REVIEW 3 major objections 4 minor 9 references
Locality, Micro- vs. Macro-, Particle Interpretations and All That: A Lagrangian Approach to the Measurement Problem
T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash
Pith's one-line read No linear auxiliary field, integrated out of a Lagrangian, can produce the nonlinear 'wavefunction energy' term of the 2017 measurement-problem proposal.
desk verdict A narrow but sound no-go for linear auxiliary-field Lagrangians, overclaimed in the abstract. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the pair (h, Ω^tΩ): a real auxiliary 'macro-field' φ coupled to the center-of-mass density h through ∫hφ, with kinetic/self term (1/2)∫(Ωφ)^2. Varying φ turns the effective Lagrangian into (1/2)∫ h (Ω^tΩ)^{-1} h, so all possible effective interactions are exactly the Green's functions of operators of the form Ω^tΩ. The argument then asks whether the squared-distance kernel ||x-y||^2 can be such a Green's function; the answer is no, because the identity (Ω^tΩ)_x ||x-y||^2 = δ(x-y) would imply that any test function f is recovered from a finite number of its moments. A secondary mechanism is the 'macro-field' concept itself: a field depending only on the center of ma
What would settle it
Exhibit a linear operator Ω, with an explicitly defined domain containing smooth compactly supported functions, that satisfies (Ω^t Ω)_x ||x-y||^2 = δ(x-y); equivalently in one dimension, an operator satisfying Ω^t Ω φ = φ''' on C_c^∞. Alternatively, construct any Lagrangian outside the normal form of Eq. (49) whose effective action reproduces the 2017 kernel (x-y)^2 and check it numerically on a two-Gaussian 'cat' state.
Extended reading notes
Core claim
On the paper's own terms, the central discovery is a no-go result. Assume a Lagrangian of the normal form L = ∫dt∫dx h(x)φ(x) - (1/2)∫dt∫dx (Ωφ(x))^2, where h is the marginal probability density of the center-of-mass coordinate and Ω is any linear operator. Varying φ and substituting back gives L = (1/2)∫dt∫dx h ((Ω^t Ω)^{-1} h)(x). Reproducing the 2017 WaveFunction Energy would require (Ω^tΩ)_x ||x-y||^2 = δ(x-y), i.e., ||x-y||^2 as the Green's function of a linear operator. The paper proves this is impossible: integrating the identity against an arbitrary f(y) yields f(x) expressed through only three moments of f, and no ordinary function space has every function determined by finitely man
Load-bearing premise
The whole conclusion rests on assuming the auxiliary field is coupled linearly and its self-interaction is quadratic in a linear operator; if that setup is broadened to nonlinear couplings, multiple fields, or another variational principle, the same effective term might reappear.
Editorial extensions
If this is right
- If the no-go is right, the 2017 nonlinear term cannot be obtained by integrating out a single auxiliary field coupled linearly to the center-of-mass density.
- The 2017 term therefore cannot be reinterpreted as the residual effect of local particle-like interactions among additional fields; nonlocality is intrinsic to the proposal.
- The one-dimensional macro-field model gives a concrete illustration: the best it can do is an effective kernel |x-y|, which has the wrong scaling and no natural reading as center-of-mass dispersion.
- The paper concludes that a micro/macro field splitting cannot reproduce its theory, and that the proper moral is the absence of a particle picture.
Reading between the lines
- The moment-based proof extends beyond the squared-distance kernel: any effective kernel that is a polynomial in y would force test functions to be determined by finitely many moments, so the same no-go should apply to all polynomial interaction kernels, not just ||x-y||^2.
- The no-go leaves open nonlinear auxiliary-field actions or multi-field constructions; a natural next test is whether a nonlinear functional of the auxiliary field can generate the WFE kernel, which would weaken the nonlocality conclusion.
- The same finite-moment test could serve as a quick diagnostic for any proposed nonlinear Schrödinger equation: write the candidate's effective kernel and check whether it overdetermines smooth compactly supported functions.
- If the 2017 proposal is accepted despite the absence of a Lagrangian derivation, the measurement problem is being resolved by embracing a fundamental nonlocality in the wavefunction equation rather than by hiding it in additional fields.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper asks whether the nonlinear 'WaveFunction Energy' (WFE) modification of the Schrödinger equation proposed by the author in 2017 can be obtained from a Lagrangian field theory by integrating out auxiliary 'macro-fields' that depend only on the center of mass. After deriving the ordinary Schrödinger equation from a variational principle, the author introduces real auxiliary fields φ± and shows in a one-dimensional model that integrating them out produces a kernel |x-y| rather than the desired |x-y|^2. The paper then attempts to obtain |x-y|^2 via a third-derivative operator, proves a one-dimensional obstruction for operators whose domain contains C∞±∞, and states a higher-dimensional 'No-Go theorem' based on a moment argument. The stated conclusion is that the 2017 theory cannot be derived from a splitting into micro- and macro-fields and cannot be given a particle interpretation.
Significance. If the broad no-go claim were established, it would be a structurally interesting result: it would tie the inability to derive a nonlinear Schrödinger modification from an auxiliary-field Lagrangian to nonlocality and to the absence of a particle interpretation. The paper has genuine strengths: the one-dimensional identity ∫ φ φ''' = 0 is a crisp, self-contained obstruction; the moment argument in Section 6 is elegant; no fitted parameters enter the no-go; and the relevant functional from the 2017 paper is exhibited explicitly. However, the significance is conditional: the theorem as stated covers only a restricted class of quadratic, single-field auxiliary actions, and the function-space hypotheses are left unspecified. The broad conclusion in the abstract and Section 6 overstates what is actually proven.
major comments (3)
- [Section 6, Eq. (55) and Abstract] The theorem proves at most that no linear operator Ω in the quadratic normal form L = ∫ h φ − (1/2)∫(Ωφ)^2 can have Green's function ||x−y||^2. It does not rule out nonlinear auxiliary-field actions, e.g. general F(φ,φ',φ'') mentioned in Section 7, or multi-field actions with non-quadratic interactions. Since the abstract and Section 6 claim 'no theory of interacting fields can generate the nonlinear terms in the 2017 paper by integrating out some of them,' the scope of the theorem is narrower than the claimed conclusion. Either narrow the claim to 'no quadratic single-field auxiliary theory with a linear kinetic operator' or provide a genuine argument covering general interacting auxiliary fields.
- [Section 6, proof of the 'Theorem'] The proof of (55) multiplies by an arbitrary f(y), integrates over y, and then interchanges the operator Ω^tΩ with the y-integration, applying it to the polynomials ||x||^2, x, and 1. This presupposes that these polynomials lie in the domain of Ω^tΩ and that the interchange is justified. The author explicitly writes 'theorem' in quotes and says 'I am not going to bother about specifying function spaces.' For a no-go result, the domain issue is load-bearing: fractional and integro-differential operators may legitimately exclude polynomials or require distributional definitions. The result remains conditional until a precise function-space setting (or a distributional formulation with an explicit test-function class) is supplied.
- [Section 7] The treatment of nonlinear auxiliary-field actions is a single failed attempt: varying F(φ,φ',φ'') makes the φ''' term drop out. This does not establish a general impossibility. A nonlinear field theory can generate effective actions that are not simply quadratic in h, so the absence of a proof for the general case is a load-bearing gap. The paper should either prove a general statement for nonlinear actions or explicitly limit the no-go to the Gaussian/quadratic case.
minor comments (4)
- [Section 4, Eq. (32)] Equation (32) writes 'ϕ−(x)+ϕ −(x)' but should be 'ϕ_-(x)+ϕ_+(x)'.
- [Introduction] Typographical errors: 'demonstate', 'intrepreted', 'modulous' (for modulus), and 'c-valued' should be clarified as 'complex-valued'.
- [References] Reference [8] (Bell 1964) appears in the reference list but is not cited in the text; either cite it where relevant or remove it.
- [Title and formatting] The title 'Micro-vs.Macro-' has inconsistent spacing; the paper would benefit from a formatting pass.
Circularity Check
No significant circularity; the no-go derivation is self-contained and does not reduce to its inputs.
full rationale
The paper's central no-go argument (Sec. 6, Eqs. 49–55) is a conditional mathematical proof: if a Lagrangian of the restricted form L = ∫ hφ − (1/2)∫(Ωφ)^2 reproduced the WFE kernel, then (Ω^tΩ)_x ||x−y||^2 = δ(x−y), which is shown to be impossible for any linear operator on a reasonable function space by a moment argument. This proof does not rely on the truth of the 2017 paper; the self-citations [1] and [3] supply only the target functional and the Barut analogy, not the proof. The no-go theorem is explicitly restricted to linear-operator Lagrangians; Sec. 7 reports only a failed attempt with nonlinear F(φ,φ′,φ″), and the function-space assumptions are left unspecified. These are scope and rigor limitations, not circularity. No fitted parameter is renamed as a prediction, and no conclusion is assumed in its own premises.
Assumptions & free parameters
free parameters (1)
- w =
unknown
assumptions (4)
- standard math Euler-Lagrange variational principle for the Schrödinger Lagrangian L in Eq. (16) yields the Schrödinger equation.
- ad hoc to paper Integrating out an auxiliary field φ can be represented by the normal form L = ∫ h φ - (1/2)∫(Ωφ)^2 with a linear operator Ω.
- domain assumption The relevant function space contains smooth functions vanishing at infinity and is not determined by finitely many moments.
- domain assumption The 2017 WFE target corresponds to kernel K(x-y)=|x-y|^2, so the effective inverse operator must have this Green's function.
invented entities (1)
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Macro-field φ(X) (with components φ_- and φ_+)
Cite this review
Pith. "Pith review of Locality, Micro- vs. Macro-, Particle Interpretations and All That: A Lagrangian Approach to the Measurement Problem." pith.science (2026). https://pith.science/paper/RL5WVRR3
@misc{pith2026250907206,
author = {Pith},
title = {Pith review of: Locality, Micro- vs. Macro-, Particle Interpretations and All That: A Lagrangian Approach to the Measurement Problem},
year = {2026},
howpublished = {\url{https://pith.science/paper/RL5WVRR3}},
note = {Machine review of arXiv:2509.07206}
}
read the original abstract
In 2017, this author proposed, as a resolution of the Measurement Problem, that terms be added to Schrodinger's wavefunction equation, rendering it nonlinear. Said equation derived from a trick employed by S. Weinberg in 1989 which may be unfamiliar to most physicists, as well as uninterpretable in terms of local ("particle") interactions. Motivated by A. O. Barut's work on electrodynamics, here I analyze which kinds of nonlinear theories can be derived from Lagrangian field-theory by integrating out some fields. The issues of "What is a local interaction?" and "Might there be Micro- and Macro-fields?" arise. In the end, I will argue that my 2017 theory cannot be given a "particle" interpretation, nor be derived from a splitting into the two categories of fields.
Reference graph
Works this paper leans on
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[1]
On Non-Linear Quantum Mechanics and the Measurement Problem I: Blocking Cats
Wick, W. D. “On Non-Linear Quantum Mechanics and the Measurement Problem I: Blocking Cats”, ArXiv 1710.03278 (October 2017)
arXiv 2017
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[2]
Weinberg, S. “Testing Quantum Mechanics”. Annals of Physics vol. 194, 336-86 (1989)
work page 1989
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[3]
Wick, W. D. “Nonlinear Nonlocal: Comparing A. O. Barut’s Theory to Mine, with special emphasis on That Dot on the Screen” ArXiv, 19 May 2025, 2505.13704
work page Pith review arXiv 2025
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[4]
Weinberg, S.The Quantum Theory of Fields. V. I, p. 28-29. Cambridge U Press, 1995
work page 1995
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[5]
Streater, R. S. and Wightman, A. S.PCT, Spin and Statistics And All That. Benjamin, New York. (1964)
work page 1964
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[6]
Nonlinear Nonlocal Classical Field Theory of Quantum Phenomena
Barut, A. O. “Nonlinear Nonlocal Classical Field Theory of Quantum Phenomena”. Int J Eng Sci, 30(10) 1469-1473 (1992)
work page 1992
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[7]
On the problem of hidden variables in quantum theory
Bell, J. S. “On the problem of hidden variables in quantum theory”. Rev. Mod. Phys. 38, 1966, reprinted inSpeakable and Unspeakable in Quantum Mechanics, Princeton, 1987
work page 1966
-
[8]
On the Einstein-Podolsky-Rosen paradox
Bell, J. S. “On the Einstein-Podolsky-Rosen paradox”. Physics 1 (1964), p. 195-200, reproduced in in Bell’s collected papers on quantum philoso- phy,Speakable and Unspeakable in Quantum Mechanics, Princeton, 1987
work page 1964
Show all 9 references
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[9]
Against ‘measurement’
Bell, J. S. “Against ‘measurement’ ”, Proceedings of 62 years of Uncer- tainty, a conference held in Erice, Italy, 5-14 August 1989, published by Microfields and Macrofields19 Plenum Publishing, New York; reprinted in Physics World, August 1990, p. 33-40
1989
Reviewed August 4, 2026 · model on record in the stance chip above.
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