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On generally covariant quantum field theory and generalized causal and dynamical structures. (English) Zbl 0655.46058
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, $${\mathcal A}({\mathfrak O})={\mathcal A}(a+\Lambda {\mathcal O}_ 1)$$. $$(a,\Lambda)\in P_+^{†}.$$
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.
Reviewer: N.D.Sengupta

##### MSC:
 46N99 Miscellaneous applications of functional analysis 81T05 Axiomatic quantum field theory; operator algebras 46L60 Applications of selfadjoint operator algebras to physics
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