We explore the interplay between the equivalence principle and a generalization of the Heisenberg uncertainty relations known as extended uncertainty principle, that comprises the effects of spacetime curvature at large distances. Specifically, we observe that, when the modified uncertainty relations hold, the weak formulation of the equivalence principle is violated, since the inertial mass of quantum systems becomes position-dependent whilst the gravitational mass is left untouched. To obtain the above result, spinor and scalar fields are separately analyzed by considering the non-relativistic limit of the Dirac and the Klein-Gordon equations in the presence of the extended uncertainty principle. In both scenarios, it is found that the ratio between the inertial and the gravitational mass is the same.

我们探讨了等价原理和被称为扩展不确定性原理的海森堡不确定性关系的推广之间的相互作用，其中包括远距离时空曲率的影响。具体来说，我们观察到，当修正的不确定性关系成立时，等效原理的弱表述被违反，因为量子系统的惯性质量变得依赖于位置，而引力质量保持不变。为了得到上述结果，在存在扩展不确定性原理的情况下，考虑狄拉克和克莱因-戈登方程的非相对论极限，分别分析旋量场和标量场。在这两种情况下，发现惯性质量和引力质量之间的比率是相同的。