A Spectral-Geometric Formulation of Extended Uncertainty Principles in Quantum Mechanics

doi:10.26565/2312-4334-2026-1-01

Balaji Padhy Department of Mathematics, Centurion University of Technology and Management, Paralakhemundi, Odisha, India https://orcid.org/0000-0002-3447-2917
B.K. Majhi Department of Mathematics, Centurion University of Technology and Management, Bhubaneswar, Odisha, India https://orcid.org/0000-0002-6800-0547
K. Navya Department of Basic Science and Humanities, Centurion University of Technology and Management, Vizianagaram, Andhra Pradesh, India https://orcid.org/0000-0001-9604-7783
K.V. Prasad Vignan’s Foundation for Science, Technology and Research (Deemed to be University), Vadlamudi, Guntur, india https://orcid.org/0009-0002-8956-4939

DOI: https://doi.org/10.26565/2312-4334-2026-1-01

Keywords: Heisenberg uncertainty principle, Quantum mechanics, Hilbert space, Operator commutators, Operator algebras

Abstract

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.

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