In local quantum field theory or AQFT one can mathematically describe over each open set $U$ of a spacetime $M$ the quantum states or observables of the theory. This structure is commonly referred as a presheaf or a copresheaf.

**Why are the states (or observables) over the open sets not a sheaf (cosheaf) structure?**

This question is motivated by the following considerations:

The net of local observables which can be roughly described as a copresheaf of (C-star algebras) on pieces of spacetime such that algebras, $A(U) $, assigned to causally disconnected regions commute inside the algebra assigned to any joint neighbourhood.

Up to this point we have by definition a copresheaf.

In order to have a sheaf we need to verify the following two conditions:

(Locality) If ($U_{i}$) is an open covering of an open set $U$, and if $s,t ∈ A(U)$ are such that $s|U_{i} = t|U_{i}$ for each set $U_{i}$ of the covering, then $s = t$

(Gluing) If ($U_{i}$) is an open covering of an open set $U$, and if for each $i$ a section $s_{i} ∈ A(U_{i})$ is given such that for each pair $U_{i},U_{j}$ of the covering sets the restrictions of $s_{i}$ and $s_{j} $ agree on the overlaps: $s_{i}|U_{i}∩U_{j} = s_{j}|U_{i}∩U_{j}$, then there is a section $s ∈ A(U)$ such that $s|U_{i} = s_{i}$ for each $i$.

The gluing condition guarantee the existence of a section $s$ which the locality condition shows it is unique.

Evidently, one of this conditions or both fail in general.

**I would be interested in a physical picture of why the sheaf conditions are not satisfied.**

To me, the locality condition is stating intuitively that if the observables coincide in every region that form an open cover, then the observables (and the qft) are the same in the open cover. The gluing condition, in the other hand establish that one is able to construct the theory just by gluing local pieces of the theory. Is there then some non local restriction that perhaps avoid us constructing the theory just from local pieces?

Are these intuitions correct?

This post imported from StackExchange Physics at 2015-09-06 15:33 (UTC), posted by SE-user yess