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A local observable in Bohmian mechanics equals the real part of a weak value, resolving a waveguide 'speed' paradox.

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2026-08-03 16:32 UTC pith:MLJ742KW

load-bearing objection Sound consolidation of Holland's local expectation values with a genuinely new but modest evolution equation; the waveguide 'resolution' is correct but already known, and one remark contains an algebraic contradiction that needs fixing. the 3 major comments →

arxiv 2512.12951 v4 pith:MLJ742KW submitted 2025-12-15 quant-ph

Actual and weak actual values in Bohmian mechanics

Weixiang Ye This is my paper
classification quant-ph
keywords Bohmian mechanicsweak valuesweak actual valuesposition postselectionevanescent wavemomentum decay ratewaveguide experimentquantum trajectories
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper formalizes a quantity it calls the weak actual value: a real-valued local expectation of an observable computed from the wave function at a particle's actual position. It proves that this quantity equals the real part of the standard weak value when the final postselection is the particle's position. Applying the identity to momentum shows that the Bohmian velocity is the real part of the momentum weak value, while the imaginary part carries amplitude-gradient information. A recent waveguide experiment that reported a 'speed' in a classically forbidden region is then shown to be measuring the imaginary part—the exponential decay rate of the evanescent field—not a particle velocity. The apparent contradiction with Bohmian mechanics dissolves once the real and imaginary parts of the weak value are separated.

Core claim

On its own terms, the paper establishes an exact correspondence: for any observable Â, the weak actual value a_w(t) = Re[ψ*(Âψ)]/|ψ|² along the Bohmian trajectory equals Re[A_w(Q(t))], the real part of the weak value with position postselection Q(t). For a single particle with wave function ψ = R e^{iS/ℏ}, the momentum weak actual value reduces to ∇S = m v, the standard Bohmian velocity. For a purely evanescent mode ψ ∝ e^{-κx}, the momentum weak value is purely imaginary, p_w = iℏκ, so the real part vanishes while |Im[p_w]|/m = ℏκ/m is the spatial decay rate. The paper applies this to the coupled-waveguide experiment: the measured 'speed' parameter ν equals |Im[p_w]|/m, not the Bohmian velo

What carries the argument

The central object is the weak actual value, defined as Re[ψ*(Âψ)]/|ψ|² evaluated at the actual configuration Q(t). Its key identity, a_w(t) = Re[A_w(Q(t))], connects it to weak measurement theory under position postselection. The argument is carried by the polar decomposition of the wave function, which splits the momentum weak value into a phase-gradient part (real, equal to m v) and an amplitude-gradient part (imaginary, proportional to ∇ ln R). The paper also derives an exact evolution equation for a_w(t), decomposing its change into quantum dynamical, convective, and probability-flow terms.

Load-bearing premise

The key assumption is that the evanescent wave in the main waveguide is a stationary, purely real exponential with no phase gradient, so that the measured decay parameter equals the imaginary part of the momentum weak value and the real part—the Bohmian velocity—is exactly zero.

What would settle it

Measure the phase of the field inside the forbidden region of the waveguide (for example, by interferometrically comparing two points along the waveguide). A nonzero phase gradient would give a nonzero real part of the momentum weak value, implying a nonzero Bohmian velocity and overturning the paper's identification of the measured 'speed' with the imaginary part alone. Conversely, if ν is found to deviate from ℏκ/m when the field is confirmed to be a real exponential, the correspondence would be refuted.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Weak-measurement reconstructions of Bohmian trajectories are directly probing the weak actual value, i.e., the real part of the position-postselected weak value.
  • The imaginary part of a weak value is not a velocity but a local amplitude-decay rate; experiments that infer 'speeds' from decay lengths are measuring imaginary parts.
  • The waveguide experiment's finite ν in the forbidden region is ℏκ/m, the evanescent decay rate, leaving the Bohmian prediction of zero particle velocity intact.
  • For spin superpositions, weak actual values can fall outside the eigenvalue range, recovering the anomalous weak-value effect within a Bohmian description.
  • The framework assigns observables contextual, configuration-dependent values, remaining compatible with no-go theorems against non-contextual hidden variables.

Where Pith is reading between the lines

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

  • A testable extension: in quantum tunneling setups where the barrier wave function is not purely real, the imaginary-part/real-part split predicts a measurable phase-gradient contribution to the weak actual value; an interferometric phase measurement in the barrier would check this.
  • The identity suggests a general dictionary—real part of weak value ↔ phase-structure observable, imaginary part ↔ amplitude-structure observable—that could be used to reinterpret other weak-value experiments, especially those involving evanescent or decaying fields.
  • If free evolution between the weak interaction and postselection is not negligible, the exact correspondence becomes approximate; quantifying this correction may matter for finite-time weak measurements in dispersive media.
  • The resolution depends on the main-waveguide field having no phase gradient; a near-step interference region would develop a nonzero real part, so the paper's conclusion applies to the asymptotic evanescent regime rather than to the immediate vicinity of the step.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper develops the notion of 'weak actual values' in Bohmian mechanics, defined as Holland's local expectation values Re[ψ*(Âψ)]/|ψ|² evaluated at the particle position. It derives an evolution equation for these quantities along Bohmian trajectories, proves that their ensemble average equals the quantum expectation value, and establishes a formal relation to the real part of the weak value under position postselection (Eq. (15)). The framework is then applied to a recent waveguide experiment, arguing that the measured 'speed' in the forbidden region is not the Bohmian particle velocity but the imaginary part of the momentum weak value, equal to the wavefunction's amplitude decay rate. The basic algebraic derivations in Appendices B.1–B.3 are straightforward and mostly correct, but the central waveguide resolution depends on an unvalidated mode ansatz, and there is an internal inconsistency in the spin section.

Significance. If the waveguide analysis is correct, the paper offers a potentially important resolution of a recent experimental challenge to Bohmian mechanics, clarifying the distinct physical roles of the real and imaginary parts of weak values. It also provides a systematic treatment of Holland's local expectation values and their connection to weak measurement theory, which could be pedagogically and conceptually useful. However, the paper's central application rests on a single-mode ansatz that is not derived from the coupled-waveguide system, and the spin-section formula (20) contradicts the paper's own scalar result (18). These issues currently prevent the manuscript from fully supporting its main claims, though they are potentially fixable.

major comments (3)
  1. [§4.5, Eq. (24)] The resolution of the waveguide experiment relies crucially on the assumption that the main-waveguide field in the forbidden region is a purely real evanescent mode ψ_m(x) ∝ e^{−κx} with no phase gradient, leading to Re[p_w]=0 and the identification ν = |Im p_w|/m = ℏκ/m. This ansatz is not derived from the coupled-mode equations, and the expression for Δ in Eq. (23) explicitly contains the inter-waveguide coupling ℏJ₀, which can in principle generate a spatially varying phase ∇S(x). If such a phase gradient is present, then Re[p_w] ≠ 0, and the measured parameter ν would include a contribution from the Bohmian velocity, undermining the claimed resolution. The paper should either derive Eq. (24) from the full coupled system, provide numerical evidence that the phase gradient is negligible in the relevant regime, or quantify a bound on ∇S that makes the identification exact enough.
  2. [§4.2, Remark 4.1 / Eq. (20)] The formula p_w = m v − (ℏ/2)∇lnρ for a Pauli spinor is inconsistent with the definition (6) and with the scalar limit Eq. (18). A direct calculation from the definition using a polar decomposition ψ = R e^{iS/ℏ} χ (with χ†χ=1) gives p_w = ∇S + ℏ Im[χ†∇χ] = m v, where v = j/ρ is the Pauli-current velocity; no osmotic term appears. In the scalar limit where χ is constant, Eq. (20) would reduce to m v − (ℏ/2)∇lnρ, which disagrees with Eq. (18) unless ∇R=0. This internal contradiction must be corrected; the osmotic term is not part of the momentum weak actual value as defined in this paper.
  3. [§3.3, Eq. (13)] The 'correspondence' with weak measurement theory is presented as a formal result, but the derivation of Eq. (15) relies on Eq. (13), which assumes that free evolution between the weak interaction at time t and the postselection at t⁺ is negligible. Mathematically, Eq. (15) is a definitional identity: a_w = Re[(Aψ)(Q)/ψ(Q)] follows immediately from definitions (6) and (14), independent of any weak-measurement protocol. The paper should state clearly that the exact relation is an algebraic identity, while the physical interpretation as a weak value inherits the approximation in Eq. (13). This distinction is important for assessing the strength of the claimed 'generalization' of earlier insights by Leavens and Matzkin, and for the wave-guide application where the finite-time interval is not explicitly controlled.
minor comments (4)
  1. [§4.1 / Appendix B.2] Equation (18) is derived correctly for a scalar wave function, but the paper should note that the same result can be read directly from the definition without invoking the polar form; the polar form is already used in the main text and appendix, but the notation R and S is not defined at first use in §4.1.
  2. [§1 / References] The introduction cites references [17–20] as 'experimental challenge' but [18], [19], [20] are comments/replies, not original experimental papers. Please clarify which references report the experiment itself.
  3. [§3.2, Eq. (10)] The display of Eq. (10) is typeset with the divergence term ∑_k ∇_k·v_k appearing as '∑ ∇_k k ⋅ v_k', which is garbled. Also, the proof in Appendix B.1 is for a single particle; the multi-particle generalization is asserted without derivation. A brief sketch of the multi-particle case would improve readability.
  4. [§4.5] The notation in Eq. (23) uses Δ = E − V0 + ℏJ0 but the sign of the coupling term and the definition of V0 should be matched with the original experiment [17]. A reader comparing with the experimental paper may find the sign conventions unclear.

Circularity Check

0 steps flagged

No significant circularity; the core identities are transparent algebraic equivalences, and the waveguide resolution rests on an explicit evanescent-mode ansatz that is a modeling assumption rather than a circular step.

full rationale

The derivation chain is self-contained. Equation (6) defines the weak actual value as Re[ψ*Âψ]/|ψ|^2; Equation (14) evaluates the position-postselected weak value as (Âψ)/ψ; Equation (15) then follows by immediate algebra. The paper states this explicitly ('Comparing this with the definition of the weak actual value (6), we immediately obtain the fundamental relation'), so the 'formal correspondence' is an exposed identity rather than a hidden circular premise. Property 1 and Theorem 2 are proved in the text and Appendix B by direct calculation from the Schrödinger equation, the continuity equation, and the definition of the Bohmian velocity. The waveguide application is also not circular in the prohibited sense: the claimed result ν = |Im[p_w]|/m = ℏκ/m follows by substituting the explicit exponential ansatz ψ_m ∝ e^{-κx} into the momentum weak value (Eqs. 24-26). That ansatz is an assumption about the mode structure, not a fitted parameter; its validity is a genuine correctness risk—coupling ℏJ0 or a position-dependent phase would make Re[p_w] ≠ 0—but this is not a reduction of the conclusion to its own input. The only self-citation, Ref. [32], appears in a 'for further theoretical analyses' list and carries no load-bearing argument. The paper's own limitation statements concern rigor at nodes, functional-analytic foundations, and QFT generalization, not the circularity of the derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

No free parameters: the paper fits nothing; the experimental ν is imported from outside as data. No invented entities: the 'weak actual value' is a relabeling of Holland's local expectation value, explicitly stated to be completely determined by ψ and Q. The load-bearing assumptions are the standard Bohmian postulates plus two paper-specific idealizations: position-eigenstate postselection with negligible intermediate evolution, and the continuum evanescent-mode model of the waveguide.

axioms (5)
  • domain assumption Quantum equilibrium hypothesis: ρ₀(Q₀) = |ψ₀(Q₀)|² (Eq. 4)
    Needed for Property 1 (ensemble average equals quantum expectation) and for the claim that nodes (|ψ|²=0) are measure-zero along trajectories. Standard Bohmian postulate.
  • domain assumption Regularity: ψ and Âψ are C² in q and C¹ in t, with ρ≠0 along the trajectory (Theorem 2 assumptions)
    Stated in Theorem 2; the evolution equation (10) holds pointwise only under these conditions and is distributional otherwise.
  • domain assumption Position eigenstate |Q(t)⟩ is an admissible postselection, with negligible free evolution between weak interaction at t and postselection at t⁺ (Eq. 13 → Eq. 14)
    Bridges Eq. (12) to Eq. (14); without it Eq. (15) is only approximate. Standard idealization in weak-measurement theory, including the use of unnormalizable position eigenstates.
  • domain assumption Evanescent main-waveguide mode is ψ_m(x) ∝ e^{−κx} with κ = √(2m|Δ|)/ℏ (Eq. 24)
    The §4.5 resolution and the identification ν = |Im p_w|/m reduce to this ansatz; a phase gradient in the actual coupled-waveguide mode would break the claimed clean separation.
  • standard math Schrödinger evolution, self-adjointness of Â, and the standard weak-value formula (Eq. 12)
    Background structure for Theorem 2 and Eq. (15); treated as unproved input from quantum mechanics.

pith-pipeline@v1.3.0-alltime-deepseek · 12063 in / 25599 out tokens · 212176 ms · 2026-08-03T16:32:26.018153+00:00 · methodology

0 comments
read the original abstract

We systematically analyze Holland's local expectation values within Bohmian mechanics, referring to them as weak actual values to emphasize their connection with weak measurement theory. We derive the exact time evolution equation for these quantities along a Bohmian trajectory and formally establish their correspondence with the real part of the weak value under position postselection. The explanatory power of this framework is demonstrated by revisiting a recent waveguide experiment: we show that the measured quantity corresponds to the imaginary part of the momentum weak value, which reflects the spatial decay of the wave field, not the particle velocity. This cleanly clarifies the distinct physical roles of the real and imaginary parts of weak values.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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Reference graph

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