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arXiv 1004.0616math-phhep-thmath.MPmath.OA
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边界量子场论的代数构造

An Algebraic Construction of Boundary Quantum Field Theory

Roberto Longo, Edward Witten

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21 pages
发表信息：
Commun.Math.Phys.303:213-232,2011
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摘要：

本文从vonNeumann代数在实数线上的局部共形网A和与A相关联的酉半群E（A）的元素V出发，建立了vonNeumann代数A_V在Minkowski半平面M_+上的局部时间平移协变边界量子场论网.当V=1时，A_V与A_+局部同构，如果A的真空特征标是可和的，则A_V与A_+局部同构。本文讨论了半群E（A）的结构。通过使用单粒子版本的Borchers定理和标准子空间分析，我们提供了Beurling-Lax定理的一个抽象模拟，它允许我们描述，特别是，单粒子希尔伯特空间上的所有酉，其第二量子化推广属于E（A^（0）），其中A^（0）是U（1）-流网。每个这样的酉函数都与一个散射函数相连，或者更一般地说，与一个对称的内函数相连。然后我们通过A^（0）的任何Buchholz-Mach-Todorov扩张得到模型族。另一类模型来自于伊辛模型。

英文摘要：

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 A of von Neumann algebras on the real line and an element V of a unitary semigroup E(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_V is locally isomorphic to A_+. We discuss the structure of the semigroup E(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 of E(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-Mach-Todorov extension of A^(0). A further family of models comes from the Ising model.

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