On generally covariant quantum field theory and generalized causal and dynamical structures (Q1109304)

Let \({\mathcal O}\) be a a region of Minkowski space \({\mathbb{M}}\), thus there corresponds one \(C^*\)-algebra A(\({\mathcal O})\), satisfying the condition if \({\mathcal O}_ 1\subseteq {\mathcal O}_ 2\subseteq {\mathbb{M}}\) then \({\mathcal A}({\mathcal O}_ 1)\subseteq {\mathcal A}({\mathcal O}_ 2)\) (isotony property). The self-adjoint elements of \({\mathcal A}({\mathcal O})\) are interpretated as observables. The motivation of the paper is to formulate in a covariant manner the axioms: (I) (a) Einstein causality: If two regions \({\mathcal O}_ 1\) and \({\mathcal O}_ 2\) are space like to each other, then [\({\mathcal A}({\mathcal O}_ 1),{\mathcal A}({\mathcal O}_ 2)]=0\), (b) Primitive causality: If \({\mathcal O}_ 2\) is a domain of dependence of \({\mathcal O}_ 1\) then \({\mathcal A}({\mathcal O}_ 2)\subseteq {\mathcal A}({\mathcal O}_ 1);\) (II) Poincaré Invariance, \({\mathcal A}({\mathfrak O})={\mathcal A}(a+\Lambda {\mathcal O}_ 1)\). \((a,\Lambda)\in P_+^{\dag}.\) The author gives an example of a generally covariant net of \(C^*\)- algebras and generalized causal relation. The main additional structure needed, is the presence of many maximal, two-sided ideals in the local algebra \({\mathcal A}({\mathcal O})\), by which one can formulate dynamical or causal structures.