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Emergent Minimal Position Uncertainty and Information Measures from Bounded Group Velocity in Condensed Matter Systems

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Preprints.org
DOI
10.20944/preprints202510.0963.v1

Fundamental kinematic constraints in condensed matter systems, such as Lieb–Robinson bounds, sound velocities, and finite bandwidths, impose strict limits on quasiparticle group velocities. We show that these ubiquitous bounds naturally lead to an emergent minimal position uncertainty, derived entirely within standard quantum mechanics without deforming the canonical commutation relations. Using a variational Sturm–Liouville formulation in momentum space, we establish a universal relation that links spatial resolu- tion limits directly to measurable system parameters. Beyond variance-based measures, we further characterize this emergent scale using Shannon entropy, Fisher information, and quantum Fisher information, thereby revealing its information-theoretic fingerprint. Representative estimates for semiconduc- tors and ultracold atomic systems yield minimal lengths on the nanometer scale, many orders of magnitude above the Planck length but fundamental within effective descriptions. Our results provide a unified perspective that connects abstract localization limits with tangible condensed matter con- straints, opening new avenues for experimental and metrological exploration of minimal-length phenomena.

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