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.* - ThreadSky

uberwensch.bsky.social • 12 hours ago

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.*

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uberwensch.bsky.social•12 hours ago

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''

uberwensch.bsky.social•12 hours ago

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

uberwensch.bsky.social•12 hours ago

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.)

coexact.bsky.social•4 hours ago

So I was thinking of detection as an elastic scattering which localizes the particle, like in the cloud chamber. But you can also consider a detector that absorbs the particle, in which case only one goes off, and my guess is that's what Haag means. I'll take a look

coexact.bsky.social•2 hours ago

Ok no, it seems that's not it, he means the same thing I do. But by 'gives no signal in n far separated detectors', he means the asymptotic statement II.4.39! So he's saying the same thing, it simply becomes unlikely for 2 detectors to ping at large distances

uberwensch.bsky.social•12 hours ago

The detectors he introduces are not local operators, but "almost local operators." Possibly the way out of this mess is to suggest that no operation, performed in any bounded region U, is *exactly* capable of verifying an occupation number, but can do so approximately insofar as U is big (??)

uberwensch.bsky.social•12 hours ago

I gotta tap out for a big I am fatigued and my brain hurts

coexact.bsky.social•4 hours ago

I was already asleep so good call ^^

uberwensch.bsky.social•12 hours ago

one last point: once we've conceded to relativize our 'particle' notion to dectectors, it opens the possibility that different answers to these questions (and different particle content) can be assigned by differently characterized detectors. this argument is developed in the paper here:

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