An algebraic construction of boundary Quantum Field Theory

We build up local, time translation covariant Boundary Quantum Field Theory nets of von Neumann algebras 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 () associated with . The case V = 1 reduces to the net + considered by Rehren and one of the authors; if the vacuum character of is summable, V is locally isomorphic to +. We discuss the structure of the semigroup (). 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 ((0)) with (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 (0). A further family of models comes from the Ising model.