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Regular one-parameter groups, reflection positivity and their application to Hankel operators and standard subspaces

Jonas Schober

TL;DR

The thesis generalizes the Borchers and Longo–Witten framework from non-degenerate standard pairs with positive generators to real regular one-parameter groups, by formulating a Lax–Phillips model and exploiting operator-valued Pick functions to generate regular one-parameter groups. It systematically develops the multiplicity space $\mathcal{K}$ as the core invariant of these groups, establishes its uniqueness and computability via kernel spaces and $M_{H^\infty}^*$-stable subspaces, and classifies regular Pick functions through Blaschke–Potapov factorization and holomorphic extendability. A key result is the multiplicativity of the multiplicity space under composition of Pick functions, with explicit degree formulas linking spectral data to the dimension of $\mathcal{K}$. The work also connects reflection positivity, Osterwalder–Schrader transforms, and Hankel operators to standard subspaces, enabling real-analytic analogues of Borchers’ and Longo–Witten’s theorems in broader AQFT contexts. Overall, it provides a comprehensive operator-theoretic framework tying RP spaces, Pick-function generators, and Lax–Phillips realizations to analyze standard subspaces and their dynamical endomorphisms.

Abstract

Standard subspaces are a well-studied object in algebraic quantum field theory (AQFT). Given a standard subspace ${\tt V}$ of a Hilbert space $\mathcal{H}$, one is interested in unitary one-parameter groups on $\mathcal{H}$ with $U_t {\tt V} \subseteq {\tt V}$ for every $t \in \mathbb{R}_+$. If $({\tt V},U)$ is a non-degenerate standard pair on $\mathcal{H}$, 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 $\mathrm{Aff}(\mathbb{R})$ 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 $(\mathcal{H},{\tt V},U)$ is a so-called real regular one-parameter group.

Regular one-parameter groups, reflection positivity and their application to Hankel operators and standard subspaces

TL;DR

The thesis generalizes the Borchers and Longo–Witten framework from non-degenerate standard pairs with positive generators to real regular one-parameter groups, by formulating a Lax–Phillips model and exploiting operator-valued Pick functions to generate regular one-parameter groups. It systematically develops the multiplicity space as the core invariant of these groups, establishes its uniqueness and computability via kernel spaces and -stable subspaces, and classifies regular Pick functions through Blaschke–Potapov factorization and holomorphic extendability. A key result is the multiplicativity of the multiplicity space under composition of Pick functions, with explicit degree formulas linking spectral data to the dimension of . The work also connects reflection positivity, Osterwalder–Schrader transforms, and Hankel operators to standard subspaces, enabling real-analytic analogues of Borchers’ and Longo–Witten’s theorems in broader AQFT contexts. Overall, it provides a comprehensive operator-theoretic framework tying RP spaces, Pick-function generators, and Lax–Phillips realizations to analyze standard subspaces and their dynamical endomorphisms.

Abstract

Standard subspaces are a well-studied object in algebraic quantum field theory (AQFT). Given a standard subspace of a Hilbert space , one is interested in unitary one-parameter groups on with for every . If is a non-degenerate standard pair on , i.e. the self-adjoint infinitesimal generator of 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 and the Longo-Witten Theorem, stating the the semigroup of unitary endomorphisms of 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 is a so-called real regular one-parameter group.
Paper Structure (40 sections, 174 theorems, 1157 equations, 1 figure)

This paper contains 40 sections, 174 theorems, 1157 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Regular one-parameter groups
  3. ROPGs and the Lax--Phillips Theorem
  4. Formulas for the multiplicity space
  5. Independence of the multiplicity space from $\mathcal{E}_+$
  6. ROPGs generated by Pick functions
  7. Regular Pick functions
  8. The multiplicity free case
  9. The general case
  10. $M_{H^\infty}^*$-stable subspaces and finite Blaschke--Potapov products
  11. The kernel space
  12. Holomorphic extensions
  13. The multiplicity space and the degree of Pick functions
  14. The degree of Pick functions and composition
  15. Reflection positivity on the Hardy space
  16. ...and 25 more sections

Key Result

Theorem 2.1.3

(Lax--Phillips)(cf. LP64NO18, Fr20) For every real/com-plex Hilbert space $\mathcal{K}$, the triple $(L^2({\mathbb R},\mathcal{K}),L^2({\mathbb R}_+,\mathcal{K}),\tau)$ is a real/complex ROPG, where the one-parameter group ${tau}: {\mathbb R} \to U(L^2({\mathbb R},\mathcal{K}))$ is the translation a Further, every real/complex ROPG $(\mathcal{E},\mathcal{E}_+,U)$ is equivalent to $(L^2({\mathbb R}

Figures (1)

  • Figure : Plot of the eigenvalues of $x \mapsto 111x$

Theorems & Definitions (409)

  • Definition 2.1.1
  • Remark 2.1.2
  • Theorem 2.1.3
  • Remark 2.1.4
  • Definition 2.1.5
  • Theorem 2.1.6
  • Theorem 2.1.7
  • Theorem 2.1.8
  • Proposition 2.1.9
  • proof
  • ...and 399 more