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arXiv:hep-th/0208162 · Physics Letters B

On uncertainty relations in noncommutative quantum mechanics

Katarzyna Bolonek-Lasoń, Piotr Kosiński

Published Oct 1, 2002• 30 citations
arXivHTMLPDF
DOI
10.1016/s0370-2693(02)02731-4

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The Heisenberg uncertainty principle is foundational to quantum mechanics, yet its standard formulation is limited to Hilbert space operator commutators. Recent advances in noncommutative geometry (NCG) allow a reformulation of quantum observables and spacetime itself using operator algebras, providing a deeper framework for uncertainty relations. In this paper, we develop a generalized uncertainty relation using spectral triples, extending the Robertson– Schrödinger inequality into the noncommutative regime. Explicit derivations are given for operator-valued distances, modified commutators, and position–momentum operators in a noncommutative configuration space. Our results reveal the emergence of a minimal measurable length scale, consistent with predictions from quantum gravity, and demonstrate that uncertainty is fundamentally geometric in origin.