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Intrinsic Heisenberg-type lower bounds on spacelike hypersurfaces in general relativity

Thomas Schürmann

Abstract

We derive intrinsic Heisenberg-type lower bounds for the canonical momentum uncertainty of scalar quantum states that are strictly localized in geodesic balls $B_Σ(p,r)$, 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 $σ_p$ by the first Dirichlet eigenvalue of the Laplace-Beltrami operator (Theorem). On weakly mean-convex balls we obtain the universal product inequality $σ_p r \ge \hbar/2$. Under the same assumption, a vector-field Barta-type argument improves this universal floor to the scale-invariant bound $σ_p r \ge π\hbar/2$, which provides a universal, foliation-independent floor. Any further sharpening of the constant requires eigenvalue-comparison results or other curvature-sensitive methods.

Intrinsic Heisenberg-type lower bounds on spacelike hypersurfaces in general relativity

Abstract

We derive intrinsic Heisenberg-type lower bounds for the canonical momentum uncertainty of scalar quantum states that are strictly localized in geodesic balls , serving as the position uncertainty, on spacelike hypersurfaces 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 by the first Dirichlet eigenvalue of the Laplace-Beltrami operator (Theorem). On weakly mean-convex balls we obtain the universal product inequality . Under the same assumption, a vector-field Barta-type argument improves this universal floor to the scale-invariant bound , which provides a universal, foliation-independent floor. Any further sharpening of the constant requires eigenvalue-comparison results or other curvature-sensitive methods.

Paper Structure

This paper contains 19 sections, 4 theorems, 88 equations.

Table of Contents

  1. Keywords.
  2. Introduction
  3. Minimal geometric setting
  4. Geodesic balls on the slice.
  5. Hilbert space on geodesic balls
  6. Momentum operator and variance decomposition
  7. An intrinsic Heisenberg-type lower bound
  8. Universal Hardy baseline on weakly mean-convex balls
  9. Compatibility with curvature-dependent bounds on spheres.
  10. Barta-type improvement under weak mean-convexity
  11. Examples and applications
  12. Example 4.1
  13. Example 4.2
  14. Example 4.3
  15. Example 4.4
  16. ...and 4 more sections

Key Result

Lemma 2.4

Let $\psi\in H^1_0(B_\Sigma(p,r))$ be normalized, $\|\psi\|_{L^2}=1$ and $\mathrm{Var}_\psi(P)$ be the geometric momentum variance defined in eq:VarGeom. Then the following properties hold:

Theorems & Definitions (15)

  • Remark 2.2: Strict localization and Dirichlet data
  • Definition 2.3
  • Lemma 2.4: Basic properties of the geometric momentum variance
  • proof
  • Remark 2.5: Physical interpretation and small-ball limit
  • Theorem 3.1: Intrinsic Heisenberg-type lower bound
  • Remark 3.2: Coordinate invariance and kinetic-energy window
  • Remark 3.3: On the ‘constant phase gradient’ condition and frame independence
  • Remark 3.4: Model saturating the bound with nonzero mean momentum
  • Definition 3.5: Weakly mean-convex domain
  • ...and 5 more