Interacting Fields and Local Covariant Renormalization

Interacting QFT on a curved background must renormalize coincident fields without relying on a preferred vacuum or global momentum space. Local covariance and causal factorization organize the allowed subtractions and their finite ambiguities.

Enter this chapter

Begin with Why Interacting Curved-Space QFT Requires Local and Covariant Renormalization to understand why ultraviolet subtraction must respect local geometry.

Helpful background. Products, Scaling Degree, and Extensions of Singular Distributions and Local Counterterms and Subdivergence Structure develop the definitions used by the chapter’s opening pages.

Topics

1. Why Interacting Curved-Space QFT Requires Local and Covariant Renormalization — Why do interacting fields on arbitrary curved backgrounds require renormalization prescriptions that are local, covariant, causal, and compatible with background embeddings?
2. Interacting Correlators and Curved-Space Time-Ordered Products — How are interacting curved-space correlators and time-ordered products defined perturbatively while controlling coincident singularities, causal ordering, and state dependence?
3. Curvature Counterterms and Composite-Operator Mixing — Which curvature-dependent local counterterms and operator-mixing coefficients are forced when interacting fields and composite insertions are renormalized on curved backgrounds?
4. Local S-Matrices and Causal Factorization — How do compactly supported interactions and causal factorization define local S-matrices when a global scattering matrix or adiabatic switching limit may not exist?
5. Running Couplings and Curvature Couplings — How do matter, nonminimal, and gravitational background couplings run under scale changes, and which beta functions are scheme dependent or physically matched?
6. Background Splitting and Perturbative Agreement — When do different splits between background and perturbation describe the same interacting curved-space theory, and what finite renormalizations enforce that agreement?
7. Interacting Stress Tensors and Ward Identities — How is the interacting stress tensor renormalized so that diffeomorphism Ward identities, conservation, operator mixing, and background response remain mutually consistent?
8. Gauge, BRST, and BV Interfaces on Curved Backgrounds — How are BRST and BV data deployed for interacting gauge fields on curved backgrounds while preserving local covariance, causal renormalization, and physical cohomology?
9. Curved-Space OPE and Local Short-Distance Expansions — How does the operator-product expansion organize interacting curved-space short-distance behavior using local geometry, state-independent coefficients, and controlled remainders?
10. Adiabatic Limits and Infrared Obstructions — When can compactly supported interacting constructions approach constant coupling, and which massless modes, long-range correlations, or geometry obstruct that adiabatic limit?
11. Interacting Scalar Theory on a Curved Background: Worked Benchmark — Can an interacting scalar theory on a simple curved background satisfy one coherent ledger for state choice, counterterms, Ward identities, observables, errors, and limits?

How the ideas fit together

Separated-point correlation functions become singular when insertions coincide. Extending them to the diagonal introduces renormalization freedom, but locality, covariance, and scaling restrict that freedom to appropriate local field and curvature terms Hollands and Wald 2001, §§ 4–5.

Causal factorization constructs perturbative local observables with compactly supported interactions. Removing the switching function everywhere is a further infrared question, not part of the ultraviolet extension. A successful local construction therefore need not yield a global interacting vacuum, and matter-loop curvature counterterms do not imply that gravitons have been quantized.

The comparison below separates the assumptions and tests needed for these different conclusions.

Domain and failure conditions

Claimed result	Domain that must be declared	Required check	Failure condition and surviving claim
Renormalized local product	Convex normal neighborhoods or globally hyperbolic region; field content; diagonal being extended	Wavefront compatibility off the diagonal and a local covariant extension with bounded scaling degree	If the unextended product is undefined, only the separated-point distribution survives.
Interacting observable	Compact support of interaction and observable; perturbative order	Causal factorization and independence from switching changes outside the relevant causal region	Failure of a constant-coupling limit does not remove the compactly supported local observable.
Counterterm classification	Dimension, symmetries, equations-of-motion quotient, boundary assumptions	Complete local scalar/tensor basis including curvature terms	A flat-space list supports only the flat or zero-curvature restriction.
Running coupling	Coupling normalization, subtraction scheme, operator basis, scale convention	Transform beta functions and operators under every finite redefinition used in the comparison	An individual beta coefficient is not comparable until the coupling coordinates are matched.
Split-independent prediction	Two free/interacting splits and the map between their propagators, states, and counterterms	Perturbative agreement through the stated order	Holding the state or finite terms fixed after changing the split proves only a mismatch of descriptions.
Stress-tensor identity	Definition by metric variation, insertion convention, contact terms, anomaly class	Distributional Ward identity, not merely conservation away from insertions	If contacts are omitted, separated-point conservation is the strongest remaining statement.
Gauge-independent observable	Gauge fixing, ghost/antifield complex, support, anomaly cohomology	Renormalized master identity and BRST-closed observable modulo exact terms	Coordinate covariance alone does not imply gauge independence.
OPE truncation	Coalescence scaling, operator basis, perturbative and engineering-dimension order, class of Hadamard states	Remainder estimate and associativity on overlapping fusion limits	A fit to one raw state-dependent correlator is not a state-independent Wilson coefficient.
Global interacting state	Geometry, spectrum or mass gap, switching sequence, state topology	Uniform infrared control beyond every fixed compact region	Local algebraic convergence alone yields a local net, not a global vacuum or KMS state.

Where to go next

Continue to Particles, Detectors, and Nonadiabatic Production to choose between mode occupation, detector response, and stress-energy observables.

References

• 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:10.1007/s002200100540.