论文标题
在有限温度下无质量玻色子间隔的相对熵
Relative entropy of an interval for a massless boson at finite temperature
论文作者
论文摘要
我们计算了与螺旋状u（1） - 电流模型中的热状态和相干激发之间的与有界间隔$ i =（a，b）$相关的Araki的相对熵，即手感玻色子的（衍生物）。为此，我们简要回顾了有关标准子空间熵以及非岩石状态（例如热状态）的相对熵的一些最新结果。特别是，最近使用Borchers和Yngvason的先前结果，Bostelmann，Cadamuro和Del Vecchio在有限的间隔$（ - \ infty，t）$的有限温度下获得了相对熵，主要是在负半线中提供模块化的统一扩张。在这里，我们找到了一个统一的旋转，以便使用完整的PSL $（2，\ mathbb {r}）$对称性，并在有限温度下获得有界间隔的模块化组，模块化的哈密顿量和相对熵$ S $。这种相对的熵既需要像Bekenstein一样的绑定和类似Qnec的绑定，但违反了$ s''\ geq 0 $。最后，我们将结果扩展到具有类似边界的$ 1+1 $尺寸的自由无质量玻色子。
We compute Araki's relative entropy associated to a bounded interval $I=(a,b)$ between a thermal state and a coherent excitation of itself in the bosonic U(1)-current model, namely the (derivative of the) chiral boson. For this purpose we briefly review some recent results on the entropy of standard subspaces and on the relative entropy of non-pure states such as thermal states. In particular, recently Bostelmann, Cadamuro and Del Vecchio have obtained the relative entropy at finite temperature for the unbounded interval $(-\infty,t)$, using previous results of Borchers and Yngvason, mainly a unitary dilation that provides the modular evolution in the negative half-line. Here we find a unitary rotation in order to make use of the full PSL$(2,\mathbb{R})$ symmetries and obtain the modular group, modular Hamiltonian and the relative entropy $S$ of a bounded interval at finite temperature. Such relative entropy entails both a Bekenstein-like bound and a QNEC-like bound, but violates $S''\geq 0$. Finally, we extend the results to the free massless boson in $1+1$ dimensions with analogous bounds.
