On revolutionizing quantum field theory with Tomita's modular theory

Hans-J├╝rgen Borchers
February 23, 2000
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 eld 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 rst 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 eld 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 ve 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.

modular theory, inclusions of von Neumann algebras, geometric modular action, algebraic quantum field theory, review