## Advances in Theoretical and Mathematical Physics

- Adv. Theor. Math. Phys.
- Volume 16, Number 2 (2012), 645-691.

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

Fabio Ciolli, Giuseppe Ruzzi, and Ezio Vasselli

#### 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 Poincaré 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.

#### Article information

**Source**

Adv. Theor. Math. Phys., Volume 16, Number 2 (2012), 645-691.

**Dates**

First available in Project Euclid: 23 January 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.atmp/1358950890

**Mathematical Reviews number (MathSciNet)**

MR3019413

**Zentralblatt MATH identifier**

1271.81095

#### Citation

Ciolli, Fabio; Ruzzi, Giuseppe; Vasselli, Ezio. Causal posets, loops and the construction of nets of local algebras for QFT. Adv. Theor. Math. Phys. 16 (2012), no. 2, 645--691. https://projecteuclid.org/euclid.atmp/1358950890