nLab
causal index set

Contents

Idea

In the Haag-Kastler approach to quantum field theory one deals with local nets indexed by bounded open regions of Minkowski spacetime. An important axiom of this approach is that of causality, which says that observables localized in spacelike separated regions of spacetime commute. A causal index set is an abstraction of the index set of bounded open regions that retains the relation induced by the concept of spacelike separation. It can be used to generalize the axiom of causality to nets with different or more general index sets as the one mentioned above. Causal index sets are needed to define the notion of a causal net of algebras.

Definition

A relation \perp on a σ\sigma-bounded poset II is called a causal disjointness relation (and a,bIa, b \in I are called causally disjoint if aba \perp b) if the following properties are satisfied:

(i) \perp is symmetric

(ii) aba \perp b and c<bc \lt b implies aca \perp c

(iii) if MIM \subset I is bounded from above, then aba \perp b for all aMa \in M implies supMbsup M \perp b.

(iv) for every aIa \in I there is a bIb \in I with aba \perp b

A poset with such a relation is called a causal index set.

Examples

One example is explained in the Idea section.

Let II be the poset of finite subspaces of a separable Hilbert space. Define \perp by orthogonality of subspaces, then (I,)(I, \perp) is a causal index set.

Let XX be a countable set and II be its finite power set, that is the collection of finite subsets of XX. Set MNM \perp N iff MN=M \cap N = \emptyset, then this defines a causal disjointness relation. This is an example of a causal complement.

Revised on June 11, 2010 09:00:38 by Toby Bartels (173.190.150.217)