# 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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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