QFT Frameworks, Object Classes, and Typed Maps

The major mathematical formulations of QFT do not begin with the same kind of object. Wightman theory begins with fields on a Hilbert space, Euclidean theory with correlation distributions or a measure, algebraic QFT with local observable algebras, locally covariant QFT with a functor, perturbative algebraic QFT with formal algebras of functionals, and factorization theory with a local-to-global cochain-level structure. A comparison is therefore a typed map with hypotheses—not an identification of terminology.

Required background. Theorem-First Claim Records supplies the claim grammar used for every arrow below. Helpful background. Categories, Functors, Natural Transformations, and Universal Properties supplies the categorical language; Wightman, Euclidean, Local-Algebraic, Constructive, and Perturbative Frameworks gives the physical comparison; Derived, Higher, and Factorization Frameworks: a Boundary Map orients the cochain-level constructions; and What Is a Quantum Field Theory? introduces the physical objects.

Six frameworks and their primitive objects

The following atlas records the minimal type information. Individual authors use variants, so a theorem must still cite its chosen definitions.

Framework	Primitive object	Typical morphism or comparison datum	Native conclusion
Wightman	$(\mathcal H,U,\Omega,\mathcal D,\{\phi_a\})$, with operator-valued tempered distributions on a common invariant dense domain	a unitary intertwining $U$, the vacuum, domains, and all smeared fields	distributional covariance, spectrum support, positivity, locality, reconstruction from vacuum functions
Euclidean or constructive	Schwinger distributions $S_n$, a characteristic functional $Z$, or a probability measure on a distribution space	pushforward of measures; equality of all moments under determinacy; analytic continuation or reconstruction under Euclidean axioms	Euclidean invariance, reflection positivity, regularity, clustering, construction of a measure
Haag–Kastler	an isotonic net $O\mapsto\mathcal A(O)$ of unital $C^*$- or von Neumann algebras, often with a state and covariance action	a compatible family of injective $*$-homomorphisms, or an isomorphism of nets preserving state and covariance	locality and inclusion relations for bounded observables
Locally covariant QFT	a covariant functor $\mathcal A:\mathbf{Loc}\to\mathbf{Alg}$	a natural transformation between functors; embeddings in $\mathbf{Loc}$ induce algebra morphisms	covariance across spacetimes, time-slice behavior, relative Cauchy evolution
pAQFT	functionals on a configuration space, their wavefront restrictions, and $\mathbb C[[\hbar,\lambda]]$-valued products	formal $*$-isomorphisms, renormalization maps, and natural transformations	coefficientwise perturbative construction and local covariance
Factorization algebra	a prefactorization assignment $U\mapsto\mathrm{Obs}(U)$ to cochain complexes, with products for disjoint opens and local-to-global descent	a natural transformation compatible with structure maps; often weak equivalence means objectwise quasi-isomorphism	support-sensitive multiplication, descent, and cohomological observables

These rows live at different levels. A Wightman field is generally unbounded; a Haag–Kastler net is built from bounded operators or abstract algebras. A locally covariant theory varies the spacetime; an ordinary net holds one spacetime fixed. A pAQFT algebra is commonly a formal power series and need not possess a nonperturbative value at $\lambda\neq0$. A factorization algebra retains cochain data that disappears after taking degree-zero cohomology.

The original algebraic formulation already emphasizes that the local assignment and its representation are separate pieces Haag and Kastler 1964, §§ 2–3, pp. 848–856. In the locally covariant formulation, the categories themselves are part of the definition: oriented, time-oriented globally hyperbolic spacetimes and causal isometric embeddings form the source, while unital algebras and injective $*$-homomorphisms form the target Brunetti, Fredenhagen, and Verch 2003, § 2, pp. 35–41. Changing either category changes the theory.

Directional arrows, not a universal equivalence

Several important maps exist, but each has a restricted domain.

Euclidean data to Wightman data

A sequence of Euclidean Green functions satisfying the selected Osterwalder–Schrader hypotheses can be analytically continued and reconstructed into Wightman distributions, a Hilbert space, a vacuum, and fields. Reflection positivity supplies a positive semidefinite form; quotienting its null space and completing gives the Hilbert space. Euclidean time translations yield a positive Hamiltonian, and analytic continuation supplies the relativistic vacuum distributions. The corrected theorem uses regularity or growth assumptions in addition to Euclidean invariance and reflection positivity Osterwalder and Schrader 1975, §§ II and IV, pp. 283–304.

This is a map on the subcategory of Euclidean data satisfying those hypotheses. It is not a map from every Euclidean-invariant measure to a unitary relativistic QFT. Nor does it say that an arbitrary Wightman model has a Euclidean measure representation of the same kind.

Wightman fields to local nets

Given Wightman fields, one may associate to a suitable region $O$ the algebra generated by bounded functions of smeared fields with support in $O$, for example Weyl operators in a free scalar theory. Isotony and locality follow after the domain, closure, and commutation questions are settled. The arrow forgets information: different field coordinatizations can generate the same observable net, charged fields need not belong to the observable algebra, and a bare net does not specify a preferred state or pointlike fields. An inverse therefore requires additional reconstruction hypotheses.

A fixed-spacetime net from a locally covariant functor

For a fixed spacetime $M$, restrict the functor $\mathcal A$ to causally suitable subregions $O\subset M$ and the inclusion embeddings. Then

$$O\longmapsto \mathcal A(O), \qquad \iota_{O_1O_2}\longmapsto\mathcal A(\iota_{O_1O_2})$$

is a net. The reverse direction is not automatic: a net on one background does not say how algebras change under embeddings into other backgrounds, nor does it determine relative Cauchy evolution. Functoriality across all source objects is extra data.

pAQFT and factorization observables

For a free theory governed by a Green-hyperbolic operator on a globally hyperbolic spacetime, Gwilliam and Rejzner construct natural transformations comparing the pAQFT and factorization-algebra models. At the classical level the relevant transformation is an isomorphism after the stated forgetful operations; at the quantum level it is an isomorphism of cochain complexes, and the time-ordered product mediates the algebra structures Gwilliam and Rejzner 2020, Theorems 3.5–3.6, pp. 122–124. The time-slice axiom is then essential to the sharper comparison of algebra structures Gwilliam and Rejzner 2020, § 3.4, pp. 127–131.

That theorem does not establish equivalence of all interacting AQFTs with all factorization algebras. Its objects, Green-hyperbolicity, free-theory scope, cochain complexes, restrictions to causally convex opens, and forgetful functors are part of the conclusion.

The massive free scalar through four incarnations

Let $m>0$. The free scalar is a useful comparison object because every arrow can be written explicitly.

Wightman incarnation. On bosonic Fock space, the smeared field $\phi(f)$ is an operator on the finite-particle domain. Its vacuum two-point distribution is

$$W_2(f,g)=\int\frac{d^4p}{(2\pi)^3}\, \theta(p^0)\delta(p^2-m^2)\,\widetilde f(-p)\widetilde g(p).$$

The support of its Fourier transform lies on the positive mass shell. Higher vacuum functions obey Wick’s rule. The operator-valued distribution and common domain are essential parts of the object, as in the Gårding–Wightman formulation Gårding and Wightman 1964, pp. 129–189.

Schwinger incarnation. In positive Euclidean metric, take the centered Gaussian measure on $\mathcal S'(\mathbb R^4)$ with covariance

$$C_m(f,g)=\int\frac{d^4p}{(2\pi)^4} \frac{\widetilde f(-p)\widetilde g(p)}{p_E^2+m^2}.$$

Its moments are the Schwinger functions. Time reflection and the positive spectral representation of $(p_E^2+m^2)^{-1}$ establish reflection positivity. Osterwalder–Schrader reconstruction returns a Wightman realization equivalent to the usual free scalar after conventions and normalizations are matched.

Weyl-net incarnation. Let $\mathscr E(M)$ be the real solution space modulo the Klein–Gordon equation with causal symplectic form $\sigma$. The Weyl algebra is generated by $W([f])$ with

$$W([f])W([g])=e^{-i\sigma([f],[g])/2}W([f]+[g]).$$

For a region $O$, define $\mathcal A(O)$ from generators with $\operatorname{supp}f\subset O$. If $O_1$ and $O_2$ are causally disjoint, the causal propagator makes $\sigma([f_1],[f_2])=0$, so the corresponding Weyl operators commute. This is locality at the bounded-algebra level.

Locally covariant incarnation. On each oriented, time-oriented globally hyperbolic spacetime $(M,g)$ assign the Klein–Gordon Weyl algebra $\mathcal A(M,g)$. A causal isometric embedding pushes compactly supported test functions forward and induces an injective algebra morphism. Compatibility with composition is a direct functorial check. A choice of state is not part of the functor unless a state space is added.

The physical comparison is continued at Wightman, Euclidean, Local-Algebraic, Constructive, and Perturbative Frameworks. The present conclusion is only that the free scalar inhabits each stated object class and that the displayed arrows are licensed under their stated hypotheses.

Failure test: one shared two-point function is insufficient

The adversarial test is to give two proposed formulations the same bidistribution $W_2$ and then ask whether framework equivalence follows. It does not. The strongest immediate conclusion is equality of that one datum. Unless both theories are known to be generalized free, $W_2$ does not determine the higher truncated vacuum functions. It also does not specify:

• the field domain and closure;
• which composite or charged fields exist;
• the local observable algebras and their completions;
• the state space or superselection sectors;
• whether a comparison is full, faithful, essentially surjective, or invertible;
• whether the correspondence respects products, adjoints, covariance, and locality.

Thus shared two-point data do not supply the missing objects, morphisms, essential-surjectivity argument, or state data. Even in a Gaussian theory, one must prove that the test-function quotient, symplectic form, representation, and state match. Equality of covariance kernels can then be an input to an equivalence theorem; it is not the theorem by itself.

Independent checks on a proposed arrow

1. Composition. Apply the map to identity morphisms and composites. A claimed functor must satisfy $\mathcal A(\psi_2\circ\psi_1)=\mathcal A(\psi_2)\circ\mathcal A(\psi_1)$.
2. Locality. For the free scalar, verify directly that the causal propagator vanishes between causally disjoint supports, hence the Weyl commutator vanishes.
3. State compatibility. Compare not only algebras but also the selected states: $\omega_B\circ F=\omega_A$ when state preservation is claimed.
4. Information loss. Exhibit two field coordinatizations generating the same bounded net. This proves that the field-to-net arrow cannot be inverted without additional data.
5. Scope. Replace the Green-hyperbolic free equation by an interacting, gauge, or non-globally-hyperbolic case and locate the first construction that fails.

Exercises

1. Restrict a functor to a net. Let $\mathcal A:\mathbf{Loc}\to\mathbf{Alg}$ be covariant and fix $M$. Prove isotony of $O\mapsto\mathcal A(O)$ for causally convex opens.

Solution

For $O_1\subset O_2$, the inclusion $\iota_{12}:O_1\hookrightarrow O_2$ is a morphism in the chosen spacetime category. The target category uses injective $*$-homomorphisms, so $\mathcal A(\iota_{12})$ identifies $\mathcal A(O_1)$ with a subalgebra of $\mathcal A(O_2)$. Functoriality makes these inclusions consistent for $O_1\subset O_2\subset O_3$. That is isotony. No state or Haag duality follows from this argument.

2. Locate the Gaussian hypothesis. Why does $W_2$ determine every $W_n$ for a generalized free scalar but not for a general Wightman theory?

Solution

For a generalized free field, all truncated functions except the two-point function vanish, so Wick’s rule expresses each even $W_n$ as a sum over pairings of $W_2$ and each odd one as zero. A general Wightman theory may have independent nonzero truncated functions of order three and higher. Equality of $W_2$ therefore leaves precisely the interaction-sensitive data undetermined.

3. Identify the missing inverse. A net is obtained from a Wightman field by bounded functional calculus. Name two pieces of information that can be lost.

Solution

The net need not retain which unbounded pointlike field generated it, and the observable net can omit charged field operators. It may also be presented without its vacuum state. Therefore an inverse needs a field-reconstruction theorem with phase-space, domain, covariance, and state hypotheses; abstract equality of nets is insufficient.

References

• Brunetti, Romeo, Klaus Fredenhagen, and Rainer Verch. “The Generally Covariant Locality Principle—A New Paradigm for Local Quantum Physics.” Communications in Mathematical Physics 237 (2003): 31–68. DOI. Open PDF.
• Gårding, Lars, and Arthur S. Wightman. “Fields as Operator-Valued Distributions in Relativistic Quantum Theory.” Arkiv för Fysik 28 (1964): 129–189. Catalog record.
• Gwilliam, Owen, and Kasia Rejzner. “Relating Nets and Factorization Algebras of Observables: Free Field Theories.” Communications in Mathematical Physics 373 (2020): 107–174. DOI. Open PDF.
• Haag, Rudolf, and Daniel Kastler. “An Algebraic Approach to Quantum Field Theory.” Journal of Mathematical Physics 5 (1964): 848–861. DOI.
• Osterwalder, Konrad, and Robert Schrader. “Axioms for Euclidean Green’s Functions II.” Communications in Mathematical Physics 42 (1975): 281–305. DOI. Open PDF.