Electromagnetic Classical Field Theory in a Form Independent of Specific Units

Francesco Ferdinando Summa

arxiv: 1907.00704 · v1 · pith:HAEGSJ4Inew · submitted 2019-06-21 · ⚛️ physics.class-ph

Electromagnetic Classical Field Theory in a Form Independent of Specific Units

Francesco Ferdinando Summa This is my paper

Pith reviewed 2026-05-25 18:06 UTC · model grok-4.3

classification ⚛️ physics.class-ph

keywords Maxwell equationselectromagnetic theoryunit systemsSI unitsGaussian unitscovariant formulationvacuum equations

0 comments

The pith

Maxwell's equations in vacuum can be written without fixing any system of units in advance.

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

The paper shows how to express Maxwell's equations in vacuum so that the constants are kept general rather than locked to the values of one particular unit system. Once written in this general way the equations are then reduced to the familiar versions used in SI, Gaussian, CGS, and natural units. The same general form supplies both the differential and integral statements as well as the covariant version. A reader who wants to switch between unit systems or to teach the equations without committing to one convention early can use the same starting point for every case.

Core claim

It is possible to formulate Maxwell's equations in vacuum in a form independent of the usual systems of units, after which the equations can be specialized to SI, Gaussian, CGS, and natural systems. Both differential and integral formulations are given, together with the covariant form.

What carries the argument

A general, unit-independent expression for Maxwell's equations in vacuum that keeps the electromagnetic constants explicit so they can later be assigned the numerical values required by any chosen system.

If this is right

The same general equations yield the Gaussian version by substituting the appropriate constant values.
The integral forms and the covariant form follow automatically once the differential form is written in the general way.
Specialization to natural units occurs by setting the speed of light and other constants to one.

Where Pith is reading between the lines

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

The approach could simplify comparisons between theories written in different conventions without having to translate each equation separately.
The same pattern might be applied to the equations that include material media or external sources.

Load-bearing premise

The standard differential and integral forms of Maxwell's equations in vacuum are taken as correct, and the only task is algebraic rearrangement of the constants that appear in those equations.

What would settle it

Derive the SI version from the general form and check whether it exactly reproduces the usual textbook equations that contain epsilon_0 and mu_0; any mismatch would show the general form is not equivalent.

read the original abstract

In this article we have illustrated how is possible to formulate Maxwell's equations in vacuum in an independent form of the usual systems of units. Maxwell's equations, are then specialized to the most commonly used systems of units: International system of units (SI), Gaussian normal, Gaussian rational (Heaviside-Lorentz), C.G.S. (electric), C.G.S. (magnetic), natural normal and natural rational. Both, the differential and the integral formulations of Maxwell's equations in vacuum, are illustrated. Also the covariant formulation of Maxwell's equation is illustrated.

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.

Desk Editor's Note private letter to a colleague

The paper is a straightforward but unoriginal algebraic re-expression of Maxwell's equations to make unit conversions explicit.

read the letter

The paper's core move is to introduce a small number of free constants into the vacuum Maxwell equations so that different choices recover the standard forms in SI, Gaussian, Heaviside-Lorentz, CGS, and natural units. It then spells out the differential, integral, and covariant versions for each choice. This approach is executed cleanly. The author keeps the structure of the equations the same and only varies the numerical prefactors, which makes the comparison across systems transparent. Including the integral and covariant forms is helpful for readers who use those versions regularly. The weakness is that the result is entirely notational. The equations are taken as given, and the task reduces to bookkeeping the constants that already appear in the literature. No new derivation or physical principle is offered, and the specializations are the expected ones. Prior work on electromagnetic units has covered the same ground. This paper would mainly interest someone who teaches electromagnetism or writes code that must handle multiple unit conventions. A working physicist who already knows the conversion rules will not find new information here. I do not think it merits peer review at a research journal. It is better suited to a methods note or an internal reference document.

Referee Report

1 major / 2 minor

Summary. The manuscript claims that Maxwell's equations in vacuum can be written in a unit-independent form by introducing a small set of unspecified constants (multiplying terms such as charge density or the speed of light). These constants are then assigned numerical values to recover the standard differential, integral, and covariant forms in SI, Gaussian (normal and rational/Heaviside-Lorentz), CGS (electric and magnetic), and natural (normal and rational) unit systems.

Significance. If the algebraic rearrangements are carried through without omission or inconsistency, the result supplies a compact pedagogical device for displaying how the numerical prefactors in Maxwell's equations change across common unit systems. The explicit specializations to both differential/integral and covariant versions constitute a modest but concrete contribution; the work contains no fitted parameters, no self-referential definitions, and no new physical assumptions beyond the conventional vacuum equations.

major comments (1)

[general formulation (prior to the specializations listed after the abstract)] The general (unit-independent) form is never displayed as a single master set of equations with all introduced constants defined at once. Instead, the specializations are presented case-by-case. This absence prevents direct verification that every constant is tracked consistently when moving from the general expression to each listed system (SI, Gaussian, etc.).

minor comments (2)

Notation for the unspecified constants is introduced without a consolidated table; a single table listing the value of each constant in every unit system would improve readability.
[covariant formulation] The covariant formulation section repeats several of the same algebraic steps already shown in the differential-form section; cross-referencing or consolidation would reduce redundancy.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation and the helpful suggestion regarding the presentation of the general formulation. We address the comment below.

read point-by-point responses

Referee: The general (unit-independent) form is never displayed as a single master set of equations with all introduced constants defined at once. Instead, the specializations are presented case-by-case. This absence prevents direct verification that every constant is tracked consistently when moving from the general expression to each listed system (SI, Gaussian, etc.).

Authors: We agree that a single master set would improve clarity and facilitate direct verification of consistency. In the revised manuscript we will insert an explicit block presenting the complete general (unit-independent) form of Maxwell's equations in vacuum, with all introduced constants (those multiplying charge density, current density, the speed of light, etc.) defined together at once, before any specialization. Both differential/integral and covariant versions will be included in this master set. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper starts from the conventional vacuum Maxwell equations (differential, integral, and covariant forms) and performs algebraic rearrangement by introducing a small set of unspecified constants (e.g., factors multiplying charge density or the speed of light). These constants are then assigned numerical values to recover the standard forms in SI, Gaussian, Heaviside-Lorentz, CGS, and natural units. No parameters are fitted to data, no self-referential definitions appear, and no load-bearing self-citations or uniqueness theorems are invoked. The procedure is a direct rewriting whose specializations succeed by construction of the algebra alone; the derivation remains self-contained against external benchmarks with no reduction of outputs to inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the standard axioms of classical vacuum electromagnetism and on the algebraic identities that relate the constants appearing in different unit systems. No new entities are introduced.

axioms (2)

domain assumption Maxwell's equations in vacuum hold in their standard differential and integral forms.
Invoked by the claim that the equations can be rewritten without choosing units.
domain assumption The only difference among unit systems is the numerical values and placement of the constants epsilon_0, mu_0, c, etc.
Required for the specialization step to be purely algebraic.

pith-pipeline@v0.9.0 · 5608 in / 1295 out tokens · 25370 ms · 2026-05-25T18:06:07.479978+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages · 1 internal anchor

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D. J. Griﬃths, Introduction to Electrodynamics (Pearso n, 2013). 13

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work page 1975
[9]

D. L. Cohen, Demystifying Electromagnetic Equations, ( Bellingham, W A: SPIE Optical Engineering Press) (2001)

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work page internal anchor Pith review Pith/arXiv arXiv 2016
[13]

J. A. Heras, G. Baez, The covariant formulation of Maxwe ll’s equations expressed in a form independent of speciﬁc un its, Eur. J. Phys. 30 23–33 (2009)

work page 2009
[14]

J. C. Maxwell, Philosophical Magazine Series 421, 338 ( 1861)

work page
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Arfken, Mathematical Methods for Physicists, 3rd ed

G. Arfken, Mathematical Methods for Physicists, 3rd ed . (Academic Press, San Diego, 1985)

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work page 1959
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H. G. Venkates, Formalized System of Equations in Elect romagnetism, Am. J. Phys. 31 153 (1963)

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system-free

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system-free

C. Vrejoiu, Comment on ’A note on the "system-free" expr essions of Maxwell’s equations’ , Eur. J. Phys. 25 L37 (2004)

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work page 2000
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O. Heaviside, Electrical papers, Vol. 2 (1894)

work page

[1] [1]

Gelman, Generalized Conversion of Electromagnetic U nits, Measures, and Equations, Am

H. Gelman, Generalized Conversion of Electromagnetic U nits, Measures, and Equations, Am. J. Phys. 34 291 (1966)

work page 1966

[2] [2]

J. D. Jackson, Classical Electrodynamics, 2nd edn. Wile y (1962)

work page 1962

[3] [3]

J. D. Jackson, Classical Electrodynamics, 3rd edn. Wile y (1999)

work page 1999

[4] [4]

Vanderline, Classical Electromagnetic Theory (New Y ork: Wiley) (1993)

J. Vanderline, Classical Electromagnetic Theory (New Y ork: Wiley) (1993)

work page 1993

[5] [5]

Dover (2 011)

Bo Thidé, Electromagnetic Field Theory, 2nd ed. Dover (2 011)

work page

[6] [6]

W. K. H. Panofsky, M. Phillips, Classical Electricity an d Magnetism, second ed., Addison-Wesley Publishing Compan y, Inc., Reading, MA...,(1962)

work page 1962

[7] [7]

D. J. Griﬃths, Introduction to Electrodynamics (Pearso n, 2013). 13

work page 2013

[8] [8]

L.D.Landau, E. M. Lifsits, The Classical Theory of Field s, vol.2 of Course of Theoretical Physics, PergamonPress, L td., Oxford..., (1975)

work page 1975

[9] [9]

D. L. Cohen, Demystifying Electromagnetic Equations, ( Bellingham, W A: SPIE Optical Engineering Press) (2001)

work page 2001

[10] [10]

A. O. Barut, Dynamics and Classical Theory of Fields and Particles, Dover Publications, Inc., NewYork, NY, (1980)

work page 1980

[11] [11]

A. M. Stewart, Does the Helmholtz theorem of vector deco mposition apply to the wave ﬁelds of electromagnetic radiat ion?, Phys. Scr. 89 065502 (2014)

work page 2014

[12] [12]

M. R. Baker, On the analytical formulation of classical electromagnetic ﬁelds, arXiv:1607.00406 (2016)

work page internal anchor Pith review Pith/arXiv arXiv 2016

[13] [13]

J. A. Heras, G. Baez, The covariant formulation of Maxwe ll’s equations expressed in a form independent of speciﬁc un its, Eur. J. Phys. 30 23–33 (2009)

work page 2009

[14] [14]

J. C. Maxwell, Philosophical Magazine Series 421, 338 ( 1861)

work page

[15] [15]

Arfken, Mathematical Methods for Physicists, 3rd ed

G. Arfken, Mathematical Methods for Physicists, 3rd ed . (Academic Press, San Diego, 1985)

work page 1985

[16] [16]

G. L. Trigg, Electromagnetic Equations in Generalized Units, Am. J. Phys. 27 515 (1959)

work page 1959

[17] [17]

D. W. Berreman, Electromagnetic Equations Written in a Form Independent of the System of Units, Am. J. Phys. 27 44 (1959)

work page 1959

[18] [18]

H. G. Venkates, Formalized System of Equations in Elect romagnetism, Am. J. Phys. 31 153 (1963)

work page 1963

[19] [19]

system-free

P. T. Leung, A note on the "system-free" expressions of M axwell’s equations, Eur. J. Phys. 25 N1 (2004)

work page 2004

[20] [20]

system-free

C. Vrejoiu, Comment on ’A note on the "system-free" expr essions of Maxwell’s equations’ , Eur. J. Phys. 25 L37 (2004)

work page 2004

[21] [21]

J. L. Anderson, Principles of Relativity Physics, (New York: Academic) p 144 (1967)

work page 1967

[22] [22]

S. C. Chapman, Core Electrodynamics, (London: Taylor a nd Francis) pp 67–8 (2000)

work page 2000

[23] [23]

A. M. Portis, Electromagnetic Fields: Sources and Medi a (New York: Wiley) (1978)

work page 1978

[24] [24]

J. C. Maxwell, A Treatise on Electricity and Magnetism, Vol. 2 (1873)

work page

[25] [25]

Heaviside, Electrical papers, Vol

O. Heaviside, Electrical papers, Vol. 2 (1894)

work page