TY - JOUR

T1 - An Algebraic Construction of Boundary Quantum Field Theory

AU - Longo, Roberto

AU - Witten, Edward

N1 - Funding Information:
Supported in part by the ERC Advanced Grant 227458 OACFT “Operator Algebras and Conformal Field Theory”, PRIN-MIUR, GNAMPA-INDAM and EU network “Noncommutative Geometry” MRTN-CT-2006-0031962.

PY - 2011/4

Y1 - 2011/4

N2 - We build up local, time translation covariant Boundary Quantum Field Theory nets of von Neumann algebras Av 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.

AB - We build up local, time translation covariant Boundary Quantum Field Theory nets of von Neumann algebras Av 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.

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U2 - 10.1007/s00220-010-1133-5

DO - 10.1007/s00220-010-1133-5

M3 - Article

AN - SCOPUS:79952484024

SN - 0010-3616

VL - 303

SP - 213

EP - 232

JO - Communications In Mathematical Physics

JF - Communications In Mathematical Physics

IS - 1

ER -