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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:

  1. (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 $s = t$

  2. (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?

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    $\begingroup$ @ACuriousMind: A presheaf either is or is not a sheaf; no additional structure is needed. So it seems to me that a good answer to this question would be one that gives an example of a presheaf occuring in LQFT that is not a sheaf, together with a (possibly quite simple) proof that it violates one of the sheaf axioms. I am sorry that I don't know enough about LQFT to provide this example. $\endgroup$ – WillO Aug 20 '15 at 3:04
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    $\begingroup$ @yuggib: When confronted with a presheaf, one should salivate (like Pavlov's dog): "Is this a sheaf?" (and if not, "why not?"). This is not "asking for more"; it's just checking one's understanding. $\endgroup$ – WillO Aug 20 '15 at 17:58
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    $\begingroup$ @WillO I do not agree...Given a presheaf, you can make it a sheaf; in addition, you may not care about the sheaf axioms. So why bother? What additional insight/result gives you to know it is a sheaf? If there is no additional information, it is just not useful to "lose time" checking... $\endgroup$ – yuggib Aug 20 '15 at 21:42
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    $\begingroup$ @yuggib: In my experience, if you confront a presheaf and are not sure whether it's a sheaf, then at some very basic level you do not understand your presheaf, and that is going to come back to bite you sooner or later. Asking whether it's a sheaf is a good (I'd say indispensable) test of whether you really understand the definition of your presheaf. $\endgroup$ – WillO Aug 20 '15 at 22:03
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    $\begingroup$ Perhaps you could describe a little mo4e about the gluing and how it could be, or could be not, compatible with Heisenberg principle and non commuutativity of observables. $\endgroup$ – arivero Sep 1 '15 at 10:49
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As you mention ncatlab I would bet that you have already revised all this... Looking in the net for old discussions and papers, it seems that the an open question was about defining the open sets beyond 1+1 dimensions. Of course (1+1) has a lot of niceties, I remember Borcherds -with 'd'- exploited very well them.

The net of open sets must be consistent with the "causal diamonds" of Haag et al. Particularly this is discussed in this thread https://golem.ph.utexas.edu/~distler/blog/archives/000987.html where Urs finishes telling that

In summary: it is not clear to me if the answer to “Should Haag-Kastler nets be taken to satisfy the co-sheaf condition?” is really “No.”

Later in https://golem.ph.utexas.edu/category/2008/11/local_nets_and_cosheaves.html someone points to the paper Generally covariant quantum field theory and scaling limits. Comm. Math. Phys. 108 (1987), no. 1, 91--115. http://projecteuclid.org/euclid.cmp/1104116359 to try to use for for a gluing property. Abstracting this paper Urs mentions than

b) It seems that for A the net of Borchers algebras, A is a co-sheaf

but still the answer is inconclusive

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