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After Urs’ post at the café about “Tricategory of conformal nets” at Oberwolfach I took a look at the paper Conformal nets and local field theory and noted that I would have to ask some trivial and boring questions about nomenclature before I could even try to get to the content.
One example is about “Haag duality”: It seems to me that we need a generalization of net index sets on the nLab that includes the bounded open sets used for the Haag-Kastler vacuum representation and the index sets used in the mentioned paper. One of the concept needed would be “causal index set”:
A relation $\perp$ on an index set (poset) $I$ is called a causal disjointness relation (and $a, b \in I$ are called causally disjoint if $a \perp b$) if the following properties are satisfied:
(i) $\perp$ is symmetric
(ii) $a \perp b$ and $c \lt b$ implies $a \perp c$
(iii) if $M \subset I$ is bounded from above, then $a \perp b$ for all $a \in M$ implies $sup M \perp b$.
(iv) for every $a \in I$ there is a $b \in I$ with $a \perp b$
A poset with such a relation is called a causal index set.
Well, that’s not completly true, because in the literature that I know there is the additionally assumtion that $I$ contains an infinite unbounded sequence and hence is not finite (that whould be a poset that is ? what? unbounded?), that is not a condition imposed on posets on the nLab.
After this definition one can go on and define “causal complement”, the “causality condition” for a net and then several notions of duality with respect to causal complements etc. all without reference to Minkowski space or any Lorentzian manifolds.
Should I create a page causal index set or is there something similar on the nLab already that I overlooked?
There’s causet, which is a bit different, but could contain what you have written (I’m not delighted with the name, but it can be renamed if need be). There doesn’t seem to be anything specific, if I just go from the page names that a search for ’causal’ on the lab gives me.
Note that it actually requires finiteness, but you could reinterpret that as having a bounded region that is both influenced by x and influences y (the over-under category) and let finiteness arise from the discrete case as usual.
Had a quick look at that page, too, after I did a search for causal…
The definition seems to contradict the idea section; for posets associated to Lorentzian manifolds the invervalls [x, y] are not finite, correct?
Another concept is the “split property”, for which I created the stubs spatial tensor product and split inclusion of von Neumann algebras.
The intervals aren’t finite, but presumably they are compact or at least bounded (true in Minkowski, but maybe not for more complicated spacetimes). In the discrete case (as at causet this would give finite intervals. If the intervals aren’t bounded, we could say that all causal paths in the interval are bounded (=finite length) as a relaxation. This may not always be the case, but seems a useful condition.
If you add causal index set and the material above, we can edit the page name later.
Created causal index set and causal complement. The next concepts should be dual and essentially dual nets with a causal index set… I prefer having the concept of causal index set on its own page and maybe make causet refer to it.
dual net of von Neumann algebras contains the definition of Haag duality and related concepts for a Haag-Kastler vacuum representation, the next step would be to introduce the link to the Haag duality in the “Conformal nets and local field theory ” paper.
(In the long run I would like to connect the concepts derived from AQFT there to the more “traditional” setting of the vacuum representation).
re #6 - that’s cool. I just thought that such a structure might be used for more than just an index set, so the title is a little specific to my ears - but as titles are malleable, it’s not a problem.
Ah, ok, I think I get your point now. I did not take that into account, because I think of this structure in the context of index sets for causal nets only. The concept that I was after is Haag duality, because that is used in the paper that caused me to start this thread in a different sense than usual.
I have added some references on classification of CFTs to conformal net
added pointer to
(more on the recently and finally established equivalence between the two axiomatizations of 2d conformal field theory: a) vertex operator algebras and b) conformal nets of observables)
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