nLab causal complement

Idea

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.

Definition

The concept of causal complement, as the name indicates, a concept related to causality on spacetimes, hence a concept associated with Lorentzian manifolds:

• On Lorentzian manifolds.

One may abstract away some of the key properties of causal complements on Lorentzian manifolds to obtain a more general concept:

• Abstract definition

On Lorentzian manifolds

Let $X$ be Lorentzian manifold (a spacetime), hence a pseudo-Riemannian manifold with one time-like dimension in each tangent space.

Then for $S \subset X$ any subset, its causal cone $J(S)$ is the subset of all points in $X$ which may be joined to a point of $S$ by a smooth curve $\gamma \colon [0,1] \to X$ which is everywere timelike or lightlike.

Definition

(causal complement of subset of Lorentzian manifold)

For $S \subset X$ a subset of a Lorentzian manifold, its causal complement $S^\perp$ is the complement of the causal cone:

$$S^\perp \;\coloneqq\; X \setminus J(S) \,.$$

The causal complement $S^{\perp \perp}$ of the causal complement $S^\perp$ is called the causal closure. If

$$S = S^{\perp \perp}$$

then the subset $S$ is called a causally closed subset.

Abstract definition

Let $X$ be an arbitrary set and $M \subset X$. An assignment

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

is called a causal complement if the following conditions hold:

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

(ii) $(\bigcup_j M_j)^{\perp} = \bigcap_j (M_j)^{\perp}$

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

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

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

Properties

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 $X$ can be defined via

$$M_1 \perp M_2 \; \text{iff} \; M_1 \subseteq (M_2)^{\perp}$$

if all sets $M$ have a causal complement and if

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

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

Related concepts

• causal structure

• causal order

• causality, microcausality

• causal locality, causal additivity

• causal complement, causal closure

• causal perturbation theory

• time-ordered product

• S-matrix