An Algebraic Construction of Boundary Quantum Field Theory

TL;DR

This paper constructs boundary quantum field theory models on the Minkowski half-plane using algebraic methods, unitaries, and conformal nets, providing a framework for new models including those related to the Ising model.

Contribution

It introduces an algebraic construction of boundary QFT nets from conformal nets and unitaries, extending previous models and describing their structure via semigroups and scattering functions.

Findings

Constructed boundary QFT nets from conformal nets and unitaries.

Described the structure of the semigroup E(A) and its relation to scattering functions.

Provided new models, including extensions of the U(1)-current and Ising models.

Abstract

We build up local, time translation covariant Boundary Quantum Field Theory nets of von Neumann algebras A_V on the Minkowski half-plane M_+ starting with a local conformal net A of von Neumann algebras on the real line and an element V of a unitary semigroup E(A) associated with A. The case V=1 reduces to the net A_+ considered by Rehren and one of the authors; if the vacuum character of A is summable A_V is locally isomorphic to A_+. We discuss the structure of the semigroup E(A). By using a one-particle version of Borchers theorem and standard subspace analysis, we provide an abstract analog of the Beurling-Lax theorem that allows us to describe, in particular, all unitaries on the one-particle Hilbert space whose second quantization promotion belongs to of E(A^(0)) with A^(0) the U(1)-current net. Each such unitary is attached to a scattering function or, more generally, to a symmetric inner function. We then obtain families of models via any Buchholz-Mach-Todorov extension of A^(0). A further family of models comes from the Ising model.