Analytical properties of standard subspaces and reflection positivity in AQFT

对于Beta带，我们看到单参数乘法群在其下边界上充当酉群。然而，它的作用在上界不是么正的，因为出现了依赖于β的指数因子。另一方面，我们发现Beta-带的Hardy空间实现为虚轴上的平移算子R的图，它在上述非酉单参数群下是不变的。下界上的这种乘法和虚轴上的平移算子R验证了正则对易关系，这些关系产生了一个范式结果：对于具有酉单参数群和满足CCR的自伴算子的Hilbert空间，得到了Hilbert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空间上的每个正自共轭算子，它的图是具有适当复结构的原始Hilbert空间的双副本中的标准子空间.在AQFT的框架下，我们考虑了二维全共形场理论，其手征分量包含足够多的不可约自同构的表示范畴.我们利用每个手征分量的合适的主场的组合来构造这类理论的Wightman场及其Hilbert结构.我们还建立了与Wightman场相对应的二维Haag-Kastler网。当U(1)电流给出两个手征分量时，我们得到了我们的结构。

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.