AQFT to Prefactorization on Time-Orderable Covers

Every AQFT supplies prefactorization products for tuples of regions that admit a causal ordering: embed each local observable into the target region and multiply in a compatible temporal order. Einstein causality proves independence of the chosen linear extension whenever two adjacent regions are causally incomparable. 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 $\rho$ such that, whenever $i<j$,

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

with the direction chosen so that earlier factors stand to the left in the product convention below. Reversing the convention reverses every product but changes no theorem. 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 linear extension

Two linear extensions of a finite partial order are connected by swaps of adjacent incomparable elements. If two components are incomparable in the causal order, their images are causally disjoint in the sense required by the construction. 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 each adjacent swap preserves the product. This proves independence of $\rho$. It also isolates the hypothesis: ordinary algebra associativity cannot replace Einstein causality because different linear extensions change the order of multiplication.

The resulting assignment is the functor from AQFTs to time-orderable prefactorization algebras proved in Benini, Perin, and Schenkel 2020, Definition 4.1 and Theorem 4.7. Additivity and Cauchy constancy are preserved in both directions ibid., Proposition 4.8.

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 double cones inside a larger causally convex $O$. Suppose $O_1$ is earlier than $O_3$, while $O_2$ is spacelike to both. Two compatible linear extensions are $(1,2,3)$ and $(2,1,3)$. Their products are

$$\mathfrak A(\iota_1)(A_1)\mathfrak A(\iota_2)(A_2)\mathfrak A(\iota_3)(A_3)$$

and the expression with the first two factors exchanged. Locality makes them equal. If $O_2$ is moved while remaining spacelike, covariance gives the corresponding coherent family. 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. Their local algebras need not commute. Causal disjointness requires the causal hull of either region to miss the other and 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. arXiv:1903.03396.
• 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.