Equivalence of Axiomatic QFT Frameworks

The question is: Under which hypotheses can Wightman, Haag–Kastler, Euclidean, perturbative algebraic, and functorial formulations of QFT be reconstructed from one another? There are rigorous equivalences and reconstruction functors between restricted object classes. There is no general theorem making all these frameworks equivalent for every interacting QFT: they retain different data, use different notions of morphism and completion, and often live at different levels—nonperturbative Hilbert-space theory versus formal deformation theory.

Evidence cutoff. 11 August 2026.

Required background. QFT classification and comparison problems supplies the requirement to compare categories rather than slogans. Frameworks, object classes, and maps supplies the domain/codomain language used below.

Helpful background. Osterwalder–Schrader reconstruction gives the Euclidean-to-Lorentzian theorem. Wightman reconstruction gives the field-from-correlators step. Wightman, Euclidean, local algebraic, constructive, and perturbative frameworks identifies what each formulation retains. Operator algebras and positive functionals supplies the GNS bridge from algebraic observables to Hilbert-space representations.

Equivalence must specify objects and retained data

An equivalence claim needs categories $\mathcal C$ and $\mathcal D$, functors

$$F:\mathcal C\rightleftarrows\mathcal D:G,$$

and natural isomorphisms—or homotopy-coherent analogues—$GF\simeq\mathrm{id}_{\mathcal C}$ and $FG\simeq\mathrm{id}_{\mathcal D}$. A map that preserves vacuum correlators but forgets charged sectors, curved-spacetime covariance, field coordinatizations, or topology on observable spaces is not automatically an equivalence of physical theories.

Framework	Primary object	Data that can be forgotten by a common comparison
Wightman	Operator-valued distributions on a Hilbert space with vacuum and Poincaré action	Choice of generating fields when passing to a local net
Haag–Kastler	Isotonic net of local operator algebras, usually in a representation	Point-field coordinatization and off-shell/time-ordered products
Euclidean OS	Euclidean Schwinger functions or measure with reflection positivity	Real-time presentation until analytic reconstruction; gauge-positive subspace can be subtle
Locally covariant/functorial AQFT	Covariant assignment of algebras to spacetimes and embeddings	A preferred vacuum representation on every spacetime
Perturbative AQFT	Formal power-series algebras and time-ordered products	Nonperturbative convergence, a positive state at finite coupling, and global phase information
Factorization/prefactorization algebra	Observables with disjoint-union products and descent structure	A Lorentzian causal product unless time ordering and time-slice structure are supplied

Reconstruction results that are established

Euclidean to Wightman. Osterwalder and Schrader reconstruct a positive-energy relativistic theory from Euclidean Green functions satisfying Euclidean covariance, symmetry, reflection positivity, regularity/growth, and clustering hypotheses Osterwalder and Schrader 1973; corrected and extended formulations appear in Osterwalder and Schrader 1975. Reflection positivity is the essential bridge to Hilbert-space positivity; Euclidean invariance and analyticity alone are insufficient.

Fields to nets and partial inverse. A Wightman theory generates local von Neumann algebras from bounded functions of smeared fields. The net can forget which point fields were chosen. Under phase-space/energy bounds, pointlike localized fields can be recovered from the net Fredenhagen and Hertel 1981, but this is an additional theorem domain rather than an inverse for arbitrary nets.

Minkowski nets to locally covariant theories. The functorial formulation makes covariance across globally hyperbolic spacetimes part of the object Brunetti, Fredenhagen, and Verch 2003. A single Minkowski net does not uniquely determine this curved-spacetime extension; naturality, time-slice behavior, and locally covariant fields are extra structure.

AQFT and prefactorization algebras. For additive, Cauchy-constant Lorentzian theories with the appropriate time-orderability hypotheses and an ordinary symmetric monoidal target, explicit comparison functors yield an equivalence Benini, Perin, and Schenkel 2020. A 2026 result generalizes the 1-categorical strategy and reduces the cochain-valued $\infty$-categorical problem to spacetime-wise problems, but explicitly leaves a key $\infty$-localization step open Benini et al. 2026.

Perturbative comparison. The observables of a perturbative AQFT form a factorization algebra under stated geometric and renormalization choices Gwilliam and Rejzner 2022. This compatibility result does not turn a formal perturbative theory into a nonperturbative Wightman theory.

Claim-by-claim assessment

Claim	Direct assessment	Decisive qualification
OS and Wightman formulations are equivalent.	Yes for the theorem’s tempered/regular Euclidean and relativistic object classes.	Gauge theories, massless infrared behavior, or non-tempered objects may require modified spaces and positivity statements.
Wightman fields and Haag–Kastler nets contain identical information.	Only with added reconstruction and completeness hypotheses.	Passing to the net forgets field coordinates; sectors and field algebras may require Doplicher–Haag–Roberts-type data beyond the observable net.
A Minkowski AQFT is the same as a locally covariant QFT.	No in general.	The latter specifies coherent behavior on a category of spacetimes, not just one object.
Lorentzian AQFT and factorization algebras are equivalent.	Proved in a restricted 1-categorical, additive, time-slice domain.	Gauge/BV examples are naturally cochain-valued; the corresponding full $\infty$-categorical equivalence remains open as of the cutoff.
Perturbative and nonperturbative axiomatic QFT are equivalent.	Not established.	Formal power series need not converge, define positive finite-coupling states, or encode nonperturbative sectors and phases.

Obstructions and method limits

Equivalence can fail through lost structure even when all shared correlation functions agree. Observable nets can share a vacuum sector while differing in charged field extensions. Euclidean gauge-fixed correlators can violate ordinary reflection positivity although gauge-invariant observables may admit a positive reconstruction. Curved-spacetime theories need not possess a preferred state, so a Wightman-style vacuum representation is not functorial data.

Categorical equivalence is also sensitive to the morphisms. If morphisms ignore embeddings, renormalization-group maps, or natural transformations of fields, the resulting theorem may be mathematically correct but answer a weaker physical question. Homotopy-coherent gauge complexes cannot generally be replaced by their cohomology before checking descent and time-slice compatibility.

Status — a network of partial equivalences, not a universal equivalence. OS/Wightman reconstruction and several algebraic/factorization comparisons are rigorous within explicit hypotheses. The all-framework claim fails as stated because object classes and retained data do not match; important gauge, curved-spacetime, and nonperturbative comparison problems remain open.

What would count as a stronger resolution

A convincing general comparison program should:

1. state each category’s objects, morphisms, topology/completion, states, and gauge equivalences;
2. construct functors in both directions and identify exactly which data are forgotten;
3. prove full faithfulness and essential surjectivity, or the appropriate derived/$\infty$-categorical analogues;
4. preserve locality, covariance, positivity, time evolution, superselection sectors, and renormalized observables; and
5. test the functors on interacting examples for which both sides exist, including gauge and curved-spacetime cases.

Research connections

The natural field context is mathematical and constructive QFT. Replica, modular, and operator-algebra methods supply the algebra/representation bridge, while Euclidean lattice and continuum extrapolation supplies concrete OS input when a measure exists. Comparisons should verify each theorem direction separately: reconstruct a Hilbert space and fields from Wightman distributions, check the full Osterwalder–Schrader hypotheses before analytically continuing Euclidean correlators, and state the categorical, locality, and descent assumptions needed for any factorization-algebra or functorial equivalence.

Literature selection and cutoff

The finite source set below was selected through targeted theorem, journal, and arXiv searches for the main reconstruction directions, the field-to-net bridge, locally covariant structure, model-independent AQFT/factorization equivalences, and the latest published extension available through 11 August 2026. Results were classified by their exact object category; illustrative correspondences without quasi-inverse or reconstruction statements were not used to set status. The selection is not exhaustive.

References

• Benini, M., Carmona, V., Grant-Stuart, A., and Schenkel, A. (2026). “On the Equivalence of AQFTs and Prefactorization Algebras.” Letters in Mathematical Physics 116, 13. DOI.
• Benini, M., Perin, M., and Schenkel, A. (2020). “Model-Independent Comparison between Factorization Algebras and Algebraic Quantum Field Theory on Lorentzian Manifolds.” Communications in Mathematical Physics 377, 971–997. DOI; arXiv:1903.03396.
• Brunetti, R., Fredenhagen, K., and Verch, R. (2003). “The Generally Covariant Locality Principle: A New Paradigm for Local Quantum Field Theory.” Communications in Mathematical Physics 237, 31–68. DOI; arXiv:math-ph/0112041.
• Fredenhagen, K., and Hertel, J. (1981). “Local Algebras of Observables and Pointlike Localized Fields.” Communications in Mathematical Physics 80, 555–561. DOI.
• Gwilliam, O., and Rejzner, K. (2022). “The Observables of a Perturbative Algebraic Quantum Field Theory Form a Factorization Algebra.” arXiv:2212.08175.
• Osterwalder, K., and Schrader, R. (1973). “Axioms for Euclidean Green’s Functions.” Communications in Mathematical Physics 31, 83–112. DOI.
• Osterwalder, K., and Schrader, R. (1975). “Axioms for Euclidean Green’s Functions II.” Communications in Mathematical Physics 42, 281–305. DOI.