Weiss Descent and Local-to-Global Observables

Weiss descent reconstructs observables on an open region from observables supported near every finite configuration of points. The reconstruction is derived: intersections, higher overlap data, and homotopies all enter. Consequently, an ordinary two-open cover can be adequate for sheaf gluing yet inadequate for factorization, and descent of observables says nothing by itself about global states or Hilbert-space representations.

Required background. Prefactorization and factorization algebras supplies the disjoint structure maps, while natural transformations and subtheory embeddings supplies the functorial language used by the descent map.

Helpful background. Locally covariant QFT as a functor gives a Lorentzian comparison, typed framework maps distinguish the objects being glued, and Čech descent supplies the ordinary model that is derived here.

Covers adapted to finite support

For an open $U\subset M$, a family $\mathfrak U=\{U_i\to U\}$ is Weiss if each finite set $S\subset U$ lies inside at least one $U_i$. This condition sees all possible finite constellations of local insertions. A useful factorizing basis consists of disks together with finite disjoint unions of disks: each configuration admits mutually disjoint neighborhoods, and refinement can shrink them independently. Costello and Gwilliam 2023, Definition 2 and §3 explains why this topology, rather than the ordinary topology alone, captures products of local operators.

For a cochain-valued prefactorization algebra $\mathcal F$, form the augmented Čech complex

$$\cdots\longrightarrow \bigoplus_{i,j,k}\mathcal F(U_i\cap U_j\cap U_k) \longrightarrow \bigoplus_{i,j}\mathcal F(U_i\cap U_j) \longrightarrow \bigoplus_i\mathcal F(U_i) \longrightarrow \mathcal F(U).$$

The alternating Čech differential combines restriction-by-inclusion maps with the internal differential of $\mathcal F$. In a derived target, direct sums and intersections are replaced by the appropriate homotopy colimit diagram. Descent requires the augmentation to be a quasi-isomorphism. Merely obtaining an isomorphism on degree-zero vector spaces can miss higher relations and gauge homotopies.

The interval calculation

Let $I=(0,1)$. An ordinary chain of overlapping intervals $I_a$ covers every point but need not contain a pair of well-separated points in a single member. To perform the promised free-field calculation, enlarge it to the Weiss family generated by finite disjoint unions of subintervals drawn from the chain. This is the precise correction that lets the cover carry multilocal observables.

For the free scalar, write the linear observable complex on $U$ as $\mathcal L_c(U)$: compactly supported test functions modulo, or resolved by, the Klein–Gordon equation. Its classical polynomial observables are

$$\operatorname{Obs}^{\mathrm{cl}}(U) =\operatorname{Sym}\bigl(\mathcal L_c(U)\bigr),$$

with the equation-of-motion differential extended as a derivation. Compact support makes $\mathcal L_c$ cosheaf-like: a partition of unity decomposes a section subordinate to the cover, and the usual overlap relations identify different decompositions. Passing to the derived symmetric algebra records finite products and their higher relations. Under the standard completed locally convex or filtered hypotheses, the resulting Čech homotopy colimit maps quasi-isomorphically to $\operatorname{Obs}^{\mathrm{cl}}(I)$. The full perturbative construction and its descent proof are given in Gwilliam and Rejzner 2023, §§3, 5.5, and 6.2.

The calculation is not a claim that a one-dimensional massless scalar has a preferred vacuum or that the global Gaussian measure exists without infrared qualifications. It reconstructs the observable cochain complex from compactly supported pieces. The physical meaning of spacelike-separated local observables is developed at Spacelike Compatibility and Local Observables.

Why the derived colimit matters

Suppose $f_i\in\mathcal L_c(U_i)$ satisfy $\sum_i f_i=0$ after extension to $I$. A naive colimit can quotient by the visible pairwise relations, but gauge complexes and equations of motion can carry relations among relations. The totalized Čech differential retains these higher syzygies. A spectral-sequence proof filters by polynomial degree: the first page reduces to descent of the linear compact-support complex, and compatibility of multiplication then propagates the quasi-isomorphism to polynomial observables. This is the mechanism used for free and perturbative multilocal observables, not a formal appeal to “locality.”

Locally finite covers require an additional functional-analytic decision. Direct sums, completed tensor products, and support conditions must be chosen so that the Čech differential and multiplication are continuous. Compactness of support makes each individual observable meet only finitely many relevant pieces, but it does not automatically make every completion exact.

Failure on an ordinary cover

Take two opens $U_L,U_R\subset I$ whose union is $I$, with neither containing a pair $(x_L,x_R)$ far to the left and right. A bilocal observable $\ell_{f_L}\ell_{f_R}$ has support near both points. Neither $\mathcal F(U_L)$ nor $\mathcal F(U_R)$ contains it. It can be assembled from a disjoint product only if the indexing family includes $V_L\sqcup V_R$ with $x_L\in V_L$ and $x_R\in V_R$. Thus ordinary Čech data miss a genuine finite-support observable, whereas the Weiss enlargement captures it.

This failure is diagnostic, not a proof that every ordinary cover fails for every theory. A special locally constant or additive theory may be recoverable from a smaller basis. The theorem must name that basis and the exact comparison map.

Exercises

Show that the family of all finite disjoint unions of relatively compact intervals in $I$ is Weiss.

Solution

Given a finite set $S=\{x_1,\ldots,x_n\}\subset I$, choose pairwise disjoint small intervals $J_a\Subset I$ around its distinct points. Their finite disjoint union belongs to the family and contains $S$.

Explain why a quasi-isomorphism of the linear descent complexes is enough to start the polynomial-degree spectral-sequence argument.

Solution

Filter $\operatorname{Sym}(\mathcal L_c)$ by polynomial degree. The associated graded is the symmetric algebra on the linear complex with no degree-lowering interaction. Over characteristic zero and with the stated exact completion, the induced map on each symmetric power is a quasi-isomorphism. Convergence of the bounded-below filtration then lifts the result to the full completed complex.

References

• Costello, Kevin, and Owen Gwilliam. “Factorization Algebra.” Encyclopedia of Mathematical Physics, 2nd ed., 2023. arXiv:2310.06137.
• Gwilliam, Owen, and Katarzyna Rejzner. “The Observables of a Perturbative Algebraic Quantum Field Theory Form a Factorization Algebra.” 2023. arXiv:2212.08175.