Representation Theory and Noncommutative Geometry, Paris

Mini-course: Algebraic Quantum Field Theory and causal homogeneous spaces 1/4 (pt.1)

17 mars 2025, 10:00

45m

Amphitheater Darboux (IHP)

Orateur

Karl-Hermann Neeb (Friedrich-Alexander-University Erlangen-Nuremberg)

Description

Lorentzian manifolds and their conformal compactifications provide the most symmetric models of spacetimes. The structures studied on such spaces in Algebraic Quantum Field Theory (AQFT) are so-called nets of operator algebras, i.e., to each open subset $\mathcal{O}$ of the space-time manifold one associates a von Neumann algebra $\mathcal{M}(\mathcal{O})$ in such a way that a certain natural list of axioms is satisfied.

We report on an ongoing project concerned with the construction of such nets on general causal homogeneous spaces $M=G/H$.

Lecture 1: Nets of operator algebras and AQFT.

We start with the translation from nets of operator algebras to nets of real subspaces, based on modular theory. We introduce real standard subspaces, discuss the Tomita-Takesaki Theorem as a key result from the modular theory of operator algebras and then describe axioms for nets of real subspaces $\mathsf{H}(\mathcal{O})$ in a unitary representation of a Lie group. These are structures than can be explored completely from the perspective of the geometry of homogeneous spaces and unitary representations.

Lecture notes are available under:
https://en.www.math.fau.de/wp-content/uploads/sites/3/2025/03/qft-lect.pdf