AQFT to Prefactorization on Time-Orderable Covers

Every AQFT supplies prefactorization products for time-orderable tuples of pairwise disjoint regions: embed each local observable into the target region and multiply with causally later factors to the left. Einstein causality proves independence of the chosen time-ordering permutation because two such permutations differ by adjacent swaps of causally disjoint regions. The construction yields a time-orderable prefactorization algebra, not products for arbitrary disjoint tuples.

Required background. Causal factorization and time-ordered products supplies the ordering rule, and factorization comparison theorems supplies the target category and exact theorem domain.

Helpful background. Haag–Kastler nets and locality supplies Einstein causality, while adiabatic limits and interacting nets explains why local interacting algebras need not determine one global adiabatic limit.

Time-orderable tuples

Let $f_i:M_i\to N$ be morphisms in $\mathbf{Loc}$ with pairwise disjoint images. The tuple $\underline f=(f_1,\ldots,f_n)$ is time-orderable if there exists a permutation $\rho\in\Sigma_n$ such that, whenever $i<j$,

$$J_N^+\bigl(f_{\rho(i)}(M_{\rho(i)})\bigr) \cap f_{\rho(j)}(M_{\rho(j)})=\varnothing,$$

With this convention, $i<j$ says that the $i$th ordered region is not causally earlier than the $j$th. Thus, for a causally related pair, the later region occurs first and its observable stands to the left in the product below; causally disjoint pairs may occur in either order. Extended regions can fail this condition even in a globally hyperbolic spacetime: different points of one region may lie to both the future and past of another.

An AQFT $\mathfrak A:\mathbf{Loc}\to\mathbf{Alg}(\mathcal C)$ assigns algebra morphisms to embeddings and satisfies Einstein causality: images associated with causally disjoint morphisms commute. Define the time-ordered factorization product by

$$\mathfrak F_{\mathfrak A}(\underline f) (a_1\otimes\cdots\otimes a_n) =\mathfrak A(f_{\rho(1)})(a_{\rho(1)}) \cdots \mathfrak A(f_{\rho(n)})(a_{\rho(n)}).$$

For one input this is the functorial map; for no inputs it is the unit. Associativity follows from the algebra product and composition in $\mathbf{Loc}$. Permutation equivariance follows because the tensor symmetry and relabeling of $\rho$ describe the same ordered list.

Independence of the time-ordering permutation

Let $\rho$ and $\rho'$ be two time-ordering permutations. Benini–Perin–Schenkel’s Lemma 4.3(iii) shows that the right permutation $\rho^{-1}\rho'$ is generated by transpositions of adjacent causally disjoint morphisms. For each such adjacent pair, Einstein causality gives

$$\mathfrak A(f_i)(a_i)\mathfrak A(f_j)(a_j) =\mathfrak A(f_j)(a_j)\mathfrak A(f_i)(a_i),$$

so every transposition preserves the product. Hence the result is independent of $\rho$, as stated in Lemma 4.6. The indispensable hypothesis is Einstein causality; associativity alone does not identify products whose factor order changes.

The definition and order-independence argument are Benini, Perin, and Schenkel 2020, Definition 4.1, Lemma 4.3(iii), and Lemma 4.6, pp. 15–17, Open PDF. The resulting functor from AQFTs to time-orderable prefactorization algebras is their Theorem 4.7; additivity and Cauchy constancy are characterized in Proposition 4.8.

On a fixed smooth globally hyperbolic spacetime, scalar pAQFT constructs renormalized local $S$-matrices as formal power series for compactly supported local Wick-polynomial interactions. Causal factorization makes relative $S$-matrices with spacelike-separated supports commute, and cutoff intertwiners assemble their generated algebras into an isotone Einstein-causal net without requiring a global adiabatic limit Brunetti and Fredenhagen 2000, eqs. (2)–(3), p. 627, and § 8, Proposition 8.1 and eqs. (60)–(67), pp. 656–657, Open PDF. If compatible choices on all spacetimes and causality-preserving embeddings define an Einstein-causal functor $\mathfrak A:\mathbf{Loc}\to\mathbf{Alg}(\mathcal C)$, then Benini, Perin, and Schenkel 2020, eq. (4.7), Lemma 4.6, and Theorem 4.7, pp. 17–18, Open PDF supplies the coherent, order-independent prefactorization products on pairwise-disjoint time-orderable tuples.

Massive scalar double cones

Take the massive free scalar Haag–Kastler net on Minkowski space with (+---) metric. Let $O_1,O_2,O_3$ be pairwise disjoint double cones inside a larger causally convex $O$, with inclusions $\iota_i:O_i\hookrightarrow O$ and observables $A_i\in\mathfrak A(O_i)$. Suppose

$$J_O^+(O_3)\cap O_1=\varnothing, \qquad J_O^+(O_1)\cap O_3\ne\varnothing,$$

while $O_2$ is causally disjoint from both. The first condition permits $O_3$ before $O_1$, whereas the second excludes the reverse order. Every time-ordering permutation must therefore put $O_3$ before $O_1$; two choices are $(3,2,1)$ and $(2,3,1)$. Their products are

$$\begin{aligned} &\mathfrak A(\iota_3)(A_3)\mathfrak A(\iota_2)(A_2)\mathfrak A(\iota_1)(A_1),\\ &\mathfrak A(\iota_2)(A_2)\mathfrak A(\iota_3)(A_3)\mathfrak A(\iota_1)(A_1). \end{aligned}$$

Einstein causality for $O_2$ and $O_3$ makes these expressions equal. Covariance compares whole configurations through a specified morphism: if $g:O\to O'$, then, with the tensor arguments understood,

$$\mathfrak A(g)\,\mathfrak F_{\mathfrak A}(\underline\iota) =\mathfrak F_{\mathfrak A}(g\circ\underline\iota).$$

It does not identify arbitrary deformations of one region. The physical microcausality statement is developed at Microcausality and Relativistic Compatibility.

The construction uses the net’s algebra product, not a Wightman time-ordered distribution at coincident points. Renormalized time ordering becomes necessary when one starts from local fields or perturbative functionals rather than already-defined bounded local algebras.

Failure on a causally cyclic tuple

Choose three disjoint extended opens such that each has points in the causal past and future of the next, producing a directed cycle among the region-level precedence relations. Global hyperbolicity forbids a closed causal curve through points, but it does not forbid this cycle formed by different points in extended sets. No permutation $\rho$ satisfies the time-orderability condition. Multiplying the three embedded algebras in an arbitrary order is possible algebraically, but different orders need not agree because the regions are not pairwise causally disjoint. The prefactorization product is therefore undefined on that tuple.

This failure is why the theorem targets time-orderable prefactorization algebras. Refining the opens into smaller orderable pieces may permit an additive reconstruction, but that is an additional descent argument, not a product on the original tuple by decree.

Exercises

Prove independence of ordering for three pairwise spacelike regions.

Solution

Every permutation is a product of adjacent transpositions. Einstein causality makes the embedded observables from each adjacent pair commute, so all six ordered products coincide.

Why is pairwise disjointness weaker than causal disjointness?

Solution

Disjoint opens can contain timelike-related points, so their local algebras need not commute. Two regions $O_i,O_j\subset O$ are causally disjoint when

$$J_O^+(O_i)\cap O_j =J_O^+(O_j)\cap O_i =\varnothing;$$

this is the hypothesis used by Einstein causality.

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: arXiv:1903.03396v2.
• 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. Open PDF: arXiv:math-ph/9903028.
• Brunetti, Romeo, Klaus Fredenhagen, and Rainer Verch. “The Generally Covariant Locality Principle: A New Paradigm for Local Quantum Field Theory.” Communications in Mathematical Physics 237 (2003): 31–68. doi:10.1007/s00220-003-0815-7.