Unifying quantum theory and gravity remains a fundamental challenge in physics. While most existing literature focuses on the ultraviolet modifications of quantum theory due to gravity, this work shows that generic infrared modifications arise when we describe quantum theory in curved spacetime. We explicitly demonstrate that the modifications to the position-momentum algebra are proportional to curvature invariants (such as the Ricci scalar and Kretschmann scalar). Our results, derived through a rigorous application of Dirac’s quantization procedure, demonstrate that infrared effects in quantum systems can be axiomatically derived. We study particle dynamics in an arbitrary curved spacetime by embedding them in a higher-dimensional flat geometry. Our approach, which involves embedding particle dynamics in a higher-dimensional flat geometry and utilizing Dirac’s quantization procedure, allows us to capture the dynamics of a particle in 4-dimensional curved spacetime through a modified position-momentum algebra. When applied to various spacetimes, this method reveals that the corrections due to the spacetime curvature are universal. We further compare our results with those derived using extended uncertainty principles. Finally, we discuss the implications of our work for black holes and entanglement.

统一量子理论和引力仍然是物理学的一个基本挑战。虽然大多数现有文献都关注由于重力而导致的量子理论的紫外线修正，但这项工作表明，当我们在弯曲时空中描述量子理论时，就会出现一般的红外修正。我们明确证明了位置动量代数的修改与曲率不变量（例如 Ricci 标量和 Kretschmann 标量）成正比。我们的结果是通过严格应用狄拉克量子化程序得出的，表明量子系统中的红外效应可以公理地推导出来。我们通过将粒子嵌入到高维平面几何中来研究任意弯曲时空中的粒子动力学。我们的方法涉及将粒子动力学嵌入到高维平面几何中并利用狄拉克的量子化过程，使我们能够通过修改的位置动量代数捕获 4 维弯曲时空中粒子的动力学。当应用于各种时空时，该方法表明时空曲率引起的修正是普遍的。我们进一步将我们的结果与使用扩展不确定性原理得出的结果进行比较。最后，我们讨论了我们的工作对黑洞和纠缠的影响。