Existence, Uniqueness, and Equivalence Claims

An existence statement names an object and proves that it satisfies a stated contract. A uniqueness statement names the equivalence relation under which alternatives agree. An equivalence statement supplies maps in both directions, together with the identities or coherent natural transformations they satisfy. These are different mathematical obligations: constructing Schwinger functions, reconstructing a Wightman theory, recovering selected observables, and proving a converse cannot be compressed into one double arrow.

Required background. The Wightman reconstruction theorem fixes the Lorentzian target and its unitary-equivalence conclusion. Haag–Kastler nets and locality supplies a distinct object class. The constructive program and cutoff removal supplies the limiting quantifiers. Helpful background. Factorization comparison theorems illustrates a hypothesis-bounded equivalence. The quantum-gravity pages on exact versus conditional statements, nonperturbative completion criteria, UV-completion claims, and fixed-theory factorization tests provide demanding comparison cases.

The anatomy of an existence claim

A useful record begins with a tuple

$$\mathcal C=(\mathsf O,d,\mathsf R,\mathsf L,\mathsf T,\mathsf A),$$

where $\mathsf O$ is the object class, $d$ the dimension and signature, $\mathsf R$ the regulators and bare parameters, $\mathsf L$ the ordered limits and their topology, $\mathsf T$ the axioms to be satisfied, and $\mathsf A$ the observables retained. For a Euclidean construction, $\mathsf O$ might be a probability measure on tempered distributions or a compatible Schwinger hierarchy. For a Lorentzian construction it might be a Wightman field on a common invariant domain or a net of operator algebras. Existence of one does not syntactically imply existence of another.

Suppose regulated correlations $S_n^{a,L}$ converge as $L\to\infty$ and then $a\to0$. The statement must say whether all $n$ converge, whether convergence is distributional or merely pointwise after smearing, whether the limits are unique, and whether the Osterwalder–Schrader hypotheses survive. Reflection positivity, Euclidean covariance, symmetry, clustering, and the growth conditions are inputs to reconstruction, not consequences of the word “continuum.” Osterwalder and Schrader construct the Hilbert space and Lorentzian fields only from a hierarchy satisfying their complete hypotheses Osterwalder and Schrader 1975, §§4–6, pp. 291–305.

Uniqueness and equivalence are typed

If two Schwinger hierarchies agree, OS reconstruction gives unitarily equivalent reconstructed theories in the theorem’s category. That is not uniqueness of a bare lattice action, of a renormalization prescription, or of a presentation by point fields. Likewise, a functor $F:\mathcal A\to\mathcal B$ being fully faithful means

$$\operatorname{Hom}_{\mathcal A}(X,Y) \simeq \operatorname{Hom}_{\mathcal B}(FX,FY),$$

but it is an equivalence only after essential surjectivity is proved. A natural transformation that is a quasi-isomorphism objectwise may identify derived observables while leaving topological completions, positive states, and unbounded-operator domains unaddressed.

The current AQFT–prefactorization comparison makes this distinction concrete. In a bicomplete closed symmetric monoidal one-category, additive time-slice theories on the specified Lorentzian site are equivalent through explicit comparison functors Benini, Carmona, Grant-Stuart, and Schenkel 2024, Theorems 3.3–3.4. For cochain-complex targets, the same paper reduces the infinity-categorical problem to spacetime-wise localization questions but leaves its key localization step open Benini et al. 2024, Open Problem 5.6 and Proposition 5.7. The proved one-categorical arrow must not be enlarged to the unresolved target.

First application: Euclidean and Wightman directions

Return to the framework comparison in Foundations. A careful Euclidean-to-Wightman claim splits into five rows:

1. regulated Euclidean objects exist;
2. their full correlation hierarchy has a unique limit in a named topology;
3. the limit satisfies the OS axioms;
4. OS reconstruction produces a Wightman theory, unique up to the theorem’s unitary equivalence;
5. a separately defined Lorentzian class admits a converse Euclidean continuation and returns to the original class.

Only the fourth row follows from the reconstruction theorem once rows two and three are supplied. The fifth is a converse theorem with its own analyticity, spectral, domain, and growth assumptions. Matching low-point functions does not recover a complete observable algebra, and agreeing perturbative series do not prove equality at finite coupling.

Failure test: the unsupported double arrow

Replace the reconstruction arrow by $\Longleftrightarrow$ while omitting the limiting topology. The first failure occurs before reconstruction: convergence of finitely many observables neither determines a measure nor supplies the full hierarchy. Even if a suitable hierarchy exists, reflection positivity can fail. If OS reconstruction succeeds, no unrestricted converse follows. The strongest surviving claim may be a one-way construction of a specified Hilbert-space theory from specified Euclidean data.

An independent check is compositional. Write every proposed equivalence as functors $F,G$ and require natural equivalences $GF\simeq\mathrm{id}$ and $FG\simeq\mathrm{id}$ in the declared categories. If only the first exists, $F$ may be fully faithful but its essential image is still a theorem obligation.

Exercises

A sequence of lattice two-point functions converges to the free massive covariance, while no higher correlations are controlled. What existence statement is licensed?

Solution

Only convergence of the specified two-point observable is licensed. One cannot infer a unique limiting probability measure, Gaussianity, OS reconstruction, or equivalence to the free Wightman theory: each requires the full compatible hierarchy or another determining construction and its hypotheses.

References

• Benini, Marco, Victor Carmona, Alastair Grant-Stuart, and Alexander Schenkel. “On the Equivalence of AQFTs and Prefactorization Algebras.” arXiv:2412.07318 (2024). arXiv.
• Benini, Marco, Marco Perin, and Alexander Schenkel. “Model-Independent Comparison between Factorization Algebras and Algebraic Quantum Field Theory on Lorentzian Manifolds.” Communications in Mathematical Physics 377 (2020): 971–997. DOI; Open PDF.
• Osterwalder, Konrad, and Robert Schrader. “Axioms for Euclidean Green’s Functions II.” Communications in Mathematical Physics 42 (1975): 281–305. Project Euclid.