Haag–Kastler axioms

Definition

Let $M = \R^{1,3}$ be Minkowski space and let $\mathcal{K}$ be the set of bounded open regions of $M$ (for instance the double cones). Two regions $O_1, O_2$ are spacelike separated if $x - y$ is a spacelike vector for all $x \in O_1$ and $y \in O_2$. Let $\mathcal{P}$ be the proper orthochronous Poincaré group. A net of local observables satisfying the Haag–Kastler axioms is an assignment $O \mapsto \mathcal{A}(O)$ of a unital C*-algebra to each $O \in \mathcal{K}$ with these properties.

1. Isotony: if $O_1 \subseteq O_2$, then $\mathcal{A}(O_1)$ is a unital C*-subalgebra of $\mathcal{A}(O_2)$. Let $\mathcal{A}$ be the C*-algebra generated by all $\mathcal{A}(O)$, called the quasilocal algebra.
2. Locality: if $O_1$ and $O_2$ are spacelike separated, then $AB = BA$ for all $A \in \mathcal{A}(O_1)$ and $B \in \mathcal{A}(O_2)$.
3. Covariance: there is a homomorphism $g \mapsto \alpha_g$ from $\mathcal{P}$ to the automorphism group of $\mathcal{A}$ with $\alpha_g(\mathcal{A}(O)) = \mathcal{A}(gO)$ for all $g$ and $O$.
4. Vacuum: there is a state $\omega$ on $\mathcal{A}$ with $\omega \circ \alpha_g = \omega$ for all $g \in \mathcal{P}$. In the GNS representation of $\omega$ on a Hilbert space $H$, the automorphisms $\alpha_g$ are implemented by a strongly continuous unitary representation $U$ of $\mathcal{P}$. The joint spectrum of the generators $P^\mu$ of the translations $U(a) = e^{i P^\mu a_\mu}$ lies in the closed forward light cone $\set{p : p^0 \ge 0, \ p_\mu p^\mu \ge 0}$ (spectrum condition).