An Algebraic Construction of Boundary Quantum Field Theory

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 of von Neumann algebras on ℝ and an element V of a unitary semigroup ε(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₊ is locally isomorphic to ε(A). We discuss the structure of the semigroup ε(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 ε(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-Mack-Todorov extension of A⁽⁰⁾. A further family of models comes from the Ising model.