1 Original Haag Kastler Axioms

We begin by restating the original Haag Kastler axioms (Haag and Kastler).
Axiom 1 (Local Algebras) The “regions” \(\mathbf{B}\) for which the correspondence

\begin{align} \mathbf{B} \rightarrow \mathfrak {U}(\mathbf{B}) \end{align}

is defined shall be the open sets with compact closure in Minkowski space, the algebras \(\mathfrak {U}(\mathbf{B})\) shall be (abstract) C*-algebras.

Axiom 2 (Isotony) If \(\mathbf{B_1} \subset \mathbf{B_2}\) then \(\mathfrak {U}(\mathbf{B_1}) \subset \mathfrak {U}(\mathbf{B_2})\). We assume in addition that one of the two following situations prevails. Either \(\mathfrak {U}(\mathbf{B_1})\) and \(\mathfrak {U}(\mathbf{B_2})\) have a common unit element, or neither of them has a unit. The first situation can be obtained from the second by formal adjunction of a unit.

Axiom 3 (Local Commutativity) If \(\mathbf{B_1}\) and \(\mathbf{B_2}\) are completely spacelike with respect to each other, then \(\mathfrak {U}(\mathbf{B_1})\) and \(\mathfrak {U}(\mathbf{B_2})\) commute.

Axiom 4 (Quasilocal Algebra) The set-theoretic union of all \(\mathfrak {U}(\mathbf{B})\) is a normed *-algebra. Taking its completion we get a C*-algebra which we denote by \(\mathfrak {U}\) and call the algebra of quasilocal observables. We maintain that \(\mathfrak {U}\) contains all observables of interest.

Axiom 5 (Lorentz Covariance) The inhomogeneous Lorentz group is represented by automorphisms \(A \in \mathfrak {U} \rightarrow A^L \in \mathfrak {U}\) such that

\begin{align} \mathfrak {U}(\mathbf{B})^L = \mathfrak {U}(\mathbf{LB}) \end{align}

where \(L\mathbf{B}\) is the image of the region \(\mathbf{B}\) under the Lorentz transformation \(L\).

Axiom 6 (Primitivity) \(\mathfrak {U}\) is primitive.