On Revolutionizing Quantum Field Theory
Borchers, Hans-Jürgen
In the book of Haag [Ha92] 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 Tomita's discovery of the theory
of modular Hilbert algebras [To67]. 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 four. Section five deals
with the structure of local algebras and with questions connected
with symmetry groups. In section six a theory of tensor product
decompositions will be presented. In the last
section problems closely connected with the modular theory
and should be treated in the future will be discussed.
Reference: J. Math. Phys. 41 (2000) 3604-3673
The published version has been abridged. The unabridged original
version is available as
ESI preprint no 773: PDF.
jh,
Last modified: Tue Jun 10 11:07:12 CEST 2008