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Zbl 1031.81547
Borchers, H.J.
On revolutionizing quantum field theory with Tomita's modular theory. (English)
[J] J. Math. Phys. 41, No.6, 3604-3673 (2000). ISSN 0022-2488; ISSN 1089-7658/e

Summary: In the book of {\it R. Haag} [Local quantum physics. Springer Verlag, Berlin (1992; Zbl 0857.46057)] about local quantum field theory the main results are obtained by the older methods of $C^*$- and $W^*$-algebra theory. A great advance, especially in the theory of $W^*$-algebras, is due to {\it M. Tomita's} discovery of the theory of modular Hilbert algebras [Quasi-standard von Neumann algebras, Preprint (1967), see also M. Takesaki (Zbl 0193.42502)]. Because of the abstract nature of the underlying concepts, this theory became (except for some sporadic results) a technique for quantum field theory only in the beginning of the nineties. In this review the results obtained up to this point will be collected and some problems for the future will be discussed at the end. In the first section the technical tools will be presented. Then in the second section two concepts, the half-sided translations and the half-sided modular inclusions, will be explained. These concepts have revolutionized the handling of quantum field theory. Examples for which the modular groups are explicitly known are presented in the third section. One of the important results of the new theory is the proof of the PCT theorem in the theory of local observables. Questions connected with the proof are discussed in Section IV. Section V deals with the structure of local algebras and with questions connected with symmetry groups. In Section VI a theory of tensor product decompositions will be presented. In the last section problems that are closely connected with the modular theory and that should be treated in the future will be discussed.

MSC 2000:: *81T05 Axiomatic quantum field theory
46L35 Classifications and factors of C*-algebras
81-02 Research monographs (quantum theory)
20C20 Modular representations and characters of groups
46L60 Appl. of selfadjoint operator algebras to physics
46N50 Appl. of functional analysis in quantum physics
81R15 Operator algebra methods

Keywords: half-sided translations; half-sided modular inclusions; modular groups; proof of the PCT theorem; symmetry groups; tensor product decompositions

Citations: Zbl 0193.42502; Zbl 0857.46057

Cited in: Zbl 1028.81036

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