2024/06/06 by J. Schober, Schober, Jonas · 1 citation
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Mathematical Analysis and Transform Methods #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2406.04241
openalex publication_date 2024/06/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Standard subspaces are a well-studied object in algebraic quantum field theory (AQFT). Given a standard subspace \tt V of a Hilbert space H, one is interested in unitary one-parameter groups on H with Ut \tt V ⊆ \tt V for every t ∈ ℝ+. If (\tt V,U) is a non-degenerate standard pair on H, i.e. the self-adjoint infinitesimal generator of U is a positive operator with trivial kernel, two classical results are given by Borchers' Theorem, relating non-degenerate standard pairs to positive energy representations of the affine group Aff(ℝ) and the Longo-Witten Theorem, stating the the semigroup of unitary endomorphisms of \tt V can be identified with the semigroup of symmetric operator-valued inner functions on the upper half-plane. In this thesis, we prove results similar to the theorems of Borchers and of Longo-Witten for a more general framework of unitary one-parameter groups without the assumption that their infinitesimal generator is positive. We replace this assumption by the weaker assumption that the triple (H,\tt V,U) is a so-called real regular one-parameter group.