Title: Entangled Harmonic Oscillators and Space-time Entanglement Show Chinese title

Title: 纠缠谐振子和时空纠缠

Abstract: The mathematical basis for the Gaussian entanglement is discussed in detail, as well as its implications in the internal space-time structure of relativistic extended particles. It is shown that the Gaussian entanglement shares the same set of mathematical formulas with the harmonic oscillator in the Lorentz-covariant world. It is thus possible to transfer the concept of entanglement to the Lorentz-covariant picture of the bound state which requires both space and time separations between two constituent particles. These space and time variables become entangled as the bound state moves with a relativistic speed. It is shown also that our inability to measure the time-separation variable leads to an entanglement entropy together with a rise in the temperature of the bound state. As was noted by Paul A. M. Dirac in 1963, the system of two oscillators contains the symmetries of O(3,2) de Sitter group containing two O(3,1) Lorentz groups as its subgroups. Dirac noted also that the system contains the symmetry of the Sp(4) group which serves as the basic language for two-mode squeezed states. Since the Sp(4) symmetry contains both rotations and squeezes, one interesting case is the combination of rotation and squeeze resulting in a shear. While the current literature is mostly on the entanglement based on squeeze along the normal coordinates, the shear transformation is an interesting future possibility. The mathematical issues on this problem are clarified. Show Chinese abstract

Abstract: 高斯纠缠的数学基础被详细讨论，以及它在相对论性扩展粒子的内部时空结构中的意义。 显示高斯纠缠与洛伦兹协变世界中的谐振子共享同一组数学公式。 因此，可以将纠缠的概念转移到需要两个组成粒子之间空间和时间分离的束缚态的洛伦兹协变图景中。 当束缚态以相对论速度运动时，这些空间和时间变量变得纠缠。 还显示，我们无法测量时间分离变量导致纠缠熵的产生以及束缚态温度的上升。 正如保罗·A·M·狄拉克在1963年所指出的，两个振子的系统包含O(3,2)德西特群的对称性，该群包含两个O(3,1)洛伦兹群作为其子群。 狄拉克还指出，该系统包含Sp(4)群的对称性，该群是两模压缩态的基本语言。 由于Sp(4)对称性包含旋转和压缩，一个有趣的情况是旋转和压缩的结合导致剪切。 虽然当前文献主要集中在基于正常坐标压缩的纠缠上，但剪切变换是一个有趣的未来可能性。 这个问题的数学问题得到了澄清。