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arxiv: 2607.01393 · v1 · pith:TRVPK7ZQnew · submitted 2026-07-01 · 🪐 quant-ph · hep-th

Coherent states in minimal-length Quantum Mechanics: inequivalent characterizations and emergent squeezing

Giuseppe Gaetano Luciano , Pasquale Bosso , Daniel Chemisana This is my paper

Pith reviewed 2026-07-03 20:01 UTC · model grok-4.3

classification 🪐 quant-ph hep-th
keywords coherent statesminimal lengthgeneralized uncertainty principleharmonic oscillatorsqueezingquantum gravityphase space
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The pith

Minimal length makes the three standard definitions of coherent states inequivalent.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper analyzes coherent states of the one-dimensional harmonic oscillator under a minimal-length deformation of quantum mechanics. It establishes that the usual equivalence between annihilation-operator eigenstates, displaced vacua, and minimum-uncertainty packets fails once the Heisenberg algebra is modified. This inequivalence produces deformed phase-space trajectories and an intrinsic squeezing that does not appear in ordinary quantum mechanics. A reader would care because the effect supplies a concrete, potentially measurable signature of Planck-scale physics in accessible oscillator systems.

Core claim

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. 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.

What carries the argument

The deformation of the Heisenberg algebra via the Generalized Uncertainty Principle, which renders the three usual coherent-state constructions inequivalent and generates emergent squeezing.

If this is right

  • Generalized coherent states evolve differently from states that saturate the GUP uncertainty relation.
  • Phase-space trajectories acquire nontrivial deformations induced by the minimal length.
  • An intrinsic squeezing mechanism appears during time evolution with no classical or standard-quantum counterpart.
  • The inequivalence supplies a unified framework for coherence in any GUP-based theory.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Precision spectroscopy or interferometry on trapped oscillators could search for the predicted squeezing as a low-energy probe of minimal length.
  • The same loss of equivalence may occur for coherent states of other deformed algebras beyond the harmonic oscillator.
  • Semiclassical limits of GUP theories would then require separate treatment of each former coherent-state class.

Load-bearing premise

The specific algebraic deformation chosen for the Generalized Uncertainty Principle correctly captures minimal-length effects from quantum gravity.

What would settle it

An experiment on a harmonic oscillator in which the three coherent-state characterizations remain equivalent or in which no extra squeezing appears beyond ordinary quantum mechanics.

Figures

Figures reproduced from arXiv: 2607.01393 by Daniel Chemisana, Giuseppe Gaetano Luciano, Pasquale Bosso.

Figure 1
Figure 1. Figure 1: FIG. 1: Plot of the ratio between the uncertainty [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Plot of [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Plot of the inner product ( [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Dispersion of minimal uncertainty states. We [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: The plots above are for different values of the [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Uncertainty profile of a maximally localized [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: for various values of the GUP parameter β. Con￾sistently with the trend displayed in [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper analyzes coherent states for the 1D harmonic oscillator in minimal-length quantum mechanics using the quadratic GUP deformation of the Heisenberg algebra, [x,p]=iħ(1+βp²). It shows that the usual equivalences among annihilation-operator eigenstates, displaced vacua, and minimum-uncertainty states are lost, leading to deformed phase-space trajectories and an intrinsic squeezing effect absent in standard QM. The work compares the dynamics of these generalized coherent states to GUP-saturating states and frames the results as a unified framework with semiclassical signatures of minimal length.

Significance. If the inequivalence and squeezing hold under the stated algebra, the results supply concrete, falsifiable distinctions between ordinary and minimal-length coherent-state dynamics that could be tested in analog systems or precision measurements. The explicit comparison of multiple state characterizations and the identification of an emergent squeezing mechanism without external driving constitute the main technical contribution.

major comments (2)
  1. [Abstract, §1] Abstract and §1: The assertion that the loss of equivalence among coherent-state characterizations 'is generically lost in the presence of a minimal length' is not supported by the manuscript. All derivations and numerical comparisons are performed exclusively for the quadratic commutator [x,p]=iħ(1+βp²); no analysis or counter-example is given for other minimal-length deformations (e.g., exponential or higher-order forms) that are also motivated by quantum-gravity approaches. Because the central claim is framed as generic rather than algebra-specific, this gap is load-bearing.
  2. [§3–4] §3–4 (definitions of the three coherent-state families and their time evolution): The manuscript does not provide an explicit check that the reported inequivalence survives a change of representation of the deformed algebra (e.g., the standard vs. the symmetric ordering). If the squeezing or trajectory deformation is representation-dependent, the dynamical conclusions would require qualification.
minor comments (2)
  1. [§2] Notation for the deformed annihilation operator and the GUP-saturating states should be introduced with a single consistent symbol set in §2 to avoid later confusion between a_β and a_GUP.
  2. [Figure captions] Figure captions for the phase-space plots should state the numerical value of βħ used and whether the trajectories are exact or obtained from a semiclassical approximation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below and indicate the revisions that will be incorporated.

read point-by-point responses

  1. Referee: [Abstract, §1] Abstract and §1: The assertion that the loss of equivalence among coherent-state characterizations 'is generically lost in the presence of a minimal length' is not supported by the manuscript. All derivations and numerical comparisons are performed exclusively for the quadratic commutator [x,p]=iħ(1+βp²); no analysis or counter-example is given for other minimal-length deformations (e.g., exponential or higher-order forms) that are also motivated by quantum-gravity approaches. Because the central claim is framed as generic rather than algebra-specific, this gap is load-bearing.

    Authors: We agree that the explicit analysis is restricted to the quadratic GUP [x,p]=iħ(1+βp²). The word 'generically' was used to emphasize the contrast with ordinary quantum mechanics rather than to claim universality across all possible deformations. To remove any ambiguity we will revise the abstract and §1 to state that the inequivalence holds for the quadratic GUP, which is the standard and most studied form. A brief remark will be added noting that other deformations remain an open direction for future work. revision: yes

  2. Referee: [§3–4] §3–4 (definitions of the three coherent-state families and their time evolution): The manuscript does not provide an explicit check that the reported inequivalence survives a change of representation of the deformed algebra (e.g., the standard vs. the symmetric ordering). If the squeezing or trajectory deformation is representation-dependent, the dynamical conclusions would require qualification.

    Authors: The three families are defined algebraically via the deformed commutator and the annihilation operator; these definitions do not depend on a particular operator ordering or representation. All explicit calculations are performed in the conventional position-space representation used throughout the GUP literature. We will add a clarifying paragraph in §3 stating that the inequivalence and squeezing are algebraic consequences of the deformed Heisenberg relation and are therefore expected to persist under equivalent representations, while noting that a full comparison with symmetric ordering lies outside the present scope. revision: partial

Circularity Check

0 steps flagged

No circularity; derivation is self-contained within the GUP algebra

full rationale

The paper derives the loss of equivalence among coherent-state characterizations and the appearance of intrinsic squeezing directly from the modified commutator [x,p]=iħ(1+βp²) by explicit construction of the relevant operators and states. No parameter is fitted to a data subset and then relabeled as a prediction, no self-citation supplies a load-bearing uniqueness theorem, and no ansatz is smuggled in via prior work. The central results are therefore independent computations inside the chosen deformed algebra rather than reductions to the paper's own inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The analysis depends on the GUP framework whose deformation strength is a free parameter and whose commutation relation is taken as the model for minimal length.

free parameters (1)
  • GUP deformation parameter β
    Controls the strength of the minimal-length correction in the deformed Heisenberg algebra.
axioms (1)
  • domain assumption Deformed commutation relation [x, p] = i ħ (1 + eta p^{2}) encodes minimal length
    This is the starting point for all subsequent analysis of coherent states.

pith-pipeline@v0.9.1-grok · 5739 in / 1181 out tokens · 34875 ms · 2026-07-03T20:01:08.745881+00:00 · methodology

discussion (0)

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Reference graph

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