We explore a new class of Non-linear GUPs (NLGUP) showing the emergence of a new non-commutative and higher derivatives quantum mechanics. Within it, we introduce the shortest fundamental scale as a UV fixed point in the NLGUP commutators [X, P] = i\hbar f(P), having in mind a fundamental highest energy threshold related to the Planck scale. We show that this leads to lose commutativity of space coordinates, that start to be dependent by the angular momenta of the system. On the other hand, non-linear GUP must lead to a redefinition of the Schrodinger equation to a new non-local integral-differential equation. We also discuss the modification of the Dyson series in time-dependent perturbative approaches. This may suggest that, in NLGUPs, non-commutativity and higher derivatives may be intimately interconnected within a unified and coherent algebra. We also show that Dirac and Klein-Gordon equations are extended with higher space-derivatives according to the NLGUP. We compute momenta-dependent corrections to the dispersion velocity, showing that the Lorentz invariance is deformed. We comment on possible implications in tests of light dispersion relations from Gamma-Ray-Bursts or Blazars, with potential interests for future experiments such as LHAASO, HAWC and CTA.

中文翻译：

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来自 GUP 的新的非交换和更高导数的量子力学

我们探索了一类新的非线性 GUP (NLGUP)，展示了新的非对易和更高导数量子力学的出现。在其中，我们将最短的基本尺度作为 NLGUP 换向器 [X, P] = i\hbar f(P) 中的 UV 固定点引入，并考虑到与普朗克尺度相关的基本最高能量阈值。我们表明这会导致空间坐标失去交换性，空间坐标开始依赖于系统的角动量。另一方面，非线性 GUP 必须导致将薛定谔方程重新定义为新的非局部积分微分方程。我们还讨论了在依赖时间的微扰方法中对戴森系列的修改。这可能表明，在 NLGUP 中，非交换性和更高阶的导数可能在统一和连贯的代数中密切相关。我们还表明，根据 NLGUP，Dirac 和 Klein-Gordon 方程扩展了更高的空间导数。我们计算了对色散速度的依赖动量的修正，表明洛伦兹不变性是变形的。我们评论了伽玛射线暴或耀变体的光色散关系测试中可能的影响，对未来的实验如 LHAASO、HAWC 和 CTA 具有潜在的兴趣。