An Algebraic Construction of Boundary Quantum Field Theory |
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Authors: | Roberto Longo Edward Witten |
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Institution: | 1. Dipartimento di Matematica, Universit?? di Roma ??Tor Vergata??, Via della Ricerca Scientifica, 1, I-00133, Roma, Italy 2. Institute for Advanced Study, School of Natural Sciences, Einstein Drive, Princeton, NJ, 08540, USA
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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. |
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