Extended Uncertainty Principle via Dirac Quantization

Unifying quantum theory and gravity remains a fundamental challenge in physics. While most existing literature focuses on the ultraviolet (UV) modifications of quantum theory due to gravity, this work shows that generic infrared (IR) 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.