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arxiv: 2603.26826 · v4 · submitted 2026-03-27 · 🪐 quant-ph

Non-Relativistic Quantum Mechanics in Multidimensional Geometric Frameworks

Dalaver H. Anjum , Shahid Nawaz , Muhammad Saleem This is my paper

Pith reviewed 2026-05-14 23:53 UTC · model grok-4.3

classification 🪐 quant-ph
keywords higher-order Schrödinger equationL^j-normed spacesgeneralized Minkowski distancepower-law dispersioninfinite potential wellbound-state energiesHeisenberg uncertainty principlej-fold conjugation
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The pith

A j-th order Schrödinger equation follows from the generalized Minkowski distance in L^j-normed spaces, yielding bound-state energies that scale as (2n+1)^j while the uncertainty principle holds.

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

The paper constructs a version of non-relativistic quantum mechanics for geometries whose dispersion relation is E proportional to |p| raised to the power j, with j tied to the dimension of the space. It begins with the generalized Minkowski distance in L^j-normed spaces and replaces the usual quadratic kinetic term with a higher-order derivative, producing a consistent j-th order Schrödinger equation. Solutions for free particles preserve plane waves, while the infinite square well yields eigenfunctions of mixed exponential, trigonometric, and hyperbolic type and energies that grow as (2n+1)^j. A j-fold conjugation rule is introduced to keep the probability density real and positive so that expectation values remain well-defined. The Heisenberg uncertainty relation is shown to survive unchanged, framing the entire structure as geometry-dependent rather than fixed.

Core claim

Starting from the generalized Minkowski distance in L^j-normed spaces, a consistent j-th order Schrödinger equation is obtained for systems obeying the power-law dispersion E ∝ |p|^j where j = N-1. Plane-wave solutions and translational invariance are retained, yet the spectral problem in an infinite well produces bound-state energies scaling as (2n+1)^j together with eigenfunctions whose form is fixed by the roots of negative unity. A probability framework based on j-fold conjugation ensures a real-valued density and consistent expectation values. The Heisenberg uncertainty principle remains intact under these generalizations.

What carries the argument

The generalized Minkowski distance in L^j-normed spaces, which replaces the quadratic kinetic energy with a j-th order spatial derivative in the Schrödinger equation.

If this is right

  • Plane-wave solutions and translational invariance survive in all j geometries.
  • Bound-state energies grow faster than linearly for j greater than 2, producing cubic and quartic spectra in 4G and 5G cases.
  • Eigenfunctions take mixed exponential-trigonometric-hyperbolic forms determined by the j-th roots of negative unity.
  • The probability density stays real and positive via j-fold conjugation, allowing standard expectation-value calculations.
  • The position-momentum uncertainty relation is unchanged despite the higher-order derivatives.

Where Pith is reading between the lines

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

  • The same construction could be applied to effective dispersion relations observed in metamaterials or engineered lattices to predict altered confinement energies.
  • Extending the formalism to time-dependent problems would test whether probability is conserved for arbitrary j.
  • The approach suggests that different spatial metrics could produce testable variants of quantum mechanics in confined systems.

Load-bearing premise

The j-fold conjugation rule produces a consistent positive-definite probability density and well-defined expectation values without violating unitarity or other quantum axioms.

What would settle it

An explicit check that the time-evolution operator fails to preserve the norm of the wave function for any j greater than 2 would show that the generalized probability framework is inconsistent.

Figures

Figures reproduced from arXiv: 2603.26826 by Dalaver H. Anjum, Muhammad Saleem, Shahid Nawaz.

Figure 1
Figure 1. Figure 1: Schematic representation of a quantum particle [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
read the original abstract

A generalized formulation of non-relativistic quantum mechanics is developed within multidimensional geometric (NG) frameworks characterized by a power-law dispersion relation \(E \propto |p|^{j}\), where \(j = N - 1\). Starting from the generalized Minkowski distance in \(L^j\)-normed spaces, the conventional quadratic kinetic structure of three-dimensional geometry is extended to higher-order spatial derivatives, yielding a consistent \(j\)-th order Schr\"odinger equation. The formalism is applied to free particles and to particles confined within a one-dimensional infinite potential well for 2G, 3G, 4G, and 5G geometries. While plane-wave solutions and translational invariance are preserved, the spectral structure is modified, with bound-state energies scaling as \((2n+1)^{j}\), leading to cubic and quartic growth in higher geometries. The corresponding eigenfunctions exhibit mixed exponential, trigonometric, and hyperbolic forms determined by the roots of negative unity. A generalized probability framework based on \(j\)-fold conjugation is introduced, ensuring a real-valued probability density and consistent expectation values. Despite these generalizations, the Heisenberg uncertainty principle is preserved. The formulation presents quantum mechanics as a geometry-dependent theory in which dispersion relations, spectral properties, and probabilistic structure emerge from the underlying spatial metric.

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

4 major / 2 minor

Summary. The paper proposes a generalization of non-relativistic quantum mechanics in L^j-normed spaces (j = N-1) derived from a generalized Minkowski distance, yielding a j-th order Schrödinger equation with power-law dispersion E ∝ |p|^j. It applies the formalism to free particles and the one-dimensional infinite well in 2G–5G geometries, reporting bound-state energies scaling as (2n+1)^j with mixed exponential-trigonometric eigenfunctions, introduces a j-fold conjugation rule for a real probability density, and asserts preservation of the Heisenberg uncertainty principle.

Significance. If the missing technical foundations were supplied, the work could represent a novel geometric extension of QM that ties dispersion and spectra directly to the underlying norm, with potential implications for higher-order operators and modified bound states. The preservation of plane-wave solutions and uncertainty relations would be noteworthy strengths. At present, however, the absence of derivations for consistency, hermiticity, and probability conservation renders the significance preliminary.

major comments (4)
  1. [Abstract] Abstract and derivation of the j-th order Schrödinger equation: the dispersion relation E ∝ |p|^j is introduced by definition from the chosen L^j metric, after which the equation is constructed to reproduce it; the reported spectral scaling therefore follows tautologically rather than from an independent calculation.
  2. [Infinite potential well application] Infinite potential well application: the bound-state energies scale as (2n+1)^j. For j=2 this yields E ∝ (2n+1)^2, which does not recover the standard infinite-well spectrum E_n ∝ n^2 (up to constants and ħ, m, L); no explicit verification that the j=2 case matches known results is provided.
  3. [Generalized probability framework] Generalized probability framework: the j-fold conjugation rule is asserted to produce a real, positive-definite probability density with conserved integral and well-defined expectation values. No derivation of the associated inner product, continuity equation, or proof that the higher-order operator is Hermitian (hence unitary evolution) for j>2 is supplied.
  4. [Heisenberg uncertainty principle] Heisenberg uncertainty principle: the claim that it remains intact is stated without an explicit calculation or proof that the generalized position and momentum operators satisfy the canonical commutation relation under the j-fold measure.
minor comments (2)
  1. [Abstract] The acronyms 2G, 3G, 4G, 5G are introduced without definition; state explicitly that they correspond to j=1,2,3,4 (or the intended mapping) on first use.
  2. [Abstract] The abbreviation NG for the geometric frameworks is not expanded; provide the full term at first appearance.

Simulated Author's Rebuttal

4 responses · 0 unresolved

We thank the referee for the careful and constructive review. We address each major comment point by point below. Where the manuscript lacks explicit derivations or verifications, we will supply them in the revised version; where a technical inconsistency is identified, we will correct it.

read point-by-point responses

  1. Referee: [Abstract] Abstract and derivation of the j-th order Schrödinger equation: the dispersion relation E ∝ |p|^j is introduced by definition from the chosen L^j metric, after which the equation is constructed to reproduce it; the reported spectral scaling therefore follows tautologically rather than from an independent calculation.

    Authors: The dispersion relation is indeed introduced from the L^j-normed geometry by construction, as this is the defining feature of the proposed framework. The j-th order Schrödinger equation is then written to be consistent with that dispersion. The bound-state spectra, however, are obtained by solving the resulting higher-order eigenvalue problem subject to boundary conditions; this step is independent of the initial definition. We will revise the abstract and the derivation section to clarify this distinction and to emphasize that the spectral results follow from explicit solution of the differential equation rather than from the dispersion alone. revision: partial

  2. Referee: [Infinite potential well application] Infinite potential well application: the bound-state energies scale as (2n+1)^j. For j=2 this yields E ∝ (2n+1)^2, which does not recover the standard infinite-well spectrum E_n ∝ n^2 (up to constants and ħ, m, L); no explicit verification that the j=2 case matches known results is provided.

    Authors: We acknowledge the discrepancy. For j=2 the formalism must recover the standard Schrödinger equation and the familiar E_n ∝ n^2 spectrum. The reported (2n+1)^j scaling appears to originate from the particular choice of roots in the characteristic equation and the resulting indexing of allowed modes. We will re-derive the spectrum for the infinite well in all geometries, explicitly verify the j=2 reduction, and correct the energy formula and associated eigenfunction discussion in the revised manuscript. revision: yes

  3. Referee: [Generalized probability framework] Generalized probability framework: the j-fold conjugation rule is asserted to produce a real, positive-definite probability density with conserved integral and well-defined expectation values. No derivation of the associated inner product, continuity equation, or proof that the higher-order operator is Hermitian (hence unitary evolution) for j>2 is supplied.

    Authors: The referee correctly notes that these foundational elements are missing. In the revision we will add a new section that (i) defines the inner product induced by the j-fold conjugation, (ii) proves hermiticity of the j-th order Hamiltonian with respect to this inner product for j>2, (iii) derives the continuity equation, and (iv) demonstrates conservation of the integrated probability density together with the definition of expectation values. revision: yes

  4. Referee: [Heisenberg uncertainty principle] Heisenberg uncertainty principle: the claim that it remains intact is stated without an explicit calculation or proof that the generalized position and momentum operators satisfy the canonical commutation relation under the j-fold measure.

    Authors: We will insert an explicit calculation showing that the generalized position and momentum operators satisfy the canonical commutation relation [X,P]=iħI when the inner product is taken with respect to the j-fold measure. This will rigorously confirm preservation of the Heisenberg uncertainty principle. revision: yes

Circularity Check

1 steps flagged

Dispersion relation introduced by definition from L^j metric, rendering spectral scaling tautological

specific steps
  1. self definitional [Abstract]
    "A generalized formulation of non-relativistic quantum mechanics is developed within multidimensional geometric (NG) frameworks characterized by a power-law dispersion relation E ∝ |p|^j, where j = N - 1. Starting from the generalized Minkowski distance in L^j-normed spaces, the conventional quadratic kinetic structure of three-dimensional geometry is extended to higher-order spatial derivatives, yielding a consistent j-th order Schrödinger equation."

    The dispersion relation E ∝ |p|^j is defined directly by the choice of L^j-normed metric. The Schrödinger equation is then constructed to reproduce exactly this dispersion, so the reported bound-state energy scaling as (2n+1)^j is a direct algebraic consequence of the input metric rather than an independent prediction from the dynamics.

full rationale

The paper starts from the generalized Minkowski distance in L^j-normed spaces, which directly encodes the power-law dispersion E ∝ |p|^j by construction. It then builds the j-th order Schrödinger equation specifically to reproduce this dispersion, so the bound-state energies scaling as (2n+1)^j and related claims follow immediately from the input metric choice rather than from an independent dynamical derivation. The j-fold conjugation probability framework is introduced as an ansatz without shown unitarity or positive-definiteness for j≠2. This produces partial circularity in the central results while leaving some geometric extensions non-circular.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim rests on the assumption that a generalized Minkowski distance in L^j space directly supplies a valid dispersion relation for a quantum theory, together with an ad-hoc j-fold conjugation rule introduced to restore real probabilities.

free parameters (1)
  • j
    Power index set equal to N-1 for N-dimensional geometry; chosen by hand to define the order of the derivative and the growth of the spectrum.
axioms (1)
  • domain assumption Generalized Minkowski distance in L^j-normed spaces defines the dispersion relation E proportional to |p|^j
    Invoked at the outset to replace the usual quadratic kinetic term.
invented entities (1)
  • j-fold conjugation no independent evidence
    purpose: To produce a real-valued probability density and consistent expectation values for the higher-order wave function
    New rule introduced because ordinary complex conjugation fails to keep the density real when j is odd or greater than 2.

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