On the connection between quantum fields and von Neumann algebras of local operators. (English) Zbl 0595.46062

Each of the two main approaches to the theory of particles, i.e. local fields and nets of local von Neumann algebras, has its own strengths and problems. In order to pool the first and to minimize the second we should know more about the relation between these two approaches. The present paper first considers a local field and a so-called AB-system (somewhat more general than a local net) action on the same Hilbert space and having the same localization - meaning that field operators and operators in the AB-system ”belonging” to causally disjoint regions of space-time commute weakly. It shows - very roughly speaking - that in this situation there is an almost canonical relation between the two structures.
The paper then defines ”intrinsic locality” for field operators and shows that any single such operator (if one exists) can be used to construct just such an AB-system as investigated in part one. Using an additional mild regularity condition - the generalized H-bound shown to be significant in other contexts, too - the AB-system so constructed turns out to be unique. This considerably reduces the problem of relating local fields to local nets.
In addition to the above the paper discusses various questions related to the main results.


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