Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

145 submissions , 122 unreviewed
3,930 questions , 1,398 unanswered
4,851 answers , 20,616 comments
1,470 users with positive rep
501 active unimported users
More ...

Why are the quantum observables defined on opens sets a presheaf and not a sheaf?

+ 6 like - 0 dislike
379 views

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?

This post imported from StackExchange Physics at 2015-09-06 15:33 (UTC), posted by SE-user yess
asked Aug 20, 2015 in Mathematics by yess (90 points) [ no revision ]
retagged Sep 6, 2015
Most voted comments show all comments
Why do you think they should be a sheaf? I mean, the definition of the assignment $\mathcal{A}$ is just that of a presheaf, where should the sheaf structure come from?

This post imported from StackExchange Physics at 2015-09-06 15:34 (UTC), posted by SE-user ACuriousMind
@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.

This post imported from StackExchange Physics at 2015-09-06 15:34 (UTC), posted by SE-user WillO
You can sheafify a presheaf, and you probably do not really need the additional axioms. So why ask more when it is not necessary?

This post imported from StackExchange Physics at 2015-09-06 15:34 (UTC), posted by SE-user yuggib
@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.

This post imported from StackExchange Physics at 2015-09-06 15:34 (UTC), posted by SE-user WillO
@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...

This post imported from StackExchange Physics at 2015-09-06 15:34 (UTC), posted by SE-user yuggib
@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.

This post imported from StackExchange Physics at 2015-09-06 15:34 (UTC), posted by SE-user WillO
@yuggib Thank you for your comment. I am aware of this. However, I would like a physical picture of what is the sheafication doing. I understand from here that somehow we are either adding or deleting the sections. But at the moment I can not grasp what does this mean in terms of the observables or states of the theory.

This post imported from StackExchange Physics at 2015-09-06 15:34 (UTC), posted by SE-user yess

1 Answer

+ 0 like - 0 dislike

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

This post imported from StackExchange Physics at 2015-09-06 15:34 (UTC), posted by SE-user arivero
answered Sep 3, 2015 by - (255 points) [ no revision ]

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysics$\varnothing$verflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...