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In the Haag-Kastler approach to axiomatic quantum field theory, it is assumed that observables are 'associated' to spacetime regions. What this actually means is that there is a map $\mathcal{A}: R \mapsto \mathcal{A}(R)$, which associates to a given region $R$ the algebra of observables $\mathcal{A}(R)$ which one may measure in $R$

I'm wondering: Is this map in some sense 'invertible'? If you hand me an observable, can I associate to it a spacetime region in a unique fashion? Is there meaningful physics in this assignment?

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First some clarification: Usually in the Haag-Kastler framework, one assumes the spacetime regions to be double cones $\mathcal{O}$ and therefore has a net of (von Neumann) algebras over these double cones, i.e. $\mathcal{O} \mapsto \mathcal{A}(\mathcal{O})$. A corresponding algebra over arbitrary spacetime regions can be achieved by taking sections of double cone algebras. Now concerning your question: The answer is no. It's not possible in general to associate a given observable to a certain spacetime region. Look for example at the Reeh-Schlieder theorem, http://en.wikipedia.org/wiki/Reeh%E2%80%93Schlieder_theorem , which would be a problem for such an inversion.

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Thanks for the answer. I see your point about the Reeh-Schlieder theorem. The thing I find confusing is that, in the case where the net of algebras is generated by operator-valued distributions, it looks like we do have a notion of support: To any smeared observable $\phi(f)$, we associate the support of $f$. For algebraic combinations of such observables, take the union of the supports. –  user1504 Apr 11 '13 at 13:24
    
You are refering to the Wightman-Garding framework, where a quantum field theory is established by a set of operator-valued distributions over Schwartz space acting on a Hilbert space. The corresponding Haag-Kastler net is obtained such that $\mathcal{A}(\mathcal{O})$ is the von Neumann algebra generated by the bounded functional calculi of fields $\phi(f)$ with $supp(f) \subset \mathcal{O}$. But still Reeh-Schlieder holds, i.e. the vacuum is cyclic and separating for $\mathcal{A}(\mathcal{O})$ for irreducible Wightman QFTs. –  AGP Apr 11 '13 at 17:47
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