QFT in Curved Spacetime and Semiclassical Gravity

QFT in curved spacetime studies quantum fields on a prescribed classical geometry; semiclassical gravity additionally lets a renormalized expectation value source a classical metric. The field includes admissible states, local observables, horizons and particle creation, cosmology, stress-tensor fluctuations, backreaction, energy inequalities, and generalized entropy. It does not supply a complete quantum theory of spacetime, and generic curved backgrounds have no preferred vacuum or particle notion.

Evidence cutoff. 11 August 2026.

Required background. Hadamard admissibility supplies the ultraviolet state condition, local field algebras and the time-slice property supplies covariant locality, and renormalized stress-tensor axioms exposes finite curvature ambiguities.

Helpful background. The semiclassical Einstein equation defines mean backreaction, vacua and representations explains why inequivalent states arise, gravity EFT supplies the expansion and counterterms, and quantum energy inequalities gives state-dependent lower bounds.

Local covariance replaces a preferred vacuum

Program	Controlled object	Validation	Limitation
locally covariant QFT	algebras, fields, states across backgrounds	covariance, causality, time-slice, microlocal spectrum	construction is best developed for perturbative or special interacting theories
stress-tensor renormalization	$\langle T_{ab}\rangle_{\rm ren}$ and anomalies	conservation, local covariance, flat/known limits	finite conserved curvature terms remain scheme parameters
horizon and cosmological particle creation	detector response, fluxes, Bogoliubov data	regularity, switching, greybody and adiabatic checks	particle number depends on observer/state and asymptotic structure
semiclassical backreaction	solutions of $G_{ab}+\cdots=8\pi G\langle T_{ab}\rangle$	order reduction, stability, stress fluctuations	mean-field validity can fail; higher-derivative runaway solutions appear
energy and entropy inequalities	smeared stress bounds, relative entropy, generalized entropy	theorem hypotheses and solvable geometries	pointwise energy positivity is false in QFT

Radzikowski’s microlocal characterization turns the Hadamard condition into a wavefront-set statement and is the key local admissibility test Radzikowski 1996, foundational theorem. Hadamard form makes local composite-field renormalization possible in locally covariant perturbation theory Hollands and Wald 2001, construction; it does not select a unique global state.

Wald’s stress-tensor axioms constrain renormalization but leave local conserved curvature ambiguities Wald 1977, foundational method. Those terms belong with the gravitational couplings and must be fixed consistently. Comparing two calculations that choose different finite terms as if they predicted different physics is a scheme error.

Curved-spacetime evidence and uncertainty cultures

Analytic work is strongest when state, geometry, boundary conditions, switching, and renormalization prescription are explicit and when flat-space, adiabatic, conservation, and trace checks agree. Numerical mode sums need uniform ultraviolet subtraction and convergence in angular/frequency cutoffs. Cosmological inference adds background history and initial-state assumptions. Analogue experiments can test kinematic mode conversion under controlled dispersion, but they do not measure gravitational backreaction.

Hawking’s stationary black-hole calculation established a thermal late-time flux under semiclassical assumptions Hawking 1975, foundational evidence. Its temperature and near-horizon kinematics are robust across many ultraviolet modifications, while finite-time formation, greybody propagation, state preparation, energy conservation, and complete evaporation require additional analysis. Quantum energy inequalities show that negative energy is allowed but constrained after smearing; they replace, rather than restore, a pointwise lower bound Fewster and Eveson 1998, theorem.

Separate the checks rather than treating agreement in one observable as validation of the whole calculation. Verify a candidate state’s Hadamard short-distance form before using it in renormalized composites; then check stress-tensor conservation, the trace anomaly, and the allowed finite renormalization terms. Detector and Hawking-response calculations should recover the appropriate KMS or detailed-balance limit and distinguish near-horizon production from greybody propagation. Mean-backreaction calculations should exhibit the perturbative order counting, apply order reduction only within that regime, and test stability under admissible state and scheme changes.

Breakdown and no-go lessons

A generic spacetime has no timelike Killing symmetry, so positive frequency and particles are not globally canonical. The semiclassical equation contains higher-derivative terms and can admit runaway solutions outside the EFT regime; order reduction is a controlled selection prescription only when the derivative expansion applies. Large stress-tensor fluctuations can make the mean source insufficient, but no single local fluctuation ratio is a universal validity criterion.

Quantum inequalities depend on field, state class, sampling, and geometry; they do not forbid every macroscopic negative-energy configuration without further assumptions. Singularity theorems and generalized-second-law proofs likewise carry causal, energy, entropy, and semiclassical hypotheses. Backreaction calculations in two-dimensional models are valuable benchmarks but cannot be transferred unqualified to four-dimensional gravity.

Entering the field

Start with a free field on a globally hyperbolic background, construct or verify a Hadamard state, and compute a local observable with conservation and flat-space checks. The gravity and cosmology pathway and scope and status module provide the entry.

This guide omits full quantum-gravity microphysics and observational cosmology not tied to a field-theoretic state or observable. It also separates analogue horizons from gravitational evidence.

Evidence scope and related assessments

The finite search used arXiv, mathematical-physics and gravity journal records, DOI registries, and targeted searches for state-dependence, runaway, and analogue-transfer limitations. Sources public through 11 August 2026 were eligible. Reassess when a backreaction calculation controls stress fluctuations in a new regime, a theorem changes generalized-entropy scope, or a proposed ultraviolet robustness result fails under a physical state or detector model.

Continue to Hawking-radiation robustness, hypotheses of the generalized second law, the analogue Hawking brief, or the QNEC scope brief.

References

• C. J. Fewster and S. P. Eveson, “Bounds on Negative Energy Densities in Flat Spacetime,” Physical Review D 58 (1998) 084010. DOI.
• S. W. Hawking, “Particle Creation by Black Holes,” Communications in Mathematical Physics 43 (1975) 199–220. DOI.
• S. Hollands and R. M. Wald, “Local Wick Polynomials and Time Ordered Products of Quantum Fields in Curved Spacetime,” Communications in Mathematical Physics 223 (2001) 289–326. DOI.
• M. J. Radzikowski, “Micro-Local Approach to the Hadamard Condition in Quantum Field Theory on Curved Space-Time,” Communications in Mathematical Physics 179 (1996) 529–553. DOI.
• R. M. Wald, “The Back Reaction Effect in Particle Creation in Curved Spacetime,” Communications in Mathematical Physics 54 (1977) 1–19. DOI.