Quantum Fields, Semiclassical Gravity, and Quantum Black Holes Source Map

From local quantum fields to quantum black holes

This map keeps distinct the results of fixed-background quantum field theory, semiclassical response, controlled lower-dimensional models, holographic calculations, information-theoretic arguments, and low-energy quantum gravity. Full treatments live in Quantum Field Theory in Curved Spacetime, Black-Hole Thermodynamics and Semiclassical Gravity, Black-Hole Information, Black Holes in String Theory and Holography, and Quantum Gravity. Each entry states the framework and approximation that make its conclusion meaningful.

Question	Source class and exact locator	What the source supports	Decisive exclusion
Which geometric and microlocal hypotheses define free-field propagation and Hadamard states?	Authoritative monograph: Bär, Ginoux, and Pfäffle 2007, Ch. 3, §§3.2–3.5, especially Thms. 3.2.11–3.2.12 and 3.4.7. Primary theorem paper: Radzikowski 1996, Thm. 5.1.	Advanced and retarded Green operators for normally hyperbolic fields on globally hyperbolic spacetimes, together with the scalar two-point wavefront-set characterization of the Hadamard condition.	These results do not choose a preferred state, construct a generic interacting or gauge theory, cover arbitrary timelike boundaries, or provide gravitational backreaction.
How does local covariance constrain renormalized observables?	Primary construction and uniqueness paper: Hollands and Wald 2001, §§2–4, §5.1 especially Thm. 5.1, and §5.2.	Local-covariant Wick powers, their existence under the stated microlocal and scaling axioms, and the finite curvature-dependent renormalization freedom.	The axioms constrain but do not uniquely fix every finite counterterm, select a quantum state, solve an interacting theory nonperturbatively, or supply a semiclassical metric.
Why are wedge modularity and an accelerated detector related but not identical statements?	Primary algebraic papers: Bisognano and Wichmann 1975, main wedge theorems, printed pp. 985–1007 and Bisognano and Wichmann 1976, main duality and modular-action theorems, printed pp. 303–321. Detector calculation: Unruh 1976, §III, Eqs. (3.1)–(3.10), pp. 883–885, and Eqs. (3.16)–(3.19), pp. 887–888.	Boost modular flow and wedge thermality under the stated Wightman-field hypotheses, plus the Planckian response with $T=a/(2\pi)$ for the idealized uniformly accelerated detector.	This is not a literal inertial heat bath, a statement for arbitrary switching or finite trajectories, a freely falling black-hole detector result, or a generic curved-horizon theorem.
What does the collapse calculation establish about Hawking radiation?	Primary calculation: Hawking 1975, §2, Eqs. (2.16)–(2.29), printed pp. 204–211, and §4, Eqs. (4.5)–(4.10), pp. 218–219, with Hawking 1976 erratum, printed p. 206. Operational derivation: Fredenhagen and Haag 1990, §§2–4, printed pp. 275–284. Greybody flux: Page 1976, §§II–IV, Eq. (4), Eqs. (11)–(12), and tables, printed pp. 199–206.	Late-time mode mixing, the thermal ratio and temperature fixed by normalized surface gravity, Hadamard short-distance input, renormalized outgoing flux, and mode-dependent transmission to infinity on a fixed collapse background.	The calculation is not a self-consistent evaporating geometry, an endpoint or information-recovery mechanism, an invariant local pair-creation history, or a spectrum at infinity without greybody factors.
When does semiclassical linear response remain controlled?	Primary analysis: Anderson, Molina-París, and Mottola 2003, §§II–III, §IV, and §V. Authoritative review: Hu and Verdaguer 2008, §§2–3.3.1 and §§4.1–5.2.	Retarded, renormalized linear response with Ward identities; a gauge-invariant stability criterion; the stated Minkowski test; and the distinction among mean-field, stochastic, and large-$N$ diagnostics.	Flat-vacuum stability above a declared cutoff is not validity on every background, control of all stress fluctuations, or a statement about ultraviolet completion.
Which results support black-hole entropy and the generalized second law?	Original proposal: Bekenstein 1973, §§II–IV, printed pp. 2334–2343. Primary proof: Wall 2012, §§2.1–2.8 and §§3–5.	The area-entropy and generalized-entropy proposal, and generalized-entropy nondecrease on causal horizons under the paper’s semiclassical, algebraic, stability, coupling, and renormalization assumptions.	The proposal alone does not derive the coefficient or count microstates, and the proof is not an unconditional law for arbitrary trapping surfaces, quantum-gravity corrections, or separately scheme-independent area and exterior entropy.
What can controlled two-dimensional backreaction models establish?	Primary model papers: Callan, Giddings, Harvey, and Strominger 1992, § model construction, Eqs. (1)–(23), printed pp. R1005–R1009 and Russo, Susskind, and Thorlacius 1992, §§1–4, printed pp. 3444–3449.	Explicit two-dimensional dilaton collapse, conformal-matter Hawking flux and large-$N$ backreaction, plus an exactly soluble modified model with a declared counterterm and boundary prescription.	Neither the CGHS setup nor the RST modification establishes a universal endpoint, four-dimensional evaporation dynamics, or generic information recovery.
What is a quantum extremal surface, and where is its prescription controlled?	Primary holographic papers: Faulkner, Lewkowycz, and Maldacena 2013, §§2–4 and Engelhardt and Wall 2015, §§2.1–2.2 and §§3–6.	The leading bulk-entanglement correction to holographic entropy and extremization of renormalized generalized entropy, with perturbative and causal checks in the stated semiclassical holographic expansion.	This is not a nonperturbative theorem for arbitrary gravity, a generic island rule, an all-orders entropy definition, or a complete Page-curve derivation.
Which assumptions drive the information-loss argument and small-corrections bound?	Original argument: Hawking 1976, complete superscattering argument, printed pp. 2460–2473. Analytic synthesis: Mathur 2009, §§4–6.	Pure-to-mixed evolution under Hawking’s semiclassical evaporation premises, and continued entanglement growth under the stated sequential-pair, subsystem-factorization, and small-corrections assumptions.	These arguments are not experimental results or theorems that full quantum gravity is nonunitary; the bound does not cover order-one changes, altered factorizations, or dynamics that violate its stepwise assumptions.
When do replica wormholes produce islands and a Page curve?	Primary gravitational-replica calculations: Almheiri et al. 2020, §§2–6 and Penington, Shenker, Stanford, and Yang 2022, §§2–6 and appendices.	Replica-wormhole saddles, generalized-entropy islands, and Page-curve behavior in specified low-dimensional gravity-plus-bath or related tractable models.	A saddle- and model-dependent result is not a theorem for every four-dimensional evaporating black hole, an unqualified replica continuation, a single-theory microscopic derivation, or state-independent interior reconstruction.
Which black-hole entropy counts and holographic limits are microscopically controlled?	Primary calculations: Strominger and Vafa 1996, complete BPS count, printed pp. 99–104 and Maldacena 1997–1998, §§1–2.	The protected D-brane count for a specified supersymmetric charged extremal black hole and the large-$N$, strong-coupling decoupling limit in which a classical anti-de Sitter bulk description is expected.	This is not a count for generic neutral, nonextremal, astrophysical black holes, equality of every index with an absolute degeneracy, or a proof of holography at finite $N$ and arbitrary coupling.
What does treating general relativity as a quantum effective field theory predict?	Primary EFT analysis: Donoghue 1994, §§II–V.	Gravitational power counting, local higher-derivative coefficients, nonanalytic loop terms, and predictive leading long-distance quantum corrections below the cutoff.	A predictive low-energy expansion is not a UV completion or strong-curvature control, and gauge- or matching-dependent metric and potential terms are not automatically invariant observables.

References

• Almheiri, Ahmed, Thomas Hartman, Juan Maldacena, Edgar Shaghoulian, and Amirhossein Tajdini. “Replica Wormholes and the Entropy of Hawking Radiation.” Journal of High Energy Physics 05 (2020) 013. DOI; arXiv:1911.12333.
• Anderson, Paul R., Carmen Molina-París, and Emil Mottola. “Linear Response, Validity of Semi-Classical Gravity, and the Stability of Flat Space.” Physical Review D 67 (2003) 024026. DOI; arXiv:gr-qc/0209075v1.
• Bär, Christian, Nicolas Ginoux, and Frank Pfäffle. Wave Equations on Lorentzian Manifolds and Quantization. EMS Monographs in Mathematics. European Mathematical Society, 2007. DOI; arXiv:0806.1036.
• Bekenstein, Jacob D. “Black Holes and Entropy.” Physical Review D 7 (1973) 2333–2346. DOI.
• Bisognano, Joseph J., and Eyvind H. Wichmann. “On the Duality Condition for a Hermitian Scalar Field.” Journal of Mathematical Physics 16 (1975) 985–1007. DOI.
• Bisognano, Joseph J., and Eyvind H. Wichmann. “On the Duality Condition for Quantum Fields.” Journal of Mathematical Physics 17 (1976) 303–321. DOI.
• Callan, Curtis G., Jr., Steven B. Giddings, Jeffrey A. Harvey, and Andrew Strominger. “Evanescent Black Holes.” Physical Review D 45 (1992) R1005–R1009. DOI; arXiv:hep-th/9111056.
• Donoghue, John F. “General Relativity as an Effective Field Theory: The Leading Quantum Corrections.” Physical Review D 50 (1994) 3874–3888. DOI; arXiv:gr-qc/9405057.
• Engelhardt, Netta, and Aron C. Wall. “Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime.” Journal of High Energy Physics 01 (2015) 073. DOI; arXiv:1408.3203v3.
• Faulkner, Thomas, Aitor Lewkowycz, and Juan Maldacena. “Quantum Corrections to Holographic Entanglement Entropy.” Journal of High Energy Physics 11 (2013) 074. DOI; arXiv:1307.2892.
• Fredenhagen, Klaus, and Rudolf Haag. “On the Derivation of Hawking Radiation Associated with the Formation of a Black Hole.” Communications in Mathematical Physics 127 (1990) 273–284. DOI; Open PDF.
• Hawking, Stephen W. “Breakdown of Predictability in Gravitational Collapse.” Physical Review D 14 (1976) 2460–2473. DOI.
• Hawking, Stephen W. “Particle Creation by Black Holes.” Communications in Mathematical Physics 43 (1975) 199–220; erratum 46 (1976) 206. DOI; erratum; Open PDF.
• Hollands, Stefan, and Robert M. Wald. “Local Wick Polynomials and Time Ordered Products of Quantum Fields in Curved Spacetime.” Communications in Mathematical Physics 223 (2001) 289–326. DOI; arXiv:gr-qc/0103074.
• Hu, Bei-Lok, and Enric Verdaguer. “Stochastic Gravity: Theory and Applications.” Living Reviews in Relativity 11 (2008) 3. DOI; arXiv:0802.0658v1.
• Maldacena, Juan. “The Large N Limit of Superconformal Field Theories and Supergravity.” Advances in Theoretical and Mathematical Physics 2 (1998) 231–252. DOI; arXiv:hep-th/9711200.
• Mathur, Samir D. “The Information Paradox: A Pedagogical Introduction.” Classical and Quantum Gravity 26 (2009) 224001. DOI; arXiv:0909.1038.
• Page, Don N. “Particle Emission Rates from a Black Hole: Massless Particles from an Uncharged, Nonrotating Hole.” Physical Review D 13 (1976) 198–206. DOI.
• Penington, Geoff, Stephen H. Shenker, Douglas Stanford, and Zhenbin Yang. “Replica Wormholes and the Black Hole Interior.” Journal of High Energy Physics 09 (2022) 002. DOI; arXiv:1911.11977.
• Radzikowski, Marek J. “Micro-Local Approach to the Hadamard Condition in Quantum Field Theory on Curved Space-Time.” Communications in Mathematical Physics 179 (1996) 529–553. DOI.
• Russo, Jorge G., Leonard Susskind, and Larus Thorlacius. “End Point of Hawking Radiation.” Physical Review D 46 (1992) 3444–3449. DOI; arXiv:hep-th/9206070.
• Strominger, Andrew, and Cumrun Vafa. “Microscopic Origin of the Bekenstein–Hawking Entropy.” Physics Letters B 379 (1996) 99–104. DOI; arXiv:hep-th/9601029.
• Unruh, William G. “Notes on Black-Hole Evaporation.” Physical Review D 14 (1976) 870–892. DOI.
• Wall, Aron C. “A Proof of the Generalized Second Law for Rapidly Changing Fields and Arbitrary Horizon Slices.” Physical Review D 85 (2012) 104049. DOI; arXiv:1105.3445v6.