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Wish I could have been there.
I spent so much time in grad school trying to wrap my head around some of this stuff without really being sure what it was I was trying to wrap my head around.
Sounds pretty awesome and dramatic. Someone correct me if Im wrong, but I believe the so-called problems of "heuristic" quantum field theory from Haag's theorem – the inequivalence of representations of the canonical commutation relations – have only to do with pathologically-constructed operators that don't correspond to anything physical. At least thats what Ive read, I should look into it myself at some point.
+Cliff Harvey: I might be wrong about this myself, because it's been so long since I looked at QFT, but I do believe that Haag's theorem is not part of the "heuristic" tradition. Haag's theorem is part of axiomatic quantum field theory (AQFT), i.e., the mathematically precise tradition. The main problem with the "heuristic tradition" ("HQFT", I'll call it), on the other hand, is (or is evidenced when) one performs renormalization (though our understanding of renormalization has greatly improved since the time I was looking at it years ago, and doesn't seem to be quite so mathematically problematic anymore - partly because of the work of David Wallace, one of the presenters at the Pittsburgh conference - because we understand how the mathematical tricks map onto certain (as it turns out reasonable) physical assumptions).

As you said, though, Haag's theorem does show that there's a problem, it's just that the problem (again, IIRC) is in AQFT. It shows that particle interactions can't be modeled in AQFT due (as you said in your post) to the inequivalence of representations of the CCRs. That's all I remember right now, though. I wish I didn't have such a terrible memory.
Thanks +So Ghosh Im pretty sure you must be right that Haag's theorem is about the AQFT setup, and the other stuff you've said too.

I also just noticed that one of the participants here, Doreen Frasier, is the author of a document I started to read about Haag's a while back.

So the main way Id like to boil down the question is in what way can we physically manifest the supposed problems Haag's theorem. Is there any at all? Based on the successes of QFT more generally, and another assessment I remember reading a while ago, I suspect that there may not be...
+Cliff Harvey: Thanks for your clarifying response. :)

You asked "In what way can we physically manifest the supposed problems in Haag's theorem?"

Haag's theorem tells us that particle interaction can't be modeled in AQFT. It's not a theorem about the world so much as a theorem about what can and can't be modeled in AQFT. In that sense, it's a purely mathematical theorem.

The physical manifestation of the problem, I guess, is that particles do interact. So, if I haven't totally misunderstood your question, we can physically manifest the problems in Haag's theorem whenever we cause particle interaction. (So Haag's theorem points to a very big problem in AQFT!)

I think, however, that you might have been asking: "Which physical assumptions of AQFT does Haag's theorem show us are problematic? I.e., which axiom or axioms is/are the source of the difficulty that leads us to Haag's theorem?" I think I may have known the answer to this question once, but I don't remember! If Wallace has written anything about this question (and I'd wager he has), I highly recommend reading what he has to say about it, more than any other philosopher of physics (as genuinely smart as Doreen Fraser is); I find him able to clear away cobwebs like no other.
I think you've reframed the question in a very productive way +So Ghosh. :]

Thanks for the recommendation, I'm definitely interested in hearing what Wallace has to say, along with the dissenters. I noticed this one on youtube, which may be enlightening: Taking particle physics seriously, David Wallace

It has a suggestive "abstract": The working assumption amongst most philosophers of QFT appears to be that algebraic QFT (AQFT), and not the "Lagrangian" QFT of the working physicist, is the proper object of philosophical and foundational study. I argue that this assumption is unmotivated, and fails to take into account important features of the post-1960s development of Lagrangian QFT. From a modern perspective the two forms of QFT are better seen as rival research programs than as variant formulations of one theory; furthermore, the Lagrangian research program is overwhelmingly supported by experiment.

Im inclined to agree, but Ill see if I find a good argument on the other side.

Let me know if you learn anything else interesting on the question!
Oh sorry, one more thing, +Cliff Harvey! :)

It looks like you were looking at Doreen's PhD thesis. Our understanding of the foundations of especially AQFT have changed quite a bit since even 2006 (one of the few places in the philosophy of physics where knowledge is rapidly increasing!), so, if possible, I'd try to find something more recent than that, regardless of whether it's by Doreen, Wallace, or someone else! :)
+Cliff Harvey: Thanks for the youtube link! It's definitely less painful to miss conferences when there are videos posted of them online. Somehow (oddly) it hadn't occurred to me that I could find such talks online! I can't wait to watch Wallace's talk! This will actually be even better for me than even being there in person, as I'll be able to pause his talk to look things up whenever and if I start to get lost (usually I need to look up an important definition or two).

And yes, I'd love to (and will) keep you posted if I learn anything on this question. Oh, and if you find any arguments on the anti-Wallace side that you find fairly convincing, I'd love to know of that too. :)

(Btw: I posted my "one more thing" comment above just as I received your own response in my stream; That's why I'm only now responding to it.)

Edit: I just started watching the talk. Embarrassing as it is, I think I must have been in the audience, though I remember nothing of the talk or of being there. I'm going to watch the whole thing to see if I see myself at any point in the audience.
The concept of a real number is precise to indefinite degree. However, this kind of precision cannot exist in a physical universe where quantum uncertainty applies. So, a simple question: Why would infinities not pop up in quantum field theories, given that they use real and complex numbers as fundamental modeling constructs?
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