Inclusions of Standard Subspaces

TL;DR

This paper studies the structure and properties of inclusions of standard subspaces in Hilbert spaces, introducing new methods and analyzing examples from conformal field theory.

Contribution

It develops novel techniques for analyzing standard subspace inclusions, independent of von Neumann algebra frameworks, and explores examples from conformal field theory.

Findings

New methods related to polarizers and Gelfand triples are introduced.

Inclusions of standard subspaces are characterized independently of von Neumann algebras.

A detailed analysis of examples from conformal field theory is provided.

Abstract

Standard subspaces are closed real subspaces of a complex Hilbert space that appear naturally in Tomita-Takesaki modular theory and its applications to quantum field theory. In this article, inclusions of standard subspaces are studied independently of von Neumann algebras. Several new methods for their investigation are developed, related to polarizers, Gelfand triples defined by modular data, and extensions of modular operators. A particular class of examples that arises from the fundamental irreducible building block of a conformal field theory on the line is analyzed in detail.