线性GUP、度量仿射引力、它们的庞加莱代数与熵界的协变织构
A covariant tapestry of linear GUP, metric-affine gravity, their Poincaré algebra and entropy bound
AI总结：
本研究构建线性GUP的协变形式与修正庞加莱代数，提出基于3-环面面积的修正协变熵界，可更精准表征贝肯斯坦普适界，还推导出让标准不确定性不等式转为等式的方程，为量子引力提供新路径。
摘要：
受度量仿射引力与相空间中线性广义不确定性原理（Generalized Uncertainty Principle，GUP）潜在关联的启发，我们构建了线性GUP的协变形式及对应的修正庞加莱代数，该代数表现出独特性质：在线性GUP提出的最小长度尺度下趋近于零。我们采用3-环面几何在协变框架下直观表征线性GUP，3-环面的面积可精确几何表征贝肯斯坦（Bekenstein）普适界。\n布索（Bousso）的方法是通过用包围物理系统的最小球体半径$R$替换史瓦西半径（$r_s$）来修正贝肯斯坦界，进而将协变熵界建立在球体面积之上；我们的思路与之不同，修正后的协变熵界由3-环面的面积描述，该面积由内半径$r_s$和外半径$R$共同决定，且因引力稳定性满足$r_s\leq R$。该方法能更精准地几何表征贝肯斯坦界，尤其对于布索界通常远大于贝肯斯坦普适界的大尺度系统效果更显著。此外，我们推导得到一个方程：当纳入3-环面协变熵界的贡献时，标准不确定性不等式可转化为等式，这为量子引力研究提供了新方向。
英文摘要：
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 3-torus geometry to visually represent linear GUP within a covariant framework. The 3-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 ($r_s$) 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 3-torus, determined by both the inner radius $r_s$ and outer radius $R$ where $r_s\leq 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 3-torus covariant entropy bound, suggesting a new avenue of quantum gravity.
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