zbMATH — the first resource for mathematics

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

46N99 Miscellaneous applications of functional analysis
81T05 Axiomatic quantum field theory; operator algebras
46L60 Applications of selfadjoint operator algebras to physics
Full Text: DOI
[1] Bannier, U.: Allgemein kovariante algebraische Quantenfeldtheorie und Rekonstruktion von Raum-Zeit. Thesis, Hamburg 1987
[2] Brattelli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics, Vol. II. Berlin, Heidelberg, New York: Springer 1981
[3] Dimock, J.: Algebras of local observables on a manifold. Commun. Math. Phys.77, 219 (1980) · Zbl 0455.58030 · doi:10.1007/BF01269921
[4] Dixmier, J.:C. Amsterdam, New York, Oxford: North-Holland 1977
[5] Dubois-Violette, M.: A generalization of the classical moment problem on *-algebras with application to relativistic quantum theory. I. Commun. Math. Phys.43, 225 (1975) · Zbl 0362.46044 · doi:10.1007/BF02345022
[6] Dyson, F.J.: Missed opportunities. Bull. Am. Math. Soc.78, 635 (1972) · Zbl 0271.01005 · doi:10.1090/S0002-9904-1972-12971-9
[7] Ekstein, H.: Presymmetry. II. Phys. Rev.184, 1315 (1969) · doi:10.1103/PhysRev.184.1315
[8] Fredenhagen, K., Haag, R.: Generally covariant quantum field theory and scaling limits. Commun. Math. Phys.108, 91 (1987) · Zbl 0626.46063 · doi:10.1007/BF01210704
[9] Friedlander, F.G.: The wave equation on a curved space-time. Cambridge, London, New York, Melbourne: Cambridge University Press 1975 · Zbl 0316.53021
[10] Haag, R., Kastler, D.: An algebraic approach to quantum field theory. J. Math. Phys.5, 848 (1964) · Zbl 0139.46003 · doi:10.1063/1.1704187
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.