Motivated by the potential connection between metric-affine gravity and linear generalized uncertainty principle (GUP) in the phase space, we develop a covariant form of linear GUP and an associated modified Poincaré algebra, which exhibits distinctive behavior, nearing nullity at the minimal length scale proposed by linear GUP. We use three-torus geometry to visually represent linear GUP within a covariant framework. The three-torus area provides an exact geometric representation of Bekenstein’s universal bound. We depart from Bousso’s approach, which adapts Bekenstein’s bound by substituting the Schwarzschild radius ( rs ) with the radius (R) of the smallest sphere enclosing the physical system, thereby basing the covariant entropy bound on the sphere’s area. Instead, our revised covariant entropy bound is described by the area of a three-torus, determined by both the inner radius rs and outer radius R where rs⩽R due to gravitational stability. This approach results in a more precise geometric representation of Bekenstein’s bound, notably for larger systems where Bousso’s bound is typically much larger than Bekensetin’s universal bound. Furthermore, we derive an equation that turns the standard uncertainty inequality into an equation when considering the contribution of the three-torus covariant entropy bound, suggesting a new avenue of quantum gravity.

中文翻译：

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线性 GUP、度量仿射引力、庞加莱代数和熵界的协变挂毯

受相空间中度量仿射引力和线性广义不确定性原理（GUP）之间潜在联系的启发，我们开发了线性 GUP 的协变形式和相关的修正庞加莱代数，它表现出独特的行为，在最小长度尺度上接近零由线性 GUP 提出。我们使用三环几何图形来直观地表示协变框架内的线性 GUP。三环面区域提供了贝肯斯坦通用界限的精确几何表示。我们与 Bousso 的方法不同，该方法通过替换 Schwarzschild 半径来适应 Bekenstein 边界（ rs ）与半径（右）包围物理系统的最小球体，从而使协变熵基于球体面积。相反，我们修改后的协变熵界是由三环面的面积来描述的，由内半径决定 rs 和外半径右在哪里 rs⩽右 由于重力稳定性。这种方法可以更精确地表示贝肯斯坦边界，特别是对于较大的系统，其中布索边界通常比贝肯斯坦通用边界大得多。此外，我们推导了一个方程，在考虑三环协变熵界的贡献时，将标准不确定性不等式转化为方程，这提出了量子引力的新途径。