Correa da Silva R, Lechner G (2023)
Publication Language: English
Publication Status: Accepted
Publication Type: Journal article, Original article
Future Publication Type: Journal article
Publication year: 2023
URI: https://link.springer.com/article/10.1007/s00220-023-04773-y
DOI: 10.1007/s00220-023-04773-y
Open Access Link: https://link.springer.com/article/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 $\mathcal{L}_{T}(H)$ 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 $\mathcal{L}_{T}(H)$ 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 $\mathcal{L}_{T}(K)\subset\mathcal{L}_{T}(H)$ 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 $K\subset H$ satisfies an $L^2$-nuclearity condition, $\mathcal{L}_{T}(K)\subset\mathcal{L}_{T}(H)$ has type III relative commutant for suitable twists $T$.
Applications of these results to localization of observables in algebraic quantum field theory are discussed.
APA:
Correa da Silva, R., & Lechner, G. (2023). Modular Structure and Inclusions of Twisted Araki-Woods Algebras. Communications in Mathematical Physics. https://dx.doi.org/10.1007/s00220-023-04773-y
MLA:
Correa da Silva, Ricardo, and Gandalf Lechner. "Modular Structure and Inclusions of Twisted Araki-Woods Algebras." Communications in Mathematical Physics (2023).
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