Canonical Uncertainty Relations for Madelung Variables in Curved Spacetime

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 $\theta$ variables and their conjugate momenta, we derive exact uncertainty principles that depend on spacetime geometry through the lapse function $N$ and spatial metric ${\gamma }_{ij}$. These relations reveal how gravitational fields modulate quantum fluctuations and provide first-principles constraints for scalar field dark matter models and stochastic quantum gravity.