Jorge Meza-Domínguez, Tonatiuh Matos

We establish fundamental uncertainty relations for the hydrodynamic variables arising from the Madelung representation of quantum fields in curved spacetime. Through canonical quantization of the density $n$ and phase $θ$ variables and their conjugate momenta, we derive exact uncertainty principles that depend on spacetime geometry through the lapse function $N$ and spatial metric $γ_{ij}$. These relations reveal how gravitational fields modulate quantum fluctuations and provide first-principles constraints for scalar field dark matter models and stochastic quantum gravity.