# Significance of the hyperfinite $III_1$ factor for axiomatic quantum field theory

Using a form of the Haag-Kastler axioms for quantum field theory (see AQFT on the nLab for more details), it is possible in quite general contexts to prove that all local algebras are isomorphic to the hyperfinite $III_1$ factor or to the tensor product of the $III_1$ factor with the center of the given local algebra.

(A local algebra is the algebra of observables that is associated to an open bounded subset of Minkowski space. The term $III_1$ factor refers to the Murray-von Neumann classification of factors of von Neumann algebras).

Also see this question on math overflow for more details.

So one could say that quantum mechanics has the $I_n$ and $I_{\infty}$ factors as playground, while QFT has the hyperfinite $III_1$ factor as playground.

My questions has two parts:

1) I would like to know about a concrete physical system where it is possible to show that the local algebras are hyperfinite $III_1$ factors, if there is one where this is possible.

2) Is there an interpretation in physical terms of the presence of the hyperfinite $III_1$ factor in QFT?

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## 2 Answers

Yngvason, J. (2005). The role of type III factors in quantum field theory. Reports on Mathematical Physics, 55(1), 135–147. (arxiv)

The Type III property says something about statistical independence. Let $\mathcal{O}$ be a double cone, and let $\mathfrak{A}(\mathcal{O})$ be the associated algebra of observables. Assuming Haag duality, we have $\mathfrak{A}(\mathcal{O}')'' = \mathfrak{A}(\mathcal{O})$. If $\mathfrak{A}(\mathcal{O})$ is not of Type I, the Hilbert space $\mathcal{H}$ of the system does not decompose as $\mathcal{H} = \mathcal{H}_1 \otimes \mathcal{H}_2$ in such a way that $\mathfrak{A}(\mathcal{O})$ acts on the first tensor factor, and $\mathfrak{A}(\mathcal{O}')$ on the second. This implies that one cannot prepare the system in a certain state when restricted to measurements in $\mathcal{O}$ regardless of the state in the causal complement. It should be noted that if the split property holds, that is there is a Type I factor $\mathfrak{N}$ such that $\mathfrak{A}(\mathcal{O}) \subset \mathfrak{N} \subset \mathfrak{A}(\widehat{\mathcal{O}})$ for some region $\mathcal{O} \subset \widehat{\mathcal{O}}$, a slightly weaker property is available: a state can be prepared in $\mathcal{O}$ irregardless of the state in $\widehat{\mathcal{O}}'$. An illustration of the consequences can be found in the article above.

Another consequence is that the Borchers property B automatically holds: if $P$ is some projection in $\mathfrak{A}(\mathcal{O})$, then there is some isometry $W$ in the same algebra such that $W^*W = I$ and $W W^* = P$. This implies that we can modify the state locally to be an eigenstate of $P$, by doing the modification $\omega(A) \to \omega_W(A) = \omega(W^*AW)$. Note that $\omega_W(P) = 1$ and $\omega_W(A) = \omega(A)$ for $A$ localised in the causal complement of $\mathcal{O}$. Type III$_1$ implies something slightly stronger, see the article cited for more details.

As to the first question, one can prove that the local algebras of free field theories are Type III. This was done by Araki in the 1960's. You can find references in the article mentioned above. In general, the Type III condition follows from natural assumptions on the observable algebras. Non-trivial examples probably have to be found in conformal field theory, but I do not know any references on the top of my head.

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Ok, I somehow missed that paper, although Yngvason is on my watch list :-) – Tim van Beek Sep 16 '11 at 7:28
Feel free to expand you answer later, even though I will accept it already. – Tim van Beek Sep 16 '11 at 7:29
Yes, I will. Have to read the paper again myself first, I don't recall the details... – Pieter Sep 16 '11 at 11:47
I updated the question, feel free to ask for more details :) – Pieter Sep 17 '11 at 0:21

Regarding the first question. As Pieter already said for a conformal net the $III_1$ property holds (if it is not $\mathbb C$). Further $e^{-\beta L_0}$ being trace class for all $\beta>0$ with $L_0$ the generator of the rotations implies the split property, which implies $\mathcal A(I)$ to be the hyperfinite $III_1$-factor.

edit The property $III_1$ and trace class implies split can be found in - D'Antoni,Longo,Radulescu. Conformal Nets, Maximal Temperature and Models from Free Probability [arXiv:math/9810003v1]

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