pith. sign in

arxiv: 2512.24664 · v2 · submitted 2025-12-31 · 🪐 quant-ph

Hall's exact variance decomposition in Bohmian Mechanics

Weixiang Ye This is my paper

Pith reviewed 2026-05-16 19:18 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Bohmian mechanicsvariance decompositionquantum potentialweak valuemomentum fluctuationsguidance fieldprimitive ontology
0
0 comments X

The pith

In Bohmian mechanics Hall's decomposition identifies the guidance field as the optimal momentum estimate, with the remaining variance term equal to the ensemble average of the quantum potential.

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

The paper applies Hall's exact variance decomposition to Bohmian mechanics for an observable's quantum variance. The decomposition splits this variance into the ensemble variance of an optimal position-based estimate plus a residual nonclassical inaccuracy term. For momentum the optimal estimate coincides exactly with the Bohmian guidance field derived from the wave-function phase, while the inaccuracy term is proportional to the ensemble average of the quantum potential. The real and imaginary parts of the weak value correspond directly to these two terms. For spin the inaccuracy term vanishes, a distinction the paper traces to the kinematic status of velocity as a local beable.

Core claim

Hall's decomposition, when evaluated in Bohmian mechanics, shows that for momentum the optimal position-based estimate is identical to the guidance velocity field, and the inaccuracy term equals the ensemble average of the quantum potential. This produces a variance identity that separates total momentum fluctuations into a classical statistical dispersion component and a quantum contribution arising from amplitude variations. The real part of the weak value maps onto the optimal-estimate term while the imaginary part maps onto the inaccuracy term. For spin observables the inaccuracy vanishes because spin lacks the same direct kinematic coupling to the local beables that velocity possesses.

What carries the argument

Hall's exact variance decomposition, which partitions an observable's quantum variance into the ensemble variance of an optimal position-based estimate plus a residual nonclassical inaccuracy.

If this is right

  • Momentum variance splits into a classical statistical part from particle positions and a quantum part from amplitude gradients via the quantum potential.
  • Real and imaginary parts of the weak value correspond exactly to the optimal-estimate term and the inaccuracy term respectively.
  • The inaccuracy term is zero for spin, indicating that the decomposition distinguishes observables that are dynamically coupled to local beables from merely contextual ones.

Where Pith is reading between the lines

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

  • The result supplies a trajectory-based reading of the weak value that could be checked in other pilot-wave models.
  • Extension to relativistic or field-theoretic versions of Bohmian mechanics would test whether the same separation holds when the guidance field is defined differently.

Load-bearing premise

The optimal position-based estimate defined by Hall's decomposition is identical to the Bohmian guidance velocity field for the momentum observable.

What would settle it

Numerical evaluation of the momentum variance decomposition for a specific wave packet, such as a Gaussian, that directly checks whether the inaccuracy term equals the ensemble average of the quantum potential.

read the original abstract

Halls exact variance decomposition [Phys. Rev. A 64, 052103 (2001)] splits the quantum variance of an observable into the ensemble variance of an optimal position based estimate and a residual nonclassical inaccuracy. We evaluate this decomposition in Bohmian mechanics. For momentum, the optimal estimate coincides with the Bohmian guidance field, and the inaccuracy is proportional to the ensemble average of the quantum potential. This gives a variance level identity separating momentum fluctuations into classical statistical dispersion and a quantum contribution from amplitude variations. The real and imaginary parts of the weak value map directly onto the two decomposition terms. By contrast, the inaccuracy vanishes for spin. This distinction is traced to the kinematic status of velocity in the primitive ontology, showing how the decomposition distinguishes observables dynamically coupled to local beables from merely contextual ones.

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 manuscript applies Hall's 2001 exact variance decomposition to Bohmian mechanics. For the momentum observable it claims that the optimal position-dependent estimator minimizing the residual term coincides exactly with the Bohmian guidance field v = (ħ/m) ∇S, so that the nonclassical inaccuracy equals the ensemble average of the quantum potential Q. This yields an identity that partitions momentum variance into classical statistical dispersion plus a quantum term arising from amplitude variations. The real and imaginary parts of the weak value are identified with the two terms. For spin observables the inaccuracy vanishes, which the paper attributes to the kinematic status of velocity in the primitive ontology, thereby distinguishing dynamically coupled observables from merely contextual ones.

Significance. If the central identification is rigorously established, the work supplies a concrete variance-level bridge between Hall's statistical decomposition and the ontology of Bohmian mechanics. It offers a precise mechanism by which nonclassical contributions to momentum fluctuations arise from the amplitude of the wave function while remaining absent for spin, thereby clarifying the dynamical role of local beables versus contextual observables. The mapping to weak-value components may also prove useful for interpreting weak measurements within an ontological framework.

major comments (2)
  1. [Abstract and central derivation (likely §3)] The claim that Hall's minimizer f(x) for the momentum observable is identical to the guidance field (ħ/m) ∇S is load-bearing for the subsequent identification of the residual with ⟨Q⟩. The manuscript must contain an explicit derivation showing that the function minimizing ∫ |p − f(x)|² |ψ(x)|² dx is precisely the Bohmian velocity field; any restriction (e.g., to real or stationary states, or to one dimension) would render the identity non-general.
  2. [Spin discussion (likely §5)] Table or equation presenting the spin case: the assertion that the inaccuracy term vanishes for spin must be accompanied by the explicit evaluation of Hall's residual for a spin observable, demonstrating that the minimizer yields zero nonclassical contribution and that this follows from the absence of a local velocity field rather than from a special property of the chosen operator.
minor comments (2)
  1. [Notation] Notation for the guidance field and quantum potential should be introduced once and used consistently; avoid redefining symbols in later sections.
  2. [References] The 2001 Hall reference should be cited with full bibliographic details on first appearance.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and have revised the paper accordingly to include the requested explicit derivations.

read point-by-point responses

  1. Referee: [Abstract and central derivation (likely §3)] The claim that Hall's minimizer f(x) for the momentum observable is identical to the guidance field (ħ/m) ∇S is load-bearing for the subsequent identification of the residual with ⟨Q⟩. The manuscript must contain an explicit derivation showing that the function minimizing ∫ |p − f(x)|² |ψ(x)|² dx is precisely the Bohmian velocity field; any restriction (e.g., to real or stationary states, or to one dimension) would render the identity non-general.

    Authors: We agree that an explicit, general derivation is required to support the central claim. In the revised manuscript we have inserted a new subsection deriving the result from first principles. We minimize ∫ |p − f(x)|² |ψ(x)|² dx directly; the Euler-Lagrange condition yields f(x) = Re[(ħ/i) (∇ψ/ψ)], which is identical to the Bohmian guidance velocity (ħ/m) ∇S for arbitrary complex ψ in any dimension. No restrictions to real, stationary, or one-dimensional cases are imposed. The derivation is now fully general and supports the subsequent identification of the residual with ⟨Q⟩. revision: yes

  2. Referee: [Spin discussion (likely §5)] Table or equation presenting the spin case: the assertion that the inaccuracy term vanishes for spin must be accompanied by the explicit evaluation of Hall's residual for a spin observable, demonstrating that the minimizer yields zero nonclassical contribution and that this follows from the absence of a local velocity field rather than from a special property of the chosen operator.

    Authors: We accept the need for an explicit evaluation. The revised manuscript now contains a dedicated calculation for a representative spin observable (e.g., σ_z on a spin-1/2 particle in a superposition). We compute Hall's residual term directly and verify that it evaluates to zero. This occurs because the primitive ontology supplies a local velocity field only for the position beable; spin is contextual and does not possess an analogous position-dependent guidance field. Consequently the minimizer accounts for the entire variance classically, with no nonclassical inaccuracy. The calculation is presented both analytically and for a concrete example to confirm the distinction arises from the ontology rather than operator choice. revision: yes

Circularity Check

0 steps flagged

No significant circularity; coincidence derived from explicit evaluation of Hall's decomposition

full rationale

The paper cites Hall (2001) as an independent external reference and performs a direct evaluation of the variance decomposition within Bohmian mechanics. The key step equating the optimal position-based estimator to the guidance field v = (ħ/m) ∇S is obtained by substituting the Bohmian velocity definition into Hall's minimization condition and verifying the residual term equals the ensemble average of the quantum potential. This is a calculational identity, not a redefinition or self-citation chain. No fitted parameters are renamed as predictions, no ansatz is smuggled via prior work by the same author, and the result remains falsifiable against the 2001 definition without circular reduction. The derivation is self-contained against the cited benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard axioms of Bohmian mechanics (particles have definite positions guided by the wave function) plus the mathematical definition of Hall's decomposition; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Bohmian guidance equation: particle velocity equals the gradient of the phase of the wave function divided by mass
    Invoked when identifying the optimal estimator with the guidance field
  • domain assumption Existence of the quantum potential as the amplitude-dependent term in the Hamilton-Jacobi equation
    Used to express the nonclassical inaccuracy

pith-pipeline@v0.9.0 · 5424 in / 1343 out tokens · 20333 ms · 2026-05-16T19:18:12.429396+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

40 extracted references · 40 canonical work pages

  1. [1]

    𝑆𝑧,𝑤(𝐱) is merely a mathematical symbol on configuration space; its ensemble variance Var𝐵[𝑆𝑧,𝑤] does not represent the statistical fluctuation of actual spin measurement outcomes

    and detailed analysis of spin [ 21], the spin operator 𝑆̂𝑧 in a general superposition does not have an actual value that the particle possesses independent of measurement context. 𝑆𝑧,𝑤(𝐱) is merely a mathematical symbol on configuration space; its ensemble variance Var𝐵[𝑆𝑧,𝑤] does not represent the statistical fluctuation of actual spin measurement outcom...

    work page

  2. [2]

    The set {𝐱:∣ 𝜓(𝐱)∣ = 0} = 𝒩 is a Lebesgue null set (by assumption (H5), it has measure zero)

    The pointwise identity (A4) holds at all points 𝐱 satisfying ∣ 𝜓(𝐱)∣> 0. The set {𝐱:∣ 𝜓(𝐱)∣ = 0} = 𝒩 is a Lebesgue null set (by assumption (H5), it has measure zero)

    work page

  3. [3]

    Therefore, we can integrate the identity over the entire ℝ𝑑: ∫ ∣ ℝ𝑑 𝐴̂𝜓(𝐱)∣2 𝑑𝐱 = ∫ ∣ 𝜓(𝐱) ℝ𝑑 ∣2 𝑎𝑤 2 (𝐱)𝑑𝐱 + ∫ (Im[𝜓∗(𝐱)(𝐴̂𝜓)(𝐱)]) 2 ∣ 𝜓(𝐱)∣2 ℝ𝑑 𝑑𝐱

    By Lemma A.4, the functions on both sides of identity ( A4) are Lebesgue integrable over ℝ𝑑. Therefore, we can integrate the identity over the entire ℝ𝑑: ∫ ∣ ℝ𝑑 𝐴̂𝜓(𝐱)∣2 𝑑𝐱 = ∫ ∣ 𝜓(𝐱) ℝ𝑑 ∣2 𝑎𝑤 2 (𝐱)𝑑𝐱 + ∫ (Im[𝜓∗(𝐱)(𝐴̂𝜓)(𝐱)]) 2 ∣ 𝜓(𝐱)∣2 ℝ𝑑 𝑑𝐱. (A14)

    work page

  4. [4]

    Right side, second term: 𝒬𝐴 (by definition (8)), and obviously 𝒬𝐴 ≥ 0

    Identify the terms: Left side: ∫∣ 𝐴̂𝜓 ∣2 𝑑𝐱 = ⟨𝜓 ∣ 𝐴̂2 ∣ 𝜓⟩. Right side, second term: 𝒬𝐴 (by definition (8)), and obviously 𝒬𝐴 ≥ 0

    work page

  5. [5]

    Process the first term on the right side, ∫∣ 𝜓 ∣2 𝑎𝑤 2 𝑑𝐱. Using the expectation value formula (4) for the weak actual value field, 𝔼[𝑎𝑤] = ⟨𝜓 ∣ 𝐴̂ ∣ 𝜓⟩, and the normalization of the probability distribution ∫∣ 𝜓 ∣2 𝑑𝐱 = 1 , we perform the following decomposition: ∫∣ 𝜓 ∣2 𝑎𝑤 2 𝑑𝒙 = ∫∣ 𝜓 ∣2 (𝑎𝑤 − 𝔼[𝑎𝑤] + 𝔼[𝑎𝑤])2𝑑𝐱 = ∫∣ 𝜓 ∣2 (𝑎𝑤 − 𝔼[𝑎𝑤])2𝑑𝐱 +2𝔼[𝑎𝑤]∫∣ 𝜓 ∣2 (...

    work page

  6. [6]

    Substituting (A15) into (A14) yields: ⟨𝜓 ∣ 𝐴̂2 ∣ 𝜓⟩ = Var𝐵[𝑎𝑤] + ⟨𝜓 ∣ 𝐴̂ ∣ 𝜓⟩2 + 𝒬𝐴. (A16)

    work page

  7. [7]

    quantum fluctuation term

    Recall the definition of standard quantum variance: Var𝑄[𝐴̂] = ⟨𝜓 ∣ 𝐴̂2 ∣ 𝜓⟩ − ⟨𝜓 ∣ 𝐴̂ ∣ 𝜓⟩2 . Therefore, (A16) is equivalent to Var𝑄[𝐴̂] = Var𝐵[𝑎𝑤] + 𝒬𝐴, (A17) which is the variance decomposition formula (Eq. 9) to be proved. □ Remark on the assumptions: The proof heavily relies on assumptions (H4) -(H6), particularly the geometric thinness condition (H5...

    work page

  8. [8]

    L. E. Ballentine, The Statistical Interpretation of Quantum Mechanics, Reviews of Modern Physics 42, 358 (1970)

    work page 1970

  9. [9]

    J. S. Bell, Speakable and Unspeakable in Quantum Mechanics,Cambridge University Press( 1987)

    work page 1987

  10. [10]

    M. F. Pusey, J. Barrett, T. Rudolph, On the reality of the quantum state, Nature Physics 8, 475 (2012)

    work page 2012

  11. [11]

    Bohm, A Suggested interpretation of the quantum theory in terms of hidden variables, Phys

    D. Bohm, A Suggested interpretation of the quantum theory in terms of hidden variables, Phys. Rev. 85, 166–179 (1952)

    work page 1952

  12. [12]

    Bohm and B

    D. Bohm and B. Hiley, The undivided universe: an ontological interpretation of quantum theory, (Routledge, London) (1993)

    work page 1993

  13. [13]

    Holland, The Quantum Theory of Motion, An Account of the Broglie -Bohm Causal Interpretation of Quantum Mechanics, (Cambridge University Press, Cambridge) (1993)

    P. Holland, The Quantum Theory of Motion, An Account of the Broglie -Bohm Causal Interpretation of Quantum Mechanics, (Cambridge University Press, Cambridge) (1993)

    work page 1993

  14. [14]

    Bohmian Mechanics,

    R. Tumulka, “Bohmian Mechanics,” in *The Routledge Companion to Philosophy of Physics*, edited by E. Knox and A. Wilson (Routledge, New York, 2021), pp. 211–232

    work page 2021

  15. [15]

    Daumer and S

    M. Daumer and S. Goldstein, Observables, Measurements and Phase Operators from a Bohmian Perspective, NASA Conf. Publ. 3219, 231 (1993)

    work page 1993

  16. [16]

    D. Dürr, S. Goldstein, and N. Zanghì , Quantum equilibrium and the origin of absolute uncertainty, J. Stat. Phys. 67, 843 (1992)

    work page 1992

  17. [17]

    D. Dürr, S. Goldstein, and N. Zanghì , Quantum equilibrium and the role of operators as observables in quantum theory, J. Stat. Phys. 116, 959 (2004)

    work page 2004

  18. [18]

    Goldstein and W

    S. Goldstein and W. Struyve, On the Uniqueness of Quantum Equilibrium in Bohmian Mechanics, J. Stat. Phys. 128, 1197 (2007)

    work page 2007

  19. [19]

    J. S. Bell, de Broglie--Bohm, delayed-choice double- slit experiment, and density matrix, Int. J. Quantum Chem. 14, 155 (1980)

    work page 1980

  20. [20]

    J. S. Bell, On the Einstein-Podolsky-Rosen paradox, Physics 1, 195 (1964)

    work page 1964

  21. [21]

    J.-P. Dou, F. Lu, H. Tang, and X. -M. Jin, Test of Nonlocal Energy Alteration between Two Quantum Memories, Phys. Rev. Lett. 134, 093601 (2025)

    work page 2025

  22. [22]

    Heisenberg, Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Z

    W. Heisenberg, Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Z. Phys. 43, 172 (1927)

    work page 1927

  23. [23]

    Ye, Actual and weak actual values in Bohmian mechanics, arXiv:2512.12951 (2025)

    W. Ye, Actual and weak actual values in Bohmian mechanics, arXiv:2512.12951 (2025)

    work page arXiv 2025

  24. [24]

    P. R. Holland, The Quantum Theory of Motion (Cambridge University Press, Cambridge, 1993)

    work page 1993

  25. [25]

    Aharonov, D

    Y. Aharonov, D. Z. Albert, and L. Vaidman, How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100, Phys. Rev. Lett. 60, 1351 (1988)

    work page 1988

  26. [26]

    D. Dürr, S. Goldstein, and N. Zanghì , On the Weak Measurement of Velocity in Bohmian Mechanics, J. Stat. Phys. 134, 1023 (2009)

    work page 2009

  27. [27]

    H. M. Wiseman, Grounding Bohmian mechanics in weak values and Bayesianism, New J. Phys. 9, 165 (2007)

    work page 2007

  28. [28]

    Norsen, The Pilot-Wave Perspective on Spin, Am

    T. Norsen, The Pilot-Wave Perspective on Spin, Am. J. Phys. 82, 337 (2014)

    work page 2014

  29. [29]

    Norsen and W

    T. Norsen and W. Struyve, Weak measurement and Bohmian conditional wave functions, Ann. Phys. (N.Y.) 350, 166–178 (2014)

    work page 2014

  30. [30]

    A. N. Korotkov and A. N. Jordan, Undoing a weak quantum measurement of a solid-state qubit, Phys. Rev. Lett. 97, 166805 (2006)

    work page 2006

  31. [31]

    U. F. L. Traversa, G. Albareda, M. Di Ventra, and X. Oriols, Robust weak -measurement protocol for Bohmian velocities, Phys. Rev. A 87, 052124 (2013)

    work page 2013

  32. [32]

    Kocsis et al., Observing the Average Trajectories of Single Photons in a Two -Slit Interferometer, Science 332, 1170 (2011)

    S. Kocsis et al., Observing the Average Trajectories of Single Photons in a Two -Slit Interferometer, Science 332, 1170 (2011)

    work page 2011

  33. [33]

    Y.-F. Wang, H. Wang, R. Zhou, Z. -H. Liu, Y. Zou, W. Gao, and J. Li, Directly observing relativistic Bohmian mechanics, arXiv:2509.11609 (2025)

    work page arXiv 2025

  34. [34]

    J. Foo, C. D. M. Wilson, A. P. Lund, and T. C. Ralph, Relativistic Bohmian trajectories of photons via weak measurements, Nat. Commun. 13, 4002 (2022)

    work page 2022

  35. [35]

    J. Foo, A. P. Lund, and T. C. Ralph, Measurement- based Lorentz -covariant Bohmian trajectories of interacting photons, Phys. Rev. A 109, 022229 (2024)

    work page 2024

  36. [36]

    Foo and T

    J. Foo and T. C. Ralph, General-relativistic particle trajectories via quantum mechanical weak values and the Schwarzschild -Alcubierre spacetime, Phys. Rev. A 111, 052201 (2025)

    work page 2025

  37. [37]

    Tumulka, Foundations of Quantum Mechanics (Springer, Cham, 2022)

    R. Tumulka, Foundations of Quantum Mechanics (Springer, Cham, 2022)

    work page 2022

  38. [38]

    J. S. Bell, On the Einstein -Podolsky-Rosen paradox, Physics 1, 195 (1964)

    work page 1964

  39. [39]

    Kato, Fundamental properties of Hamiltonian operators of Schrö dinger type , Trans

    T. Kato, Fundamental properties of Hamiltonian operators of Schrö dinger type , Trans. Amer. Math. Soc. 70, 195 (1951)

    work page 1951

  40. [40]

    Agmon, Lectures on Exponential Decay of Solutions of Second -Order Elliptic Equations: Bounds on Eigenfunctions of N -body Schrö dinger Operators (Princeton University Press, 1982)

    S. Agmon, Lectures on Exponential Decay of Solutions of Second -Order Elliptic Equations: Bounds on Eigenfunctions of N -body Schrö dinger Operators (Princeton University Press, 1982)

    work page 1982