Analytical properties of standard subspaces and reflection positivity in AQFT

AQFT 中标准子空间的分析性质和反射正性

• 批准号: 22KF0082
• 负责人: 河東 泰之
• 金额: $ 1.47万
• 依托单位: The University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for JSPS Fellows
• 财政年份: 2023
• 资助国家: 日本
• 起止时间: 2023-03-08 至 2024-03-31
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-22KF0082/
• 关键词: 場の量子論; 代数的場の量子論; 反転正値性; 作用素環; 共形場理論

For the beta-strip, we see that the one-parameter group of multiplication acts as a unitary on its lower boundary. However, its action is not unitary on the upper boundary since an exponential factor depending on beta appears. On the other hand, we find a realization of the Hardy space of the beta-strip as a graph of a translation operator R on the imaginary axis, which is invariant under the above non-unitary one-parameter group. Such multiplication on the lower boundary and the translation operator R on the imaginary axis verify canonical commutation relations, which produce a normal form result: for a Hilbert space equipped with a unitary one parameter group and a self-adjoint operator which satisfy a CCR produce a realization of the Hilbert space as L2-functions, the one-parameter group as translations and the self-adjoint operator as multiplication. Similarly, we show that for every positive self-adjoint operator on a Hilbert space, its graph is a standard subspace in a double copy of the original Hilbert space endowed with a suitable complex structure.In the framework of AQFT, we consider full conformal field theories in 2-dimensions whose chiral components admit representation categories with enough irreducible automorphisms.We construct the Wightman fields and their Hilbert structure for such theories using a suitable combination of primary fields for each chiral component. We also build the 2-dimensional Haag-Kastler net that corresponds to the Wightman fields. We worked out our construction when the two chiral components are given by the U(1)-current.

对于β带，我们看到乘法的单参数群在其下边界上充当酉群。然而，它的行动是不是单一的上边界，因为一个指数因子取决于β出现。另一方面，我们找到了β带的哈代空间的一个实现，它是一个平移算子R在虚轴上的图，它在上述非酉单参数群下是不变的。这种下边界上的乘法和虚轴上的平移算子R验证了正则对易关系，这产生了一个标准形式的结果：对于一个配备有酉单参数群和满足CCR的自伴算子的希尔伯特空间，产生了希尔伯特空间作为L2函数的实现，单参数群作为平移，自伴算子作为乘法。类似地，我们证明了对于Hilbert空间上的每一个正自伴算子，它的图都是原Hilbert空间的一个双重副本中的一个标准子空间，并赋予了一个合适的复结构。我们考虑在2中的完全共形场论，维数的手征分量允许具有足够的不可约自同构的表示范畴。我们构造了这样的Wightman场及其Hilbert结构理论使用每个手征分量的初级场的适当组合。我们还建立了二维Haag-Kastler网，对应于Wightman场。当两个手征分量由U（1）-流给出时，我们给出了我们的结构。

项目成果

C*-algebras associated to homeomorphisms twisted by vector bundles over finite dimensional spaces

与有限维空间上向量丛扭曲的同态相关的 C* 代数

• 发表时间: 2023
• 期刊: Trans. Amer. Math. Soc.
• 作者: M. S. Adamo;D. E. Archey;M. Forough;M. C. Georgescu;J. A Jeong;K. R. Strung;M. G. Viola
• 通讯作者: M. G. Viola