Can the double slit experiment distinguish between QM interpretations?

[LittleSchwinger]

The problem with AQFT, however is that for decades nobody was able to describe interacting particles in (1+3) dimensions!

I think here you are discussing the lack of a rigorous existence proof, correct me if I am wrong. If so I think one should distinguish between work in rigorous QFT and algebraic field theory, as I don't think what you mention is really a problem for C*-algebra methods.

Algebraic Field Theory involves using C*-algebras to extract physical predictions. This can be done at a non-rigorous level, just as in normal particle physics applications of QFT one uses operator-valued distributions without concerning oneself with their operator domain or smearing class of functions.

Recent papers by Witten on the arxiv about how deSitter space uses a Type $II_{1}$ algebra or many papers on QFT in curved spacetime are examples were C*-algebras are used in a non-rigorous or semi-rigorous way. In these "physicist" approaches to C*-algebras it has been possible to handle particles in 3+1 dimensions for around forty years and in the last ten years extract the same physical information one would from normal Feynman diagram techniques. It's become increasingly necessary to use C*-algebraic methods for discussing issues of quantum information in curved spacetime once we lack the symmetries giving a preferred Hilbert space structure or anything like particle states or global states.

Existence proofs in 3+1D are a separate issue and are equally an issue/non-issue for regular operator or path integral approaches to field theory as they are for C*-algebra methods.

 

[vanhees71]

Indeed, that's the point. We have to use the, unfortunately mathematically non-rigorous "descriptions" of QFT as effective theories using perturbative methods and (regularization and) renormalization. I think in this respect the more abstract and also more general $\text{C}^*$-algebra methods also belong to this class of "physicists' treatment".

I think to discuss the physics content of QFT AQFT is of little use, and one must rely on these non-rigorous but from a physicist's point of view very successful, descriptions.

 

[LittleSchwinger]

I think to discuss the physics content of QFT AQFT is of little use, and one must rely on these non-rigorous but from a physicist's point of view very successful, descriptions.

I take it here you are using AQFT to denote "rigorous use of C*-algebras", for example by people like Haag, as distinct from physicists using C*-algebras in a non-rigorous way as one finds in Witten's papers and the literature on QFT in curved spacetime.

If so, yes certainly it is very difficulty to extract physical results from Haag-Kastler type axiomatic set ups. It's not much different from rigorous non-Relativistic QM where results on operator domains and results like Kato's theorem usually don't give you much physical information.

Here's some sample papers of a physicist usage:
https://arxiv.org/abs/2206.10780
https://arxiv.org/abs/2301.07257

As I mentioned above the reason physicists have to use C*-algebras in such set ups is that the normal Hilbert space approach involves certain assumptions that fail in the general curved spacetime case.

 

[vanhees71]

I don't know much about QFT in curved spacetimes. I was more referring to usual QFT in Minkowski spacetime, but I guess it's even more difficult to find a rigorous formulation than in flat spacetime.

 

[LittleSchwinger]

I don't know much about QFT in curved spacetimes. I was more referring to usual QFT in Minkowski spacetime, but I guess it's even more difficult to find a rigorous formulation than in flat spacetime.

Yes definitely. For a brief explanation if we take the path integral approach, then in the Minkowski case the space of scalar, spinor, tensor fields to integrate over on Euclidean* space have a very nice and well studied linear structure. In the curved spacetime case there are very few such results and one even lacks a single unique space of fields or canonical measures on them and there can be subtleties in defining the Riemannian continuation of a general spacetime.

*Rigorously the path integral has to be in a Riemannian space as you might be aware.

 

[vanhees71]

You mean you study Euclidean QFT in the flat-spacetime case, i.e., the Wick rotated imaginary time to make the metric Euclidean rather than Lorentian. That's of course a simplifying step to formulate the path integral (though also the path-integral formulation is not entirely mathematically rigorous but also relies on regularization-renormalization and perturbation theory for the interacting case) and the trouble then is to do the analytic continuation back to real time quantities, which is everything than simple. I guess in curved spacetimes it's even more complicated, if not impossible. I've to read more about this!

 

[LittleSchwinger]

You mean you study Euclidean QFT in the flat-spacetime case, i.e., the Wick rotated imaginary time to make the metric Euclidean rather than Lorentian

Yes indeed, it's the only way to make the Path Integral rigorous as one can prove the path integral doesn't exist rigorously in the Lorentzian case.
In a rigorous approach regularization-renormalization shows up in the fact that the interacting measure $\mathcal{D}\mu[\phi]$ and the free measure $\mathcal{D}\nu[\phi]$ are relatively singular.
The curved spacetime case is exactly more difficult for the analytic continuation reasons you mentioned.