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

September 22, 2011

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
Poincar\'e 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.

open access link

Adv. Th. Math. Phys. 16 (2012) 1-47

Keywords:

gauge field theories