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arxiv: 1907.05786 · v2 · pith:Y2UBIUPLnew · submitted 2019-07-12 · 🧮 math-ph · math.MP· quant-ph

Lectures on Quantum Mechanics for mathematicians

Alexander Komech This is my paper

Pith reviewed 2026-05-24 22:07 UTC · model grok-4.3

classification 🧮 math-ph math.MPquant-ph
keywords quantum mechanicsnonlinear PDEsHamiltonian systemsattractorsMaxwell-Schrödingerwave-particle dualitydiffraction
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The pith

Quantum mechanical postulates arise dynamically from attractors of nonlinear Hamiltonian PDEs.

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

The lectures introduce the mathematical structure of quantum mechanics to readers with a background in analysis and PDEs. They focus on interpreting the postulates of stationary orbit transitions, wave-particle duality, and probability through a dynamical lens. The proposed mechanism relies on conjectures about the long-time behavior of solutions to nonlinear Hamiltonian partial differential equations, which have been established for various model systems. These ideas are illustrated with calculations of electron diffraction amplitudes and the Aharonov-Bohm phase shift using the Kirchhoff approximation.

Core claim

The paper proposes that the main quantum postulates receive a dynamical interpretation as consequences of attractors in nonlinear Hamiltonian PDEs. This interpretation is supported by results on model equations since 1990, but remains a conjecture for the Maxwell-Schrödinger system describing real physical interactions.

What carries the argument

Conjectures on attractors of nonlinear Hamiltonian partial differential equations, providing a dynamical mechanism for quantum phenomena.

If this is right

  • If correct, transitions between quantum states correspond to the approach of solutions toward specific attractors in the PDE evolution.
  • Wave-particle duality and probabilistic outcomes follow from the structure and stability properties of these attractors.
  • Scattering and interference effects, such as diffraction and Aharonov-Bohm shifts, can be derived from the underlying PDE dynamics via approximations like Kirchhoff's.

Where Pith is reading between the lines

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

  • This approach could bridge the gap between classical nonlinear wave equations and quantum theory without invoking additional postulates.
  • Verification for Maxwell-Schrödinger would require new analytical or numerical methods for the coupled system.
  • Similar attractor conjectures might apply to other quantum field theories or relativistic systems.

Load-bearing premise

The attractor conjectures extend from model Hamiltonian PDEs to the Maxwell-Schrödinger equations that govern physical quantum systems.

What would settle it

Demonstration that solutions to the Maxwell-Schrödinger equations do not converge to attractors corresponding to observed quantum stationary states or interference patterns.

Figures

Figures reproduced from arXiv: 1907.05786 by Alexander Komech.

Figure 1
Figure 1. Figure 1: Convergence to stationary states. 14.2 Global attraction to solitons For “generic” translation-invariant equations, the long-time asymptotics of all finite energy solutions is the convergence to solitons ψ(x, t) ∼ ψ±(x − v±t), t → ±∞, (14.3) where the convergence holds in local seminorms in the comoving frame of reference, that is, in L 2 (|x−v±t| < R) for any R > 0. Such soliton asymptotics were proved in… view at source ↗
Figure 2
Figure 2. Figure 2: Convergence to stationary orbits. The global attraction was proved under the assumption that the equations are “strictly nonlinear”. For linear equations, the attraction can fail if the discrete spectrum consists at least of two points. Remark 14.2. Let us comment on the term generic in the results of the previous section and in Conjecture (13.2). Namely, this conjecture means that the asymptotics (13.2) h… view at source ↗
Figure 3
Figure 3. Figure 3: Diffraction by double-slit. 17 On diffraction of electrons and Aharonov–Bohm shift Abstract We calculate the amplitude of diffraction for the electron beams in the framework of the Kirchhoff approx￾imation applying the limiting amplitude and the limiting absorption principles, and the Sommerfeld radiation condition. The Aharonov–Bohm Ansatz and the corresponding shift of the diffraction pattern are justifi… view at source ↗
Figure 4
Figure 4. Figure 4: Shift in magnetic field of the set-up of [PITH_FULL_IMAGE:figures/full_fig_p031_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Magnetic chain and the film. Moreover, the estimate (17.27) implies that β := max x∈T˜ |b(x)| ≤ C max |x|≤R(T˜) |A(x)| ≤ C1kBkH1(R3) (17.32) where R(T˜ ) is the radius of a ball which contains the tube T˜. The splitting (17.30) allows us to factorize the operator H(k) as follows: H(k) = e − e i~c ϕ(x)Hb(k)e e i~c ϕ(x) , Hb(k) := [∇ + e i~c b(x)]2 + k 2 . (17.33) Now the theory [31, 32] implies the existenc… view at source ↗
read the original abstract

The main goal of these lectures -- introduction to Quantum Mechanics for mathematically-minded readers. The second goal is to discuss the mathematical interpretation of the main quantum postulates: transitions between quantum stationary orbits, wave-particle duality and probabilistic interpretation. We suggest a dynamical interpretation of these phenomena based on the new conjectures on attractors of nonlinear Hamiltonian partial differential equations. This conjecture is confirmed for a list of {\it model Hamiltonian nonlinear} PDEs by the results obtained since 1990 (we survey sketchy these results). However, for the Maxwell--Schr\"odinger equations this conjecture is still an {\it open problem}. We calculate the diffraction amplitude for the scattering of electron beams and Aharonov--Bohm shift via the Kirchhoff approximation.

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

1 major / 2 minor

Summary. The manuscript consists of lectures introducing quantum mechanics to mathematically-minded readers. Its primary aim is to offer a dynamical interpretation of key quantum postulates—transitions between stationary orbits, wave-particle duality, and probabilistic interpretation—based on conjectures regarding attractors of nonlinear Hamiltonian PDEs. These conjectures are surveyed as confirmed for various model PDEs since 1990 but are explicitly noted as remaining open for the Maxwell-Schrödinger equations. The lectures also include calculations of electron diffraction amplitudes and the Aharonov-Bohm shift using the standard Kirchhoff approximation.

Significance. If the central conjecture holds for the Maxwell-Schrödinger system, the work would supply a dynamical mechanism linking nonlinear PDE attractors to quantum phenomena, providing a mathematical foundation for the postulates. The survey of model cases and the independent diffraction calculations add value as a pedagogical resource. However, the open status for the physically relevant equations means the interpretation remains suggestive rather than established, reducing its immediate significance for resolving foundational issues in QM.

major comments (1)
  1. Abstract: The suggested dynamical interpretation of quantum phenomena for physical systems rests on the conjecture for the Maxwell-Schrödinger equations, which the manuscript states is an open problem with support limited to surveyed model cases. No derivation or new evidence is provided to bridge this gap, making the interpretation conjectural for the load-bearing physical application.
minor comments (2)
  1. Abstract: 'we survey sketchy these results' is grammatically awkward and should be revised to 'we sketchily survey these results' or equivalent for clarity.
  2. Abstract: Inconsistent LaTeX formatting for 'Schrödinger' (uses Schröedinger with escaped quote) should be standardized.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed reading and the recommendation of minor revision. The manuscript is a set of lectures whose primary purpose is pedagogical, with the dynamical interpretation presented explicitly as a conjecture supported by model cases but open for the Maxwell-Schrödinger system. We address the single major comment below.

read point-by-point responses

  1. Referee: Abstract: The suggested dynamical interpretation of quantum phenomena for physical systems rests on the conjecture for the Maxwell-Schrödinger equations, which the manuscript states is an open problem with support limited to surveyed model cases. No derivation or new evidence is provided to bridge this gap, making the interpretation conjectural for the load-bearing physical application.

    Authors: The manuscript already states in the abstract and introduction that the conjecture remains an open problem for the Maxwell-Schrödinger equations and is confirmed only for the surveyed model PDEs. The lectures do not claim to provide a derivation or new evidence for the physical case; their goal is to introduce the conjectural dynamical mechanism to mathematically-minded readers and to survey the existing results on model systems. The interpretation is therefore presented as conjectural for the physically relevant equations, consistent with the referee's observation. We do not believe a change to the abstract is required, as the current wording already qualifies the status of the conjecture. revision: no

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents a conjecture on attractors of nonlinear Hamiltonian PDEs as new, explicitly states it is confirmed only for model systems since 1990 and remains an open problem for the Maxwell-Schrödinger equations of physical interest. Diffraction and Aharonov-Bohm calculations are performed separately via the standard Kirchhoff approximation. No load-bearing step reduces by definition, fitted parameter, or self-citation chain to its own inputs; the work is self-contained with explicit acknowledgment of open status.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper adds one central conjecture linking quantum postulates to attractors; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • ad hoc to paper Attractors of nonlinear Hamiltonian PDEs provide the dynamical mechanism for quantum stationary states, transitions, and wave-particle duality
    This is the new conjecture proposed in the abstract as the basis for the dynamical interpretation.

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