Causal posets, loops and the construction of nets of local algebras for QFT

We provide a model independent construction of a net of C∗-algebras satisfying the Haag–Kastler axioms over any spacetime manifold. Such a net, called the net of causal loops, is constructed by selecting a suitable base $K$ encoding causal and symmetry properties of the spacetime. Considering $K$ as a partially ordered set (poset) with respect to the inclusion order relation, we define groups of closed paths (loops) formed by the elements of $K$. These groups come equipped with a causal disjointness relation and an action of the symmetry group of the spacetime. In this way, the local algebras of the net are the group C∗-algebras of the groups of loops, quotiented by the causal disjointness relation. We also provide a geometric interpretation of a class of representations of this net in terms of causal and covariant connections of the poset $K$. In the case of the Minkowski spacetime, we prove the existence of Poincaé covariant representations satisfying the spectrum condition. This is obtained by virtue of a remarkable feature of our construction: any Hermitian scalar quantum field defines causal and covariant connections of $K$. Similar results hold for the chiral spacetime $S^1$ with conformal symmetry.

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Published 18 January 2013