LCQFT, Nets, Fields, and Factorization: Typed Comparisons

Locally covariant functors, fixed-spacetime nets, natural fields, and factorization structures encode different maps. There are useful directional constructions between them, but the converses require covariance across backgrounds, additivity, time-slice, or time-orderability hypotheses. A typed comparison prevents a construction on one spacetime from being promoted to a generally covariant theory.

Required background. QFT Frameworks: Object Classes and Maps supplies directional comparison grammar. OS and Wightman Comparison: Directions and Failure Modes separates Euclidean reconstruction from Lorentzian algebraic input. From Pointlike Fields to Nets and Affiliated Operators treats the field-to-net direction. Locally Covariant QFT as a Functor fixes the principal source and target.

Helpful background. Categories, Functors, and Natural Transformations supplies the categorical language. Derived, Higher, Factorization, and Boundary Maps locates factorization structures in a wider local-to-global setting.

From one LCQFT to a net and natural fields

Fix a Loc object $M$. Let $O$ range over causally convex globally hyperbolic open subsets and write $\iota_O:M|_O\to M$. Evaluation of a locally covariant theory gives the net

$$O\longmapsto\mathcal A_M(O) =\operatorname{im}\mathcal A(\iota_O)\subseteq\mathcal A(M).$$

Functoriality gives isotony, and Einstein causality gives commutation for causally disjoint regions. A Cauchy region generates the full algebra when the theory obeys time-slice. Brunetti, Fredenhagen, and Verch make this passage precise in Brunetti, Fredenhagen, and Verch 2003, Proposition 2.3, pp. 9–11.

A natural field $\Phi:\mathcal D\Rightarrow\mathcal A$ evaluates to smeared operators $\Phi_M(f)$. Under appropriate common-domain, closure, and strong-commutativity hypotheses, their bounded functions may generate the net. The reverse direction is not unique: a net can admit several field coordinatizations, and not every net element is canonically attached to a pointlike field.

For the free Klein–Gordon functor, this realizes the requested application of Local Field Algebras, Causality, and the Time-Slice Property: evaluate on one spacetime, extract the causally convex net, retain the natural smeared field, and use only the prefactorization products justified below.

Time-orderable prefactorization products

On a fixed Lorentzian manifold, an algebraic QFT with Einstein causality gives a time-orderable prefactorization algebra. For pairwise disjoint embeddings whose images can be ordered so that later regions do not lie in the causal past of earlier ones, multiply the transported observables in that temporal order:

$$\mathcal F(f_1,\ldots,f_k)(A_1\otimes\cdots\otimes A_k) =\mathcal A(f_1)(A_1)\cdots\mathcal A(f_k)(A_k).$$

Einstein causality makes the result independent of swapping causally disjoint adjacent entries. This construction is Benini, Perin, and Schenkel 2020, Definition 4.4 and Theorem 4.7, pp. 15–17. It does not license products for arbitrary overlapping or non-time-orderable families.

The converse becomes an equivalence only on restricted categories: Cauchy-constant additive AQFTs correspond to Cauchy-constant additive time-orderable prefactorization algebras under the hypotheses of Benini, Perin, and Schenkel 2020, Theorem 5.1, pp. 17–19. Dropping additivity or Cauchy constancy blocks that result.

Directional comparison table

Input	Construction	Output	Missing converse data
LCQFT $\mathcal A:\mathsf{Loc}\to\mathsf{Phys}$	evaluate on causally convex subspacetimes of $M$	fixed-$M$ local net	one net does not encode maps between different spacetimes
natural field $\Phi:\mathcal D\Rightarrow\mathcal A$	evaluate and take controlled bounded functions	field-generated subnet	a net need not select a unique field family
causal AQFT on one Lorentzian manifold	ordered multiplication on time-orderable disjoint embeddings	time-orderable prefactorization algebra	equivalence needs additivity and Cauchy constancy
Euclidean Schwinger hierarchy	OS reconstruction under reflection positivity, regularity, symmetry, and clustering hypotheses	Lorentzian Wightman theory	LCQFT covariance across curved backgrounds is not reconstructed automatically

The last row is deliberately separate. Osterwalder–Schrader reconstruction changes signature and reconstructs a Hilbert-space theory under analytic hypotheses; evaluating an LCQFT changes neither signature nor representation. No theorem here identifies the two processes.

For the free scalar, every arrow can be checked on test functions. The LCQFT uses extension by zero along admissible embeddings; its fixed-background net restricts this operation to regional inclusions; the natural field evaluates the generator class $[f]$; and the prefactorization product multiplies images for a time-orderable disjoint family. These four operations agree where their domains overlap because they descend from the same functor, but their domains remain different.

Additivity is the local-to-global input in the conditional factorization converse. Without it, a prefactorization algebra can carry compatible products on disjoint regions while withholding global observables not generated from smaller pieces. Cauchy constancy is separately needed to match the Lorentzian time-slice structure. Neither condition follows from the existence of disjoint products.

Failure boundary and independent check

Starting from a single Minkowski Haag–Kastler net, one cannot recover the values $\mathcal A(N)$ on arbitrary curved spacetimes or maps $\mathcal A(\psi)$ between them. Poincaré covariance supplies only automorphisms associated with Minkowski symmetries, not cross-background covariance. This is the adversarial nonconverse.

For any claimed comparison, write the source objects and arrows on both sides. Then test identity and composition, causal disjointness, and the stated additivity/time-slice conditions. If the construction has no place to evaluate a morphism $M\to N$ with $M\neq N$, it is not yet an LCQFT.

Exercise

Why is ordinary multiplication unsuitable as a prefactorization product for two arbitrary spacelike-overlapping embeddings?

Solution

A prefactorization product is assigned to a disjoint family of embeddings. Overlap can identify degrees of freedom and gives no tensor-product input with independent supports. Even for disjoint regions, the Lorentzian construction uses time-orderability; causality proves independence only of exchanges that are causally disjoint. Arbitrary overlapping inputs require different operator-product or renormalized time-ordering data.

References

• 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.
• 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.