# Extensions of DHR superselection theory to long range forces

For Haag-Kastler nets $M(O)$ of von-Neumann algebras $M$ indexed by open bounded subsets $O$ of the Minkowski space in AQFT (algebraic quantum field theory) the DHR (Doplicher-Haag-Roberts) superselection theory treats representations that are "localizable" in the following sense.

The $C^*-$algebra

$$\mathcal{A} := clo_{\| \cdot \|} \bigl( \bigcup_{\mathcal{O}}\mathcal{M}(\mathcal{O}) \bigr)$$

is called the quasi-local algebra of the given net.

For a vacuum representation $\pi_0$, a representation $\pi$ of the local algebra $\mathcal{A}$ is called (DHR) admissible if $\pi | \mathcal{A}(\mathcal{K}^{\perp})$ is unitarily equivalent to $\pi_0 | \mathcal{A}(\mathcal{K}^{\perp})$ for all double cones $K$.

Here, $\mathcal{K}^{\perp}$ denotes the causal complement of a subset of the Minkowski space.

The DHR condition says that all expectation values (of all observables) should approach the vacuum expectation values, uniformly, when the region of measurement is moved away from the origin.

The DHR condition therefore excludes long range forces like electromagnetism from consideration, because, by Stokes' theorem, the electric charge in a finite region can be measured by the flux of the field strength through a sphere of arbitrary large radius.

In his recent talk

• Sergio Doplicher: "Superselection structure in Local Quantum Theories with (neutral) massless particle"

at the conference Modern Trends in AQFT, it would seem that Sergio Doplicher announced an extension of superselection theory to long range forces like electromagnetism, which has yet to be published.

I am interested in any references to or explanations of this work, or similar extensions of superselection theory in AQFT to long range forces. (And of course also in all corrections to the characterization of DHR superselection theory I wrote here.)

And also in a heads up when Doplicher and coworkers publish their result.

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## 2 Answers

Now I wish I had paid more attention while I was sitting in the audience. :-) Unfortunately, I'm not familiar enough with the original DHR analysis to have retained more than just the broad outlines of the arguments anyway.

With that disclaimer, I do (imperfectly) recall some apparently important points. Doplicher drew attention to the parallels between their new analysis and the analysis of Buchholz and Fredenhagen (CMP 84 1, doi), which relied only on spacelike wedges for a notion of localization, rather than the double diamonds of DHR. Starting from wedges, localization properties can be refined to spacelike cones and, under fortuitous circumstances, to arbitrarily small bounded regions. On the other hand, the new analysis makes use of localization in future pointed light cones. The analogs of spacelike cones are now played by hyperbolic cones, which are thickenings of cones defined on 3-hyperbolids asymptotic to the given light cone. I'm afraid I cannot be more specific, but this notion seems to have come up independently in hyperbolic 3-geometry.

As to the results, I recall that they are very similar to the results of the previous DHR or BF analyses. In particular, no exotic statistics appear and only the standard (para)bose and (para)fermi cases are possible. I can't recall any result that is different from the previous analyses (though that could be just my memory).

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Well, that's a good start :-) But I'll leave the question open for now. – Tim van Beek Sep 28 '11 at 9:49

Slides by Buchholz on this project are available here: http://www.univie.ac.at/qft-lhc/?page_id=10

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Thanks for the tip! I am curious about Bucholz's conlcusion "Origin of infrared difficulties can be traced back to unreasonable idealization of observations covering all of Minkowski space". I remember asking about this idealization in all QFT calculations in an introductory class, but had no idea that this would reappier in such a context :-) – Tim van Beek Oct 7 '11 at 13:54
I think the key physical insights are that 1) for an observer the relevant part of Minkowski spacetime where he can perform measurements (observables) is its causal future (future lightcone $V_+$) and 2) photons from the past lightcone $V_-$ cannot enter $V_+$ which provides, in a sense, a geometric infrared cutoff. But of course, if you want to compare measurements for different observers, you need to consider all of Minkowski space. – Eric Oct 10 '11 at 7:21
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