REVIEW 3 major objections 5 minor 18 references
This paper claims that the uncertainty bound ΔrΔk ≥ 5/2 is universal across classical light beams, coherent states of the quantized field, and single-photon wave functions, with identical saturating fields.
Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →
2026-08-04 04:55 UTC pith:4WCFR6GK
load-bearing objection The main claim is credible and the explicit saturating fields are new, but the proof of the global minimum is sketched, not shown. the 3 major comments →
Uncertainty relations in classical and quantum theories of electromagnetism
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The central discovery is that the lower bound on the product of the spatial and wavevector spreads, ΔrΔk ≥ 5/2, is universal across classical electromagnetism, quantum coherent states, and single-photon quantum theory. Using the Riemann–Silberstein vector and measuring spatial extent by the second moment of the energy density, the authors reduce the variational search for the minimum to an eigenvalue equation of a three-dimensional harmonic oscillator; its ground state gives the constant 5/2 and explicit Gaussian-times-polynomial functions that saturate the inequality. In the photon case, those saturating functions are the photon wave functions obtained from the positive-frequency part of th
What carries the argument
The carrying object is the Riemann–Silberstein vector F = D/√(2ε0) + iB/√(2μ0), whose squared modulus is proportional to the electromagnetic energy density. The paper uses this density to define variances Δr² and Δk² via second moments, then derives the variational equations for the spectral amplitudes f±(k). A substitution reduces these to the eigenvalue equation of a harmonic oscillator on k-space, whose lowest eigenvalue is 5/2; the associated eigenfunction, after transformation back to r-space, gives the saturating fields, including explicit Dawson-function expressions for photon wave functions.
Load-bearing premise
The load-bearing premise is that the second moment of the energy density is the right measure of spatial spread; if one adopts a different operational definition of spread, such as the earlier center-of-energy operators, the universal 5/2 bound changes and the 'same form' claim no longer holds.
What would settle it
Compute Δr² and Δk² from the definitions (8)–(9) for the saturating field (23) and check they give 5a²/2 and 5/(2a²). More decisively, run a numerical variational minimization of Δr²Δk² over all square-integrable Riemann–Silberstein vector fields with finite energy; any field with product below 25/4 would falsify the claimed bound, and any independent exact minimization yielding a different infimum would do the same.
If this is right
- Any classical light beam, coherent state, or single-photon state must satisfy ΔrΔk ≥ 5/2 under energy-density variances, giving a sharp design constraint for nanophotonic confinement.
- The three theories share the same saturating wave forms, so an experiment that measures the product close to 5/2 cannot distinguish classical from quantum light on that basis.
- The uncertainty product becomes ΔrΔp ≥ (5/2)ℏ when wavevector is converted to momentum, recovering the quantum form without quantum postulates.
- The saturating photon wave functions are given in closed form (Dawson functions and exponentials), making the bound directly testable.
- Since the bound is derived purely from Maxwell equations, it is a property of wave propagation, not of measurement disturbance.
Where Pith is reading between the lines
- Because the same bound appears in a purely classical theory, this inequality alone cannot serve as a quantumness witness; claims that use it to certify quantum behavior need additional assumptions.
- The universality hinges on the energy-density variance convention; under the earlier center-of-energy operator definition the constant is 1 + √5/2, so a family of uncertainty relations exists depending on what one calls spatial spread.
- A natural next step is to test the bound experimentally by measuring second-moment variances of engineered beams and single-photon states; observing a product below 5/2 would disprove the claim, and approaching 5/2 would confirm the saturating forms.
- The same variational technique might extend to other wave equations, such as massive relativistic fields, to see whether the 5/2 constant is specific to massless helicity-1 fields.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper derives sharp uncertainty relations for the spatial and wavevector spreads of light, defined through the second moments of the energy density and its Fourier transform. It claims a universal bound ΔrΔk ≥ 5/2 in three settings: classical Maxwell fields, coherent states of the quantized field, and single-photon wave functions. The universal bound is obtained by a variational calculation in the Riemann–Silberstein representation, with a specific family of saturating fields given explicitly. The paper also provides closed-form photon wave functions that saturate the bound and argues that the result demonstrates the wave-theoretic, rather than quantum, origin of these uncertainty relations.
Significance. If the central claim is correct, the result is significant: it identifies a common sharp uncertainty product and identical minimizing states across classical, coherent, and single-photon descriptions, making the wave-theoretic nature of the uncertainty relation concrete. The paper's explicit saturating fields and closed-form photon wave functions are valuable and checkable, and the coherent-state reduction is conceptually clean. The main caveat is that the bound is tied to the chosen variance convention (energy-density second moment); the authors acknowledge that an alternative center-of-energy definition gives a different constant. The result is therefore a statement about the chosen measure of spatial spread rather than a purely convention-independent property.
major comments (3)
- [Section 2, Eq. (11)] The central identity expressing Δr² in terms of f±(k) is quoted with only 'One finds' and no derivation. This formula is the foundation of the variational equations (13)–(15), so the reader cannot independently verify the variational principle without repeating a substantial calculation. Please supply a derivation, at least in an appendix, starting from F̃(k)=e(k)f(k), using |∇e|²=1/k⊥² and the appropriate first-order angular term. This is a load-bearing step because all subsequent results inherit Eq. (11).
- [Section 2, Eqs. (13)–(15)] The global lower bound is not fully proven. The text restricts to f(k)=k⊥g(k)/k and states that 'any angular dependence increases the eigenvalue' without proof. This assertion is load-bearing: the claimed bound (1) is over all square-integrable f(k), and Sections 3 and 4 inherit the same variational structure. The reduction to the three-dimensional harmonic oscillator (14)–(15) establishes the lowest eigenvalue only within the single angular channel l=1,m=0. To make the argument complete, the authors should prove that the angular part of the operator in (13) has no eigenvalues below the value corresponding to this channel, e.g., by showing that the relevant angular operator is bounded below by 2 on all admissible functions. Without such an argument, the paper establishes a stationary point, not a global minimum.
- [Section 4, photon case] The photon uncertainty relation is presented as following directly from the classical calculation, but the identification of the photon wave function with the positive-frequency Riemann–Silberstein part and the use of the energy-density second moment (8) are conventions. The authors note in the Introduction that the earlier center-of-energy operator gave 1+√5/2; under that alternative definition the 'same form' claim would not hold. This convention-dependence should be stated as a limitation of the universality claim, not only as a motivation for the present choice. The explicit claim that all three theories share the same minimum uncertainty product is true only within the chosen variance convention.
minor comments (5)
- [Section 3, after Eq. (27)] Typo: 'the factor √ℏc does affect our uncertainty relations' should read 'does not affect'.
- [Section 3, Eq. (28)] The normal-ordered energy density should be :F†(r,t)·F(r,t):, not :F†·F†:.
- [Section 2, Eq. (11)] The displayed formula has unbalanced parentheses in the last term; please correct the typography.
- [Section 2, Eq. (22)] The denominator '4kπ^{3/2}' is confusing; writing '4π^{3/2} k' would be clearer.
- [Section 4, Eq. (39)] The lengthy photon wave functions are stated without derivation. A brief derivation or an appendix showing how (38) leads to (39) would improve reproducibility.
Circularity Check
No circularity: the classical variational derivation is self-contained, and the photon and coherent-state cases reduce to it by explicit calculation rather than by self-citation.
full rationale
The paper's central derivation is the classical variational problem in Section 2. The variance definitions (8) and (9) are fixed, and equation (11) is presented as a direct computation of Δr² in terms of f±(k); the variational equations (13)–(15) then yield the bound through an explicit eigenvalue calculation. No parameter is fitted to data and then relabeled as a prediction. The coherent-state case (Section 3) is explicitly an identity: because coherent-state expectation values of normal-ordered field operators reproduce the same f±(k) expressions as the classical fields, equations (31)–(32) are the same as (8)–(9), and the bound follows by substitution. This is a constructional identity, not a circularity. The single-photon case (Section 4) cites the authors' prior [15] for the same bound, but then states 'The uncertainty relation (1) in the case of photons follows directly from our results for the classical electromagnetic field' and independently computes saturating functions (39). Thus the self-citation is contextual, not load-bearing. The skeptical concern that the reduction f(k)=k⊥g(k)/k and the assertion 'any angular dependence increases the eigenvalue' are not fully proven is a correctness gap in establishing the global lower bound, not a circularity by construction. Similarly, Eq. (11) being 'given without derivation' is an omitted derivation, not a circular step. The paper's convention dependence (energy-density spread vs. center-of-energy operator) is openly acknowledged in the introduction and does not amount to defining the result in terms of itself. Overall, the derivation chain is self-contained and not circular.
Axiom & Free-Parameter Ledger
axioms (5)
- domain assumption Maxwell's equations for the free field are equivalent to i∂_t F = c ∇×F for the Riemann–Silberstein vector F (Eq. 3).
- ad hoc to paper Spatial spread of the field is measured by the normalized second moment of the energy density, Δr²=∫r²F*·F/∫F*·F (Eq. 8), rather than by a standard position operator.
- domain assumption The minimum of the product is attained by a spherically symmetric function; 'any angular dependence increases the eigenvalue' is asserted without proof (Section 2, after Eq. 14).
- domain assumption Single-photon wave functions are the positive-frequency parts of the Riemann–Silberstein vector (Eqs. 34–35), and the same variance definitions apply (Section 4).
- domain assumption Coherent-state expectation values of normally ordered field operators are obtained by replacing creation/annihilation operators with the classical amplitudes f± (Section 3).
Cite this review
Pith. "Pith review of Uncertainty relations in classical and quantum theories of electromagnetism." pith.science (2026). https://pith.science/paper/4WCFR6GK
@misc{pith2026260528906,
author = {Pith},
title = {Pith review of: Uncertainty relations in classical and quantum theories of electromagnetism},
year = {2026},
howpublished = {\url{https://pith.science/paper/4WCFR6GK}},
note = {Machine review of arXiv:2605.28906}
}
read the original abstract
Sharp uncertainty relations restricting the values of variances in the position space and in the momentum (wavevector) space are derived. They have the same form $\Delta r\Delta k\ge 5/2$ in the classical theory of light beams, in the quantum theory of coherent light beams, and in the quantum theory of individual photons.
Reference graph
Works this paper leans on
-
[1]
Wu H, Yang L, Xu P, Gong J, Guo X, Wang P and Tong L 2022 Photonic nanolaser with extreme optical field confinement Phys. Rev. Lett.129013902
2022
-
[2]
Yang L, Zhou Z, Wu H, Dang H, Yang Y, Gao J, Guo X, Wang P and Tong L 2023 Generating a sub-nanometer-confined optical field in a nanoslit waveguiding mode, Advanced Photonics 5046003
2023
-
[3]
Phys.43172
Heisenberg W 1927 ¨Uber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik Z. Phys.43172
1927
-
[4]
Yang Y-H , Liu X-Z, Zheng X-Z., Jiang J-L, Yang X, Fei S-M , Ma Z, Wang Z and Luo M-X 2025 Experimental characterization of quantumness using the uncertainty principle, coherence, and nonlocality Phys. Rev. Lett.135020201
2025
-
[5]
Bialynicki-Birula I and Bialynicka-Birula Z 2012 Heisenberg uncertainty relations for photons Phys. Rev. A86022118
2012
-
[6]
of the New York Akad
Dragoman D 2025 On the photon–classical electromagnetic field relationship Ann. of the New York Akad. Sci.154411
2025
-
[7]
Phys.22579
Silberstein L 1907 Elektromagnetische Grundgleichungen in bivectorieller Behandlung Ann. Phys.22579
1907
-
[8]
Bialynicki-Birula I 1994 On the wave function of the photon Acta Phys. Polon. A8697
1994
-
[9]
Bialynicki-Birula I 1996 Photon wave function Progress in Optics36ed E Wolf (Elsevier Amsterdam) (arXiv:quant-ph/0508202)
Pith/arXiv arXiv 1996
-
[10]
Bialynicki-Birula I and Bialynicka-Birula Z 2013 The role of the Riemann-Silberstein vector in classical and quantum theories of electromagnetism J. Phys. A: Math. Theor.46053001
2013
-
[11]
Opt.19125201
Bialynicki-Birula I and Bialynicka-Birula Z 2017 Quantum-mechanical description of optical beams J. Opt.19125201
2017
-
[12]
Phys.21119502
Bialynicki-Birula I and Bialynicka-Birula Z 2019 Heisenberg uncertainty relation for relativistic electrons New J. Phys.21119502
2019
-
[13]
2025 Mathematica Version 14.3 Champaign IL
Wolfram Research Inc. 2025 Mathematica Version 14.3 Champaign IL
2025
-
[14]
Rev.1312766
Glauber R J 1963 Coherent and incoherent states of the radiation field Phys. Rev.1312766
1963
-
[15]
Phys.28014506
Bialynicki-Birula I and Bialynicka-Birula Z 2026 Universal Heisenberg uncertainty relations for all massless particles New J. Phys.28014506
2026
-
[16]
Math.40 149
Wigner E 1939 On unitary representations of the inhomogeneous Lorentz group, Ann. Math.40 149
1939
-
[17]
Opt.13064014
Bialynicki-Birula I and Bialynicka-Birula Z 2011 Canonical separation of angular momentum of light into its orbital and spin partsJ. Opt.13064014
2011
-
[18]
Aiello A 2022 One more time on the helicity decomposition of spin and orbital optical currents J. Phys. A: Math. Theor.55244004
2022
This paper was first reviewed by deepseek-v4-flash on August 4, 2026.
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.