Intrinsic Heisenberg-type lower bounds on spacelike hypersurfaces in general relativity
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912876354076672 |
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| author | Schürmann, Thomas |
| author_facet | Schürmann, Thomas |
| contents | 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, $σ_p \ge \hbar\sqrt{λ_1}$, 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, $σ_p r \ge π\hbar/2$, depending only on the proper radius $r$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_01628 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Intrinsic Heisenberg-type lower bounds on spacelike hypersurfaces in general relativity Schürmann, Thomas General Relativity and Quantum Cosmology Mathematical Physics Quantum Physics 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, $σ_p \ge \hbar\sqrt{λ_1}$, 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, $σ_p r \ge π\hbar/2$, depending only on the proper radius $r$. |
| title | Intrinsic Heisenberg-type lower bounds on spacelike hypersurfaces in general relativity |
| topic | General Relativity and Quantum Cosmology Mathematical Physics Quantum Physics |
| url | https://arxiv.org/abs/2510.01628 |