Speaker: Vincenzo Morinelli (University of Rome, Tor Vergata)

Title: Covariant homogeneous nets of standard subspaces

Time/Date: 4:45-6:15pm, November 19 (Thu.), 2020

Room: This will be an online seminar on Zoom. Please ask Kawahigashi for the Zoom link.

Abstract: In Algebraic Quantum Field Theory (AQFT), a canonical algebraic construction of the fundamental free field models was provided by Brunetti Guido and Longo in 2002. The Brunetti-Guido-Longo (BGL) construction relies on the identification of spacetime regions called wedges and one-parameter groups of Poincaré symmetries called boosts, the Bisognano-Wichmann property and the CPT-theorem. The last two properties make geometrically meaningful the Tomita-Takesaki theory. The fundamental step is to associate to an (anti-)unitary positive energy representation of the Poincaré group on the one-particle Hilbert space a local net of real subspaces (of the one-particle space) indexed by wedge regions. Then the von Neumann algebra net corresponding to the free field is obtained via second quantization.

In this talk we recall this fundamental structure and explain how the one-particle picture can be generalized. The BGL-construction can start just by considering the Poincaré symmetry group and forgetting about the spacetime. Then it is natural to ask what kind of Lie groups can support a one-particle net and in general a QFT. Given a Z₂-graded Lie group we define a local poset of abstract wedge regions. We provide a classification of the simple Lie algebras supporting abstract wedges in relation with some special wedge configurations. This allows us to exhibit an analog of the Haag-Kastler axioms for one-particle nets undergoing the action of such general Lie groups without referring to any specific spacetime. This set of axioms supports a first quantization net obtained by generalizing the BGL-construction. The construction is possible for a large family of Lie groups and provides several new models.

Based on the joint work with Karl-Hermann Neeb (FAU Erlangen-Nürnberg)