Modular Structure and Inclusions of Twisted Araki-Woods Algebras

• Published in: Communications in mathematical physics Vol. 402; no. 3; pp. 2339 - 2386
• Main Authors: Correa da Silva, Ricardo, Lechner, Gandalf
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2023; Springer Nature B.V
• Subjects: Algebra; Classical and Quantum Gravitation; Complex Systems; Inclusions; Mathematical and Computational Physics; Mathematical Physics; Modular structures; Operators (mathematics); Physics; Physics and Astronomy; Quantum field theory; Quantum Physics; Quantum theory; Relativity Theory; Subspaces; Symmetry; Theoretical
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-023-04773-y

In the general setting of twisted second quantization (including Bose/Fermi second quantization, S -symmetric Fock spaces, and full Fock spaces from free probability as special cases), von Neumann algebras on twisted Fock spaces are analyzed. These twisted Araki-Woods algebras L T ( H ) depend on the twist operator T and a standard subspace H in the one-particle space. Under a compatibility assumption on T and H , it is proven that the Fock vacuum is cyclic and separating for L T ( H ) if and only if T satisfies a standard subspace version of crossing symmetry and the Yang-Baxter equation (braid equation). In this case, the Tomita-Takesaki modular data are explicitly determined. Inclusions L T ( K ) ⊂ L T ( H ) of twisted Araki-Woods algebras are analyzed in two cases: If the inclusion is half-sided modular and the twist satisfies a norm bound, it is shown to be singular. If the inclusion of underlying standard subspaces K ⊂ H satisfies an L 2 -nuclearity condition, L T ( K ) ⊂ L T ( H ) has type III relative commutant for suitable twists T . Applications of these results to localization of observables in algebraic quantum field theory are discussed.