Perturbative AQFT and Causal Renormalization

Causal perturbation theory constructs interactions by extending distributions at coincident points. This chapter connects that local analytic problem to renormalization freedom, formal interacting observables, and algebraic local nets.

Enter this chapter

Read causal factorization and the functional domain before Epstein–Glaser induction. The formal power-series setting matters throughout. Helpful background. Products, scaling degree, and distribution extensions supplies the analytic extension theorem. Counterterms, subdivergences, and locality supplies the physical renormalization comparison. Adiabatic limits and infrared obstructions fixes the boundary between local construction and global switching limits.

Topics

1. Causal Factorization and Time-Ordered Products — Determines time-ordered products away from coincident configurations.
2. Local and Microcausal Functionals with Peierls Brackets — States the functional regularity needed for causal brackets and contractions.
3. Epstein–Glaser Induction — Builds successive perturbative orders from lower-order causal data.
4. Scaling Degree and Extension of Distributions — Bounds extension ambiguities by scaling degree.
5. Time-Ordered Products, Causal Wick Expansion, and Renormalization — Combines diagonal extensions with Wick expansion and normalization conditions.
6. Renormalization Freedom and the Stückelberg–Petermann Group — Organizes changes of prescription as local formal maps.
7. Perturbative AQFT and the Bogoliubov Map — Constructs retarded interacting observables from relative S-matrices.
8. Perturbative Agreement and Background Independence — Compares quadratic splittings and controlled changes of background.
9. Ward Identities and Anomalous Obstructions — Separates removable normalization defects from cohomological obstructions.
10. Adiabatic Limits and Algebraic Interacting Nets — Distinguishes local switching independence from a global infrared limit.
11. Nonperturbative Local C*-Net Developments: Construction and Scope — Identifies what separate operator-algebra constructions actually establish.
12. Formal-Power-Series Interacting Constructions, States, and Scope — States the formal theory’s positivity, convergence, and representation limits.

How the ideas fit together

Causality fixes distributions off diagonals; extension to the diagonals supplies local counterterms Epstein and Glaser 1973, §§ 3–4; Brunetti and Fredenhagen 2000, Theorems 5.2–5.3. Admissible prescriptions are related by a local formal renormalization map Brunetti, Dütsch, and Fredenhagen 2009, Theorem 4.1. The functional space also needs closure: equicausal estimates address failures of smoothness left by pointwise wavefront restrictions Hawkins, Rejzner, and Visser 2026, §§ 4–7 (PDF). None of these steps proves convergence of the perturbation series.

What each hypothesis licenses

Objects, hypotheses, licensed conclusions, and boundary tests

Object and domain	Essential hypotheses	Licensed conclusion	Excluded converse or extension	Adversarial check
Equicausal functionals and Peierls kernel	Normally hyperbolic linearization; compact functional support; Hörmander-compatible microcausal cones plus equicontinuity for every derivative family.	The retarded and advanced variations and the Peierls bracket are defined and remain smooth on the declared functional domain.	Pointwise microcausality alone need not give smooth closure, and locality does not make arbitrary distributional smearings admissible.	Use the regular-functional counterexample to smooth closure, or smear on a characteristic surface so covectors sum to zero.
Local S-matrix $S(F)$ and coefficients $T_n$	Compact supports, a specified causal order, normalized unit, and causal factorization.	Factorization in ordered configurations and inductive determination of $T_n$ away from coincidence diagonals.	No factorization formula is licensed for overlapping causally unordered supports.	Choose two overlapping supports with neither later than the other and attempt to swap them.
Off-diagonal distribution $t^0$	Finite scaling degree at the diagonal and lower-order extensions already compatible on partial diagonals.	An extension exists with controlled local ambiguity; it is unique only below the codimension threshold.	Existence does not imply uniqueness at $\operatorname{sd}(t^0)\ge d$, nor does choosing extensions independently preserve causal induction.	Extend $(\Delta_F)^2$ in four dimensions and exhibit its delta-function ambiguity.
Renormalized $T_n$ and comparison map $Z$	Causal Wick expansion, locality, additivity, field independence, covariance, scaling, and the same declared basic axioms for two prescriptions.	Finite differences are local diagonal counterterms organized by a Stückelberg–Petermann transformation.	A momentum-dependent nonlocal redefinition is not licensed, and a change of Hadamard normal ordering must be re-expanded consistently.	Insert a nonlocal kernel into $Z$ and test additivity for disjointly supported interactions.
Bogoliubov map, quadratic background shift, or Ward differential	Compact interaction; controlled Møller maps for perturbative agreement; local anomaly consistency and an admissible counterterm class for Ward identities.	Formal retarded observables, equivalence for the proved background split, and restoration of a Ward identity exactly when its local class is trivial.	The formal series need not converge; arbitrary noncompact metric changes are not covered; consistency does not force an anomaly class to vanish.	Treat the series as norm convergent, use an uncontrolled kinetic shift, or cancel one current divergence while breaking consistency.
Local interacting algebra $\mathfrak A_g(O)$	Switchings agree on a causal neighborhood of $O$ and relative-S intertwiners obey the cocycle relation.	The local algebra is independent of switching up to canonical isomorphism and fits into an isotonic local net.	Local stabilization does not imply a global constant-coupling S-matrix, vacuum, KMS state, or massless adiabatic limit.	Let a massless switching sequence grow and observe infrared-dependent coefficients while fixed bounded regions remain equivalent.
Formal state or separately constructed local operator algebra	For a formal state, ordered-series positivity and coefficientwise microlocal conditions; for a factorizing model, regular-strip analyticity and modular nuclearity; for a dynamical C*-algebra, its exact defining relations.	A normalized positive formal functional, or the particular nonperturbative local-net conclusion proved by the separate theorem.	Negative numerical truncations do not refute formal positivity; wedge locality alone does not prove compact intersections; neither construction proves general four-dimensional interacting existence.	Evaluate $1-2\lambda$ at $\lambda=1$, remove the regularity strip, or demand an unspecified vacuum representation.

Structured table data (JSON).

Review the chapter

An off-diagonal distribution has scaling degree strictly below the diagonal’s codimension. Is there a local extension ambiguity at that scaling degree?

Solution

No. The extension preserving that scaling degree is unique. Delta distributions on the diagonal and their derivatives have scaling degree at least the codimension, so none can be added while retaining the stated bound.

Where to go next

Continue to microlocal QFT for the singularity criteria behind the allowed contractions and locally covariant composite fields.

References

• Brunetti, Romeo, Michael Dütsch, and Klaus Fredenhagen. “Perturbative Algebraic Quantum Field Theory and the Renormalization Groups.” Advances in Theoretical and Mathematical Physics 13 (2009): 1541–1599. DOI; Open PDF.
• Brunetti, Romeo, and Klaus Fredenhagen. “Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds.” Communications in Mathematical Physics 208 (2000): 623–661. DOI.
• Epstein, Henri, and Vladimir Glaser. “The Role of Locality in Perturbation Theory.” Annales de l’Institut Henri Poincaré A 19, no. 3 (1973): 211–295. EuDML record.
• Hawkins, Eli, Kasia Rejzner, and Berend Visser. “A Novel Class of Functionals for Perturbative Algebraic Quantum Field Theory.” Revised 2026. Open manuscript, arXiv:2312.15203.