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On the Impossibility of Obtaining Time-Independent, Three-Dimensional, Spherically-Symmetric Densities of Confined Systems of Relativistically Moving Constituents

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Pith's one-line read No legitimate three-dimensional, time-independent, spherically-symmetric density exists for confined systems of relativistically moving constituents; only two-dimensional light-front transverse densities satisfy quantum mechanics, the…

desk verdict Solid constructive 2D light-front densities, but the universal no-go claim for 3D densities is a definitional choice, not a theorem. read the letter →

arxiv 2507.14388 v1 pith:4JI3RL5O submitted 2025-07-18 hep-ph nucl-exnucl-th

Gerald A. Miller This is my paper
classification hep-phnucl-exnucl-th
keywords light-frontformalisminfinitemomentumframespatialdensitiesformfactorsgravitationaluncertaintyprincipleBreitaxial-vectordensity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that for systems whose constituents move relativistically, there is no legitimate way to define a time-independent, three-dimensional, spherically-symmetric density from form-factor data. The only densities that satisfy the quantum-mechanical rule that probability is the square modulus of a wave function, the uncertainty principle, and Poincaré invariance are two-dimensional transverse densities obtained in the light-front, infinite-momentum frame. The paper demonstrates that the Breit-frame interpretation, Abel transforms, Wigner distributions, and spherically-symmetric wave packets of vanishing spatial extent each violate at least one of these restrictions. If correct, this invalidates standard extractions of three-dimensional proton charge, mass, and pressure distributions.

What carries the argument

The central device is a light-front wave packet with fixed plus-momentum $P^+$ and Gaussian transverse width $\sigma$, combined with the Drell-Yan condition $\Delta^+ = 0$. Integrating the expectation value of a local operator over $x^-$ and then taking $P^+ \to \infty$ before $\sigma \to 0$ removes the time ($x^+$) dependence and leaves the two-dimensional Fourier transform $\rho_O(\mathbf{x}_\perp) = \int \frac{d^2\Delta_\perp}{(2\pi)^2} F_O(\Delta_\perp^2) e^{-i\Delta_\perp \cdot \mathbf{x}_\perp}$. The order of limits is the load-bearing step: it lets the wave packet be localized while keeping the boost kinematic, producing wave-packet-independent densities only for operators whose matrix elements contain no transverse momentum $\mathbf{P}_\perp$.

What would settle it

Construct an explicitly relativistic two-body bound state (for example, via the Bethe-Salpeter equation) with known form factors, form a spherically-symmetric wave packet of finite width, and compute the three-dimensional expectation-value density $\rho(t, r)$. If any such computation yields a stable, time-independent, spherically-symmetric $\rho(r)$ whose three-dimensional Fourier transform reproduces the form factor for all momentum transfers without taking $P^+ \to \infty$, the no-go claim would be falsified; a simpler check is to find a specific form factor and wave packet where $\rho(t, r)$ does not vanish for $t > r$.

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Extended reading notes

Core claim

An author claims that a legitimate spatial density must be the expectation value of a local operator in a wave packet that can be localized arbitrarily well, must be independent of the wave packet, and must be time-independent. Under these criteria, the only viable densities are two-dimensional: with $\Delta^+ = 0$ and $P^+ \to \infty$, the $x^+$-dependence of the $x^-$-integrated density disappears, leaving $\rho_O(\mathbf{x}_\perp)$ as the two-dimensional Fourier transform of the corresponding form factor $F_O(\Delta_\perp^2)$. This yields new transverse densities for the axial-vector form factor $G_A$, the induced pseudoscalar term $G_P$, and all three gravitational form factors $A(t)$, $J(t)$, and $D(t)$, plus a mass density from the trace of the energy-momentum tensor. The author further shows that each existing route to a three-dimensional density fails: the Breit frame uses different internal wave functions for initial and final states and has no meaningful position operator; Wigner distributions have infinite position and momentum fluctuations; Abel transforms presuppose spherical symmetry that Lorentz contraction forbids; and spherically-symmetric wave packets of vanishing spatial extent produce densities that vanish for $t > r$.

Load-bearing premise

The paper assumes that a legitimate density must be time-independent and identical for every sufficiently localized wave packet; this requirement is presented as a postulate rather than derived from quantum mechanics.

Editorial extensions

If this is right

  • The transverse charge, axial, mass, and pressure densities extracted in the light-front frame are the only densities consistent with the stated postulates; quoted three-dimensional proton radii should be re-expressed as two-dimensional transverse radii.
  • The new axial-vector density $\rho_A(\mathbf{x}_\perp)$, the two-dimensional Fourier transform of $G_A$, provides a concrete prediction for antineutrino-scattering and lattice determinations of $G_A$.
  • The mass density $m(\mathbf{x}_\perp)$ from the trace of the energy-momentum tensor integrates to $2M^2$ and can be compared with lattice QCD n-pole parametrizations (dipole, tripole, quadrupole) of the gravitational form factors.
  • The induced pseudoscalar density $\rho_P(\mathbf{x}_\perp)$ is strongly anisotropic, reflecting the pion-pole term in $G_P$, and offers a testable signature that distinguishes pseudoscalar from axial contributions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the no-go result is accepted, widely used three-dimensional extractions such as Abel tomography of proton mechanical properties should be reframed as purely two-dimensional statements; comparing three-dimensional and two-dimensional radii via geometric factors such as $\sqrt{2/3}$ becomes meaningless because the three-dimensional densities do not exist.
  • The definitional core could be relaxed: one could instead treat time-dependent densities as physically meaningful (for example, describing the response to a localized probe), in which case the spherical wave-packet densities of Ref. [92] would carry real dynamical information rather than being dismissed as illegitimate.
  • The same criteria may apply to any relativistic confined system, including heavy quarkonia or high-momentum nuclei, so three-dimensional densities in those contexts should be scrutinized in the same way.
  • A direct experimental-consistency test: measure $G_A$ and the gravitational form factors at a future electron-ion collider and check whether the two-dimensional Fourier transforms are positive-definite and independent of the wave-packet width; violation would signal beyond-the-paper issues.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper argues that a time-independent, three-dimensional, spherically symmetric density cannot be defined for confined systems of relativistically moving constituents, and that light-front two-dimensional densities are the only ones compatible with standard quantum-mechanical probability, the uncertainty principle, and Poincaré invariance. Section II derives a general expression for x^- -integrated expectation values of local operators, carefully takes the limits P^+ -> infinity and sigma -> 0 in that order, and applies it to obtain two-dimensional densities for the axial-vector current, the gravitational form factors A, J, D, and the trace-combination mass density. The subsequent sections critique the Breit-frame interpretation, Wigner-distribution constructions, the Abel transformation, and spherical wave packets of vanishing spatial extent. The central claim is that all known 3D density extractions fail these basic requirements.

Significance. The Section II derivation is careful and the treatment of the order of limits is a genuine strength; the new two-dimensional axial and mass densities, together with the explicit lattice-input illustrations, are useful results if the framework is accepted. The critiques of the Breit frame and of the particular unnormalized Wigner construction are also valuable and should be preserved. However, the central no-go statement is not as general as the title and abstract assert: the definition of a 'legitimate' density is introduced by fiat in Section II, and the Section VI vanishing-density result is an artifact of taking the zero-width limit before time evolution. The paper therefore establishes a conditional impossibility under a specific, restrictive definition, rather than a universal no-go theorem.

major comments (3)
  1. [Sec. II, Eqs. (6)-(9)] The no-go conclusion depends on a definition that is not derived from the stated postulates. Equation (7) and the instruction to take sigma -> 0 only after all momentum integrals define 'legitimate' as a wave-packet expectation value that is completely localized, time-independent, and independent of the packet. This is a reasonable and useful criterion, but it is an additional axiom. Alternative proposals, such as finite-width wave-packet densities evaluated at a fixed equal-time surface or Wigner-type quasi-distributions, are not shown to be internally inconsistent; they are excluded by the chosen standard. Since the universal impossibility claim rests on this choice, the title and abstract should be weakened, or a separate argument should be added showing that any acceptable density must satisfy this definition.
  2. [Sec. VI, Eqs. (45)-(49)] The claimed pathology that spherical wave packets with vanishing spatial extent produce densities that vanish almost everywhere as time increases is generated by taking the point-like limit before evolving in time. For any initially localized quantum state, spreading makes rho(t,r) tend to zero for fixed r; the delta-function support at r=t in Eq. (49) is precisely the zero-width artifact of the probe state. This does not invalidate a density extracted at t=0 from a finite-width packet, nor does it demonstrate that no legitimate three-dimensional density can be defined. The conclusion of Section VI therefore overstates what the calculation shows.
  3. [Sec. IV, Eqs. (34)-(40)] The Wigner-distribution critique identifies an important property of the specific construction in Ref. [82]: the average position and momentum vanish by parity while the variances in both are infinite. This is a valid criticism of that object. However, it does not rule out all phase-space or Wigner-type definitions; the infinite variances follow from the unnormalized plane-wave-state construction used there. To support the 'all known methods' claim, the paper would need to treat more general constructions or prove a general lower bound on the variance product.
minor comments (4)
  1. [Sec. VII] The summary incorrectly assigns sections: the Abel transformation is discussed in Section V, not Section IV, and the Wigner distribution in Section IV, not Section V. These cross-references should be corrected.
  2. [Fig. 1 caption] The caption contains a typographical error: 'Solid- ~rho_A(b), Dashed-b~rho_A(b)' should read something like 'Solid: b rho_A(b); dashed: rho_A(b)' to match the plotted quantities.
  3. [Eqs. (12)-(13)] The notation for the mean-square transverse radius is inconsistent: Eq. (12) uses <x_perp^2>_O while the following sentence uses <x_perp^2 O>; a uniform notation would improve readability.
  4. [Fig. 4] The first panel of Fig. 4 appears to repeat the label '(a)' four times, and the axis label 'rho_G(0.3,t)' is repeated; the figure should be cleaned.

Circularity Check

2 steps flagged · score 6.0 of 10

The impossibility claim is largely self-definitional: 'legitimate' is defined (Sec. II) to require complete localization plus time-independence, and the Sec. VI 'densities vanish almost everywhere' is the causal spreading (light-cone delta of Eq. 49) of that point-localized input, not an independent disproof of 3D densities.

  1. self definitional [Sec. II, definition of a 'legitimate' density (passage after Eq. (7)); summarized in Sec. VII.]
    "A wave function that is completely localized at the origin is needed to obtain a density depending only on internal structure. Localization may be achieved by considering wave packets with arbitrarily narrow width in the position representation, though this width must be kept non-zero until after all other calculations have been performed [37]."

    The paper's headline claim—'not possible to legitimately obtain a density that depends on r and is independent of time'—is an unpacking of this definition, not a corollary of the three restrictions named in the abstract. 'Legitimate' is defined to demand both complete localization (sigma -> 0) and time-independence; but a completely localized state spreads causally, so the two requirements cannot coexist in 3D. The 2D light-front density qualifies only because the P+ -> infinity limit is taken before sigma -> 0, killing the x+ dependence in Eq. (9).

  2. self definitional [Sec. VI A, Eqs. (44)-(49) and Fig. 4.]
    "Next, I follow Ref. [92], and take their limit of the vanishing spatial extent of the wave packet. This leads to the result: rho(t,r) = integral d^3q/(2pi)^3 e^{-iq.r} integral_{-1}^{1} dalpha 1/2 F((alpha^2-1)q^2)e^{i alpha |q| t} ... lim_{t->infinity} rho(t,r) = 0 ... At large times, t, the density vanishes except at r = t ... rho(t,r) ~ 1/(8 pi r t) delta(r-t)."

    The claimed pathology—densities vanishing almost everywhere as time increases—is the generic causal evolution of the point-localized state built into the input. Any completely localized wave packet expands at light speed, and Eq. (49)'s delta(r-t) is the light-cone Green's function of a point source: the Fourier-space image of the zero-width assumption, not a property of form factors or of confined constituents. Presenting this as the abstract's headline 'Furthermore' and concluding 'there is no time-independent spherically-symmetric density' feeds the assumption's own consequence back as a disproof: legitimate => complete localization => causal spreading => no static 3D density. The vanishing is a zero-width artifact, so this negative result reduces by construction to the Sec.

full rationale

This paper's positive programme is not circular: the two-dimensional densities of Sec. II are self-contained derivations from the light-front wave-packet formalism, and the axial, stress, angular-momentum, and trace densities are Fourier transforms of input form factors (dipole fits from Ref. [68]; lattice gravitational form factors from Ref. [2])—applications, not fitted predictions. The self-citations (Refs. [36,37,38] for the D(t)/J(t) densities, the Breit-frame 'infinite term,' and the Abel-transform failure) are published, parameter-free derivations whose assumptions do not include the target no-go result, so under the present rules they are real evidence and do not by themselves raise the score. The circularity sits in the central negative claim. The abstract attributes the impossibility to 'the quantum mechanical definition of probability, the uncertainty principle and Poincaré invariance,' but the derivation actually uses a fourth, unstated premise: the Sec. II definition of a legitimate density as the expectation value in a completely localized wave packet, required to be both wave-packet-independent and time-independent. Once that standard is adopted, the 3D no-go is essentially the statement that a point-localized state spreads—the uncertainty principle restated—and the Sec. VI finding that rho(t,r) vanishes for t > r, with the light-cone delta of Eq. (49), is the causal evolution of the zero-width input, a point-source artifact presented as a new disproof. The conclusion that 'there is no time-independent spherically-symmetric density' is thus an unpacking of the chosen meaning of 'legitimate' rather than a consequence of the three named restrictions alone. The score of 6 reflects this partial circularity: the negative result reduces in part by construction, while the positive 2D results and the by-case critiques (Breit-frame wave-function frame-dependence, infinite Wigner fluctuations, Lorentz-contraction argument against Abel tomography) retain independent content. It is not an 8 because the paper is transparent about defining 'legitimate' and computes the failures of the extant methods instead of merely asserting them.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central no-go argument rests on definitional choices (density = |Psi|^2 and wave-packet independence) rather than on fitted parameters; the only numerical inputs are illustrative. No new particles, forces, or dimensions are introduced.

free parameters (3)
  • Axial dipole mass M_A = 1.014 GeV
    From MINERvA axial form factor fit [68], used only for illustrative plots in Sec. IIA. The central no-go argument does not depend on it.
  • Axial charge G_A(0) = 1.2723
    Input from beta decay and neutrino scattering, used for illustration in Sec. IIA.
  • Lattice n-pole form factor parameters for A, J, D = n = 2, 3, 4 fits from Hackett et al. [2]
    Used in Sec. IIB to plot mass density m(x_perp); the central claim is independent of these fits.
assumptions (5)
  • domain assumption Probability density is |Psi|^2 with the same wave function appearing in bra and ket.
    Stated in the Introduction as the foundational postulate; underlies rejection of Breit-frame overlap of different wave functions.
  • ad hoc to paper A legitimate spatial density must be independent of the wave packet and obtained in the limit of complete localization (sigma -> 0 after P+ -> infinity).
    Introduced in Sec. II as the criterion for a density depending only on internal structure. This is the load-bearing definitional premise for the no-go result.
  • domain assumption Only kinematic transverse boosts are used, with Delta+ = 0 (Drell-Yan frame) and P+ -> infinity to remove x+ dependence.
    Standard light-front quantization assumption used in Eqs. (6)-(9).
  • domain assumption A state at rest must have vanishing average momentum and finite fluctuations of position and momentum.
    Used in Sec. IV to reject Wigner distribution approach because <K^2> and <r^2> diverge.
  • standard math Single-nucleon completeness relation Eq. (5) and plane-wave matrix elements define form factors.
    Used to connect local operator matrix elements to form factors.

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Cite this review

Pith. "Pith review of On the Impossibility of Obtaining Time-Independent, Three-Dimensional, Spherically-Symmetric Densities of Confined Systems of Relativistically Moving Constituents." pith.science (2026). https://pith.science/paper/4JI3RL5O

@misc{pith2026250714388,
  author       = {Pith},
  title        = {Pith review of: On the Impossibility of Obtaining Time-Independent, Three-Dimensional, Spherically-Symmetric Densities of Confined Systems of Relativistically Moving Constituents},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4JI3RL5O}},
  note         = {Machine review of arXiv:2507.14388}
}
read the original abstract

The quantum mechanical definition of probability, the uncertainty principle and Poincare invariance provide strong basic restrictions on the ability to define spatial densities associated with form factors describing the properties of confined systems of relativistically moving constituents. Despite this, many papers ignore one or more of these restrictions. Here I show how to obtain time-independent, two-dimensional densities that are consistent with the stated restrictions. This is done using the light-front, infinite momentum frame formalism. Two-dimensional density interpretations of the axial-vector form factor and all three gravitational form factors are obtained. Additionally, an expression of a two-dimensional mass density related to the trace of the energy momentum tensor is obtained. I also show that all known methods for finding three-dimensional densities: using the Breit frame, Abel transformations, Wigner distributions and spherically-symmetric wave packets with vanishing spatial extent violate the basic restrictions in different manners. Furthermore, the use of the latter leads to densities that vanish almost everywhere in space as time increases from an initial value.

Figures

Figures reproduced from arXiv: 2507.14388 by the authors.

Figure 1
Figure 1. FIG. 1. Solid- [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Mass (squared) distribution [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

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