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.
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.
Various theories that aim at unifying gravity with quantum mechanics suggest modifications of the Heisenberg algebra for position and momentum. From the perspective of quantum mechanics, such modifications lead to new uncertainty relations that are thought (but not proven) to imply the existence of a minimal observable length. Here we prove this statement in a framework of sufficient physical and structural assumptions. Moreover, we present a general method that allows us to formulate optimal and state-independent variance-based uncertainty relations. In addition, instead of variances, we make use of entropies as a measure of uncertainty and provide uncertainty relations in terms of min and Shannon entropies. We compute the corresponding entropic minimal lengths and find that the minimal length in terms of min entropy is exactly 1 bit.
Several approaches to quantum gravity suggest the emergence of a fundamental minimal length at the Planck scale. In quantum mechanics, this feature is naturally encoded through deformations of the Heisenberg algebra, leading to the Generalized Uncertainty Principle (GUP). While the phenomenological implications of GUP have been extensively explored, a consistent characterization of coherent states in minimal-length quantum mechanics remains elusive. In this work, we present a systematic analysis of coherent states for the one-dimensional harmonic oscillator. We show that the canonical equivalence among their standard characterizations - as eigenstates of the annihilation operator, displaced vacuum states and minimum-uncertainty wave packets - is generically lost in the presence of a minimal length. We then investigate the dynamical and semiclassical consequences of this inequivalence by comparing the evolution of generalized coherent states with that of states saturating the GUP. In particular, we demonstrate that minimal-length effects induce nontrivial deformations of phase-space trajectories and give rise to an intrinsic squeezing mechanism with no counterpart in ordinary quantum mechanics. These results establish a unified framework for coherence in GUP-based quantum theories and identify distinctive semiclassical signatures of minimal-length physics, opening a new avenue for probing quantum-gravitational effects.
Several quantum gravity theories predict a minimal length at the order of magnitude of the Planck length, under which the concepts of space and time lose their physical meaning. In quantum mechanics, the insurgence of such a minimal length can be described by introducing a modified position-momentum commutator, which in turn yields a generalized uncertainty principle, where the uncertainty on position measurements has a lower bound. The value of the minimal length is not predicted by theories and must be estimated experimentally. In this paper, we address the quantum bound to the estimability of the minimal uncertainty length by performing measurements on a harmonic oscillator, which is analytically solvable in the deformed algebra induced by the deformed commutation relations.
Uncertainty relations represent a foundational principle in quantum mechanics, imposing inherent limits on the precision with which \textit{mechanically} conjugate variables such as position and momentum can be simultaneously determined. This work establishes analogous relations for \textit{thermodynamically} conjugate variables -- specifically, a classical intensive parameter θ and its corresponding extensive quantum operator O^ -- in equilibrium states. We develop a framework to derive a rigorous thermodynamic uncertainty relation for such pairs, where the uncertainty of the classical parameter θ is quantified by its quantum Fisher information Fθ. The framework is based on an exact integral representation that relates Fθ to the autocorrelation function of operator O^. From this representation, we derive a tight upper bound for the quantum Fisher information, which yields a thermodynamic uncertainty relation: ΔθΔO≥kBT with ΔO≡∂θ⟨O^⟩Δθ and T is the system temperature. The result establishes a fundamental precision limit for quantum sensing and metrology in thermal systems, directly connecting it to the thermodynamic properties of linear response and fluctuations.
Quantum gravity theories predict a minimal length at the order of magnitude of the Planck length, under which the concepts of space and time lose every physical meaning. In quantum mechanics, the insurgence of such minimal length can be described by introducing a modified position-momentum commutator, which in turn yields a generalized uncertainty principle, where the uncertainty on the position measurement has a lower bound. The value of the minimal length is not predicted by theories and must be evaluated experimentally. In this paper, we address the quantum bound to estimability of the minimal uncertainty length by performing measurements on a harmonic oscillator, which is analytically solvable in the deformed algebra of the Hilbert subspace.
The 1995 article by Achim Kempf, Gianpiero Mangano, and Robert B. Mann, “Hilbert Space Representation of the Minimal Length Uncertainty Relation,” constitutes one of the most important formal constructions through which a nonzero minimal uncertainty in position can be incorporated into quantum mechanics. Rather than representing a fundamental length merely as a discrete lattice imposed upon ordinary space, Kempf, Mangano, and Mann modify the canonical commutation relation and construct a Hilbert- space representation in which arbitrarily precise physical localization is excluded. Their framework yields a minimal position uncertainty, modifies the mathematical status of exact-position states, introduces maximally localized states and a quasi-position representation, and, in multidimensional formulations, leads to noncommuting position coordinates. The original article was published in Physical Review D, volume 52, pages 1108–1118, DOI https://doi.org/10.1103/PhysRevD.52.1108 (Kempf, Mangano, and Mann, 1995). This article develops a critical–propositional analysis of that framework in dia- logue with the Theory of Objectivity (TO), particularly its Seven Absolute Truths, its interpretation of logical Nothingness, the concept of individualizing fields, the onto- logical status attributed to infinity, the role of boundaries and interfaces, nonhuman relational observation, compositional emergence, and the transcendent element under- stood in TO as knowledge or information produced in atomic relations and considered equivalent to atomic radiation. Special attention is given to the foundational TO work A Esfera Perfeita (The Perfect Sphere), because the KMM minimal-localization structure provides an especially useful mathematical dialogue partner for a TO conception of a limiting physical scale. The central thesis developed here is deliberately restricted. Kempf, Mangano, and Mann do not demonstrate the Theory of Objectivity, do not derive its cosmogony, do not establish the individual magnetic field asserted by its Second Absolute Truth, and do not reproduce the triadic relational condition associated with its Fifth Absolute Truth. Nevertheless, their work provides a productive structural analogy for several TO concepts: finite physical localizability, distinction between formal mathematical objects and physically realizable states, existence of a forbidden domain below a localization threshold, generation of geometrical properties by fundamental algebraic relations, and conversion of fundamental constraints into potentially observable consequences. The strongest scientific use of KMM by TO would therefore not consist in claiming retrospective confirmation. TO should instead attempt to derive, from its own modal and cosmogonic architecture, an effective deformation parameter and a nonzero localization scale. If such a derivation were independently obtained, the resulting theory could produce quantitative deviations from ordinary quantum mechanics and could thereby enter genuine empirical contact. The analysis consequently proposes a formal research program in which modal axioms are connected to operator algebra, spectral corrections, propagation relations, information constraints, and experimentally bounded parameters. Later developments in generalized uncertainty principle research are also considered, including qualifications of the KMM algebra concerning Jacobi identities and spin (Fadel and Maggiore, 2022). KMM should therefore be understood as a historically and mathematically impor- tant model of minimal-length quantum mechanics, rather than as an experimentally established universal description of fundamental physics. Keywords: Theory of Objectivity; minimal length; generalized uncertainty principle; Kempf–Mangano–Mann algebra; modal ontology; Perfect Sphere; quantum gravity; physical localizability; information; atomic radiation; noncommutative geometry; In- ductive Effects; cosmology; modal necessity; operational bridges.