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arXiv:2510.01628 · Classical and Quantum Gravity

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

Thomas Schürmann

Published Jan 20, 2026• 0 citations
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Abstract

Abstract In quantum theory on curved backgrounds, Heisenberg’s uncertainty principle is usually discussed in terms of ensemble variances and flat-space commutators. Here we take a different, preparation-based viewpoint tailored to sharp position measurements on spacelike hypersurfaces in general relativity. A projective localization is modeled as a von Neumann–Lüders projection onto a geodesic ball B Σ ( r ) of radius r on a Cauchy slice ( Σ , h ) , with the post-measurement state described by Dirichlet data. Using DeWitt-type momentum operators adapted to an orthonormal frame, we construct a geometric, coordinate-invariant momentum standard deviation σ p and show that strict confinement to B Σ ( r ) enforces an intrinsic kinetic-energy floor. The lower bound is set by the first Dirichlet eigenvalue λ 1 of the Laplace–Beltrami operator on the ball, σ p ⩾ ℏ λ 1 , and is manifestly invariant under changes of coordinates and foliation. A variance decomposition separates the contribution of the modulus | ψ | from phase-gradient fluctuations and clarifies how the spectral geometry of ( Σ , h ) controls momentum uncertainty. Assuming only minimal geometric information, weak mean-convexity of the boundary yields a universal, scale-invariant Heisenberg-type product bound, σ p r ⩾ π ℏ / 2 , depending only on the proper radius r .

DOI
10.1088/1361-6382/ae3afb

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Abstract In quantum theory on curved backgrounds, Heisenberg’s uncertainty principle is usually discussed in terms of ensemble variances and flat-space commutators. Here we take a different, preparation-based viewpoint tailored to sharp position measurements on spacelike hypersurfaces in general relativity. A projective localization is modeled as a von Neumann–Lüders projection onto a geodesic ball B Σ ( r ) of radius r on a Cauchy slice ( Σ , h ) , with the post-measurement state described by Dirichlet data. Using DeWitt-type momentum operators adapted to an orthonormal frame, we construct a geometric, coordinate-invariant momentum standard deviation σ p and show that strict confinement to B Σ ( r ) enforces an intrinsic kinetic-energy floor. The lower bound is set by the first Dirichlet eigenvalue λ 1 of the Laplace–Beltrami operator on the ball, σ p ⩾ ℏ λ 1 , and is manifestly invariant under changes of coordinates and foliation. A variance decomposition separates the contribution of the modulus | ψ | from phase-gradient fluctuations and clarifies how the spectral geometry of ( Σ , h ) controls momentum uncertainty. Assuming only minimal geometric information, weak mean-convexity of the boundary yields a universal, scale-invariant Heisenberg-type product bound, σ p r ⩾ π ℏ / 2 , depending only on the proper radius r .
The Heisenberg Uncertainty Principle is traditionally formulated within the mathematical idealization of perfectly flat Euclidean space. However, physical reality dictates thatspace is inherently curved by gravitational fields, cosmological dynamics, and perpetualquantum vacuum fluctuations. This paper proposes a novel theoretical framework that generalizes three fundamental uncertainty principles—Position-Momentum, Energy-Time, andAngle-Angular Momentum—by directly incorporating spatial curvature on a cosmologicalscale. We introduce a continuous spatial modulating function, f(x), governed by the netspatial curvature R(x), representing the dynamic interplay between attractive and repulsive geometric forces. To rigorously preserve the quantum mechanical axiom that physicalobservables correspond to Hermitian operators, we employ Weyl symmetrization to construct modified differential operators for momentum and angular momentum. Through exactmathematical derivations utilizing wave packet phase-space dynamics, a modified Ehrenfesttheorem, and the resolution of topological boundary discontinuities via mapping to a Universal Covering Space with strict Dirichlet boundary conditions, we establish absolute lowerbounds for these uncertainties. Ultimately, we demonstrate that the minimum bound ofquantum uncertainty is not a rigid universal constant, but is strictly positive and dynamically dictated by local geometric curvature, providing a robust bridge between quantummechanics and spacetime geometry.Keywords: Generalized Uncertainty Principle; Spatial Curvature; Weyl Symmetrization; Hermitian Operators; Modified Ehrenfest Theorem; Universal Covering Space; Cosmological Scale.PACSNumbers: 03.65.Ta(Foundationsof quantummechanics); 04.60.-m (Quantum gravity);03.65.Ca (Formalism).MSC2020 Codes: 81P05 (General and philosophical foundations of quantum mechanics);81S05 (Canonical quantization, commutation relations and statistics)
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Quantum spatial uncertainty is usually introduced as a fundamental property of microscopic systems through the Heisenberg uncertainty relation between position and momentum. In this work, we investigate whether a Heisenberg scale uncertainty bound can emerge from classical higher dimensional geometry without initially assuming wave functions or noncommuting quantum operators. We consider ordinary three dimensional space as a projection of a higher dimensional configuration space containing one compact fourth spatial dimension q 4 with characteristic length L. The compact sector is described by a hidden phase space distribution ρ(q 4 ,p 4 ), where curvature and warping of the compact geometry generate a position- dependent compact momentum p 4 (q 4 ) and therefore a nonzero compact phase space covariance. To connect the hidden compact sector with observable variables, we introduce an explicit compact to three dimensional projection rule. Applying the classical Cauchy-Schwarz inequality to the projected observable variables yields a conditional lower bound on the measurable position-momentum product without any using wave function. The resulting uncertainty scale is determined by the parameter such as compact covariance, the phase overlap between the curvature-induced momentum modulation and the projection coordinate and a geometric compact action scale derived from the higher-dimensional gravitational sector. The compact gravitational action leads to an effective uncertainty scale of the form [Formula: see text] (c is light velocity, L is size of compact dimension and G 4 is gravity constant). The proposed framework provides a compact-geometric mechanism capable of generating a Heisenberg-scale projected uncertainty bound from higher dimensional classical geometry.
We revisit the Extended Uncertainty Principle (EUP) from an operational viewpoint, replacing wavefunction-based widths with apparatus-defined position constraints such as a finite slit of width Δx or a geodesic ball of radius R. Using Hermitian momentum operators consistent with the EUP algebra, we prove a sharp lower bound on the product of momentum spread and preparation size in one dimension and show that it reduces smoothly to the standard quantum limit as the deformation vanishes. We then extend the construction to dimensions two and three on spaces of constant curvature and obtain the corresponding bound for spherical confinement, clarifying its geometric meaning via an isometry to S2 and S3. The framework links curvature-scale effects to operational momentum floors and suggests concrete tests in diffraction, cold-atom, and optomechanical settings.
In classical and relativistic thermodynamics, the proper-time accumulation rate of a physical ensemble approaches coordinate time as thermal motion is quenched (⟨𝕋⟩ → c as T → 0). In this paper, we demonstrate that when spatial confinement is imposed, the quantum mechanical ground state prevents physical clocks from ever reaching this classical zero-entropy baseline. Applying the chronokinetic conservation principle v² + 𝕋² = c² to quantum bound states, Heisenberg's uncertainty principle (Δx Δp ≥ ℏ/2) enforces an irreducible motional variance ⟨p²⟩ = σ_p² > 0 even at absolute zero temperature (T = 0 K). This quantum momentum variance drains the chronokinetic velocity, establishing an inescapable zero-point proper-time floor: ⟨𝕋⟩_ZPM < c For a confined harmonic oscillator (such as a laser-cooled ion in a Paul trap ground state, n = 0), this geometric tax generates an exact fractional clock retardation: (Δν / ν_0)_ZPM = -(3 ℏ ω_trap) / (4 M_ion c²) We extend this analysis across quantum bound-state systems, evaluating the geometric proper-time budget for the relativistic 3D harmonic oscillator, the hydrogenic electron ground state, and confined field cavities. This work demonstrates that quantum spatial localization inescapably retards proper-time accumulation, establishing a direct structural bridge connecting quantum momentum uncertainty, relativistic kinematics, and fundamental limits in ultra-stable optical clock metrology.