• DOI:10.1007/s00220-010-1133-5
• Corpus ID: 115174261

An Algebraic Construction of Boundary Quantum Field Theory

@article{Longo2010AnAC,
  title={An Algebraic Construction of Boundary Quantum Field Theory},
  author={Roberto Longo and Edward Witten},
  journal={Communications in Mathematical Physics},
  year={2010},
  volume={303},
  pages={213-232}
}

• R. Longo, E. Witten
• Published 5 April 2010
• Mathematics
• Communications in Mathematical Physics

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.