → 2 → 3 → 7. The volume gives physical constructions, representative calculations, and failure tests; theorem-first formulations and proofs continue in Mathematical QFT.

Mean and fluctuating geometry. Follow Chapters 7 → 8 → 9 → 10, keeping the in-in contour and loop content explicit. The output progresses from a renormalized one-point function to causal response, a stress two-point function, and controlled induced metric correlations; none of these alone defines a complete quantum state of geometry.

Entropy and focusing. Follow Chapters 7 → 9 → 11 → 12. State the surface, algebra, regulator, counterterms, averaging curve, and theorem domain before using generalized entropy or a focusing inequality. Islands, replica wormholes, and holographic Page curves are downstream topics rather than automatic consequences of a stationary generalized entropy.

Gravitational EFT. Use the general EFT framework, then Chapters 8 → 9 → 13 → 19. Matter-loop effective actions and metric-loop EFT calculations share curvature operators but have different functional measures and expansion parameters.

Cosmological fields and correlators. Follow Chapters 1 → 2 → 4 → 7 → 14 → 16. Add Chapter 17 for de Sitter infrared dynamics or Chapter 18 for analytic reconstruction. The initial state, contour, gauge-invariant variable, ultraviolet subtraction, and late-time limit must remain visible throughout.

Cosmological transitions. Bring running particle-physics inputs from Gauge Theories and the Standard Model, flat-space bounce control from Nonperturbative Dynamics, and thermal or transport inputs from Thermal and Nonequilibrium QFT. Chapter 15 adds the gravitational saddle and expansion-history integration without relabeling those inputs.

Six recurring calculations

The volume revisits six systems so that added structure can be seen rather than merely named.

1. Background to local observable: causal geometry → field operator → algebra → Hadamard state → local subtraction → finite curvature terms → conservation and scheme comparison.
2. Detector to horizon flux: worldline and coupling → switching and smearing → response → state regularity → horizon geometry → greybody scattering → asymptotic energy flux.
3. Mean to fluctuations: renormalized stress → causal effective action → constraints → mean equation → linear response → noise kernel → gauge-invariant metric correlation.
4. Entropy to focusing: surface and accessible algebra → regulator and counterterms → modular or replica variation → GSL domain → QEI or QNEC hypotheses → qualified focusing statement.
5. Expanding background to correlator: FLRW field and state → canonical gauge-invariant mode → in-in contour → interactions → loop and secular control → observable interpretation.
6. Low-energy result to handoff: requested observable → curvature, energy, coupling, loop, duration, and fluctuation hierarchy → remainder estimate → exact reason a stronger framework is or is not required.

The invariant check changes with the route: commutator support and symplectic flux for fields, positivity and wavefront structure for states, conservation for local stress, retarded support for backreaction, gauge-invariant observables for metric fluctuations, Wronskians for cosmological modes, and factorization or flat-space limits for cosmological analytic objects.

Conventions that travel with a result

The site uses the (+---) Lorentzian metric, $\hbar=c=k_{\mathrm B}=1$, the curvature convention

$$[\nabla_\mu,\nabla_\nu]V^\rho =R^\rho{}_{\sigma\mu\nu}V^\sigma,$$

and the measure $\mathrm d^d x\sqrt{-g}$. For a real scalar we write

$$P_\xi=\Box+m^2+\xi R, \qquad \xi_{\mathrm{conf}}=-\frac{d-2}{4(d-1)},$$

so $\xi_{\mathrm{conf}}=-1/6$ in four dimensions. The causal propagator is $E=G_{\mathrm{ret}}-G_{\mathrm{adv}}$ and the field commutator is $[\Phi(f),\Phi(h)]=-iE(f,h)\mathbf1$. Equivalently, sources that define $\Delta=-E$ write the commutator as $i\Delta$.

A page states dimension, orientation, field normalization, state, boundary conditions, contour, gauge, regulator, subtraction scale, and order of limits wherever they affect the result. Stress-tensor variation signs, Euclidean continuation, extrinsic curvature, and normal orientation are always local data.

Much of the standard literature uses the opposite metric or Riemann-tensor sign. A translation is complete only when it maps the action, wave operator, curvature coupling, Green function, and stress tensor together and then recovers an invariant checkpoint such as causal support, a positive detector probability, a conserved flux, an anomaly coefficient in a fixed basis, or a dimensionless observable. The treatments of Birrell and Davies 1982, Chapters 3–6 and Parker and Toms 2009, Chapters 2–7 are broad reference spines, but individual pages translate their formulas rather than importing signs silently.

Scientific stopping rules

• A Hadamard statement establishes short-distance admissibility, not preferred-state selection.
• A detector response establishes the response of the stated device and trajectory, not an observer-independent particle population.
• A KMS relation establishes a thermal correlation property for a specified flow and state, not by itself an asymptotic stress-energy flux.
• A renormalized stress tensor is defined only after finite local curvature freedom and conservation conditions are recorded.
• A matter determinant is not a graviton loop; an in-out effective action is not automatically a causal mean equation.
• A stochastic metric correlation is a controlled approximation to selected correlators, not a complete quantum-gravity state.
• A stationary generalized entropy is not by itself an island prescription, and an energy inequality carries its smearing and geometric hypotheses.
• A cosmological wavefunction coefficient, an in-in correlator, a boundary object, and a flat-space amplitude are related by explicit transformations, not identical by notation.
• A low-energy constraint may test a proposed ultraviolet completion without constructing or uniquely selecting one.

Mutable observational limits, disputes, analogue experiments, transition forecasts, and current de Sitter or bootstrap assessments belong in dated Research treatments. Executable scans and benchmarks should be distributed with their code, inputs, checks, and claim boundaries. The durable pages here state the calculations and evidence ceilings needed to interpret such outputs.

Exercises

1. Classify a backreaction calculation

A calculation evaluates a matter one-loop determinant on a prescribed FLRW metric, varies the resulting in-out action, and calls the result “one-loop quantum gravity with causal backreaction.” Identify the three classification errors and the minimum repair.

Solution

The loop is a matter loop because the metric was not integrated over; curvature in the output does not change the internal degrees of freedom. The in-out variation computes an in-out matrix-element equation and need not be real or causal for expectation values. Finally, evaluating the determinant on a prescribed metric is not self-consistent backreaction. The repair is to name the result a matter effective action, construct the corresponding in-in or retarded mean equation, specify compatible state–geometry initial data and finite curvature terms, and solve while checking constraints and stability. Metric and ghost loops would be a separate gravitational-EFT calculation.

2. Diagnose a horizon claim

A stationary detector outside a black hole has a KMS response at a redshifted temperature. Does this establish a Hawking flux at future null infinity?

Solution

No. The detector result fixes a state, trajectory, coupling, switching limit, and time flow. An asymptotic flux additionally requires the global state, collapse or eternal geometry, mode propagation, greybody transmission, possible superradiance, and the renormalized stress-energy observable at infinity. A Hartle–Hawking equilibrium state, for example, can give thermal local responses without the same net outgoing flux as the Unruh state.

References

• Birrell, N. D., and P. C. W. Davies. Quantum Fields in Curved Space. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 1982. doi:10.1017/CBO9780511622632.
• Donoghue, John F. “General Relativity as an Effective Field Theory: The Leading Quantum Corrections.” Physical Review D 50 (1994): 3874–3888. doi:10.1103/PhysRevD.50.3874.
• Hollands, Stefan, and Robert M. Wald. “Quantum Fields in Curved Spacetime.” Physics Reports 574 (2015): 1–35. doi:10.1016/j.physrep.2015.02.001.
• Hu, B. L., and Enric Verdaguer. “Stochastic Gravity: Theory and Applications.” Living Reviews in Relativity 11 (2008): 3. doi:10.12942/lrr-2008-3.
• Parker, Leonard, and David Toms. Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2009. doi:10.1017/CBO9780511813924.
• Wald, Robert M. Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics. Chicago Lectures in Physics. University of Chicago Press, 1994. Publisher record.