We study the modular Hamiltonian associated with a Gaussian state on the Weyl algebra. We obtain necessary/sufficient criteria for the local equivalence of Gaussian states, independently of the classical results by Araki and Yamagami, Van Daele, Holevo. We also present a criterion for a Bogoliubov automorphism to be weakly inner in the GNS representation. The main application of our analysis is the description of the vacuum modular Hamiltonian associated with a time-zero interval in the scalar, massive, free QFT in two spacetime dimensions, thus complementing recent results in higher space dimensions. In particular, we have the formula for the local entropy of a one dimensional Klein-Gordon wave packet and Araki's vacuum relative entropy of a coherent state on a double cone von Neumann algebra. Besides, we derive the type III_1 factor property. Incidentally, we run across certain positive selfadjoint extensions of the Laplacian, with outer boundary conditions, seemingly not considered so far.
翻译:我们在Weyl 代数上研究与高斯状态相关的单元汉密尔顿仪。 我们获得高斯州本地等值的必要/足够标准, 独立于阿拉基和山上、 范代尔、 霍莱沃的经典结果。 我们还提出了一个标准, 使博格利乌夫的自制性在 GNS 代表中处于微弱的内在状态。 我们分析的主要应用是描述真空单元汉密尔顿仪, 在两个时空尺度上与时零间隔相伴, 大规模免费的QFT, 从而补充了空间方面的最新结果 。 特别是, 我们有一个单维的克莱因- 哥尔登波包和阿拉基的真空相对同步状态的公式, 在一个双锥形的Neumann 代数布拉上。 此外, 我们得出了三一因子属性。 顺便说, 我们跨越了拉普拉西亚的某些积极的自相连接的扩展, 外缘条件似乎远未被考虑。