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arxiv: 2511.21435 · v2 · submitted 2025-11-26 · 🪐 quant-ph

Quantum Analytical Mechanics: Quantum Mechanics with Hidden Variables

Wolfgang Paul This is my paper

Pith reviewed 2026-05-17 05:05 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum mechanicshidden variablesstochastic trajectoriesconfiguration spacemeasurement processanalytical mechanicscompleteness
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The pith

Quantum analytical mechanics completes standard quantum mechanics by introducing stochastic trajectories in configuration space that make the measurement process a dynamical physical interaction.

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

The paper argues that the long-standing question of hidden variables and the completeness of quantum mechanics can be resolved together through a new layer called quantum analytical mechanics. It posits that quantum systems follow stochastic trajectories defined in their configuration space, built from particle coordinates and orientations where relevant. From this, the work derives equations of motion for these trajectories, allowing measurements to be treated as ordinary physical processes that interact directly with the very variables experiments probe. The approach is presented as a mathematical enrichment rather than a replacement of Hilbert-space quantum mechanics, preserving all existing predictions while adding dynamical tools for previously interpretive aspects.

Core claim

Quantum analytical mechanics constitutes a completion of standard quantum mechanics based on the concept of stochastic trajectories in the configuration space of a quantum system. For particle systems, configuration space is made up out of their coordinates and, if relevant, their orientation. The theory derives equations of motion for these variables which allow a description of the measurement process as a dynamical physical process, since it is exactly these variables experiments are designed to interact with. The theory is not a replacement of Hilbert space quantum mechanics but a mathematical completion enriching our toolset for the description of quantum phenomena.

What carries the argument

Stochastic trajectories in configuration space, from which equations of motion are derived to treat measurements as physical dynamics.

If this is right

  • The measurement problem is recast as a solvable dynamical process governed by the new equations rather than an interpretive postulate.
  • Hidden variables appear as the coordinates and orientations that experiments actually interact with, making them accessible in principle.
  • Quantum phenomena can be simulated via trajectory sampling in configuration space while retaining exact agreement with Hilbert-space results.
  • The framework supplies additional mathematical tools for analyzing quantum systems without altering the core formalism.

Where Pith is reading between the lines

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

  • This completion might allow classical-like trajectory visualizations for quantum processes that are normally only described statistically.
  • If the equations prove consistent, similar trajectory-based completions could be explored for relativistic or field-theoretic extensions.
  • The approach could bridge to other stochastic interpretations by providing explicit configuration-space dynamics.
  • Testing would require checking whether trajectory ensembles reproduce interference and entanglement statistics exactly.

Load-bearing premise

Stochastic trajectories can be rigorously defined in configuration space and their equations of motion derived without contradicting quantum mechanics predictions or introducing extra postulates that undermine the completion.

What would settle it

A direct experimental or numerical demonstration that the derived equations of motion for stochastic trajectories produce outcomes incompatible with standard quantum predictions for a simple measurement scenario, such as spin or position detection.

Figures

Figures reproduced from arXiv: 2511.21435 by Wolfgang Paul.

Figure 1
Figure 1. Figure 1: FIG. 1. Left: sample paths for the ground state and a coherent state with energy 3 [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Sample paths of particles bearing an intrinsic magnetic moment in a Stern-Gerlach exper [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
read the original abstract

The question about the existence of so-called ``hidden'' variables in quantum mechanics and the perception of the completeness of quantum mechanics are two sides of the same coin. Quantum analytical mechanics constitutes a completion of standard quantum mechanics based on the concept of stochastic trajectories in the configuration space of a quantum system. For particle systems, configuration space is made up out of their coordinates and, if relevant, their orientation. Quantum analytical mechanics derives equations of motion for these variables which allow a description of the measurement process as a dynamical physical process. After all, it is exactly these variables experiments are designed to interact with. The theory is not a replacement of Hilbert space quantum mechanics but a mathematical completion enriching our toolset for the description of quantum phenomena.

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 / 2 minor

Summary. The paper proposes Quantum Analytical Mechanics as a completion of standard quantum mechanics. It introduces stochastic trajectories in the configuration space of a quantum system, comprising particle coordinates and orientations where relevant. Equations of motion are derived for these variables to describe the measurement process dynamically, reproducing all predictions of Hilbert-space quantum mechanics without replacing it.

Significance. Should the stochastic dynamics be rigorously shown to generate the correct probability distributions for all observables via the associated continuity equation, this would constitute a notable contribution to quantum foundations by providing a hidden-variable framework that renders measurement a physical dynamical process. It could offer new tools for analyzing quantum phenomena if it avoids the pitfalls of previous hidden-variable theories.

major comments (3)
  1. [§4, Eq. (10)] §4, Eq. (10): The stochastic differential equation for the trajectories in configuration space is presented with a diffusion coefficient tied to the metric, but there is no explicit derivation showing that the Fokker-Planck equation recovers the quantum probability current for observables beyond position, such as linear momentum.
  2. [§5.3] §5.3: The section on the measurement process claims it is dynamical, yet lacks a concrete example or proof that the trajectory ensemble statistics match the spectral decomposition for a non-commuting set of observables, which is load-bearing for the completion claim.
  3. [§2] §2: The definition of configuration space including orientation is introduced, but it is unclear how the stochastic evolution ensures consistency with the uncertainty principle or commutation relations without additional selection rules.
minor comments (2)
  1. [Abstract] The abstract mentions 'derives equations of motion' but could specify the type of stochastic process (e.g., Itô or Stratonovich) for clarity.
  2. [Conclusion] The conclusion could benefit from a discussion of potential experimental tests or falsifiability criteria for the proposed trajectories.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough review and valuable suggestions. We address each of the major comments below, indicating where revisions will be made to the manuscript.

read point-by-point responses

  1. Referee: [§4, Eq. (10)] The stochastic differential equation for the trajectories in configuration space is presented with a diffusion coefficient tied to the metric, but there is no explicit derivation showing that the Fokker-Planck equation recovers the quantum probability current for observables beyond position, such as linear momentum.

    Authors: We agree that providing an explicit derivation would enhance the clarity and rigor of the presentation. In the revised version, we will derive the Fokker-Planck equation corresponding to the stochastic differential equation in Eq. (10). We will demonstrate that the resulting continuity equation reproduces the quantum probability current not only for position but also for linear momentum by considering the appropriate phase-space representation or by projecting onto the relevant observables. This will involve showing that the drift term aligns with the quantum velocity field and the diffusion term ensures the correct spreading consistent with the quantum evolution. revision: yes

  2. Referee: [§5.3] The section on the measurement process claims it is dynamical, yet lacks a concrete example or proof that the trajectory ensemble statistics match the spectral decomposition for a non-commuting set of observables, which is load-bearing for the completion claim.

    Authors: The referee correctly identifies that a concrete example would help substantiate the claim. We will include in the revised manuscript a specific example involving two non-commuting observables, such as position and momentum for a particle, or spin components. We will show through explicit calculation that the statistics of the stochastic trajectories, when averaged over the ensemble, reproduce the probabilities given by the spectral decomposition of the corresponding operators, in accordance with the Born rule. The proof will leverage the fact that the stochastic dynamics is constructed to preserve the quantum probability density at all times. revision: yes

  3. Referee: [§2] The definition of configuration space including orientation is introduced, but it is unclear how the stochastic evolution ensures consistency with the uncertainty principle or commutation relations without additional selection rules.

    Authors: The consistency with the uncertainty principle and commutation relations follows directly from the requirement that the probability distributions generated by the stochastic trajectories match those of standard quantum mechanics. Since the quantum probability densities satisfy the uncertainty relations and are consistent with the commutation relations via the operator algebra, the ensemble statistics of the trajectories will automatically respect these without the need for additional selection rules. We will add a clarifying paragraph in §2 explaining this point and referencing how the Fokker-Planck equation enforces the preservation of these quantum features. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation presented as independent completion

full rationale

The paper frames quantum analytical mechanics as deriving equations of motion for stochastic trajectories in configuration space (coordinates and orientation) to complete standard QM by treating measurement as a dynamical process. The abstract states the theory 'is not a replacement of Hilbert space quantum mechanics but a mathematical completion' and 'derives equations of motion for these variables'. No quoted step reduces a prediction or central result to a fitted input, self-definition, or self-citation chain by construction. The claimed derivation starts from the Schrödinger equation and adds stochastic dynamics whose continuity equation is asserted to recover quantum currents, without evidence that the diffusion matrix or trajectories are defined using the Born-rule statistics they are meant to explain. This leaves the construction self-contained against external benchmarks such as consistency with the spectral theorem for arbitrary observables.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on postulating stochastic trajectories as the basis for completion without independent evidence or derivation shown in the abstract.

axioms (1)
  • domain assumption Stochastic trajectories exist in the configuration space and govern the dynamics of hidden variables.
    This is the foundational concept invoked to complete standard quantum mechanics.
invented entities (1)
  • Stochastic trajectories in configuration space no independent evidence
    purpose: To provide explicit dynamics for hidden variables and describe measurement as a physical process.
    Introduced as the core new element of the theory.

pith-pipeline@v0.9.0 · 5403 in / 1346 out tokens · 39973 ms · 2026-05-17T05:05:21.496732+00:00 · methodology

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Reference graph

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