I simply don't believe this. It undermines either the idea of a single-particle space or a particle detector if a single-particle state can make two remote detectors click. Why call either of them those things?
There must be something wrong about the measurement semantics you have in mind.
I don't think so? It's still an eigenstate of the particle number operator. Two remote detectors are extremely unlikely to both click if they are far apart in terms of the compton wavelength. So if you have a lot of detectors strewn about (say, a cloud chamber), you see localized responses
Ok, so c.f. Duncan the situation is far worse than I realized. If N(U) and N(V) do not commute, then it's not just that a "single particle" detected in U can also be detected in V. Rather, experiments to learn the occupation number both of U and of V *cannot even be designed.*
Those occupation numbers do not simultaneously exist! We can only learn one of N(U) or N(V), and then we can't even partition the particle-count we found there into a cover of U by U', U''
By contrast, Haag says this in this text (pg 93) which seems to explicitly contradict what you've said. I need to read closer to determine if he has adopted a different measurement semantics (identifies a different mathematical expression or condition as
the probability or certainty that his detectors click), or if his definition of n-particle subspace differs from the standard notion. But, I think its the former. (Finding it difficult to internalize what he's saying in the lead-up to this paragraph.)
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There must be something wrong about the measurement semantics you have in mind.