Curved-Spacetime AQFT, Stress Tensor, and Energy Inequalities

Curved-spacetime QFT links causal propagation to local observables and sampled energy bounds. The chapter makes explicit which extra hypotheses are needed before these results can enter a focusing argument or semiclassical field equation.

Enter this chapter

Start with the free algebra and Hadamard states. Keep the geometry, coupling, state class, and sampling curve fixed when comparing later inequalities. Helpful background. Renormalized stress-tensor axioms and ambiguities provides the physical observable. Quantum energy inequalities provides the sampled-energy motivation. The semiclassical Einstein equation provides the backreaction problem.

Topics

1. Algebraic Free Fields on Curved Spacetimes — Constructs free algebras from causal propagation on a given geometry.
2. Existence, Deformation, and Gluing of Hadamard States — Proves state existence without selecting a universal preferred vacuum.
3. The Renormalized Stress Tensor: Conservation and Ambiguities — Classifies stress-tensor conservation conditions and finite local freedom.
4. Trace Anomalies, Local Covariance, and Scaling — Relates scaling to anomalous trace terms.
5. Absolute and Difference Quantum Energy Inequalities — Separates reference-state comparisons from absolute lower bounds.
6. Worldline, Spacetime-Volume, and Null-Energy Bounds — Explains why changing the sampling support changes the theorem.
7. Averaged Null Energy and QNEI Geometric Hypotheses — States the additional geometry required for null-energy results.
8. Quantum Energy Inequalities and Singularity-Theorem Interfaces — Tracks the extra assumptions needed to infer focusing.
9. Semiclassical Einstein Equation: Existence and Stability Results — Identifies existence and stability statements for the coupled mean-field equation.
10. Interacting pAQFT on Curved Spacetimes — Places curved-background interactions in their formal perturbative setting.
11. Boundaries, Nonglobally Hyperbolic Spacetimes, and State Obstructions — Explains what boundary conditions and causal structure must be supplied.

How the ideas fit together

The common starting point is causal propagation and a Hadamard state. Renormalized local fields retain finite curvature freedom Hollands and Wald 2001, Theorems 5.1–5.2. Positivity can then bound a specified smeared energy observable, but the bound belongs to its field, sampler, and geometry Fewster 2012, §§ 1–3 (PDF). A timelike-worldline bound does not automatically extend to a null curve, and a valid stress tensor alone does not prove that the semiclassical equation has a stable solution.

Hypotheses and licensed conclusions

Domains, required hypotheses, directional results, and failure tests in curved-spacetime AQFT

Object and domain	Required hypotheses	Licensed result	Excluded converse or upgrade	Adversarial check
Real free field on a boundaryless spacetime	Global hyperbolicity; formally self-adjoint Green-hyperbolic operator; quotient by the field-equation image	Nondegenerate symplectic space, CCR algebra, causal covariance, and time-slice	The algebra does not select a representation, vacuum, or Hadamard state	Retain $P C_0^\infty$ and exhibit the radical of the proposed symplectic form
Quasifree Klein–Gordon state	Globally hyperbolic metric; positive ultrastatic endpoint covariance or equivalent construction; Cauchy transport; propagation of singularities	Existence of Hadamard states on the target spacetime	Existence supplies neither stationarity nor a preferred natural state	Demand invariance on a future metric with no timelike Killing field
Renormalized scalar stress tensor in four dimensions	Hadamard state; local covariance; conservation; scaling and regularity; fixed subtraction scale and boundaryless domain	Finite state-dependent expectation plus conserved local curvature freedom; conformal trace anomaly	Conservation alone does not give uniqueness, and the $\Box R$ anomaly coefficient is not universal	Add a finite $R^2$ counterterm and track the shifted trace and gravitational coupling
Sampled energy or null contraction	Declared field and dimension; Hadamard state class; exact timelike or null domain; smooth sampler and normalization	Timelike free-field QEI; two-dimensional chiral null bound; theorem-specific volume bounds	No state-independent pointwise bound or general compact null-geodesic QEI in four dimensions	Narrow a timelike sampler, or take an infinite boost while holding the wrong normalization fixed
Complete null average or focusing argument	Affine completeness and limiting prescription; theorem-specific achronality, causal tube, field, state, initial contraction, and global causality	ANEC or a focal/incompleteness conclusion only under the named theorem’s full inputs	ANEC does not bound a finite negative pulse or imply a singularity without geometry	Truncate the geodesic, or use a noncontracting Minkowski congruence
Semiclassical equation or interacting perturbation	Matched renormalized couplings; conserved Hadamard source; state–geometry data; gauge and causal response; or compact interaction and formal series	Special FLRW existence results, controlled linear stability, or local pAQFT algebras order by order	No generic nonlinear well-posedness, convergent perturbation series, or global interacting vacuum follows	Count Planckian runaway roots outside the approximation, or set the coupling constant globally without an infrared theorem
Scalar field with timelike boundary	Fixed positive self-adjoint extension; boundary-preserving maps; boundary-adapted microlocal state condition	Extension-dependent dynamics, propagator, algebra, and ground state in the proved static model	The bulk differential expression does not determine a unique theory, and one Robin model is not generic boundary covariance	Compare two admissible extensions and their different spectra

Structured table data (JSON).

Review the chapter

Why can a timelike sampling inequality not simply be evaluated on a null worldline?

Solution

The pullback and positivity estimates depend on the sampling geometry. A null limit can change the wavefront transversality and destroy uniform bounds. One needs a separate null theorem or a controlled limiting argument with its own hypotheses.

Where to go next

Continue to factorization algebras for another organization of local observables, or return to locally covariant QFT for the background-response structure.

References

• Fewster, Christopher J. “Lectures on Quantum Energy Inequalities.” In Quantum Field Theory and Gravity, edited by Felix Finster et al., 97–156. Basel: Birkhäuser, 2012. DOI; 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; Open PDF.