Title: Antiunitary representations and modular theory,
Abstract: Antiunitary representations of Lie groups take values in the group of unitary
and antiunitary operators on a Hilbert space H. In quantum physics, antiunitary
operators implement time inversion or a PCT symmetry, and in the modular theory
of operator algebras they arise as modular conjugations from cyclic separating
vectors of von Neumann algebras. We survey some of the key concepts at the
borderline between the theory of local observables (Quantum Field Theory (QFT)
in the sense of Araki--Haag--Kastler) and modular theory of operator algebras
from the perspective of antiunitary group representations. Here a central point
is to encode modular objects in standard subspaces V in H which in turn are in
one-to-one correspondence with antiunitary representations of the
multiplicative group R^x. Half-sided modular inclusions and modular
intersections of standard subspaces correspond to antiunitary representations
of Aff(R), and these provide the basic building blocks for a general theory
started in the 90s with the ground breaking work of Borchers and Wiesbrock and
developed in various directions in the QFT context. The emphasis of these notes
lies on the translation between configurations of standard subspaces as they
arise in the context of modular localization developed by Brunetti, Guido and
Longo, and the more classical context of von Neumann algebras with cyclic
separating vectors. Our main point is that configurations of standard subspaces
can be studied from the perspective of antiunitary Lie group representations
and the geometry of the corresponding spaces, which are often fiber bundles
over ordered symmetric spaces. We expect this perspective to provide new and
systematic insight into the much richer configurations of nets of local
observables in QFT.