An Algebraic Construction of Boundary Quantum Field Theory

Abstract

We build up local, time translation covariant Boundary Quantum Field Theory nets of von Neumann algebras ${\mathcal A_V}$ on the Minkowski half-plane M ₊ starting with a local conformal net ${\mathcal A}$ of von Neumann algebras on ${\mathbb R}$ and an element V of a unitary semigroup ${\mathcal E(\mathcal A)}$ associated with ${\mathcal A}$. The case V = 1 reduces to the net ${\mathcal A_+}$ considered by Rehren and one of the authors; if the vacuum character of ${\mathcal A}$ is summable, ${\mathcal A_V}$ is locally isomorphic to ${\mathcal A_+}$. We discuss the structure of the semigroup ${\mathcal E(\mathcal A)}$. 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 ${\mathcal E(\mathcal A^{(0)})}$ with ${\mathcal A^{(0)}}$ 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-Mack-Todorov extension of ${\mathcal A^{(0)}}$. A further family of models comes from the Ising model.