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arXiv:gr-qc/0408098 · General Relativity and Gravitation

Exact uncertainty approach in quantum mechanics and quantum gravity

Michael J. W. Hall

Published Sep 1, 2005• 39 citations
arXivHTMLPDF
DOI
10.1007/s10714-005-0131-y

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We propose that Heisenberg's uncertainty principle is a consequence of gravity. While quantum mechanics and gravity are widely considered incompatible, we show that Heisenberg's uncertainty principle, which is a central tenet in quantum mechanics, in fact, arises from non-diminishing finite gravity between a particle and another identical particle. We observe that it is impossible to know the position or momentum of the particle with unlimited precision, unless the gravity is infinite or zero, respectively, between the particles. We derive upper and lower bounds on the uncertainties in position and momentum of the particle, based on the uncertainty principle emerging from gravity. Accordingly, the quantum-classical boundary is identified, beyond which gravity behaves classically.
Gravity might obey its own version of quantum uncertainty. In quantum mechanics, you can't measure position without disturbing momentum—a trade-off MIT confirmed again in their 2025 double-slit experiment. We found something analogous for gravity. When gravity causes a tiny object to lose its quantum behavior, two things happen: interference patterns fade, and the object heats up. But these effects trade off exactly. Gravity cannot act in secret. What it takes from coherence, it deposits as heat. Their sum is always one. This gives experimenters a clean test: rotate the detector angle and watch coherence and heating swap places. The technology already exists in labs using glass beads floating in laser light. Previous proposals to test quantum-gravity complementarity required entangling two masses via gravity—never achieved. Ours needs just one particle and standard detectors. This would be the first direct test of how gravity respects quantum complementarity.
A minimum length scale of the order of Planck length is a feature of many models of quantum gravity that seek to unify quantum mechanics and gravitation. Recently, Perivolaropoulos in his seminal work (Perivolaropoulos in Phys. Rev. D 95:103523, 2017) predicted the simultaneous existence of minimal and maximal length measurements of quantum gravity. More recently, we have shown that both measurable lengths can be obtained from position-dependent noncommutativity (Lawson in J. Phys. A Math.Theor. 53:115303, 2020). In this paper, we present an alternative derivation of these lengths from non-Hermitian position-dependent noncommutativity. We show that a simultaneous measurement of both lengths form a family of discrete spaces. In one hand, we show the similarities between the maximal uncertainty measurement and the classical properties of gravity. On the other hand, the connection between the minimal uncertainties and the non-Hermicity quantum mechanic scenarios. The existence of minimal uncertainties are the consequences of non-Hermicities of some operators that are generators of this noncommutativity. With an appropriate Dyson map, we demonstrate by a similarity transformation that the physically meaningfulness of dynamical quantum systems is generated by a hidden Hermitian position-dependent noncommutativity. This transformation preserves the properties of quantum gravity but removes the fuzziness induced by minimal uncertainty measurements at this scale. Finally, we study the eigenvalue problem of a free particle in a square-well potential in these new Hermitian variables.