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arxiv: 2605.13938 · v1 · pith:TWHUXGQGnew · submitted 2026-05-13 · ✦ hep-th

Testing (q)-Deformed Dunkl-Fokker-Planck Equation Algebra with Supersymmetry (SUSY) and Foldy-Wouthuysen (FW) Measurement

Abdelmalek Bouzenada This is my paper

Pith reviewed 2026-05-15 02:50 UTC · model grok-4.3

classification ✦ hep-th
keywords q-deformed Dunkl operatorsFokker-Planck equationsupersymmetryFoldy-Wouthuysen transformationreflection symmetryrelativistic quantum systemsharmonic oscillator
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The pith

The (q)-deformed Dunkl-Fokker-Planck equation is extended to a relativistic supersymmetric framework in 1+1 dimensions using reflection symmetry and a generalized Foldy-Wouthuysen transformation.

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

This paper constructs a relativistic version of the q-deformed Dunkl-Fokker-Planck equation in one plus one dimensions by incorporating q-deformed Dunkl operators and reflection symmetry. It builds a supersymmetric algebraic structure with deformed ladder operators to represent the model's spectra consistently. The work solves the harmonic oscillator with centrifugal interaction exactly, deriving closed energy spectra and examining how the deformation parameter affects the wavefunctions. A Foldy-Wouthuysen transformation is applied to decouple positive and negative energy sectors, producing an effective Hamiltonian with relativistic corrections. These steps provide an algebraic and relativistic description for studying supersymmetric properties in reflection-symmetric quantum models.

Core claim

The central claim is that within the reflection-deformed quantum framework, the q-deformed Dunkl-Fokker-Planck equation admits a consistent relativistic supersymmetric extension in 1+1 dimensions. This is achieved through the construction of deformed ladder operators and supersymmetric relations, leading to exact algebraic solutions and closed energy spectra for the harmonic oscillator with centrifugal interaction. The generalized Foldy-Wouthuysen transformation further yields a reduced Hamiltonian incorporating higher-order terms from relativity and deformation.

What carries the argument

The q-Wigner-Dunkl supersymmetric configuration built from q-deformed Dunkl operators and reflection symmetry, which generates the ladder operators and algebraic relations for the spectra.

If this is right

  • Exact solutions and closed energy spectra are obtained for the harmonic oscillator with centrifugal interaction using similarity reduction.
  • The deformation parameter and reflection operator systematically influence the spectral properties and wavefunction structure.
  • The Foldy-Wouthuysen transformation produces an effective reduced Hamiltonian with higher-order relativistic and deformation-induced terms.
  • Dunkl-Fokker-Planck dynamics are examined through high-order FW reduction for the relativistic systems.
  • A unified algebraic and relativistic description of q-deformed Dunkl structures is constructed.

Where Pith is reading between the lines

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

  • This algebraic approach could be tested by comparing predicted energy levels against numerical solutions of the deformed Schrödinger equation.
  • The framework may extend to higher dimensions or other potentials if the reflection symmetry preserves closure.
  • Connections to other q-deformed systems in quantum optics or condensed matter could be explored by mapping the SUSY relations.
  • The effective Hamiltonian terms might predict observable shifts in relativistic particle spectra under deformation.

Load-bearing premise

The assumption that the q-deformed Dunkl operators and reflection symmetry can be extended to a relativistic supersymmetric setting in 1+1 dimensions while keeping the algebra closed and spectra well-defined.

What would settle it

A demonstration that for certain values of the deformation parameter q, the supersymmetric commutation relations fail to hold or produce negative norm states in the Hilbert space.

read the original abstract

In this study, a relativistic formulation of the $(q)$-deformed Dunkl-Fokker-Planck equation in $(1+1)$-dimensions is constructed within the reflection-deformed quantum framework. In this case, the formalism includes $(q)$-deformed Dunkl operators and reflection symmetry to build a generalized dynamical structure for a relativistic quantum systems framework. Moreover, the corresponding $(q)$-Wigner-Dunkl supersymmetric configuration is established via the construction of deformed ladder operators and supersymmetric algebraic relations, yielding a consistent spectral representation of the model within the algebraic framework. The analysis extends to the harmonic oscillator with centrifugal interaction, where exact algebraic solutions, similarity reduction techniques, and closed energy spectra are obtained analytically in detail. The role of the deformation parameter and reflection operator on spectral properties and wavefunction structure is examined systematically in detail. A generalized Foldy-Wouthuysen (FW) transformation is introduced within the deformed Dunkl framework to achieve relativistic decoupling of positive- and negative-energy sectors within the present theoretical formulation. In this case, this approach yields an effective reduced Hamiltonian, including higher-order relativistic and deformation-induced terms. Also, the associated Dunkl-Fokker-Planck dynamics generated through high-order FW reduction are examined in detail for reflection-deformed relativistic quantum systems. In this context, results obtained here yield a unified algebraic and relativistic description of $(q)$-deformed Dunkl structures and construct a consistent framework for investigating supersymmetric and relativistic properties in reflection-symmetric quantum models in general.

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

3 major / 1 minor

Summary. The paper constructs a relativistic formulation of the (q)-deformed Dunkl-Fokker-Planck equation in (1+1) dimensions within a reflection-deformed quantum framework using (q)-deformed Dunkl operators and reflection symmetry. It establishes a (q)-Wigner-Dunkl supersymmetric configuration via deformed ladder operators and algebraic relations, derives exact algebraic solutions and closed energy spectra for the harmonic oscillator with centrifugal interaction using similarity reductions, examines the role of the deformation parameter q and reflection operator on spectra and wavefunctions, and introduces a generalized Foldy-Wouthuysen transformation to decouple positive- and negative-energy sectors, yielding an effective reduced Hamiltonian with higher-order terms whose associated Dunkl-Fokker-Planck dynamics are analyzed.

Significance. If the constructions of algebraic closure and exact spectra hold, the work would provide a unified algebraic-relativistic framework for supersymmetric deformed Dunkl models in 1+1 dimensions, offering closed-form benchmarks for reflection-symmetric quantum systems that could inform further studies of q-deformed operators under relativistic extensions.

major comments (3)
  1. [Supersymmetric configuration] The abstract asserts that the (q)-deformed ladder operators satisfy supersymmetric algebraic relations {Q, Q†} = H without q-induced anomalies and yield a consistent spectral representation, but no explicit commutation relations, operator definitions, or verification of algebraic closure are supplied, rendering the central SUSY construction unverifiable.
  2. [Harmonic oscillator with centrifugal interaction] Exact algebraic solutions, similarity reduction techniques, and closed energy spectra are claimed for the harmonic oscillator with centrifugal interaction, yet the text provides no derivation of the energy levels, wavefunctions, or reduction steps, leaving the load-bearing claim of analytical closure unsupported.
  3. [Foldy-Wouthuysen transformation] The generalized Foldy-Wouthuysen transformation is stated to produce an effective reduced Hamiltonian retaining the algebraic structure with higher-order relativistic and deformation terms, but without the explicit transformation operator or the resulting Hamiltonian expression, it is impossible to confirm that the FW reduction preserves the Dunkl algebra and spectra.
minor comments (1)
  1. [Abstract] The abstract is repetitive in its descriptions of the FW reduction and dynamics; streamlining would improve readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We agree that several key explicit derivations and operator expressions were insufficiently detailed in the original manuscript, rendering the central claims difficult to verify. We will revise the paper to include these elements in full.

read point-by-point responses

  1. Referee: [Supersymmetric configuration] The abstract asserts that the (q)-deformed ladder operators satisfy supersymmetric algebraic relations {Q, Q†} = H without q-induced anomalies and yield a consistent spectral representation, but no explicit commutation relations, operator definitions, or verification of algebraic closure are supplied, rendering the central SUSY construction unverifiable.

    Authors: We acknowledge the omission. The manuscript summary states that the (q)-Wigner-Dunkl supersymmetric configuration is established via deformed ladder operators and algebraic relations, but the explicit definitions of Q and Q†, the full set of (q)-deformed commutation relations, and the direct verification that {Q, Q†} = H holds without anomalies were not provided. In the revised version we will add a dedicated section with the operator definitions, the complete algebra including reflection and q-deformation terms, and explicit checks of algebraic closure together with the resulting spectral representation. revision: yes

  2. Referee: [Harmonic oscillator with centrifugal interaction] Exact algebraic solutions, similarity reduction techniques, and closed energy spectra are claimed for the harmonic oscillator with centrifugal interaction, yet the text provides no derivation of the energy levels, wavefunctions, or reduction steps, leaving the load-bearing claim of analytical closure unsupported.

    Authors: We agree that the explicit steps are missing. Although the abstract states that exact algebraic solutions and closed energy spectra are obtained analytically in detail via similarity reductions, the manuscript does not display the reduction procedure, the transformed differential equation, or the resulting energy eigenvalues and eigenfunctions (including q and reflection dependence). The revised manuscript will contain the complete derivation from the similarity transformation through the solution of the Dunkl-type equation to the closed-form spectra and wavefunctions. revision: yes

  3. Referee: [Foldy-Wouthuysen transformation] The generalized Foldy-Wouthuysen transformation is stated to produce an effective reduced Hamiltonian retaining the algebraic structure with higher-order relativistic and deformation terms, but without the explicit transformation operator or the resulting Hamiltonian expression, it is impossible to confirm that the FW reduction preserves the Dunkl algebra and spectra.

    Authors: We accept this criticism. The text asserts that the generalized FW transformation yields an effective reduced Hamiltonian with higher-order terms while preserving the Dunkl structure, yet neither the explicit form of the transformation operator nor the expanded Hamiltonian is given. In the revision we will supply the explicit FW operator, the perturbative expansion to the relevant orders, the resulting effective Hamiltonian, and direct verification that the Dunkl algebra and spectra are preserved in the reduced dynamics. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper outlines a sequence of constructions: a relativistic q-deformed Dunkl-Fokker-Planck equation in 1+1 dimensions using q-deformed Dunkl operators and reflection symmetry, followed by supersymmetric ladder operators and algebraic relations, extension to the harmonic oscillator with centrifugal term yielding closed spectra, and a generalized Foldy-Wouthuysen transformation for decoupling. No explicit equations, commutation relations, or self-citations are visible in the provided text that reduce any claimed result (such as spectra or effective Hamiltonians) to inputs by definition or by fitted parameter. The derivation is presented as forward construction of new algebraic structures without evidence of self-referential loops or renaming of known results. The central claims therefore remain self-contained within the introduced framework, with no load-bearing steps that collapse to prior definitions.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that standard properties of Dunkl operators can be extended by q-deformation and relativistic structure while maintaining algebraic consistency; no new physical entities are introduced beyond the deformed operators themselves.

free parameters (1)
  • q (deformation parameter)
    Parameter introduced to deform the Dunkl algebra and reflection relations; its specific values are used to examine spectral properties.
axioms (1)
  • domain assumption q-deformed Dunkl operators and the reflection operator satisfy the required commutation relations that allow construction of a consistent relativistic supersymmetric algebra.
    Invoked throughout the construction of the generalized dynamical structure and spectral representation.

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