causal complement



The concept of a causal complement is suitable to establish a causal disjointness relation on index sets where the indices are subsets of a given set. This generalizes the concept of causal complement of subsets of the Minkowski spacetime, see for example Haag-Kastler vacuum representation.


Let XX be an arbitrary set and MXM \subset X. An assignment

MM X M \mapsto M^{\perp} \subset X

is called a causal complement if the following conditions hold:

(i) MM M \subseteq M^{\perp\perp}

(ii) ( jM j) = j(M j) (\bigcup_j M_j)^{\perp} = \bigcap_j (M_j)^{\perp}

(iii) MM =M \bigcap M^{\perp} = \emptyset

A set MM is causally closed iff M=M M = M^{\perp\perp}.

The set M M^{\perp\perp} is the causal closure of MM.


Causal complements are always causally closed. The intersection of two causally closed sets is again a causally closed set. The causal complement of a set may be empty.

A causal disjointness relation on an index set of subsets of a given set XX can be defined via

M 1M 2iffM 1(M 2) M_1 \perp M_2 \; \text{iff} \; M_1 \subseteq (M_2)^{\perp}

if all sets MM have a causal complement and if

(iv) there is a sequence (Y n) n=1 (Y_n)_{n=1}^{\infty} of mutually different subsets with Y n Y_n^{\perp} \neq \emptyset and Y n=X\bigcup Y_n = X.

The latter condition is needed to get a σ\sigma-bounded poset; the σ\sigma-boundedness is part of the definition of a causal index set.

Revised on June 11, 2010 09:02:48 by Toby Bartels (