## Murray-von Neumann classification of local algebras in Haag-Kastler QFT

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The Haag-Kastler approach to quantum field theory (QFT) is one of the oldest approaches to rigorously define what a QFT is, it deals with nets of operator algebras: You start with a spacetime and assign von Neumann algebras (or $C^*$-algebras, but my question is about the von Neumann algebra situation only) to certain subsets of the spacetime subject to certain axioms. I am interested in results about the Murray-von Neumann classification of these algebras, i.e. which kind of factors can occur in the central decomposition (the decomposition of the algebra as a direct integral of factors).

To make this more precise, here is an example: One can define vacuum representations on Minkowski spacetime, for details please see

Haag-Kastler vacuum representation on the nLab.

A net of a vacuum representation is said to satisfy duality, or be a dual net, if one has $\mathcal{M}(\mathcal{O}') = \mathcal{M}'(\mathcal{O})$, put in words: the algebra of the causal complement of a bounded open set $\mathcal{O}$ is the commutant of the algebra associated with $\mathcal{O}$. Then it is a theorem that algebras associated to diamonds can only have factors of type $III_1$ in their central decomposition.

1. Is the assumption of duality necessary or is causality enough? Is Haag duality enough? (Haag duality means that the duality condition does not have to hold for all algebras associated to bounded open regions, but for diamonds only).
2. What are the necessary assumptions to deduce that algebras associated to diamonds are a factor of type $III_1$, i.e. have trivial center? What are the necessary assumptions to get that these algebras are hyperfinite?
3. Are there similar results about the factor decomposition of algebras associated to more general subsets than diamonds of the Minkowski space, like open bounded subsets?
4. Are there similar results about more general spacetimes, like globally hyberbolic ones?
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## 1 Answer

There is a nice overview about algebraic quantum field theory by Halverson and Müger, which covers some of the stuff I mention below and can be found at

http://arxiv.org/abs/math-ph/0602036

Concerning your question(s): Having a factor of type III means that every (non-zero) projection in it is Murray-von Neumann equivalent to the identity. In the Haag-Kastler approach Borchers came up with a slightly weaker notion of "type III-ness", which is sometimes called 'property B'.

Definition: Let $\mathcal{O} \rightarrow \mathcal{A}(\mathcal{O})$ be a net of von Neumann algebras on some common Hilbert space $H$. $\mathcal{A}$ has property B if for any two double cones $\mathcal{O}_1$ and $\mathcal{O}_2$ such that $\overline{\mathcal{O}_1}\subset \mathcal{O}_2$ and for any non-zero projection $E \in \mathcal{A}(\mathcal{O}_1)$ there is an isometry $V \in \mathcal{A}(\mathcal{O}_2)$, such that $VV^* = E$ and $V^*V= 1$ (in other words: $E$ is equivalent to $1$ in the algebra $\mathcal{A}(\mathcal{O}_2)$).

The point is the following theorem proven by Borchers in

A remark on a theorem of B. Misra, H.J. Borchers, Communications in Mathematical Physics, Volume 4 (5), page 315-323.

Theorem: Let $\mathcal{O} \rightarrow \mathcal{A}(\mathcal{O})$ be a net of von Neumann algebras satisfying microcausality, the spectrum condition, and weak additivity. Then the net $\mathcal{A}$ satisfies property B.

So, what are these notions?

• microcausality means that the algebras associated to spacelike separated double cones commute
• additivity means that for any double cone $\mathcal{O}$ the set of all $\mathcal{A}(\mathcal{O} + x)$ for $x \in T$, where $T$ denotes the translation group generates the associated (universal) $C^*$-algebra (I am not quite sure, what weak means in the statement).
• the spectrum condition means there is a subset $T_+ \subset T$ with $T_+ \cap (-T_+) = {0}$ and the spectrum of the unitary representation of $T$ is contained in $T_+$.

Since these are all motivated by physics, they are all kind of natural to demand for nets of von Neumann algebras in AQFT.

Anyway there are also some results concerning the type of local algebras. For example, if you take wedge shaped regions $W$ in Minkowski spacetime, then it is shown in

On the Net of von Neumann algebras associated with a Wedge and Wedge-causal Manifolds, H.J. Borchers, http://www.lqp.uni-goettingen.de/papers/09/12/09120802.html

that the local algebra associated to $W$ is actually of type $III_1$ for nets satisfying similar assumptions motivated by physics.

Furthermore there is the paper

The Universal Structure of Local Algebras, Buchholz, D., D'Antoni and Fredenhagen, K., Communications in Mathematical Physics 111 (1), page 123-135

in which it is shown that if you assume that your net is derived from a Wightman QFT and satisfies asymptotic scale invariance and nuclearity, then the local algebras associated to double cones are of the form $\mathfrak{R} \otimes Z(\mathcal{O})$, where $\mathfrak{R}$ is the unique hyperfinite type $III_1$-factor and $Z(\mathcal{O})$ is the center of the local algebra $\mathcal{A}(\mathcal{O})$.

So, this and the fact that in many examples of nets you can check directly that the local algebras are of type $III_1$ is the justification for physicists to assume this to be the case in everything physically relevant.

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Thanks, that is the kind of answer I was looking for. I knew that in Minkowski spacetime the algebra of the wedge is also of type $III_1$, but did not know the paper of Borchers about "wedge-local manifolds". I'm also interested in the consequences that physicists think this fact implies, but did not ask explicitly because I considered it not to be appropriate here (it's not about mathematics). I did not mark this as the accepted answer because I hope to get further input :-) – Tim van Beek Apr 16 2010 at 12:03
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