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Antiunitary representations and modular theory

Volume 113 / 2017

Karl-Hermann Neeb, Gestur Ólafsson Banach Center Publications 113 (2017), 291-362 MSC: Primary 22E45; Secondary 81R05, 81T05. DOI: 10.4064/bc113-0-16


Antiunitary representations of Lie groups take values in the group of unitary and antiunitary operators on a Hilbert space $\mathcal{H}$. 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 $V\subseteq \mathcal{H}$ which in turn are in one-to-one correspondence with antiunitary representations of the multiplicative group $\mathbb R^\times$. Half-sided modular inclusions and modular intersections of standard subspaces correspond to antiunitary representations of $\operatorname{Aff}(\mathbb R)$, 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.


  • Karl-Hermann NeebDepartment Mathematik
    Universität Erlangen-Nürnberg
    Cauerstrasse 11
    91058 Erlangen, Germany
  • Gestur ÓlafssonDepartment of Mathematics
    Louisiana State University
    Baton Rouge, LA 70803, USA

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