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

TL;DR

This paper constructs a model-independent net of C*-algebras for quantum field theory over any spacetime, using causal loops and posets, and proves the existence of covariant representations in Minkowski and chiral spacetimes.

Contribution

It introduces a novel construction of nets of local algebras based on causal loops and posets, applicable to various spacetimes, and establishes covariant representations satisfying physical conditions.

Findings

Existence of Poincaré covariant representations in Minkowski spacetime.

Construction of causal and covariant connections from scalar quantum fields.

Applicability to chiral spacetime with conformal symmetry.

Abstract

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.