2025/10/02 by Schürmann, Thomas
#FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Mathematical Physics (math-ph) #Quantum Physics (quant-ph)
paper · doi:10.48550/arxiv.2510.01628
We derive intrinsic Heisenberg-type lower bounds for the canonical momentum uncertainty of scalar quantum states that are strictly localized in geodesic balls BΣ(p,r), serving as the position uncertainty, on spacelike hypersurfaces (Σ,h) of arbitrary Lorentzian spacetimes. The estimate depends only on the induced Riemannian geometry of the slice; it is independent of the lapse, shift, and extrinsic curvature, and controls the canonical momentum variance/uncertainty σp by the first Dirichlet eigenvalue of the Laplace-Beltrami operator (Theorem). On weakly mean-convex balls we obtain the universal product inequality σp r ≥ ℏ/2. Under the same assumption, a vector-field Barta-type argument improves this universal floor to the scale-invariant bound σp r ≥ πℏ/2, which provides a universal, foliation-independent floor. Any further sharpening of the constant requires eigenvalue-comparison results or other curvature-sensitive methods.