Canonical Uncertainty Relations for Madelung Variables in Curved Spacetime

TL;DR

This paper derives fundamental uncertainty relations for Madelung variables in curved spacetime, showing how gravity influences quantum fluctuations and constraining scalar field dark matter models.

Contribution

It introduces exact uncertainty principles for hydrodynamic variables in quantum fields within curved spacetime, incorporating gravitational effects.

Findings

Uncertainty relations depend explicitly on spacetime geometry.

Gravity modulates quantum fluctuations of Madelung variables.

Provides constraints relevant for scalar field dark matter and quantum gravity.

Abstract

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.